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- ... Bone Reports 20 (2024) 101735 Contents lists available at ScienceDirect Bone Reports journal homepage: www.elsevier.com/locate/bonr Am I big boned? Bone length scaled reference data for HRpQCT measures of the radial and tibial diaphysis in White adults Stuart J. Warden a, b, *, Robyn K. Fuchs b, c, Ziyue Liu b, d, Katelynn R. Toloday a, Rachel Surowiec e, Sharon M. Moe b, f a Department of Physical Therapy, School of Health and Human Sciences, Indiana University, Indianapolis, IN, United States of America Indiana Center for Musculoskeletal Health, Indiana University, IN, United States of America College of Osteopathic Medicine, Marian University, Indianapolis, IN, United States of America d Department of Biostatistics, School of Medicine, Indiana University, Indianapolis, IN, United States of America e Department of Biomedical Engineering, Purdue University, Indianapolis, IN, United States of America f Division of Nephrology and Hypertension, Department of Medicine, School of Medicine, Indiana University, Indianapolis, IN, United States of America b c A R T I C L E I N F O A B S T R A C T Keywords: Bone allometry Bone strength Cortical bone Normative data Osteoporosis Cross-sectional size of a long bone shaft influences its mechanical properties. We recently used high-resolution peripheral quantitative computed tomography (HRpQCT) to create reference data for size measures of the radial and tibial diaphyses. However, data did not take into account the impact of bone length. Human bone exhibits relatively isometric allometry whereby cross-sectional area increases proportionally with bone length. The consequence is that taller than average individuals will generally have larger z-scores for bone size outcomes when length is not considered. The goal of the current work was to develop a means of determining whether an individual's cross-sectional bone size is suitable for their bone length. HRpQCT scans performed at 30 % of bone length proximal from the distal end of the radius and tibia were acquired from 1034 White females (age = 18.0 to 85.3 y) and 392 White males (age = 18.4 to 83.6 y). Positive relationships were confirmed between bone length and cross-sectional areas and estimated mechanical properties. Scaling factors were calculated and used to scale HRpQCT outcomes to bone length. Centile curves were generated for both raw and bone length scaled HRpQCT data using the LMS approach. Excel-based calculators are provided to facilitate calculation of z-scores for both raw and bone length scaled HRpQCT outcomes. The raw z-scores indicate the magnitude that an individual's HRpQCT outcomes differ relative to expected sex- and age-specific values, with the scaled z-scores also considering bone length. The latter enables it to be determined whether an individual or population of interest has normal sized bones for their length, which may have implications for injury risk. In addition to providing a means of expressing HRpQCT bone size outcomes relative to bone length, the current study also provides centile curves for outcomes previously without reference data, including tissue mineral density and moments of inertia. 1. Introduction Bone strength is influenced by the amount and quality of material present in addition to how the material is distributed (Fuchs et al., 2019). The distribution of bone material is colloquially referred to as bone structure or size and is often assessed via cross-sectional bone images acquired using 3D imaging modalities such as computed to mography and magnetic resonance imaging. High-resolution peripheral quantitative computed tomography (HRpQCT) is a powerful imaging modality capable of providing non-invasive measures of bone cross- sectional properties, along with volumetric bone mineral density (vBMD) and micro-finite element (FE) estimates of bone strength (Whittier et al., 2020). HRpQCT is principally used to assess structure at sites rich in trabecular bone (e.g., distal radius), with outcomes predicting incident fracture (Mikolajewicz et al., 2020; Samelson et al., 2019) and revealing the effects of aging, disease, and intervention (Lespessailles et al., 2016). However, there is growing interest in assessing more proximal sites containing a higher proportion of cortical bone (Cheung et al., 2014; Hughes et al., 2018; Kazakia et al., 2014; O'Leary et al., 2021; Orwoll * Corresponding author at: Department of Physical Therapy, School of Health and Human Sciences, Indiana University, 1140 W. Michigan St., CF-120, Indian apolis, IN 46202, United States of America. E-mail address: stwarden@iu.edu (S.J. Warden). https://doi.org/10.1016/j.bonr.2024.101735 Received 9 December 2023; Accepted 4 January 2024 Available online 6 January 2024 2352-1872/ 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). S.J. Warden et al. Bone Reports 20 (2024) 101735 et al., 2022; Patsch et al., 2013; Warden et al., 2022a; Warden et al., 2021a). Most bone loss during aging occurs from within the cortical compartment (Zebaze et al., 2010), and assessment of cortical bone-rich diaphyseal sites may provide unique insight into bone changes occurring in disease states and with lifestyle and pharmaceutical interventions. We recently used a second-generation HRpQCT scanner to create reference data for cortical bone outcomes obtained at 30 % of bone length proximal from the distal end of the radius and tibia (Warden et al., 2022b). The data can be used to calculate z-scores to indicate the number of standard deviations an individual's outcomes vary from ageand sex-matched median outcomes. However, the reference data did not take into account the impact of bone length on cross-sectional bone size. There has long been a fascination with the relationship between bone length and cross-sectional size. Galileo (Galilei, 1638) predicted that bones in different sized animals would exhibit positive allometry. That is, he predicted bone size would increase to a relatively greater extent than length in order to maintain the same strength. More recently, skeletal allometry in mammals, including humans, has been reported to be more modest with the relationship between bone length and crosssectional size being closer to isometric (Biewener, 2005; Ruff, 1984). Specifically, the bones of taller people are generally expanded versions of bones of people who are shorter, with bone cross-sectional area increasing proportionally as bone length increases. The net result is that taller-than-average individuals will generally have larger z-scores for bone size outcomes using currently available reference HRpQCT data, and vice versa for shorter individuals. The goal of the current study was to develop a means of determining whether an individual's cross-sectional bone size is suitable not only for their age and sex, but also their bone length. The primary aims were to: 1) explore the relationship between bone length and HRpQCT measures of radial and tibial diaphysis size in adults and 2) generate bone length scaled age- and sex-specific reference data for the HRpQCT measures across adulthood. In doing so, the current study also aimed to provide centile curves for outcomes previously without reference data, including tissue mineral density and moments of inertia. The ultimate goal was to provide calculators to enable the computation of subject-specific per centiles and z-scores for both raw and bone length scaled HRpQCT outcomes. maneuvers without arms from standardized chair (seat height = 45 cm) were performed to assess physical function, as we have previously detailed (Warden et al., 2022c). Self-reported physical function was assessed using the physical functioning domain of the National Institutes of Health Patient Reported Outcomes Measurement Information System (PROMIS-PF), performed via computerized adaptive testing. PROMIS scores are standardized and expressed as T-scores with a population mean of 50 and standard deviation of 10 (Cella et al., 2007). Spine and total hip aBMD were assessed by dual-energy x-ray absorptiometry (DXA) (Norland Elite; Norland at Swissray, Fort Atkinson, WI). 2.2. High-resolution peripheral quantitative computed tomography (HRpQCT) The non-dominant arm and contralateral leg were imaged using an HRpQCT scanner (XtremeCT II; Scanco Medical, Bruttisellen, Switzerland). Phantoms were imaged daily to confirm scanner stability. Bone length was measured in triplicate using a segmometer (Realmet Flexible Segmometer, NutriActiva, Minneapolis, MN) and as described by Bonaretti et al. (Bonaretti et al., 2017). Per convention, ulna length (mm) was measured as a surrogate for radial length. The skin overlying the distal apex of the ulnar styloid was marked and the elbow placed on a rigid surface. The Euclidean distance between the surface and styloid mark was measured. Tibial length (mm) was measured between skin marks placed at the distal tip of the medial malleolus and medial knee joint line. Short-term precision for repeat mark placement and length measures of the ulna and tibia in 15 individuals tested on two consec utive days showed root mean square standard deviations of 2.8 mm (1.1 %) and 6.3 mm (1.7 %), respectively. Scans were acquired and reconstructed as previously described (Warden et al., 2021a). Subjects laid supine with their limb immobilized using an anatomically formed carbon fiber cast. The scanner operated at 68 kVp and 1.47 mA to acquire 168 slices (10.2 mm of bone length) with a voxel size of 60.7 m. After performance of a scout view, reference lines were placed at the medial edge of the distal radius articular surface and center of the distal tibia joint surface. Scan stacks were centered 30 % of bone length proximal to the reference lines. The 30 % location was chosen as it is accessible in most individuals when using the manufac turer's standard forearm and leg casts on a second generation HRpQCT scanner. Assessment of more proximal sites requires use of custom casts and different reference line landmarks to stay within the z-axis limits of the scanner. Scans were scored for motion artifacts on a standard scale of 1 (no motion) to 5 (discontinuities in the cortical shell) (Sode et al., 2011). Scans scoring 3 were repeated when time permitted. Scans with a motion artifact of 4 or 5 were excluded from analyses. A manufacturer provided evaluation script using a dual threshold technique was used to contour the outer periosteal surface and inner trabecular/medullary compartment. Manufacturer provided evaluation scripts were used to obtain outcomes (Table 1). A standard cortical bone script was used to obtain CtvBMD, CtAr, CtPm, CtTh, and CtPo. The script utilized a low-pass Gaussian filter (sigma 0.8, support 1.0 voxel) and fixed thresholds of 320 and 450 mgHA/cm3 to extract trabecular and cortical bone, respectively. Only cortical outcomes were recorded as diaphyseal sites contain limited trabecular bone. The manufacturer's bone midshaft evaluation script was used with a low-pass Gaussian filter (sigma 0.8, support 1.0) and outer threshold of 450 mgHA/cm3. The evaluation provided outcomes for the whole bone (i.e. cortical and any trabecular bone) as the script was run with a single outer contour and without an inside clock-wise (i.e. negative excluding) contour. Outcomes obtained were TotDen (identified as Mean1 in the manu facturer's script), TMD (identified as Mean2 in the manufacturer's script), TA, BA, BA/TA, IMIN, IMAX, and pMOI. Stiffness and failure load were estimated by FE analysis (Scanco Medical FE software version 1.13). Each voxel within the segmented images was assigned a modulus of 10 GPa and Poisson's ratio of 0.3. Axial compression was applied and failure load estimated when 5 % of elements exceeded 1 % strain (Arias- 2. Methods 2.1. Participants HRpQCT scans were performed on 1856 adults (aged 18 years) between 12/2017 and 12/2022 within the Musculoskeletal Function, Imaging, and Tissue Resource Core (FIT Core) of the Indiana Center for Musculoskeletal Health's Clinical Research Center (Indianapolis, Indi ana). Participants were recruited to the core by investigators seeking standardized musculoskeletal outcomes for their research subjects, as well as from the local community via self-referral. The FIT Core has Institutional Review Board approval from Indiana University to assess all-comers who provide written informed consent. To be eligible for inclusion in the current dataset, participants were required to: 1) self-identify as being of White ancestry, 2) be ambulatory and 3) have no self-reported diabetes, liver or kidney disease, past or present history of cancer, thyroid disorders, cystic fibrosis, or rare bone disease (e.g., osteopetrosis or X-linked hypophosphatemia). Individuals with the later conditions were excluded due to their overrepresentation in the FIT Core cohort resulting from investigator-initiated trials and their known or potential impact on HRpQCT outcomes (van den Bergh et al., 2021). Height (m) and weight (kg) were measured without shoes using a calibrated stadiometer (Seca 264; Seca GmbH & Co., Hamburg, Ger many) and scale (MS140-300; Brecknell, Fairmont, MN), respectively. Grip strength using a JamarPlus+ digital hand dynamometer (Sammons Preston, Bolingbrook, IL) and time to complete five sit-to-stand 2 S.J. Warden et al. Bone Reports 20 (2024) 101735 2005). Participant characteristics were described according to decade stage of life (1829, 3039, 4049, 5059, 6069, and 70+ yrs). Spearman partial correlation controlling for age was used to assess the relationship between HRpQCT outcomes and bone length. Scaling factors were calculated for HRpQCT outcomes exhibiting a Spearman partial correlation (on age) with bone length of R2 0.05. Scaling factors were calculated using the simple allometric linear regression model, Y = X, applied in natural logarithmic form as: Table 1 HR-pQCT outcomes. Outcome Abbreviation Units Description Total density (Mean1)b TotDen mgHA/ cm3 Tissue mineral density (Mean2)b TMD mgHA/ cm3 Cortical vBMDa CtvBMD mgHA/ cm3 Total areab TA mm2 Bone areab BA mm2 Cortical areaa CtAr mm2 Bone area/total areab BA/TA % Cortical perimetera CtPm mm Cortical thicknessa Cortical porositya CtTh mm CtPo % Minimum second moment of areab IMIN mm4 Maximum second moment of areab IMAX mm4 Polar moment of inertiab pMOI mm4 Stiffnessc kN/mm Failure loadc N Average density of all voxels within the periosteal contour, including tissue and voids (i.e. medullary cavity and pores) Average density of all voxels within the periosteal contour with a density >450 mgHA/cm3 (excludes medullary cavity and pores) Average density of all voxels within the segmented cortical compartment (includes pores) Area within the periosteal contour, including tissue and voids (i.e. medullary cavity and pores) Area within the periosteal contour with a density >450 mgHA/cm3 (excludes medullary cavity and pores) Average cross-sectional area of the segmented cortical compartment (includes pores) Proportion of voxels within the periosteal contour with a density >450 mgHA/cm3 Average length of the outer periosteal surface within the segmented cortical compartment Thickness of cortical bone including any pores Percentage of void voxels from the total cortical voxels in the segmented cortical compartment Estimated ability of all the components within the outer contour with a density >450 mg/ cm3 to resist bending on the direction of least bending resistance Estimated ability of all the components within the outer contour with a density >450 mg/ cm3 to resist bending on the direction of most bending resistance Estimated ability of all the components within the outer contour with a density >450 mg/ cm3 to resist torsional loading Total reaction force divided by the applied displacement within the finite model Failure load indirectly estimated from linear finite element model loge Y = loge + loge X + loge with Y the HRpQCT outcome of interest, X the predictor variable (i.e., bone length), the scaling exponent or power (scaling factor), the proportionality constant, and the multiplicative error. HRpQCT out comes exhibiting a Spearman partial correlation with bone length of R2 < 0.05 (i.e., <5 % of the variation explained) were considered to have low explanatory value. Scaling factors were used to scale HRpQCT outcomes as: / Yscaled = Yraw (X/X0 )SF with Yscaled the scaled value for the HRpQCT outcome, Yraw the original measured value for the HRpQCT outcome, X the bone length, X0 the sexspecific median bone length within the study population, and SF the scaling factor () from the linear regression model. Bone length (X) was normalized to sex-specific median bone length (X0) so that the scaled value retained the same units and were in a similar value range as the original measured value for the HRpQCT outcome. Sex-specific reference centile curves for raw and scaled HRpQCT outcomes were generated using the LMS method (Cole and Green, 1992) with R package GAMLSS (version 5.2.0) (Rigby and Stasinopoulos, 2005). The LMS method uses Box-Cox transformation to achieve normality at a given age (Box-Cox Cole and Green [BCCG] distribution). Nonparametric smooth curves are fit to the parameter values across the age range using penalized likelihood with penalty on the second derivatives. Centile curves and z-scores were calculated from the estimated parameter curves. LMS-derived z-scores are not suited for identifying extreme values because the LMS transformation method to achieve normality constrains maximum obtainable z-scores. Modified z-scores are provided for scores greater than +2 to address this. In modified zscores, the HRpQCT outcome is expressed relative to the sex- and agematched median in units of half the distance between 0 and + 2 zscores, as per the approach used for growth charts (Centers for Disease Control and Prevention, n.d.). 3. Results 3.1. Participant and scan characteristics Scans from 1426 participants (1034 females, 392 males) were included following exclusion of 430 participants due to: 1) race not White (n = 247, including 87 with a self-reported ineligible disease) and 2) race White, but self-reported ineligible disease or illness (n = 183). The final cohort included females and males ranging in age from 18.0 to 85.3 yrs. and 18.4 to 83.6 yrs., respectively. Participant characteristics stratified by decade of age are detailed in Table 2. Total hip and/or spine aBMD t-score was 1 to 2.5 and 2.5 in 336 (32.5 %) and 20 (1.9 %) females, respectively. One hundred fifty (38.3 %) and 21 (5.4 %) males had a total hip and/or spine aBMD t-score 1 to 2.5 and < 2.5, respectively. HRpQCT reference data in females was generated from 1023 and 919 scans of the radius and tibia, respectively (Table 2). Lower scan numbers than participants was due to: 1) scan not performed due to time con straints (n = 5 radius and 59 tibia scans); 2) excessive motion artifact (n = 26 radius and 9 tibia scans), and; 3) participant size (e.g. leg too large for the carbon fiber cast; leg too long to place the reference line and scan a Acquired using the manufacturer's standard cortical bone script, a low-pass Gaussian filter (sigma 0.8, support 1.0 voxel) and fixed thresholds of 450 and 320 mgHA/cm3 for cortical and trabecular bone, respectively. b Acquired using the manufacturer's bone midshaft evaluation script, a lowpass Gaussian filter (sigma 0.8, support 1.0 voxel) and fixed threshold of 450 mgHA/cm3. c Acquired using the manufacturer's FE analysis with voxels assigned a modulus of 10 GPa and Poisson's ratio of 0.3. Failure estimated when 5 % of elements exceeded 1 % strain. Moreno et al., 2019). 2.3. Statistical analyses All statistical analyses were performed for females and males sepa rately as skeletal proportions and cross-sectional properties differ across sexes independent of height and weight (Kun et al., 2023; Nieves et al., 3 S.J. Warden et al. Bone Reports 20 (2024) 101735 Table 2 Participant characteristics stratified by decade of agea. Characteristic Females n Height (m) Ulna length (cm) Tibia length (cm) Weight (kg) BMI (kg/m2) Physical function Grip strength (kg) 5 sit-to-stand test (s) PROMIS-PF (T-score) Bone densitometry Spine aBMD z-score Total hip aBMD z-score HRpQCT scans included (n) Radial diaphysis Tibial diaphysis Males n Height (m) Ulna length (cm) Tibia length (cm) Weight (kg) BMI (kg/m2) Physical function Grip strength (kg) 5 sit-to-stand test (s) PROMIS-PF (T-score) Bone densitometry Spine aBMD z-score Total hip aBMD z-score HRpQCT scans included (n) Radial diaphysis Tibial diaphysis Age group (yrs) 1829 3039 4049 5059 6069 70+ 184 1.66 (1.611.70) 25.5 (24.426.4) 37.0 (35.438.3) 71.5 (59.076.8) 25.9 (21.727.7) 113 1.66 (1.621.70) 25.4 (24.626.2) 36.9 (35.737.9) 71.4 (60.079.6) 25.9 (21.629.5) 126 1.64 (1.611.68) 25.3 (24.526.1) 36.5 (34.838.0) 74.1 (60.487.1) 27.6 (21.933.0) 224 1.63 (1.601.67) 25.3 (24.426.3) 36.7 (35.137.9) 74.1 (60.683.6) 27.7 (22.831.5) 286 1.63 (1.591.67) 25.2 (24.326.0) 36.7 (35.238.1) 73.5 (62.382.5) 27.8 (23.231.1) 101 1.62 (1.581.65) 25.3 (24.625.9) 36.5 (35.137.9) 69.3 (59.476.9) 26.5 (22.729.3) 27.9 (23.532.0) 8.6 (7.19.9) 58.0 (52.463.5) 29.5 (26.032.5) 8.7 (7.210.1) 57.1 (51.263.5) 26.8 (22.331.1) 9.0 (7.410.5) 54.7 (48.759.6) 25.3 (21.628.3) 9.3 (7.710.6) 52.5 (47.157.8) 23.9 (21.026.8) 10.3 (8.611.5) 50.5 (47.154.7) 21.8 (17.725.0) 10.9 (9.012.8) 48.6 (44.653.6) 0.37 ( 0.320.95) 0.76 (0.061.51) 0.42 ( 0.130.90) 0.64 (0.061.24) 0.32 ( 0.531.13) 0.47 ( 0.411.30) 0.04 ( 0.790.78) 0.27 ( 0.570.94) 0.39 ( 0.521.23) 0.27 ( 0.360.89) 1.07 (0.421.85) 0.66 (0.131.18) 182 162 112 103 123 115 224 195 283 257 99 87 105 1.79 (1.741.83) 28.2 (27.128.8) 40.1 (39.041.5) 85.2 (73.991.6) 26.5 (23.528.6) 45 1.77 (1.741.81) 27.6 (26.828.7) 39.1 (38.040.7) 85.2 (75.193.0) 27.3 (24.630.7) 37 1.78 (1.731.84) 27.9 (26.729.0) 39.5 (37.741.0) 88.4 (79.395.8) 28.0 (24.730.8) 41 1.76 (1.721.83) 27.9 (27.028.9) 39.4 (37.841.2) 88.8 (76.9100.3) 28.5 (24.431.4) 95 1.78 (1.741.81) 28.1 (26.929.2) 40.0 (38.341.3) 88.2 (76.197.2) 27.8 (24.230.4) 69 1.75 (1.701.79) 27.8 (26.928.9) 39.5 (37.640.7) 87.0 (75.895.8) 28.4 (25.131.4) 47.6 (40.755.5) 8.5 (7.09.6) 60.1 (55.864.7) 47.4 (40.454.9) 8.2 (6.19.6) 58.7 (53.166.2) 48.3 (43.053.2) 8.8 (7.39.4) 57.1 (50.564.5) 44.6 (42.048.6) 9.0 (7.39.9) 53.3 (47.760.6) 41.3 (32.948.1) 9.8 (7.911.1) 53.4 (49.257.3) 31.6 (24.839.2) 11.7 (8.913.2) 48.0 (45.251.6) 0.05 ( 0.630.59) 0.04 ( 0.860.80) 0.29 ( 0.900.05) 0.28 ( 1.10.25) 0.07 ( 1.020.55) 0.12 ( 0.800.49) 0.06 ( 0.580.72) 0.09 ( 0.810.52) 0.90 (0.011.66) 0.25 ( 0.550.88) 0.56 ( 0.221.19) 0.12 ( 0.680.30) 104 90 44 41 37 32 40 37 91 81 68 55 aBMD = areal bone mineral density; BMI = body mass index; HRpQCT = high-resolution peripheral quantitative computed tomography; PROMIS-PF = physical function domain of the National Institues of Health Patient-Reported Outcomes Measurement Information System. a Data are median (interquartile range), except for frequencies. stack within the constraints of the z-axis of the scanner; presence of a local tomography artifact due to absorbing tissue outside of field of view) (n = 47 tibia scans). Reference data in males was generated from 384 and 336 scans of the radius and tibia, respectively (Table 2). Lower scan numbers than par ticipants was due to: 1) scan not performed due to time constraints (n = 6 radius and 31 tibia scans); 2) excessive motion artifact (n = 2 radius and 1 tibia scans) and; 3) participant size (n = 24 tibia scans). load) (all partial correlations = 0.280 to 0.452) (Table 3). Bone length explained 7.9 % to 20.5 % of the variance in the estimated mechanical properties. The highest scaling factors at both sites and in both sexes were for bone length's relationship with the estimated ability to resist bending (IMIN, IMAX) and torsion (pMOI) (all scaling factors = 1.652 to 2.113). Estimated ability to resist compression forces (stiffness and failure load) scaled to length at both the radius and tibia with scaling factors ranging from 0.934 to 1.019 in females and 0.685 to 0.984 in males. 3.2. Correlations and scaling factors 3.3. Centile curves There were negative relationships between density outcomes (Tot Den, TMD, CtvBMD) and bone length at both the radius and tibia in females and males (all partial correlations = 0.191 to 0.055) (Table 3). However, correlations did not rise to the level of R2 0.05 required for density data to be scaled to bone length. Similarly, bone length had low value in explaining radial and tibial BA/TA, CtTh or CtPo in either sex (all R2 < 0.05). Bone length explained 8.7 % to 13.6 % of the variance in radius areas (TA, BA, CtAr) and 9.9 % to 19.3 % of the variance in tibial areas (TA, BA, CtAr) in both sexes (Table 3). Areas in females scaled to length with scaling factors ranging from 0.934 to 1.009 for both the radius and tibia. Areas in males scaled to length with scaling factors ranging from 0.685 to 0.888 at the tibia and 0.856 to 0.984 at the radius. There were positive relationships at both sites and in each sex be tween bone length and the estimated ability to resist bending (IMIN, IMAX), torsion (pMOI), and compression forces (stiffness and failure Centile curves for raw CtvBMD, CtTh and CtPo outcomes, which did not satisfactorily scale to bone length, have previously been published (Warden et al., 2022b). Similar curves for TotDen, TMD, and BA/TA at the radius and tibia are presented in Supplemental Files 1 and 2, respectively. The fitted median centile curve for TMD peaked between 35 and 40 years of age in both sexes before declining thereafter. The decline in females was more precipitous, especially between 40 and 60 years of age. HRpQCT raw values for outcomes correlating with bone length at R2 0.05 (TA, BA, CtAr, CtPm, IMIN, IMAX, pMOI, stiffness, failure load) were converted to scaled values using the scaling factors and the ratio of the individual's bone length to sex specific median bone length (females = 25 cm for radius and 37 cm for tibia; males = 28 cm for radius and 40 cm for tibia). For example, for a female with a radial bone length of 24.2 cm and raw IMIN outcome of 514 mm4, the scaled IMIN would be 4 S.J. Warden et al. Bone Reports 20 (2024) 101735 females and an additional 82 radius (+27 %) and 88 (+35 %) tibia scans in males. Beyond an expanded dataset, the current study provides cen tile curves for HRpQCT outcomes previously without reference data, including TotDen, TMD, TA, BA, BA/TA, IMIN, IMAX, and pMOI. In addition, we explored the relationship between bone length and HRpQCT outcomes, developed a means of scaling outcomes for bone length, and generated centile curves for bone length adjusted outcomes. To facilitate the utility of the latter curves, Excel-based calculators (Supplementary Files 5 and 6) were developed to calculate age- and sexmatched percentiles and z-scores for both raw and bone length adjusted outcomes. Many of the HRpQCT outcomes at the radial and tibial diaphysis in both sexes were related to bone length, consistent with established in terrelationships between bone length and cross-sectional size (Biewener, 2005; Ruff, 1984). Size outcomes (TA, BA, CtAr, CtPm) positively correlated with bone length confirming that individuals with longer bones also had wider bones. As bones with larger cross-sectional size have material distributed further from bending axes, bone length also correlated with estimates of bone rigidity and strength (IMIN, IMAX, pMOI, stiffness, failure load). There were no relationships between bone length and TotDen, TMD, CtvBMD, and BA/TA as the later HRpQCT outcomes are already expressed relative to bone size. There was no relationship between CtTh and bone length. This is consistent with previous work (Bjrnerem et al., 2013) and likely re flects a means of minimizing the energy costs of larger and heavier bones. Bending resistance increases to the fourth power of the radius of a bone. By placing material further from its central axis and increasing its radius, a larger bone is disproportionately stronger for the same mass and energy cost. The more distant distribution of material relatively thins the cortex such that CtTh does not correspondingly increase with the increase in size of longer bones. Despite the relatively thinner CtTh, compressive strength is preserved via the increase in CtAr as bone length increases (Seeman, 2003). Scaling factors were calculated for outcomes for which bone length explained at least 5 % of variance (i.e., R2 0.05). There is no accepted cut-off in the literature. Our cut-off was chosen based on the rationale that a lesser relationship implied bone length had limited explanatory value, and is a more liberal cut-off than (for example) the 10 % cut-off implemented when adjusting DXA Z-scores for height in children (Zemel et al., 2011). Outcomes expressed in linear dimensions (e.g., CtPm) exhibited negative allometry, with scaling factors around 0.5 indicating disproportionately lower increases in size relative to in creases in bone length. Relatively isometric allometry (scaling factors around one) was observed for outcomes expressed in squared linear dimensions (e.g., TA, BA, CtAr), indicating bone cross-sectional area measures increased proportionally as bone length increased. Outcomes to the fourth power of linear dimensions (e.g., IMIN, IMAX, pMOI) exhibited positive allometry with scaling factors around two. The latter indicates estimated bone rigidity had disproportionately greater in creases relative to increases in bone length. The scaling factors were used to scale HRpQCT outcomes to bone length, and centile curves were generated for both raw and bone length scaled HRpQCT data. The centile curves can be used to calculate zscores. Excel-based calculators are provided to facilitate this process (Supplementary Files 5 and 6). The raw and scaled z-scores indicate the magnitude that an individual's HRpQCT outcomes differ relative to ex pected sex- and age-specific values. The difference between the two zscores being that the scaled z-score also considers bone length. The consideration of bone length enables it to be determined whether an individual has normal sized bones for their bone length. For example, consider a 43-year-old female with a tibial length of 33.5 cm and tibial pMOI of 15,000 mm4. Their raw z-score for pMOI using the Excel-based calculator equates to 0.738 indicating they have lower than expected torsional rigidity for their sex and age. This may be sug gestive of reduced cross-sectional bone development and an increased risk of injury, such as a bone stress injury (Warden et al., 2021b). Table 3 Spearman partial (on age) correlation between bone length and HRpQCT out comes, and scaling factors (SF) for outcomes exhibiting correlation with an R2 > 0.05. Site and HRpQCT outcome Female Spearman Male R 2 SF Spearman R2 SF Radius TotDen TMD CtvBMD TA BA CtAr BA/TA CtPm CtTh CtPo IMIN IMAX pMOI Stiffness Failure load 0.067 0.109 0.088 0.376 0.369 0.367 0.048 0.356 0.170 0.010 0.418 0.325 0.371 0.370 0.363 0.004 0.012 0.008 0.141 0.136 0.135 0.002 0.127 0.029 0.010 0.175 0.105 0.138 0.137 0.132 1.009 0.934 0.934 0.503 2.095 1.852 1.951 0.966 0.934 0.120 0.144 0.145 0.329 0.303 0.296 0.082 0.329 0.138 0.008 0.353 0.284 0.320 0.291 0.280 0.014 0.021 0.021 0.108 0.092 0.087 0.007 0.108 0.019 <0.001 0.125 0.081 0.103 0.085 0.079 0.984 0.866 0.856 0.530 1.989 1.858 1.912 0.865 0.829 Tibia TotDen TMD CtvBMD TA BA CtAr BA/TA CtPm CtTh CtPo IMIN IMAX pMOI Stiffness Failure load 0.055 0.191 0.182 0.439 0.409 0.406 0.002 0.444 0.207 0.027 0.393 0.443 0.452 0.424 0.432 0.003 0.036 0.033 0.193 0.168 0.165 0.000 0.197 0.043 <0.001 0.154 0.196 0.205 0.179 0.187 0.999 0.959 0.949 0.544 1.831 2.113 2.009 1.007 1.019 0.218 0.209 0.222 0.419 0.328 0.314 0.147 0.172 0.070 0.159 0.374 0.397 0.418 0.318 0.328 0.048 0.044 0.049 0.176 0.107 0.099 0.022 0.415 0.005 0.025 0.140 0.158 0.175 0.101 0.107 0.888 0.702 0.685 0.508 1.652 1.755 1.713 0.700 0.719 514.5/(24.2/25)2.095 = 550.7 mm4. Centile curves fitted to scaled TA, BA, pMOI, and failure load out comes are presented for the radius (Fig. 1) and tibia (Fig. 2). Centile curves fitted to scaled CtAr, CtPm, IMIN, IMAX, and stiffness outcomes are provided in Supplemental Files 3 (radius) and 4 (tibia). 3.4. Percentile and z-score calculator, and centile curve plotter Excel-based calculators were developed for both the radius (Sup plemental File 5) and tibia (Supplemental File 6). Entry of basic de mographic information (sex, date of birth, scan date, and bone length) and one or more HRpQCT outcome (Fig. 3A) results in plotting of sexspecific centile curves (Fig. 3B). The centile curves are based on the curves fitted using the LMS approach fitted to raw data (TotDen, TMD, CtvBMD, Ct.Th) or bone-length scaled data (TA, BA, CtAr, CtPm, IMIN, IMAX, pMOI, stiffness, failure load), depending on whether the outcome was related to bone length at R2 0.05. Beneath each curve, the raw entered and bone length scaled value for the HRpQCT outcome is pro vided along with the associated z-score and percentile (Fig. 3C). The raw z-score and percentile are derived from curves fitted to the raw reference data, whereas the scaled z-score and percentile are derived from curves fitted to the scaled reference data. 4. Discussion The current study expands our previously published reference data for HRpQCT outcomes at the cortical bone rich radial and tibial di aphyses (Warden et al., 2022b). We included data in the current analyses from an additional 173 radius (+20 %) and 171 tibia (+23 %) scans in 5 S.J. Warden et al. Bone Reports 20 (2024) 101735 Fig. 1. Bone length scaled data and fitted centile curves for total area (A, E), bone area (B, F), polar moment of inertia (C, G), and estimated failure load (D, H) at the radial diaphysis for females (top row) and males (bottom row). Fig. 2. Bone length scaled data and fitted centile curves for total area (A, E), bone area (B, F), polar moment of inertia (C, G), and estimated failure load (D,H) at the tibial diaphysis for females (top row) and males (bottom row). However, the individual also has a tibial length that is lower than the median 37 cm for females. When bone length is considered, a scaled pMOI of 18,314 mm4 is calculated (15,000/[33.5/37]2.009) and a scaled z-score of 0.002 obtained indicating relatively normal torsional ri gidity for their bone length. Scaled HRpQCT outcomes will be higher and lower than raw HRpQCT outcomes for individuals with bone lengths shorter and longer than median values in the current reference cohort, respectively. This is because scaled values were normalized to sex-specific median bone length. The latter was performed so that scaled outcomes would retain the same units and be within the same range as raw outcomes. However, the scaling approach in our cohort does raise the question of whether the bone lengths in our cohort are representative. Percutaneous measures of ulna (used as a surrogate for radius length) and tibia length are increasingly being performed prior to HRpQCT to enable the scanning region to be positioned relative to bone length, which has advantages over scanning at a fixed distance offset (Bonaretti et al., 2017; Ghasem-Zadeh et al., 2017; Okazaki et al., 2021; Shanb hogue et al., 2015). However, measured mean or median bone lengths are either not reported (Shanbhogue et al., 2015), not dichotomized by sex (Bonaretti et al., 2017; Shanbhogue et al., 2015) or acquired in a race and/or ethnicity with a different stature (Okazaki et al., 2021), negating the ability to compare to lengths acquired in the current study. GhasemZadeh et al. (Ghasem-Zadeh et al., 2017) did report forearm lengths in White females (25.7 cm) and males (28.1 cm) which compare favorably to our measured lengths of 25 cm and 28 cm in females and males, respectively. Similarly, comparable lengths of 24.7 cm and 27.5 cm have been reported in another cohort of White females and males, respec tively (Madden et al., 2012), and the 40 cm tibial length in males in our study matches the 40.2 cm measured in 18-year-old U.S. military re cruits (Nieves et al., 2005). In the absence of a wealth of percutaneously measured bone length data in the literature, a feasible proxy is to compare the heights of our individuals to population norms. Height and bone length are closely related, so much so that bone length is frequently used to estimate an individual's height. The average height within each decade of age in our cohort (Table 2; females = 1.621.64 m, males = 1.751.79 m) matches that of the U.S. White adult population (females = 1.62 m, males = 1.77 cm) (Fryar et al., 2021). The comparable height provides confidence that our measured median bone lengths are representative of the broader U. S. White population. We assessed other outcomes to explore the comparability of our 6 S.J. Warden et al. Bone Reports 20 (2024) 101735 principally developed for distal bone sites with a greater proportion of trabecular bone. The ability of the model to estimate failure load in other loading directions and at the cortical bone rich diaphysis remains to be established. Finally, we had more limited inclusion of males and older (age >70 yrs) adults and our data are specific to White individuals living in the Midwest of the United States. HRpQCT outcomes vary by race potentially requiring the generation of separate reference data for other races (van den Bergh et al., 2021). In summary, the current study expands our previous dataset by providing reference data for additional HRpQCT outcomes, including TotDen, TMD, TA, BA, BA/TA, IMIN, IMAX, and pMOI. More importantly, the study provides a means of scaling outcomes for bone length and provides reference data for bone length adjusted outcomes. The refer ence data enable HRpQCT outcomes in an individual (or population of interest) to be expressed relative to the reference cohort to determine if they are big boned for their age, sex and bone length. Supplementary data to this article can be found online at https://doi. org/10.1016/j.bonr.2024.101735. CRediT authorship contribution statement Stuart J. Warden: Conceptualization, Data curation, Formal anal ysis, Funding acquisition, Investigation, Methodology, Project admin istration, Supervision, Writing original draft, Writing review & editing. Robyn K. Fuchs: Conceptualization, Data curation, Formal analysis, Writing review & editing. Ziyue Liu: Formal analysis, Writing review & editing. Katelynn R. Toloday: Investigation, Writing review & editing. Rachel Surowiec: Data curation, Investi gation, Methodology, Writing review & editing. Sharon M. Moe: Conceptualization, Funding acquisition, Supervision, Writing review & editing. Declaration of competing interest Fig. 3. Screenshots of the Excel-based calculator for tibia outcomes (available in Supplemental File 5). Following entry of basic demographic information and one or more HRpQCT outcome (A), centile curves are plotted (B), and sex- and age-specific raw and bone length scaled z-scores and percentiles are calcu lated (C). None. Data availability Data will be made available on request. cohort to the broader population, including DXA-derived bone out comes, performance on physical function tests, and self-reported phys ical function (Table 1). DXA z-scores at the hip and spine were slightly higher than zero in females and approximated zero in males. Grip strength in our cohort according to decade stage of life mirrored refer ence values for individuals residing in the U.S. (Wang et al., 2018), whereas time to complete five sit-to-stand maneuvers matched or was slightly slower (Bohannon, 2006; Bohannon et al., 2010). Self-reported physical function (PROMIS-PF T-score) was slightly above the popula tion mean of 50, depending on sex and decade stage of life. These cu mulative data suggest our cohort had slightly above-to-normal general bone health and equivalent or a slightly higher level of functioning than the general U.S. population. Our study has several strengths, but it is also not without limitations. Data were obtained at a single center and variability in machine per formance at other centers may influence outcomes. The outcomes are specific to the sites scanned and the scanning, segmentation, and anal ysis procedures used. We used the manufacturer's bone midshaft eval uation script with a single outer contour and without an inner clockwise (i.e., negative excluding) contour. This means that outcomes using this script (including TotDen and TMD) include any trabecular bone present at the scan sites. This approach was selected as we wanted to include any trabecularized cortical bone, which increases with age (Zebaze et al., 2010). The micro-finite element model used to estimate bone strength is specific to axial compressive loading and was Acknowledgements This contribution was made possible by support from the National Institutes of Health (NIH/NIAMS P30 AR072581), and the Indiana Clinical Translational Science Award/Institute (NCATS UL1TR00252901). 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CT muscle density, D3Cr muscle mass, and body fat associations with physical performance, mobility outcomes, and mortality risk in older men. J. Gerontol. A Biol. Sci. Med. Sci. 77 (4), 790799. https://doi.org/10.1093/gerona/ glab266. 8 ...
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- ... LODAY CONSTRUCTIONS ON TWISTED PRODUCTS AND ON TORI arXiv:2002.00715v1 [math.AT] 3 Feb 2020 ALICE HEDENLUND, SARAH KLANDERMAN, AYELET LINDENSTRAUSS, BIRGIT RICHTER, AND FOLING ZOU Abstract. We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology. Introduction When one studies commutative rings or ring spectra, important homology theories are topological Hochschild or its higher versions. These are specific examples of the Loday construction, whose definition relies on the fact that commutative ring spectra are enriched in simplicial sets: for a simplicial set X and a commutative ring spectrum R one can define the tensor X R as a simplicial spectrum whose n-simplices are ^ R. xXn By slight abuse of notation X R also denotes the commutative ring spectrum that is the geometric realization of this simplicial spectrum. This recovers topological Hochschild homology of R, THH(R), when X = S 1 , and higher topological Hochschild homology, THH[n] (R), for higher dimensional spheres S n . Tensoring satisfies several properties [8, VII, 2, 3], two of which are: If X is a homotopy pushout, X = X1 hX0 X2 , then the tensor product of R with X splits as a homotopy pushout in the category of commutative ring spectra which is the derived smash product: (X1 hX0 X2 ) R (X1 R) L (X0 R) (X2 R). A product of simplicial sets X Y gives rise to an iterated tensor product: (X Y ) R X (Y R). This last expression does not, however, imply that calculating the homotopy groups of (XY )R is easy. In particular, if one iterates the trace map from algebraic K-theory to topological Hochschild homology n times, one obtains a map K (n) (R) = K(K(. . . (K (R)) . . .)) (S 1 . . . S 1 ) R. | {z } | {z } n n Since iterated K-theory is of interest in the context of chromatic red-shift, one would like to know as much about (S 1 . . . S 1 ) R as possible. Date: February 4, 2020. 2000 Mathematics Subject Classification. Primary 18G60; Secondary 55P43. Key words and phrases. torus homology, (higher) Hochschild homology, (higher) topological Hochschild homology, stability, twisted Cartesian products. 1 In some good cases, the homotopy type of X R only depends on the suspension of X in the sense that if X Y , then one has X R Y R. This property is called stability. Stability for instance holds for Thom spectra R that arise from an infinite loop map to the classifying space BGL1 (S) (see Theorem 1.1 of [25]), or for R = KU or R = KO [14, 4]. One can also work relative to a fixed commutative ring spectrum R and consider commutative R-algebra spectra A and ask whether X R A only depends on the homotopy type of X. In this paper, we will often work with coefficients: we look at pointed simplicial sets X and place a commutative A-algebra spectrum C at the basepoint of X. In other words, when X is pointed then the inclusion of the basepoint makes X R A into a commutative A-algebra and we can look at LR X (A; C) = (X R A) A C, the Loday construction with respect to X of A over R with coefficients in C. We call the pair (A; C) stable if the homotopy type of LR X (A; C) only depends on the homotopy type of X. Note that the ring R is not part the notation when we say that (A; C) is stable although the question depends on the choice of R, so the context should specify the R we are working over. We call the commutative R-algebra A multiplicatively stable as in [14, R Definition 2.3] if X Y implies that LR X (A) LY (A) as commutative A-algebra spectra. If A is multiplicatively stable, then for any cofibrant commutative A-algebra C, the pair (A; C) is stable (see [14, Remark 2.5]). We investigate several algebraic examples, i.e., commutative ring spectra that are Eilenberg Mac Lane spectra of commutative rings. For instance we show that the pairs (HQ[t]/tm ; HQ) are not stable for all m > 2, extending a result by Dundas and Tenti [7]. We also prove integral and mod-p versions of this result. Work of Berest, Ramadoss and Yeung implies that the homotopy types of LHk X (HA; Hk) and Hk LX (HA) only depend on the homotopy type of X if k is a field and if A is a commutative Hopf algebra over k. We generalize this result to commutative Hopf algebra spectra. Moore introduced twisted cartesian products as simplicial models for fiber bundles. We develop a Serre type spectral sequence for Loday constructions of twisted cartesian products where the twisting is governed by a constant simplicial group. As a concrete example we compute the Loday construction with respect to the Klein bottle for a polynomial algebra over a field with characteristic not equal to 2. Content. In Section 1 we recall the definition of the Loday construction and fix notation. Section 2 contains the construction of a spectral sequence for the homotopy groups of Loday constructions with respect to twisted cartesian products. Our results on commutative Hopf algebra spectra can be found in Section 3. In Section 4 we prove that truncated polynomial algebras of the form Q[t]/tm and Z[t]/tm for m > 2 are not multiplicatively stable by comparing the Loday construction of tori to the Loday construction of a bouquet of spheres corresponding to the cells of the tori. We also show that for 2 6 m < p the Fp -algebra Fp [t]/tm is not stable. Acknowledgements. We thank the organizers of the third Women in Topology workshop, Julie Bergner, Angelica Osorno, and Sarah Whitehouse, and also the Hausdorff Institute of Mathematics for their hospitality during the week of the workshop. We thank the Hausdorff Research Institute for Mathematics, grants nsf-dms 1901795 and nsf-hrd 1500481AWM ADVANCE, and the Foundation Compositio Mathematica for their support of the workshop. We thank Maximilien Peroux for help with coalgebras in spectra, Inbar Klang for a helpful remark about norms, Mike Mandell for a helpful discussion on En -spaces, Jelena Grbic for pointing out [23] to us, and Thomas Nikolaus for -category support. AL was supported by Simons Collaboration Grant 359565. The last two authors thank the Department of Mathematics at Indiana University for its hospitality and BR thanks the Department of Mathematics at Indiana University for support as a short-term research visitor in 2019. 2 1. The Loday construction: basic features We recall some definitions concerning the Loday construction and we fix notation. For our work we can use any good symmetric monoidal category of spectra whose category of commutative monoids is Quillen equivalent to the category of E -ring spectra, such as symmetric spectra [12], orthogonal spectra [17] or S-modules [8]. As parts of the paper require us to work with a specific model category we chose to work with the category of S-modules. Let X be a finite pointed simplicial set and let R A C be a sequence of maps of commutative ring spectra. Definition 1.1. The Loday construction with respect to X of A over R with coefficients in C is the simplicial commutative augmented C-algebra spectrum LR X (A; C) given by ^ LR (A; C) = C A n X xXn \ where the smash products are taken over R. Here, denotes the basepoint of X and we place a copy of C at the basepoint. The simplicial structure of LR X (A; C) is straightforward: Face maps di on X induce multiplication in A or the A-action on C if the basepoint is involved. Degeneracies si on X correspond to the insertion of the unit maps A : R A over all n-simplices which are not hit by si : Xn1 Xn . As defined above, LR X (A; C) is a simplicial commutative augmented C-algebra spectrum. In the following we will always assume that R is a cofibrant commutative S-algebra, A is a cofibrant commutative R-algebra and C is a cofibrant commutative A-algebra. This ensures that the homotopy type of LR X (A; C) is well-defined and depends only on the homotopy type of X. Remark 1.2. When R A C is a sequence of maps of commutative rings, we can of course use the above definition for HR HA HC. The original construction by Loday [15, Proposition 6.4.4] used O C A xXn \ instead with the tensors taken over R as the n-simplices in LR X (A; C). This algebraic definition also makes sense if R is a commutative ring and A C is a map of commutative simplicial R-algebras. It continues to work if R is a commutative ring and A C is a map of graded-commutative R-algebras, with the n-simplices defined as above, but the maps between them require a sign correction as terms are pulled past each othersee [21, Equation (1.7.2)]. An important case is X = S n . In this case we write THH[n],R (A; C) for LR S n (A; C); this is the higher order topological Hochschild homology of order n of A over R with coefficients in C. Let k be a commutative ring, A be a commutative k-algebra, and M be an A-module. Then we define THH[n],k (A; M ) := LHk S n (HA; HM ). If A is flat over k, then THHk (A; M ) = HHk (A; M ) [8, Theorem IX.1.7] and this also holds for [n],k higher order Hochschild homology in the sense of Pirashvili [21]: THH[n],k (A; M ) = HH (A; M ) if A is k-flat [4, Proposition 7.2]. Given a commutative ring A and an element a A, we write A/a instead of A/(a). 3 2. A spectral sequence for twisted cartesian products We will start by letting R A be a map of commutative rings and we study Loday constructions LR B (A ) over a finite simplicial set B, where indicates a twisting by a discrete group G that acts on A via ring isomorphisms. This construction can be adapted as in Definition 1.1 and Remark 1.2 to allow coefficients in an A-algebra C if B is pointed, and to the case where R A is a map of commutative ring spectra, or R is a commutative ring and A is a graded-commutative R-algebra or a simplicial commutative R-algebra. If we have a twisted cartesian product (TCP) in the sense of [18, Chapter IV] E( ) = F B where the fiber F is a simplicial R-algebra and the simplicial structure group G acts on F by simplicial R-algebra isomorphisms, it is possible to generalize this definition of the Loday construction to allow twisting by a simplicial structure group, as expained in Definition 2.1 below. We show an example where such a TCP arises: if we start with a TCP of simplicial sets E( ) = F B with twisting in a simplicial structure group G acting on F simplicially on the left and with a map of commutative rings R A, we can use that twisting to construct a TCP with fiber equal R to the simplicial commutative R-algebra LR F (A) and with the structure group G acting on LF (A) by R-algebra isomorphisms. In that situation, we get that LR (A) = LR (LR (A) ), B E( ) F R R which generalizes the fact that for a product, LR F B (A) = LB (LF (A)). If the structure group G is discrete, i.e., if G is a constant simplicial group, LR E( ) (A) can be written as a bisimplicial set and we get a spectral sequence for calculating its homotopy groups. Definition 2.1. Let B be a finite simplicial set, R be a commutative ring, and A be a commutative R-algebra (or a graded-commutative R-algebra, or a simplicial commutative R-algebra). Let G be a discrete group acting on A from the left via isomorphisms of R-algebras, and let be a function from the positive-dimensional simplices of B to G so that (2.2) (b) (di b) (si b) (s0 b) = = = = [ (d0 b)]1 (d1 b) (b) (b) eG for for for for q > 1, b Bq , i > 2, q > 1, b Bq , i > 1, q > 0, b Bq , and q > 0, b Bq . The twisted Loday construction with respect to B of A over R twisted by is the simplicial com mutative (resp., graded-commutative, or bisimplicial commutative) R-algebra LR B (A ) given by O R LR A B (A )n = LBn (A) = bBn where the tensor products are taken over R, with O O Y d0 fb = gc with gc = (b)(fb ), cBn1 bBn di O bBn si O bBn fb = fb = O b:d0 b=c gc with gc = cBn1 O Y fb for 1 i n, and b:di b=c hd with hd = dBn+1 Y fb for 0 i n. b:si b=d We should think of the copy of A sitting over a simplex b Bn as sitting over its 0th vertex, and of (b) as translating between the A over bs 0th vertex and the A over bs 1st vertex. 4 Lemma 2.3. The definition above makes LR B (A ) into a simplicial set. Proof. To check this we need only check the relations involving d0 , since the ones that do not involve work in the same way that they do in the usual Loday construction. For j > 1, we get d0 dj = dj1 d0 because in both terms, for any c Bn2 we get the product over all b Bn with d0 dj b = dj1 d0 b = c of terms that are either (b)(fb ) or (dj b)(fb ). These are the same by the condition in Equation (2.2) above. For j = 1, we get the product over all b Bn with d0 d1 b = d0 d0 b = c of terms that are either (d1 b)(fb ) or (d0 b) (b)(fb ), which again agree by Equation (2.2). We get d0 s0 = id since (s0 b) = eG , and d0 si = si1 do for i > 0 since for those i, (si b) = (b). Following Moore, May considers the following simplicial version of a fiber bundle [18, Definition 18.3]: Definition 2.4. Let F and B be simplicial sets and let G be a simplicial group which acts on F from the left. Let : Bq Gq1 for all q > 0 be functions so that d0 (b) (di+1 b) (si+1 b) (s0 b) = = = = [ (d0 b)]1 (d1 b) di (b) si (b) eq for for for for q > 1, b Bq , i 1, q > 1, b Bq , i 0, q > 0, b Bq , and q > 0, b Bq . The twisted Cartesian product (TCP) E( ) = F B is the simplicial set whose n-simplices are given by E( )n = Fn Bn , with simplicial structure maps (i) d0 (f, b) = ( (b) d0 f, d0 b), (ii) di (f, b) = (di f, di b) i > 0, and (iii) si (f, b) = (si f, si b) i 0. These structure maps satisfy the necessary relations to be a simplicial set because of the conditions that satisfies. Definition 2.5. If R is a commutative ring and E( ) = C B is a TCP as in Definition 2.4 where C is a commutative simplicial R-algebra and the simplicial group G acts on C by R-algebra isomorphisms (that is, for every q 0, the group Gq acts on the commutative R-algebra Cq by R-algebra isomorphisms) then we can use the twisting to define the twisted Loday construction with respect to B of C over R, twisted by , O R Cn LR B (C )n = LBn (Cn ) = bBn N with twisted structure maps given on monomials bBn fb , with fb Cn for all b Bn , by O Y O (b)(d0 fb ), gc with gc = d0 fb = cBn1 bBn (2.6) di O bBn si O bBn fb = fb = O b:d0 b=c gc with gc = cBn1 O Y di fb for 1 i n, and b:di b=c hd with hd = dBn+1 Y b:si b=d 5 si fb for 0 i n. Note that there are two sets of simplicial structure maps being used, those of C inside and those of B outside. This looks like the diagonal of a bisimplicial set, but since our twisting : Bq+1 Gq explains only how to twist elements in Cq , this is not the case unless the structure group G is a discrete group, viewed as a constant simplicial group. If the structure group G is discrete, there is overlap between Definition 2.1 and Definition 2.5. The simplicial commutative R-algebra case of Definition 2.1 actually gives a bisimplicial set: we use only the simplicial structure of B in the definition and if A also has simplicial structure, that remains untouched. The diagonal of that bisimplicial set agrees with the constant simplicial group case of Definition 2.5. Given any TCP of simplicial sets E( ) = F B as in Definition 2.4 and a map R A of commutative rings, we can construct LR F (A) B which is a TCP of commutative simplicial algebras R-algebras as in Definition 2.5 using the same structure group G and twisting function : Bq Gq1 . We use the simplicial left action of Gn on Fn which we denote by (g, f ) 7 gf to obtain a left action by simplicial R-algebra isomorphisms R Gn LR Fn (A) LFn (A) O O (g, af ) 7 ag1 f . (2.7) f Fn f Fn Since the original action of Gn on Fn was a left action, this is a left action. In the original monomial, the f th coordinate is af . After g Gn acts on it, the f th coordinate is bf = ag1 f . After h Gn acts on the result of the action of g, the f th coordinate is bh1 f = ag1 h1 f , which is the same as the result of acting by hg on the monomial. Proposition 2.8. If E( ) = F B is a TCP and R A is a map of commutative rings, and we use the simplicial set twisting function to construct a simplicial R-algebra twisting function to obtain a TCP LR F (A) B as above, we get that LR (A) = LR (LR (A) ). B E( ) F This uses the definition of the Loday construction of a simplicial algebra twisted by a simplicial group in Definition 2.5. Proposition 2.8 generalizes the well-known fact that for a product of simplicial sets, LR (A) = LR (LR (A)). F B B F R R Proof. Both LR E( ) (A) and LB (LF (A) ) have the same set of n-simplices for every n > 0: O O O O A= A ( A). = eE( )n bBn (f,b)Fn Bn f Fn We have to show that the simplicial structure maps agree with respect to this identification. For 1 6 i 6 n, for any choice of elements x(f,b) A, O O y(g,c) di x(f,b) = (g,c)Fn1 Bn1 (f,b)Fn Bn where y(g,c) = Y x(f,b) = Y b:di b=c (f,b):(di f,di b)=(g,c) Y f :di f =g x(f,b) . The internal product on the right-hand side is what we get from di on LR F (A) and the external product is what we get from di of LR , so this agrees with the definition in Equation (2.6). B 6 The proof that the si , 0 6 i 6 n agree is very similar. The interesting case is that of d0 . For any choice of elements x(f,b) A, the boundary d0 associated to LR E( ) (A) satisfies O O (2.9) d0 x(f,b) = y(g,c), (f,b)Fn Bn (g,c)Fn1 Bn1 where y(g,c) = Y x(f,b) = Y b:d0 b=c (f,b):d0 (f,b)=(g,c) Y f : (b)d0 f =g x(f,b) . R From the LR B (LF (A)) point of view, by Equation (2.6). O O O Y O d0 ( x(f,b) ) = (b)d0 x(f,b) bBn f Fn f Fn cBn1 b:d0 b=c = O Y O Y cBn1 b:d0 b=c = cBn1 b:d0 b=c = O (b) O Y gFn1 f :d0 f =g O Y x(f,b) gFn1 f :d0 f = (b)1 g Y (g,c)Fn1 Bn1 b:d0 b=c which is exactly what we got in (2.9). Y x(f,b) f :d0 f = (b)1 g x(f,b) , If G is a discrete group and E( ) is constructed using G, then for every q > 0 there is a function : Bq G satisfying the conditions listed in Equation (2.2) and G acts simplicially on F on the left. Theorem 2.10. If E( ) = F B is a TCP where the twisting is by a constant simplicial group G and if R A is a map of commutative rings so that (LR F (A)) is flat over R, then there is a spectral sequence (2.11) 2 R R Ep,q = p ((LR B ( LF (A) ))q ) p+q (LE( ) (A)). Here, LR F (A) is a graded commutative R-algebra. For any fixed p and q, we consider the degree R R R q part of LR Bp ( LF (A) ), (LBp ( LF (A) ))q . This forms a simplicial abelian group which in degree R p is (LR Bp ( LF (A)))q , with simplicial structure maps induced by those of B with the twisting by R , and p ((LR B ( LF (A) ))q ) denotes its pth homotopy group. The flatness assumption above is for instance satisfied if R is a field. Proof. Since the twisting is by a constant simplicial group G, we are able to form a bisimplicial R-algebra O O (2.12) (m, n) 7 A. bBm f Fn In the n-direction, the simplicial structure maps dFi and sFi will simply be the simplicial structure R maps of the Loday construction LR F (A) applied simultaneously to all the copies of LF (A) over all 7 B the b Bn . In the m direction, dB i and si are the simplicial structure maps of the twisted Loday construction, as in Equation (2.2) in Definition 2.1. These commute exactly because the simplicial structure maps in G are all equal to the identity. For any choice of xb LR F (A)n for all b Bm , O O O Y F xb = dB dFi (xb ) = (b)dFi (xb ) dB 0 di 0 bBm while dFi dB 0( O bBm cBm1 b:d0 b=c bBm xb ) = dFi which is the same since O Y cBm1 b:d0 b=c (b) xb = O Y dFi ( (b) xb ), cBm1 b:d0 b=c dFi ( (b) xb ) = di ( (b)) dFi (xb ) = (b) dFi (xb ). R R Note that since the twisting is by a constant simplicial group, LR E( ) (A) = LB (LF (A) ) is exactly the diagonal of the bisimplicial R-algebra in Equation (2.12). We use the standard result (see for instance [9, Theorem 2.4 of Section IV.2.2]) that the total complex of a bisimplicial abelian group with the alternating sums of the vertical and the horizontal face maps is chain homotopy equivalent to the usual chain complex associated to the diagonal of that bisimplicial abelian group. Since we know that the realization of the diagonal is homeomorphic to the double realization of the bisimplicial abelian group, in order to know the homotopy groups of the double realization of a bisimplicial abelian group, we can calculate the homology of its total complex with respect to the alternating sums of the vertical and the horizontal face maps. Filtering by columns gives an E 2 spectral sequence calculating the homology of the total complex associated to a bisimplicial abelian group consisting of what we get by first taking vertical homology and then taking horizontal homology. In the case of the bisimplicial abelian group we have in Equation (2.12), Pn the vertical qth homology of the columns will be the qth homology with respect to i=0 (1)i dFi of the complex O LR F (A) and this is isomorphic to q obtain N bBm q bBm R LF (A) . Since O bBm LR =( F (A) we assumed that (LR F (A)) is flat over R, we O (LR F (A)))q . bBm N Here, the subscript q denotes the degree q part of the graded abelian group bBm (LR F (A)). N R Moreover, the effect of the horizontal boundary map on bBm (LF (A)) is the boundary of the twisted Loday construction, with the action of G on the graded-commutative R-algebra (LR F (A)) R induced by that of G on the commutative simplicial R-algebra LF (A). As the boundary map preserves internal degree, we get the desired spectral sequence. 2.1. Norms and finite coverings of S 1 . The connected n-fold cover of S 1 given by the degree n map can be made into a TCP as follows. Let B = S 1 be the standard simplicial circle and Cn = hi be the cyclic group of order n with generator . The twisting function : Sq1 Cn sends the non-degenerate simplex in S11 to and is then determined by Equation (2.2). Let F = Cn , viewed as a constant simplicial set, and let Cn act on F from the left. Then E( ) = F B is in fact another simplicial model of S 1 with n non-degenerate 1-simplices. Therefore, LR (A) LR1 (A) and (LR (A)) = HHR (A) E( ) E( ) S 8 R n is the constant commutative for every commutative R-algebra A. In this case, LR FA = A simplicial R-algebra, with the Cn -action given by (a1 an ) = an a1 an1 . As LR F (A) is a constant simplicial object, we obtain that ( AR n , = 0, L R (A) = F 0, > 0. If A is flat over R, the spectral sequence of Equation (2.11) is 2 R n R Ep,q = p (LR ) )q p+q LR E( ) (A) = HHp+q (A). S 1 (A But here, the spectral sequence is concentrated in q-degree zero, and hence it collapses, yielding p (LR1 (AR n ) ) = HHR (A). p S LSE( ) (A) With Proposition 2.8 we can identify if A is a commutative ring spectrum and we recover the known result (see for instance [2, p. 2150]) that THHCn (NeCn A) THH(A). (2.13) 1 Here, THHCn (A) = NCSn (A) is the Cn -relative THH defined in [2, Definition 8.2], where NeCn A is the Hill-Hopkins-Ravanel norm. See also [1, Definition 2.0.1]. The identification in (2.13) is an 1 1 instance of the transitivity of the norm: NCSn NeCn A NeS A. 2.2. The case of the Klein bottle. For the Klein bottle we compute the homotopy groups of the Loday construction of the polynomial algebra k[x] for a field k using our TCP spectral sequence and we confirm our answer using the following pushout argument. We assume that the characteristic of k is not 2, so 2 is invertible in k. Note that the Klein bottle can be represented as a homotopy pushout K (S 1 S 1 ) hS 1 D 2 . Since the Loday construction converts homotopy pushouts of simplicial sets into homotopy pushouts of commutative algebra spectra, we obtain LkK (k[x]) LkS 1 S 1 (k[x]) L Lk S1 (k[x]) LkD2 (k[x]). Homotopy invariance of the Loday construction yields that LkD2 (k[x]) = k[x], and as LkS 1 (k[x]) = HHk (k[x]) is well known to be isomorphic to k[x] (x) as a graded commutative k-algebra, we get that LkS 1 S 1 (k[x]) = LkS 1 (k[x]) k[x] LkS 1 (k[x]) = k[x] (xa , xb ) where the indices a and b allow us to distinguish between the generators emerging from each of the circles Sa1 Sb1 . Let Sc1 represent the circle along which we glue the disk, and call the corresponding generator in dimension one for the Loday construction over it xc . Let Sa1 denote the circle that Sc1 will go twice around in the same direction and Sb1 denote the circle that it will go around in opposite directions. So we have a projection K Sb1 . We can calculate LkK (k[x]) with a Tor spectral sequence whose E 2 -page is (2.14) 2 E, = Lk (k[x]) Tor, S 1 LkS 1 S 1 (k[x]), LkD2 (k[x]) k[x](xc ) (k[x] (xa , xb ), k[x]). = Tor, We need to understand the LkS 1 (k[x])-module structure on k[x] (xa , xb ), so we need to understand the map k[x] (xc ) k[x] (xa , xb ). 9 Since k[x] in both cases is the image of the Loday construction on a point, we know that x on the left maps to x on the right. If we map Sc1 to Sa1 Sb1 and then collapse Sa1 to a point, we end up with a map Sc1 Sb1 that is contractible, so if we only look at the xb part of the image of xc in (xa , xb ) (that is, if we augment xa to zero) we get zero. We deduce that k[x](xc ) Tor, (k[x] (xa , xb ), k[x]) (x ) k[x] = Tor, (k[x], k[x]) Tor, c ((xa ), k) Tork, ((xb ), k) (x ) = k[x] Tor, c ((xa ), k) (xb ). (x ) In order to calculate Tor, c ((xa ), k), we map Sc1 to Sa1 Sb1 and then collapse Sb1 to a point. This gives a map Sc1 Sa1 that is homotopic to the double cover of the circle as depicted below. We consider elements of LSc1 (k[x]), which we think of as built on the top circle, and of LSa1 (k[x]), which we think of as built on the bottom circle, and write them as sums of tensor monomials of ring elements with subscripts indicating the simplex each ring element lies over. Under this map, we have 1 vr1 x0 := 1s0 v0 1s0 v1 x0 11 7 1s0 v x x1 := 1s0 v0 1s0 v1 10 x1 7 1s0 v x Then d0 d1 maps these elements to the following: 0 r x0 7 1v0 xv1 xv0 1v1 x1 7 xv0 1v1 1v0 xv1 v0 Note that the sum of the images under d0 d1 is zero, and so x0 + x1 is a cycle with one copy of x in simplicial degree 1, which is what xc should be. Monomials that put the copy of x over s0 vi are the image under d0 d1 +d2 of monomials that put one copy of x over s20 vi , so do not contribute to the homology, and all cycles not involving those and involving only one copy of x are multiples of x0 + x1 , and so x0 + x1 represents xc . But we r v know that xa is represented by 1s0 v x , so we get that that xc 7 2xa . We take the standard resolution of k as a (xc )-module: ... xc // (xc ) xc // (xc ) Since we saw above that xc 7 2xa , tensoring () (xc ) (xa ) yields 2xa ... // (xa ) 2xa // (xa ) (x ) Since we assume that 2 is invertible in k, we get that Tor c ((xa ), k) = k, and so when 2 is invertible in k, the spectral sequence in Equation (2.14) has the form E2 = k[x] k (xb ) = k[x] (xb ), , and therefore also collapses for degree reasons, yielding Lk (k[x]) = k[x] (x). K Remark 2.15. In fact we have shown that LkK (k[x]) = LkS 1 (k[x]) and that the projection K Sb1 induces this isomorphism. 10 Now we want to get the same result using our TCP spectral sequence for A = k[x]. We will use the following simplicial model for the Klein bottle: K = (I S 1 )/(0, t) (1, flip(t)) where flip is the reflection of the circle about the y-axis. If we use the same model of the circle with two vertices and two edges that we used in the double cover picture above but we reverse the orientation on 0 so that both edges go top to bottom, this is a simplicial map preserving v0 and v1 and exchanging the i . The flip map induces a map on (LkS 1 (k[x]) = k[x] (x) sending x 7 x and x 7 x. The fact that x 7 x comes from the fact that it is the image of the Loday construction over a point. Using the same notation and argument as before, with the different orientation on 0 , x can be represented by x0 x1 , so exchanging the i sends x to x. The nontrivial twist : S 1 C2 = hi maps the non-degenerate 1-cell S11 to and is then determined by Equation (2.2), yielding (2.16) d0 (a0 a1 . . . an ) = a0 a1 a2 . . . an . The TCP spectral sequence (2.11) in this case takes the form ! 2 k k = p+q LkK (k[x]) Ep,q = p LS 1 (LS 1 (k[x])) q and since LkS 1 (k[x]) = k[x] (x), 2 Ep,q = p LkS 1 k[x] (x) ! , q which is the pth homotopy group of the simplicial k-vector space whose p-simplices are LkSp1 (k[x] (x) ) . q For each p, LkS 1 (k[x](x)) LkS 1 (k[x])k LkS 1 ((x)), and so LkS 1 (k[x](x) ) LkS 1 (k[x])k p p p LkS 1 ((x) ). We can think of this tensor product of simplicial k-algebras as the diagonal of a bisimplicial abelian group, and again by [9, Theorem 2.4 of Section IV.2.2] the total complex of a bisimplicial abelian group with the alternating sums of the vertical and the horizontal face maps is chain homotopy equivalent to the usual chain complex associated to the diagonal of that bisimplicial abelian group. But in this case of a tensor product, the total complex was obtained by tensoring together two complexes, and since we are working over a field its homology is the tensor product of the homology of the two complexes, so k LS 1 k[x] (x) LkS 1 ((x) ) . = LkS 1 (k[x]) The first factor is just the Hochschild homology of k[x]. It sits in the 0th row of the E 2 term since x has internal degree zero, and gives us (LkS 1 (k[x]) = k[x] (x)) concentrated in positions (0, 0) and (1, 0). All spectral sequence differentials vanish on it for degree reasons, and so it will just contribute k[x] (x) to the E term. The second factor in the E 2 term is the twisted Hochschild homology for (x). To calculate it, we can use the normalized chain complex and therefore we only have to consider non-degenerate elements, which means that we only have two elements to take into account in any given simplicial degree: 11 p-degree 0 1 2 ... 1 1 x 1 x x . . . x x x x x x . . . Elements of the form x . . . x will map to zero under the Hochschild boundary map. We need to consider the odd and even cases of differentials on elements of the form 1 x . . . x. The di maps in the twisted and untwisted Hochschild complex are all the same except d0 , which incorporates the twisting action of . Therefore we have d(1 (x)2k ) = x2k + (1)2k (1)x2k = 2x2k d(1 (x)2k+1 ) = x2k+1 + (1)2k+1 (1)x2k+1 = 2x2k+1 . Here, the first 1 comes from the action on x as in (2.16) and the extra 1 in brackets come from passing the one-dimensional x past an odd or an even number of copies of itself. Since we are assuming that 2 is invertible in k, we get that the second part of the E 2 term has only k left in degree 0. So, if 2 is invertible in k, then the entire E 2 term is just k[x] (x) in the 0th row, and the TCP spectral sequence collapses and confirms that Lk (k[x]) = k[x] (x). K 3. Hopf algebras in spectra We start by describing what we mean by the notion of a commutative Hopf algebra in the -category of spectra, Sp. We consider the -category CAlg of E -ring spectra. Definition 3.1. A commutative Hopf algebra spectrum is a cogroup object in CAlg. Hopf algebra spectra are fairly rare, so let us list some important examples. Example 3.2. If G is a topological abelian group, then the spherical group ring S[G] = +G equipped with the product induced by the product in G, the coproduct induced by the diagonal map G G G, and the antipodal map induced by the inverse map from G to G is a commutative Hopf algebra spectrum. This follows from the fact that the suspension spectrum functor + : S Sp is a strong symmetric monoidal functor. Here S denotes the -category of spaces. Example 3.3. If A is an ordinary commutative Hopf algebra over a commutative ring k and A is flat as a k-module then the Eilenberg-Mac Lane spectrum HA is a commutative Hopf algebra spectrum over Hk because the canonical map HA Hk HA H(A k A) is an equivalence. We use the fact that the category of commutative ring spectra is tensored over unpointed topological spaces and simplicial sets in a compatible way [8, VII, 2, 3]. If U denotes the category of unbased (compactly generated weak Hausdorff) spaces and X U , then for every pair of commutative ring spectra A and B there is a homeomorphism of mapping spaces ([8, VII, Theorem 2.9]) (3.4) CS (X A, B) = U (X, CS (A, B)). Here, CS denotes the (ordinary) category of commutative ring spectra in the sense of [8]. By [16, Corollary 4.4.4.9], (3.4) corresponds to an equivalence of mapping spaces of -categories (3.5) CAlg(X A, B) S(X, CAlg(A, B)). See also [22, 2] for a detailed account on tensors in -categories. 12 If we consider a commutative Hopf algebra spectrum H, then the space of maps CS (H, B) has a basepoint: the composition of the counit map H S followed by the unit map S B is a map of commutative ring spectra. The functor that takes an unbased space X to the topological sum of X with a point + is left adjoint to the forgetful functor the category of pointed spaces, Top , to spaces, so we obtain a homeomorphism (3.6) U (X, CS (H, B)) = Top (X+ , CS (H, B)) and correspondingly, an equivalence in the context of -categories (3.7) S(X, CAlg(H, B)) S (X+ , CAlg(H, B)). For path-connected spaces Z, May showed that the free En -space on Z, Cn (Z), is equivalent to n n Z [19, Theorem 6.1]. Segal extended this result to spaces that are not necessarily connected. He showed that for well-based spaces Z there is a model of the free E1 -space, C1 (Z), as follows: The spaces C1 (Z) and C1 (Z) are homotopy equivalent, C1 (Z) is a monoid, its classifying space BC1 (Z) is equivalent to (Z) [24, Theorem 2], and thus, C1 (Z) BC1 (Z) is a group completion. We can apply this result to Z = X+ because X+ is well-based, thus BC1 (X+ ) (X+ ). Note that BC1 (X+ ) (X+ ). Nikolaus gives an overview about group completions in the context of -categories [20]. He shows that for every E1 -monoid M , the map M BM gives rise to a localization functor of cateories in the sense of [16, Definition 5.2.7.2], such that the local objects are grouplike E1 -spaces. In particular, there is a homotopy equivalence of mapping spaces [16, Proposition 5.2.7.4] MapE1 (BC1 (X+ ), Y ) MapE1 (C1 (X+ ), Y ) if Y is a grouplike E1 -space. Here, E1 denotes the -category of E1 -spaces. If H is a commutative Hopf-algebra, then the space CAlg(H, B) is a grouplike E1 -space. Therefore, by using Equations (3.5) and (3.7), we obtain a chain of homotopy equivalences CAlg(X H, B) S(X, CAlg(H, B)) S (X+ , CAlg(H, B)) MapE1 (C1 (X+ ), CAlg(H, B)) MapE1 (BC1 (X+ ), CAlg(H, B)) MapE1 ((X+ ), CAlg(H, B)). If (X+ ) (Y+ ) is an equivalence of pointed spaces, then (X+ ) (Y+ ) as grouplike E1 -spaces and therefore we get a homotopy equivalence CAlg(X H, B) CAlg(Y H, B). Applying the Yoneda Embedding to the above equivalence yields the following result: Theorem 3.8. If H is a commutative Hopf algebra spectrum and if (X+ ) (Y+ ) is an equivalence of pointed spaces, then there is an equivalence X H Y H in CAlg. Remark 3.9. If X is a pointed simplicial set, then the suspension (X+ ) is equivalent to (X) S 1 . Therefore, if X and Y are pointed simplicial sets, such that (X) (Y ) as pointed simplicial sets, then we also obtain an equivalence between (X+ ) and (Y+ ). Segals result also works for larger n than 1. If two spaces are equivalent after an n-fold suspension, then an En -coalgebra structure on a Hopf algebra is needed for the Loday construction to be equivalent on these two spaces. There are indeed interesting spaces that are not equivalent after just one suspension, but that need iterated suspensions to become equivalent: 13 Christoph Schaper [23, Theorem 3] shows that for affine arrangements A one needs at least a (A + 2)-fold suspension in order to get a homotopy type that only depends on the poset structure of the arrangement. Here, A is a number that depends on the poset data of the arrangement, namely the intersection poset and the dimension function. For homology spheres, the double suspension theorem of James W. Cannon and Robert D. Edwards [6, Theorem in 11] states that the double suspension 2 M of any n-dimensional homology sphere M is homeomorphic to S n+2 . Here, a single suspension does not suffice unless M is an actual sphere. 4. Truncated polynomial algebras One way of showing that a commutative R-algebra spectrum A is not multiplicatively or linearly stable is to prove that the homotopy groups of the Loday construction LR T n (A) differ from those of W n R k W W L n (A), as in [7]. Here, we write (n) S for the k -fold -sum of S k . Indeed, there is a Sk k=1 k (nk) homotopy equivalence n _ _ (T n ) ( S k ). k=1 (n) k If A is augmented over R, then for proving that R A is not multiplicatively or additively stable, it suffices to show that R W W LR k (A; R). T n (A; R) 6 L n k=1 (n) S k See [14, 2] for details and background on different notions of stability. In the following we restrict our attention to Eilenberg-Mac Lane spectra of commutative rings and we will use this strategy to show that none of the commutative Q-algebras Q[t]/tm for m > 2 can be multiplicatively stable. We later generalize this to quotients of the form Q[t]/q(t) where q(t) is a polynomial without constant term and to integral and mod-p results. Pirashvili determined higher order Hochschild homology of truncated polynomial algebras of the form k[x]/xr+1 additively when k is a field of characteristic zero [21, Section 5.4] in the case of odd spheres. A direct adaptation of the methods of [4, Theorem 8.8] together with the flowchart from [5, Proposition 2.1] yields the higher order Hochschild homology with reduced coefficients for all spheres. See also [7, Lemma 3.4]. Proposition 4.1. For all m > 2 and n > 1 ( Q (xn ) Q[yn+1 ], [n],Q HH (Q[t]/tm ; Q) = Q[xn ] Q (yn+1 ), if n is odd, if n is even. In both cases Hochschild homology of order n is a free graded commutative Q-algebra on two generators in degrees n and n + 1, respectively, and the result does not depend on m. We will determine for which m and n we get a decomposition of the form Q m m W W (4.2) L Q k (Q[t]/t ; Q). T n (Q[t]/t ; Q) = L n k=1 (nk) S Note that the right-hand side is isomorphic to n O O L Q (Q[t]/tm ; Q) Sk k=1 (n) k where all unadorned tensor products are formed over Q. Thus, if we have a decomposition as in m (4.2), then we can read off the homotopy groups of LQ T n (Q[t]/t ; Q) with the help of Proposition 4.1. 14 Expressing Q[t]/tm as the pushout of the diagram t7tm Q[t] // Q[t] t70 Q allows us to express the Loday construction for Q[t]/tm , now viewed as a commutative HQ-algebra spectrum, as the homotopy pushout of the diagram LHQ T n (HQ[t]; HQ) t7tm // LHQ T n (HQ[t]; HQ) t70 HQ and so HQ m L LHQ T n (HQ[t]/t ; HQ) LT n (HQ[t]; HQ) LHQ (HQ[t];HQ) HQ. Tn As Q[t] is smooth over Q, LHQ T n (HQ[t]; HQ) is stable [7, Example 2.6]. So we can write HQ W LHQ T n (HQ[t]; HQ) L n k=1 W (nk) Sk (HQ[t]; HQ). Again, we obtain an isomorphism Wn L Q k=1 W n k ( ) (Q[t]; Q) = Sk n O O k=1 [k],Q HH (Q[t]; Q) n k ( ) and with the help of [5, Proposition 2.1] we can identify the terms as follows: ( Q[xk ], if k is even , [k],Q HH (Q[t]; Q) = Q (xk ), if k is odd. Lemma 4.3. There is an isomorphism of graded commutative Q-algebras Wn L Q k=1 W LQ T n (Q[t];Q) (nk) Sk Q (Q[t]/tm ; Q) = LT n (Q[t]; Q) Tor (Q, Q). Proof. We already know that (4.4) Wn L Q k=1 W (nk) (Q[t]/tm ; Q) = Sk n O O k=1 [k],Q HH (Q[t]/tm ; Q) = n k n O O k=1 ( ) gFQ (xk , yk+1 ) n k ( ) where gFQ (xk ) denotes the free graded commutative Q-algebra generated by an element xk in degree k and gFQ (xk , yk+1 ) denotes the free graded commutative Q-algebra generated by an element xk in degree k and an element yk+1 in degree k + 1. Nn N n gFQ (xk ), we obtain that As LQ k=1 T n (Q[t]; Q) = ( ) k LQ (Q[t];Q) (Q, Q) Tor T n = n+1 O O =2 ( gFQ (y ) n 1 ) and hence the tensor product of the two gives a graded commutative Q-algebra isomorphic to (4.4) 15 Q Let A denote the graded commutative Q-algebra LQ T n (Q[t]; Q) and B denote LT n (Q[t]; Q) viewed as an A -module via a morphism of graded commutative Q-algebras f : A B . Lemma 4.5. Let f1 : A B be the morphism f1 = B A where A : A Q is the augmentation that sends all elements of positive degree to zero and where B : Q B is the unit map of B . Let f2 : A B be any map of graded commutative algebras such that there is ,fi an element x An with n > 0 such that f2 (x) = w 6= 0. Let TorA , (B , Q) denote the graded Tor-groups calculated with respect to the A -module structure on B given by fi . Then A ,f2 ,f1 dimQ (Tor, (B , Q))n < dimQ (TorA , (B , Q))n L ,fi A ,fi where (TorA , (B , Q))n = r+s=n Torr,s (B , Q). Proof. Let P Q be an A -free resolution of Q. We want to choose P efficiently, in the following sense: since QLis concentrated in degree zero and A0 = Q, we can choose P0 to be A . Then we nj choose P1 = jI1 A with the minimal possible number of copies of A in each suspension degree, beginning from the bottom (that is, the only reason we add a new n A is if there is a class in P1 that has not yet been L hit bynjthe suspensions of A in lower dimensions that we already have) to guarantee that d1 : jI A0 P0 is injective, and moreover M M ker(d1 : nj A0 P0 ) nj ker(A ). jI1 jI1 And of course we need Im(d1 : P1 P0 ) = ker(A : A Q) and similarly for higher . For every > 0 we choose P with M P = nj A jI so that d : L nj jI A0 P1 is injective and moreover M M ker(d : nj A0 P1 ) nj ker(A ). jI Then we get jI M Im(d : P P1 ) = ker(d1 : P1 P2 ) nj ker(A ). jI1 The Tor groups we want are the homology groups of M M B A P = B A nj A n j B = jI jI with respect to the differential id d for either A -module structure. As f1 : A B factors through the augmentation, we claim that the differentials in the chain complex B A P with the A -module structure given by f1 L are trivial: they are of the form L id d where d is the differential of P . As d sends every nj 1 jI nj A to something in jI1 nj ker(A ), M (id d)(b A nj 1) Q{b} A nj ker(A ) = 0 jI1 for all b B . Hence A ,f1 Tor,s (B , Q) = ( M nj B )s = jI M jI 16 nj Bs . ,f1 In particular, we have of course TorA 0,s (B , Q) = Bs for all s. For the A -module structure on B given by f2 we obtain that ,f2 TorA 0, (B , Q) = B A Q but here, the tensor product results in a nontrivial quotient of B . Recall that we assumed that f2 (x) = w 6= 0. The element w 1 B A Q is trivial because the degree of x is positive and hence A (x) = 0: w 1 = f2 (x) 1 = 1 A (x) = 1 0 = 0. Therefore, A ,f1 ,f2 dimQ TorA 0,n (B , Q) < dimQ Tor0,n (B , Q). ,f2 The other Tor-terms in total degree n of the form TorA r,s (B , Q) with r + s = n are subquotients of M nj Bs jIr and hence for all (r, s) with r + s = n and r > 0 we obtain A ,f1 ,f2 dimQ TorA r,s (B , Q) 6 dimQ Torr,s (B , Q). Note that if f : A B factors through the augmentation A Q then A TorA (B , Q) = B Tor (Q, Q). We use Lemma 4.5 to prove the following result. Theorem 4.6. Let n > 2. Then Q n W dimQ n LQ T n (Q[t]/t ; Q) < dimQ n L n k=1 W n k ( ) Sk (Q[t]/tn ; Q). In particular, for all n > 2 the pair (Q[t]/tn ; Q) is not stable and Q Q[t]/tn is not multiplicatively stable. The n = 2 case of Theorem 4.6 was obtained earlier by Dundas and Tenti [7]. Before we prove the theorem, we state the following integral version of it: Corollary 4.7. For all n > 2 the pair (Z[t]/tn ; Z) is not stable and Z Z[t]/tn is not multiplicatively stable. Proof of Corollary 4.7. If for some n > 2 the pair (Z[t]/tn ; Z) were stable, then in particular LZT n (Z[t]/tn ; Z) = LZWn k=1 W (nk) S k (Z[t]/tn ; Z). Localizing at Z \ {0} would then imply Q n W L Q T n (Q[t]/t ; Q) = L n k=1 in contradiction to Theorem 4.6. W (nk) S k (Q[t]/tn ; Q) 17 4.1. Proof of Theorem 4.6. We prove Theorem 4.6 by identifying an element in A of positive degree that is sent to a nontrivial element of B . More precisely, we will show that the map that sends t to tn sends the indecomposable element in n LQ S n (Q[t]; Q) up to a unit to the element dt1 . . . dtn n LQ (Q[t]; Q). S 1 ...S 1 We consider both elements as elements of n LQ T n (Q[t]; Q) via the inclusions of summands Q n L Q (Q[t]; Q) n LQ T n (Q[t]; Q) n LS n (Q[t]; Q). S 1 ...S 1 In the following we consider T n as the diagonal of an n-fold simplicial set where every ([p1 ], . . . , [pn ]) ()n is mapped to Sp11 . . . Sp1n . Then LQ T n (Q[t]; Q) can also be interpreted as the diagonal of an n-fold simplicial Q-vector space with an associated n-chain complex. By abuse of notation we still denote this n-chain complex by LQ T n (Q[t]; Q). We use the following notation concerning the n-chain complex LQ T n (Q[t]; Q): 0m = (0, 0, . . . , 0) and 1m = (1, 1, . . . , 1) are the vectors containing only 0 or 1, respectively, repeated m times. A vector V Nn is viewed as a multi-degree of an element in the n-chain complex. A vector v Nn for which 0n 6 v 6 V in every entry can be thought of as specifying a coordinate in the multi-matrix of an element in multi-degree V. We call the ith entry of a vector v Nn the ith place in v. It is always assumed that V = 1n if not otherwise specified. Each element of LQ T n (Q[t]; Q) in degree V = (v1 , . . . , vn ) is a multi-matrix of dimension (v1 + 1, . . . , vn + 1) with entries in Q[t] at coordinates v 6= 0n and an entry in Q at coordinate 0n . xv for x Q[t] and v Nn is the multi-matrix with term x at coordinate v and 1 at other coordinates. We say a term is trivial if it is 1 in all its coordinates. Therefore xv yw for x, y Q[t] and v, w Nn is the product of xv and yw in degree V of LQ T n (Q[t], Q) regarded as an n-simplicial ring. Explicitly, if v 6= w, it is the multi-matrix with x at coordinate v, y at coordinate w, and 1 elsewhere; if v = w, it is the multi-matrix with xy at coordiante v and 1 elsewhere. Suppose that C is an n-chain complex with differentials d1 , . . . , dn in the n different directions, then the total chain complex Tot(C ) has differential in component (v1 , . . . , vn ) given by d= n X (1)v1 +...+vi1 di . i=1 Pvi In our case we will have each di = j=0 (1)j di,j where di,j : Cv1 ,...,vn Cv1 ,...,vi 1,...vn is the face map. We are interested in low degrees, especially in 1n . Any vi = 1 will imply di = 0 since the di are cyclic differentials and Q[t] is commutative. This allows us to eliminate the di from d. We have the following three lemmas about homologous classes and tori of different dimensions: Lemma 4.8 (Split Moving Lemma). Let a, b be coordinates in degree 1n1 (that is, in 22. . .2dimensional matrices). Then x(a,1) y(b,1) x(a,0) y(b,1) + x(a,1) y(b,0) . Proof. Their difference is a boundary of an element of degree (1n1 , 2): d(x(a,1) y(b,2) ) = (1)n1 dn (x(a,1) y(b,2) ) = x(a,0) y(b,1) x(a,1) y(b,1) + x(a,1) y(b,0) . 18 For example, when n = 2, a = 0, b = 1, the 1 x 1 x d = 1 1 y 1 difference is 1 1 x 1 x + . y 1 y y 1 Let b be a coordinate of a multi-matrix of an element in degree 1nm such that b 6= 0nm . For any multi-matrix c in degree W Nm , we can form the following multi-matrix in degree (W, 1nm ) Nn : at coordinate (a, 0nm ); ca c(,0) y(0,b) has terms yb at coordinate (0m , b); 1 elsewhere. Lemma 4.9. The following is a chain map: Q Tot(LQ T m (Q[t], Q)) Tot(LT n (Q[t], Q)); c 7 c(,0) y(0,b) . Proof. Clearly di (c(,0) y(0,b) ) = di c(,0) y(0,b) for 0 6 i 6 m. But since the multi-degree of c(,0) y(0,b) is V = (W, 1nm ) Nn and whenever vi = 1, di = di,0 di,1 = 0, we also get di (c(,0) y(0,b) ) = 0, for m < i 6 n. This lemma also applies when y(0,b) is replaced by another multi-matrix that has more than one nontrivial term, as long as the nontrivial terms are all in coordinates of the form (0m , b) for b in degree 1nm and b 6= 0nm . It has the following immediate corollary: Lemma 4.10 (Orthogonal Moving Lemma). Let b be a coordinate in degree 1nm such that b 6= 0nm . Let c, c be elements in multi-degree W Nm . If c c in multi-degree W, then c(,0nm ) y(0m ,b) c(,0nm ) y(0m ,b) in multi-degree (W, 1nm ) Conceptually, the moving lemmas tell us how to move the nontrivial elements x, y in certain multi-matrices to lower coordinates. They are stated for a special case for simplicity, but of course they work for any permulation of copies of Nn in the statement. The split moving lemma says that if we have xv and yw where the coordinates share a 1 in a particular place, the 1s can be moved to coordinate 0 separately. The orthogonal moving lemma says that the x in xv and the y in yw can be moved separately if they are supported in orthogonal tori (that is, have their nontrivial entries in different coordinates). Proposition 4.11. Let v and w be two coordinates of degree 1n . (1) If v and w are both 0 in the ith place for some 1 6 i 6 n, then xv yw 0. In particular, if v 6= 1n , then xv 0. (2) In general, xv y w X xv y w , v 6v,w 6w, v +w =1n where the sum is taken over all coordinates v and w such that They are place-wise no greater than v and w respectively; They take 1 in complementary places. 19 (3) For k > 1 and n > 1, we have the following homologous relation: X k (t )1n k Y tw i w1 ,...,wk 6=0n , i=1 w1 +...+wk =1n In particular, if k = n and we let ei denote the coordinate that has 1 at the ith place and 0 at other places, we get n (4.12) (t )1n n! n Y tei . i=1 Also, if k > n, this gives us (tk )1n 0 Proof. The class in (1) is a cycle because everything is in multi-degree 1n is a cycle; it is nullhomologous because it is in the image of the degeneracy si,0 in the ith place. For (2) we write |v| for the sum of the places of the vector v. We induct on |v| + |w|. Notice that a coordinate v of degree 1n is just a sequence of length n of 0s and 1s and |v| is just the number of 1s in it. For |v| + |w| 6 n, there are two cases: One is that v and w are both 0 in one place. Then the claim holds because the right-hand side is the empty sum and the left-hand side is 0 by part (1). The other case is that v + w = 1n . Then the claim also holds because the right-hand side has only one copy that is exactly the left-hand side. Assume that the claim is true for |v| + |w| 6 m where m > n and suppose now |v| + |w| = m + 1. Since m + 1 > n + 1, v and w have to be both 1 in some place. Without loss of generality, we assume that v = (v0 , 1), w = (w0 , 1) where v0 , w0 6 1n1 . By the Split Moving Lemma (Lemma 4.8), xv yw x(v0 ,0) yw + xv y(w0 ,0) . Since |(v0 , 0)| + |w| = |v| + |(w0 , 0)| = m, by inductive hypothesis we have that X X xv y w x(v0 ,0) yw + xv y(w0 ,0) v0 6v0 ,w 6w, (v0 ,0)+w =1n = X v 6v,w0 6w0 , v +(w0 ,0)=1n xv y w . v 6v,w 6w, v +w =1n For (3) we order the pair (k, n) by the lexicographical ordering. We induct on (k, n). When k = 1, the claim is trivially true. Suppose the claim is true for all pairs less than (k, n) where k > 2. Taking v = w = 1n , x = t and y = tk1 in part (2), we get that X X (4.13) (tk )1n tw1 (tk1 )v = tw1 (tk1 )v . w1 6=0n w1 +v =1n w1 +v =1n 20 The second step above uses that t0n = 0 because t is 0 in the Q[t]-module Q. Let m = |v |. By the inductive hypothesis, we have (4.14) (t k1 )1m k Y X twi w2 ,...,wk 6=0m , i=2 w2 +...+wk =1m For each wi which is a coordinate of degree 1m , we add in 0 in places where v is 0 to make it a coordinate of degree 1n . Denote it by wi . Then the Orthogonal Moving Lemma (Lemma 4.10), (4.13) and (4.14) combine to k X Y k (t )1n tw i . w1 ,...,wk 6=0n , i=1 w1 +...+wk =1n Qn For any n > 2, we call t1n the diagonal class and denote it by n . We call i=1 tei the volume form and denote it by voln . If we include S 1 T n as the ith coordinate and identify the first Hochschild homology group with the Kahler differentials, the generator dt of HHQ 1 (Q[t]; Q) maps to the generator we call dti in the Loday construction of the torus. In this sense voln corresponds to the degree-n class dt1 . . . dtn . Proof of Theorem 4.6. By Equation (4.12) we know that the map t 7 tn induces a map on L Q T n (Q[t]; Q), that sends the diagonal class, n , to n!voln . Hence, by Lemma 4.5 we know that Q n n W W dimQ n (LQ k (Q[t]/t ; Q)). T n (Q[t]/t ; Q)) < dimQ n (L n n S k=1 (k) In particular, Q n n W W (LQ k (Q[t]/t ; Q)). T n (Q[t]/t ; Q)) (L n n S k=1 (k) Remark 4.15. For the non-reduced Loday construction LQ T n (Q[t]), parts (1) and (2) of Proposition 4.11 are still true. Part (3) will become k (t )1n k Y X tw i w1 +...+wk =1n i=1 and Equation (4.12) is no longer true. 4.2. Q[t]/tm on T n for 2 6 m < n. We know that for Q[t]/tn we get a discrepancy between n of the Loday construction on the n-torus and that of the bouquet of spheres that correspond to the cells of the n-torus. We use this to first show that Q[t]/tm causes a similar discrepancy for 2 6 m < n. Proposition 4.16. Let 2 6 m 6 n. Then Q m W m L Q T n (Q[t]/t ; Q) m L n k=1 W n k ( ) Sk (Q[t]/tm ; Q). Proof. We consider the Tor-spectral sequence LQ (Q[t];Q) Tn Tor, Q m ( LQ T n (Q[t]; Q), Q) LT n (Q[t]/t ; Q) 21 Q m where the LQ T n (Q[t]; Q)-module structure on LT n (Q[t]; Q) is induced by t 7 t . The m-chain (m) complex C := LQ T m (Q[t]; Q) can be considered as an n-chain complex whose m + 1, . . . , ncoordinates are trivial. Then (m) C (n) = LQ T m (Q[t]; Q) C := LQ T n (Q[t]; Q) (n) is a sub-n-complex of C . We know that m 7 m!volm in the homology of the total complex of (m) (n) C and hence the same is true in C . Therefore the map Q m L Q T n (Q[t]; Q) m LT n (Q[t]; Q) m that is induced by t 7 tm is nontrivial and by Lemma 4.5 the dimension of m LQ T n (Q[t]/t ; Q) is strictly smaller than the dimension of Wn m L Q k=1 W n k ( ) Sk (Q[t]/tm ; Q). 4.3. Quotients by polynomials without constant term. Let q(t) = am tm + . . . + a1 t Q[t]. Then we can still write Q[t]/q(t) as a pushout Q[t] t7q(t) // Q[t] t70 // Q[t]/q(t) Q hence the above methods carry over. Proposition 4.17. Let m0 be the smallest natural number with 1 6 m0 6 m with am0 6= 0. Then Q W m0 LQ T m0 (Q[t]/q(t); Q) m0 L m0 k=1 W (mk0 ) S k (Q[t]/q(t); Q). Q Proof. If m0 = 1, then t HHQ 1 (Q[t]; Q) maps to (q(t)) HH1 (Q[t]; Q) under the map t 7 q(t). In the module of Kahler differentials this element corresponds to a1 dt + 2a2 tdt + . . . + mam tm1 dt but all these summands are null-homologous except for the first one. So t 7 a1 t 6= 0 and this, along with Lemma 4.5, proves the claim. We denote by m0 (q(t)) the element (q(t))1m0 . If m0 > 1, then the diagonal element m0 (t) maps to m X m0 (q(t)) = ai m0 (ti ) i=m0 and this is homologous to (m0 )!am0 volm0 + terms of higher t-degree by (4.12). Hence m0 (t) maps to a nontrivial element and again Lemma 4.5 gives the claim. 22 4.4. Truncated polynomial algebras in prime characteristic. We know that for commutative Hopf algebras A over k the Loday construction is stable, so Loday constructions of truncated polynomial algebras of the form Fp [t]/tp have the same homotopy groups when evaluated on an n-torus and on the corresponding bouquet of spheres. However, we show that there is a discrepancy for truncated polynomial algebras Fp [t]/tn for 2 6 n < p. Theorem 4.18. Assume that 2 6 n < p and n 6 m, then F F (LTpm (Fp [t]/tn ; Fp )) (LWpm k=1 In particular for all 2 6 n < p the pair (Fp [t]/tn ; F p) W m k ( ) Sk (Fp [t]/tn ; Fp )). is not stable. Proof. We consider the case m = n. The cases n < m follow by an argument similar to that for Proposition 4.16. As Fp [t] is smooth over Fp , we know that Fp Fp [t] is stable, so that F F (LTpn (Fp [t]; Fp )) = (LWpn k=1 [k],Fp and HH W n k ( ) (Fp [t]; Fp )) = Sk n O O k=1 [k],Fp HH (Fp [t]; Fp ) n k ( ) (Fp [t]; Fp ) is calculated in [4, 8] so that we obtain [k],Fp HH (Fp [t]; Fp ) = Bk+1 B B where B1 = Fp [t] and Bk+1 = Tor,k (Fp , Fp ) where the grading on Tor,k (Fp , Fp ) is the total F [2],F grading. Thus in low degrees this gives HH p (Fp [t]; Fp ) = Fp (t) with |t| = 1, HH p (Fp [t]; Fp ) = N F (0 t) with |0 t| = 2. As F (0 t) Fp [k t]/(k t)p we can iterate the result. = p i>0 p [n],F Note that in HHn p (Fp [t]; Fp ) there is always an indecomposable generator of the form 0 . . . 0 t or 0 0 . . . 0 t in degree n and we call this generator n . We also obtain a volume class F F voln := t1 . . . tn n LSp1 ...S 1 (Fp [t]; Fp ) n LTpn (Fp [t]; Fp ). The results from Proposition 4.11 work over the integers. If n < p, then n! is invertible in Fp and therefore the class n maps to n!voln . An argument analogous to Lemma 4.5 finishes the proof. References [1] Katharine Adamyk, Teena Gerhardt, Kathryn Hess, Inbar Klang, Hana Jia Kong, Computational tools for twisted topological Hochschild homology of equivariant spectra, preprint arXiv:2001.06602. [2] Vigleik Angeltveit, Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill, Tyler Lawson, and Michael A. Mandell, Topological cyclic homology via the norm, Doc. Math. 23 (2018), 21012163. [3] Yuri Berest, Ajay C. Ramadoss and Wai-Kit Yeung, Representation homology of spaces and higher Hochschild homology, Algebr. Geom. Topol. 19 (2019), no. 1, 281339. [4] Irina Bobkova, Ayelet Lindenstrauss, Kate Poirier, Birgit Richter, Inna Zakharevich On the higher topological Hochschild homology of Fp and commutative Fp -group algebras, Women in Topology: Collaborations in Homotopy Theory. Contemporary Mathematics 641, AMS, (2015), 97122. [5] Irina Bobkova, Eva Honing, Ayelet Lindenstrauss, Kate Poirier, Birgit Richter, Inna Zakharevich, Splittings and calculational techniques for higher THH , Algebr. Geom. Topol. 19 (2019), no. 7, 37113753. [6] James W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), no. 1, 83112. [7] Bjrn Ian Dundas, Andrea Tenti, Higher Hochschild homology is not a stable invariant, Math. Z. 290 (2018), no. 1-2, 145154. [8] Anthony D. Elmendorf, Igor Kriz, Michael A. Mandell, J. Peter May, Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Mathematical Surveys and Monographs, 47. American Mathematical Society, Providence, RI, (1997), xii+249 23 [9] Paul G. Goerss, John F. Jardine, Simplicial homotopy theory, Modern Birkhauser Classics, Birkhauser Verlag 2009, xv+510 pp. [10] Gemma Halliwell, Eva Honing, Ayelet Lindenstrauss, Birgit Richter, Inna Zakharevich, Relative Loday constructions and applications to higher THH-calculations, Topology Appl. 235 (2018), 523545. [11] Lars Hesselholt, Ib Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), no. 1, 29101. [12] Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149208. [13] Michael Larsen, Ayelet Lindenstrauss, Cyclic homology of Dedekind domains. K-Theory 6 (1992), no. 4, 301334. [14] Ayelet Lindenstrauss, Birgit Richter, Stability of Loday constructions, preprint arXiv:1905.05619. [15] Jean-Louis Loday, Cyclic Homology, Second edition. Grundlehren der Mathematischen Wissenschaften 301. Springer-Verlag, Berlin, (1998), xx+513 pp. [16] Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies vol. 170, 2009. [17] Michael A. Mandell, J. Peter May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp. [18] J. Peter May, Simplicial objects in algebraic topology, Vol. 11. University of Chicago Press, 1992. [19] J. Peter May, The geometry of iterated loop spaces, Lectures Notes in Mathematics, Vol. 271. SpringerVerlag, Berlin-New York, 1972. viii+175 pp. [20] Thomas Nikolaus, The group completion theorem via localizations of ring spectra, expository notes, available at https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Group_completion.pdf [21] Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. Ecole Norm. Sup. (4) 33 (2000), no. 2, 151179. [22] Nima Rasekh, Bruno Stonek, Gabriel Valenzuela, Thom spectra, higher THH and tensors in -categories, preprint arXiv:1911.04345. [23] Christoph Schaper, Suspensions of affine arrangements, Math. Ann. 309 (1997), no. 3, 463473. [24] Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213221. [25] Christian Schlichtkrull, Higher topological Hochschild homology of Thom spectra, J. Topol. 4 (2011), no. 1, 161189. Department of Mathematics, University of Oslo, Box 1053, Blindern, NO - 0316 Oslo, Norway E-mail address: aliceph@math.uio.no Department of Mathematics, Michigan State University, 619 Red Cedar Rd, East Lansing, MI 48840, USA E-mail address: klander2@msu.edu Department of Mathematics, Indiana University, 831 East 3rd Street, Bloomington, IN 47405, USA E-mail address: alindens@indiana.edu Fachbereich Mathematik der Universitat Hamburg, Bundesstrae 55, 20146 Hamburg, Germany E-mail address: birgit.richter@uni-hamburg.de Department of Mathmatics, University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA E-mail address: foling@math.uchicago.edu 24 ...
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- ... Oral Oncology 49 (2013) 93101 Contents lists available at SciVerse ScienceDirect Oral Oncology journal homepage: www.elsevier.com/locate/oraloncology DSG3 as a biomarker for the ultrasensitive detection of occult lymph node metastasis in oral cancer using nanostructured immunoarrays Vyomesh Patel a, Daniel Martin a, Ruchika Malhotra b, Christina A. Marsh a, Colleen L. Doi a, Timothy D. Veenstra f, Cherie-Ann O. Nathan c, Uttam K. Sinha d, Bhuvanesh Singh e, Alfredo A. Molinolo a, James F. Rusling b, J. Silvio Gutkind a, a Oral and Pharyngeal Cancer Branch, National Institute of Dental and Craniofacial Research, National Institutes of Health, Bethesda, MD 20892-4330, United States Departments of Chemistry and Cell Biology, University of Connecticut, Storrs, CT, United States Department of Otolaryngology, Head and Neck Surgery, Louisiana State University Health Sciences Center, Shreveport, LA, United States d Department of Otolaryngology, Head and Neck Surgery, University of Southern California, Keck School of Medicine, Los Angeles, CA, United States e Laboratory of Epithelial Cancer Biology, Head and Neck Service, Memorial Sloan-Kettering Cancer Center, New York, NY, United States f Laboratory of Proteomics and Analytical Technologies, Science Applications International Corporation-Frederick, Inc., National Cancer Institute, Frederick, MD, United States b c a r t i c l e i n f o Article history: Received 26 April 2012 Received in revised form 19 July 2012 Accepted 1 August 2012 Available online 23 September 2012 Keywords: DSG3 Head and neck cancer Desmosomes Biomarker Sentinel lymph nodes Nanosensors Immunoarray Lymph nodes metastasis Proteomics SCC s u m m a r y Objectives: The diagnosis of cervical lymph node metastasis in head and neck squamous cell carcinoma (HNSCC) patients constitutes an essential requirement for clinical staging and treatment selection. However, clinical assessment by physical examination and different imaging modalities, as well as by histological examination of routine lymph node cryosections can miss micrometastases, while false positives may lead to unnecessary elective lymph node neck resections. Here, we explored the feasibility of developing a sensitive assay system for desmoglein 3 (DSG3) as a predictive biomarker for lymph node metastasis in HNSCC. Materials and methods: DSG3 expression was determined in multiple general cancer- and HNSCC-tissue microarrays (TMAs), in negative and positive HNSCC metastatic cervical lymph nodes, and in a variety of HNSCC and control cell lines. A nanostructured immunoarray system was developed for the ultrasensitive detection of DSG3 in lymph node tissue lysates. Results: We demonstrate that DSG3 is highly expressed in all HNSCC lesions and their metastatic cervical lymph nodes, but absent in non-invaded lymph nodes. We show that DSG3 can be rapidly detected with high sensitivity using a simple microuidic immunoarray platform, even in human tissue sections including very few HNSCC invading cells, hence distinguishing between positive and negative lymph nodes. Conclusion: We provide a proof of principle supporting that ultrasensitive nanostructured assay systems for DSG3 can be exploited to detect micrometastatic HNSCC lesions in lymph nodes, which can improve the diagnosis and guide in the selection of appropriate therapeutic intervention modalities for HNSCC patients. Published by Elsevier Ltd. Introduction With more than 500,000 new cases annually, squamous cell carcinomas of the head and neck (HNSCC) represent one of the ten most common cancers globally,1 and result in more than 11,000 deaths each year in the US alone.2 The 5 year survival of newly diagnosed HNSCC patients is 50%, and despite new treatment approaches, it has improved only marginally over the past decades.3 HNSCC has a high propensity to metastasize to loco Corresponding author. Address: Oral and Pharyngeal Cancer Branch, National Institute of Dental and Craniofacial Research, NIH, 30 Convent Drive, Building 30, Room 211, Bethesda, MD 20892-4330, United States. Tel.: +1 301 496 6259; fax: +1 301 402 0823. E-mail address: sg39v@nih.gov (J.S. Gutkind). regional lymph nodes due to the presence of a rich lymphatic network and the overall high number of lymph nodes in the neck region.38 Even in patients without clinical evidence of lymph node involvement (N0), there is a high incidence of occult lymph node metastasis, ranging from 10% to 50%.4,5,7 The diagnosis of cervical lymph node metastasis an essential requirement for clinical staging and treatment,9 and is now widely accepted as the most important factor in HNSCC prognosis.3,5,6,10 However, due to limitations in the accurate diagnosis of lymph node metastasis, patients with clinically negative nodes often undergo elective neck resection or radiation,11,12 with the consequent associated morbidity and adverse impact in the quality of life.12 Clinical assessments of lymph node metastases include physical examination, imaging modalities such as computed tomography (CT), magnetic resonance imaging (MRI), ultrasonography, and 1368-8375/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.oraloncology.2012.08.001 Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 94 V. Patel et al. / Oral Oncology 49 (2013) 93101 [18F]-2-uorodeoxyglucose positron emission tomography scans (PET).13,14 However, poor spatial resolution, false positive detection of reactive lymph nodes, and limited sensitivity under 5 mm size,1518 can add to potential false negative results. Histopathological, immunohistochemical (IHC), and molecular approaches to evaluate sentinel lymph node biopsies have improved the detection rate of metastatic disease in some cancers,19,20 but histopathology-based methods can often miss micrometastases, while more sensitive techniques such as IHC and real-time PCR for validated cancer markers are time consuming and require stringent handling procedures and technical expertise. A recent proteomic analysis of parafn embedded normal oral mucosa and HNSCC lesions revealed a very high abundance of Desmoglein 3 (DSG3) in both non-neoplastic epithelium and cancer lesions.21 DSG3 is a transmembrane glycoprotein involved in cell-to-cell adhesion that is exclusively expressed in stratied epithelium.22 These observations prompted us to explore whether the assessment of DSG3 protein levels could be used to investigate the presence of malignant squamous epithelial cells in cervical lymph nodes, and hence serve as a predictive biomarker for metastasis. In this regard, high sensitivity electrochemical immunoassays have recently gained acceptance in biomedicine.23 For example, we have developed immunosensors based on nanostructured electrodes coupled to microuidics and multilabel strategies to achieve highly sensitive detection of protein cancer biomarkers in serum.24,25 We have combined these strategies into a simple microuidic immunoarray26,27 and here explore the suitability of this platform for the rapid and sensitive detection of DSG3 protein. We show that this system can be used to rapidly detect and quantify DSG3 in frozen human tissue sections, distinguishing between clinically positive and negative cervical lymph nodes. Overall, these studies may help develop point-of-care procedures aiding in the diagnosis of invaded lymph nodes in HNSCC patients, thereby facilitating educated decisions regarding appropriate therapeutic intervention modalities, and decreasing the morbidity often associated with HNSCC. Materials and methods Reagents, antibodies, and cell culture All chemicals and reagents were from SigmaAldrich (St. Louis, MO), unless indicated. The following antibodies: goat-anti-human DSG3 [AF1720]; mouse-anti-human DSG3 [MAP1720], biotin labeled goat-anti-human DSG3 [BAF1720], recombinant human DSG3 Fc Chimera protein [1720-DM], were from R&D Systems (MN, USA). The mouse anti-human DSG3 antibody [32-6300] from Invitrogen (MA, USA), and rabbit-anti-cytokeratin Wide Spectrum Screening [N1512] from Dako (CA), were used for immunohistochemistry (IHC). The a-tubulin antibody [11H10] was from Cell Signaling Technology (MA, USA). Biotinylated peroxidase and streptavidin coated magnetic beads were from Invitrogen. Antirabbit and anti-mouse biotinylated secondary antibodies were from Vector, Burlingame, CA, US. HN12, HN13 and HN30 cells were described previously.28 Cal27 and Jurkat cells were from ATCC (VA); and primary human cells from Lonza (MD). See Supplemental information for additional information. Human clinical tissues and tissue microarrays (TMAs), immunohistochemistry and immunouorescence Formalin xed, parafn-embedded, and freshly frozen HNSCC and lymph node samples were obtained anonymized with Institutional Review Board approval. Five lm sections from all tissues underwent standard H&E staining for histopathological evaluation and immunostaining. Tissue microarrays used include TMA MC2081 US (Biomax, MD) with 208 representative cases of colorectal, breast, prostate and lung cancers, and normal tissue; TMA LC810 (Biomax, MD), consisting of 40 cases of different types of lung cancers with their matched metastatic lymph nodes (total 80 tissue cores); and the Head and Neck Tissue Microarray Initiative, including 317 HNSCC cases.29 Tissue processing and analysis are described in detail in Supplemental information. All slides were scanned at 400 magnication using an Aperio CS Scanscope (Aperio, CA) and quantied using the available Aperio algorithms. Immunodetection of DSG3 was quantied according to percent of tumor cells stained (125%, 2650%, 5175%, or 76100%).29 For immunouorescence, 10 lm cryosections were immunostained with goat-anti-human DSG3 (AF1720), mouse-anti-vimentin and DAPI containing. See Supplemental information for additional details. Western blot analysis of cell and tissue extract, and microuidic immunoarrays systems for DSG3 A detailed description of the procedures used for tissue lysate preparation, SDSPAGE gel analysis and Western blotting, and the fabrication of the microuidic immunoarrays made of gold nanoparticles layered with glutathione are described in detail in the Supplemental information. Briey, the immunoarrays consisting of eight sensor elements, made of gold nanoparticles layered with glutathione, were rst coated with the capture antibody and transferred to a microuidic chamber. In parallel, biotinylated horseradish peroxidase and a biotinylated secondary antibody were attached to streptavidin-coupled magnetic beads and collected with a magnet. Next, 5 ll of 5750 fg/mL of recombinant DSG3 protein standards or 4 ll tissue extract were diluted 1:6000 in RIPA buffer and added to the bioconjugates. The bioconjugates with captured proteins were then magnetically separated, washed, resuspended in a nal volume of 110 lL, and immediately injected into the microuidic channel housing the immunoarrays. At this step, the ow was stopped, incubated for 20 min, washed, and hydroquinone solution was passed through the channel. The amperometric signal was developed by injecting 50 lL of 100 lM H2O2. Tissue lysates used for Western blot analysis and microuidic immunoarrays were made from primary tumors (n = 4), lymph node ( ) (n = 3), and lymph node (+) (n = 3). Results In a previous proteome-wide analysis of HNSCC progression, we noted that DSG3 was highly expressed in normal oral mucosa and HNSCC lesions.21 To further investigate a possible use for DSG3 as a predictive biomarker, we rst assessed DSG3 expression by immunohistochemistry in an independent cohort of human normal and malignant oral squamous tissues. By H&E histological evaluation, normal squamous epithelium shows a dened basement membrane with layers of differentiating keratinocytes, whereas in the malignant counterpart, this organized pattern is lost (Fig. 1A). Normal tissues sections stained for DSG3 show that it is predominantly expressed in the basal and suprabasal layers of the normal squamous epithelium, while in SCC DSG3 expression is restricted to cancer cells. Stromal cells were negative. We next evaluated DSG3 expression in a HNSCC tissue microarray (TMA) containing 317 evaluable cores.29 DSG3 was readily detected in all HNSCC cores and localized to tumors cells (Fig. 1B). Within these cases, well-differentiated carcinomas (n = 120) had the highest percentage of DSG3-positive cells (90%). The moderate- (n = 119), and poorlydifferentiated (n = 66) cores showed slightly lower proportion of DSG-reactive cells (80% and 70%, respectively), the remaining 12 cores consisting of corresponding to non-squamous tissues were Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 95 V. Patel et al. / Oral Oncology 49 (2013) 93101 Normal HNSCC H&E A 100 100 100 DSG3 100 B DSG3 immunostaining WD % positive cancer cells C 100 100 n: 120 119 66 76-100 51-75 26-50 1-25 80 60 40 20 0 PD 30 10 1 3 0.3 Recombinant DSG3 (ng) 100 Tubulin PD DSG3 100 HN13 HN30 CAL27 HaCaT Total cell lysates JURKAT HMVEC LEC HUVEC HN12 MD D MD 0.1 WD Figure 1 Validation of DSG3 expression in normal and malignant HNSCC. Normal oral mucosa biopsies and HNSCC were evaluated for DSG3 by IHC. (A) DSG3 is expressed throughout the normal epithelium, but is stronger in the basal and parabasal layers. Diffuse expression was seen in the epithelial component of all HNSCC. (B) Representative well (WD), moderate (MD) and poorly (PD) differentiated HSCC cases are shown. (C) DSG3 was expressed in all tumors regardless of differentiation, with increased expression in WD cases. Numbers of cases analyzed is depicted. (D) Total cell extracts from non-squamous (Jurkat, HMVEC, LEC, HUVEC) and oral-squamous (HN12, HN13, HN30, Cal27), and epidermal-squamous (HaCaT) were processed for Western blot analysis. Native DSG3 and its glycosylated forms were readily detected in squamous cells extracts, while absent from the non-squamous counterparts. These levels were compared with human recombinant DSG3 that was processed in a background of Jurkat cell lysate. Tubulin staining indicates equivalent loading and protein integrity. negative for DSG3 (Fig. 1C). The data demonstrates that DSG3 is highly expressed in human oral squamous epithelium and HNSCC. We next sought to assess in vitro the specicity of the epithelial expression of DSG3 in a panel of squamous and non-squamous model cells. The latter included Jurkat cells (immortalized T lymphocyte cells), HMVEC (skin human microvascular endothelial cells), LEC (lymphatic endothelial cells), and HUVEC (human umbilical vein endothelial cells). HaCaT cells are squamous, nonoral immortalized epidermal keratinocytes, while the oral squamous cell carcinoma lines included HN12, HN13, HN30 and Cal27.28 DSG3 was readily detected in all squamous oral cancer cell lines and HaCaT cells, with lower levels in HN12 and higher in Cal27 (Fig. 1D). No expression for DSG3 was observed in any of four non-squamous lines, while levels of a-tubulin indicated equal loading as well as protein integrity. The data seem to indicate that DSG3 is exclusively expressed in squamous epithelial-derived cells. To further validate the specicity of DSG3 expression, we evaluated TMAs containing cores representing the four most common cancers (breast, lung, prostate, and colon cancer) for DSG3 levels, and scored cases based on the presence or absence of DSG3 expression. As seen in Fig. 2A), DSG3 is poorly expressed in breast and prostate cancers, as well in adenocarcinoma of the lung (ADC), which likely reects the glandular epithelial cell origin of these human malignancies. In colon carcinoma, the expression was variable, and in all cases the pattern of expression was diffused and not the characteristic membrane lace-like pattern. In contrast, Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 96 V. Patel et al. / Oral Oncology 49 (2013) 93101 A TMA major cancer types Lung ADC Breast n: 42 47 16 45 37 13 298 100 + - Lung 100 m Colon % Cases 80 100 m 60 40 20 100 m OSCC Lung SCC Lung ADC Prostate Breast Prostate OSCC Rectum 0 100 m Colon 100 m 100 m Lung cancer B TMA lung cancer 100 SCC Met 100 m 100 m SCLC ADC n=30 % positive tumors SCC 80 60 40 20 100 m n=42 100 m n=6 0 SCC ADC SCLC Figure 2 Immunoreactivity of DSG3 in common tumor types. (A) Multi-tumor TMAs (lung, breast, colon, prostate), and an oral specic TMA were assessed for DSG3 expression by IHC, and the staining scored for the presence of specic staining as (+) or ( ). Most squamous cell lung cancers stained positive for DSG3, but very few of the adenocarcinomas gave positive reaction. All cores from the OSCC TMA scored positive. (B) In lung cancer, DSG3 expression was positive in most squamous cell carcinomas (SCC) including lymph node metastasis (SCC Met), while few cases of adenocarcinomas (ADC) gave positive reaction, and all small cell lung carcinoma samples (SCLC), were negative. DSG3 is highly expressed in tumors derived from cells of squamous epithelial origin, such as lung squamous carcinoma (SCC) and additional oral squamous carcinoma (OSCC) that were included in these arrays, showing a membrane localized staining. All stainings were scored blindly and tabulated (Fig. 2A). All cancers of squamous origin (oral and lung SCC) were strongly positive for DSG3 expression. As lung cancers include SCC, ADC, and small cell lung carcinomas (SCLCs), we examined further the specicity of DSG3 expression in these distinct lung cancer lesions. All lung SCC show strong membrane localized staining in both the primary tumor and metastasis, while ADC and SCLC show marginal to no DSG3 expression, as reected by scoring their corresponding tissue cores (Fig. 2B). Collectively, our results indicate the high specicity of DSG3 expression in oral and lung SCC lesions. Based on the observation that DSG3 is highly expressed in HNSCC, we wanted to determine if the presence of this protein in cervical lymph nodes of the neck region could be used as a predictive biomarker of HNSCC invasion. To this end, we evaluated formalin-xed, parafn-embedded and anonymized tissue sections of non-metastatic (N0) or metastatic (N+) human cervical lymph node biopsies from patients diagnosed with HNSCC for expression of DSG3 and cytokeratin, a squamous-specic protein marker. Negative lymph nodes were negative (Fig. 3A) whereas clusters of tumor cells stained positive for membrane-localized DSG3 (inset, top right), can be seen throughout the invaded lymph node (N+), indicating the metastatic spread of squamous tumor cells of the primary tumor lesions from the oral cavity (Fig. 3A). Noteworthy, small clusters of 23 isolated tumor cells, constituting micrometastases were readily detected by the presence of DSG3 protein, and this size tumor island could potentially be missed by histopathological evaluation (Inset). Next, we screened a larger cohort of metastatic and non-metastatic human cervical lymph nodes for cytokeratin and DSG3 expression (n = 35). All metastatic (n = 30), but not non-metastatic cases (n = 5) were positive for DSG3. Serial sections stained for H&E and cytokeratin conrmed the epithelial nature of the malignant DSG3-positive cells. (Fig. 3B, and low magnication of a whole lymph node in Suppl. Fig. 1). This indicates that DSG3 expression can help identify small numbers of malignant squamous tumor cells in lymph nodes, and hence the metastatic nature of the primary lesion. The sensitivity and specicity of this detection suggests that DSG3 may hold promise for accurately detecting micrometastasis in cervical lymph nodes in newly diagnosed HNSCC patients. Our previous study adapted amperometric sandwich immunoassays to a microuidic system for the ultrasensitive, multiplexed detection of secreted biomarker proteins.26 Here, we used a similar strategy for detecting DSG3 in complex tissue extracts using the microuidic immunoassay system. As shown in Fig. 4A, DSG3 Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 97 V. Patel et al. / Oral Oncology 49 (2013) 93101 A N- 100m B N+ 100m N+ 100m DSG3 N+ 50m 100m H&E N+ 50m DSG3 50m 100m N- 100m DSG3 50m CK 50m DSG3 50m Figure 3 Specic detection of DSG3 in human cervical lymph nodes. (A) Formalin xed and parafn embedded tissue sections of non-metastatic (N ) and metastatic lymph nodes (N+) show DGS3 expression only in N+, with the staining localized to the malignant squamous cells (n = 30). All N cases were negative (n = 5). B. The epithelial specicity of DSG3 immunoreactivity was further conrmed using simultaneous cytokeratin (CK) staining. A representative example is shown, whereby the H&E stained tumor island is matched with CK and DSG3 expression, with no non-specic staining. An example of a N case stained for DSG3 is shown. capture antibodies are attached to up 8 sensor elements, and streptavidin-coated paramagnetic beads (MB) loaded with 400,000 biotin-HRPs and thousands of secondary biotin-labeled antibodies to DSG3 (Ab2) are used to capture the protein off-line. After washing and magnetic separation, the MBs that have bound DSG3 (DSG3MB) are injected into the microuidic channel. Incubation at stopped ow allows the sensor antibodies to capture DSG3-MBs, and amperometric signals are developed by injecting hydrogen peroxide to activate HRP and hydroquinone to mediate the amperometic oxidation, resulting in peak currents proportional to DSG3 concentration (Fig. 4B). Noteworthy, the entire assay from incubation of sample with Ab2-MB-HRP to measurement takes 50 min. Remarkably, using this approach we were able to accurately and reproducibly detect DSG3 protein at levels down to 5 fg mL 1 in complex tissue extracts, with minimal non-specic binding. We next used these protocols for capturing DSG3 from clinical samples of human HNSCC. Desmosomes are notoriously insoluble, and multiple buffers tested, RIPA buffer afforded excellent solubility and retaining antigenicity of DSG3 extracted directly from cryosections of HNSCC and lymph node tissue. Protein extracts were made from a single 10 lm cryosection from each sample and used as input. Picogram levels of DSG3 protein were detected in all tumor samples (T14), and this was conrmed by Western blot analysis, where high levels of the protein were also detected in total cell extracts that were used for these analyses (Fig. 4C and D). The specicity of DSG3 identication was further conrmed by simultaneous uorescence microscopy of DSG3 and vimentin, as a stromal marker, in frozen sections of a series of metastatic and non-metastatic lymph nodes. As seen in Fig. 5A, only vimentin (red) was identied in non-metastatic lymph nodes (in blue, nuclear DAPI staining), whereas all metastatic lymph nodes showed pockets of very strong staining for DSG3 (green). To further explore the sensitivity of the method, we decided to analyze metastatic lymph node tissues in which the number of malignant epithelial cells was known. For this, H&E stained slides of each case was scanned, analyzed histologically, the malignant areas identied, and the number of cells quantied using the Aperio nuclear algorithm (Aperio, Vista, CA). The number of cancer cells per lymph node section is indicated in Table 1. No tumor cells were be present in the negative lymph nodes, while all three metastatic lymph nodes (13) evaluated had differing number of tumor cells. Noteworthy, positive lymph node 1 had less than Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 98 V. Patel et al. / Oral Oncology 49 (2013) 93101 A B Ab1 DSG3 (fg mL-1) 40 750 500 250 30 DSG3 PDDA HRP Ab2 I, nA 100 AuNPs 50 20 25 5 10 10 0 0 1 m streptavidin coated magnetic bead -10 0 400 800 1200 1600 t, s Ab2 -MB-HRP 40 35 y = -7.4659 + 14.987log(x) R = 0.99315 30 I, nA 25 20 15 10 5 Voltage + H2O2 HQ amperometric signal 0 100 10 C 1000 [DSG3], pg mL-1 1000 [DSG3], fg mL-1 D Primary tumor T1 T2 T3 T4 100 10 DSG3 1 0.1 0.01 Tubulin Figure 4 Rapid and ultrasensitive detection of DSG3 in human HNSCC samples using nanosensors. (A) Scheme used for the ultrasensitive detection by the microuidic immunoarray showing a single sensor in the array with capture DSG3 antibodies attached. Proteins are captured off-line on Ab2-magnetic bead (MB)-HRP bioconjugates , and after magnetic separation and washes, the MBs are injected into the immunoarray containing 8 sensors. A single immunoarray sensor is depicted. Following incubation, amperometric signals are generated by applying 0.3 V versus Ag/AgCl to the sensors by injecting a mixture of HRP-activator H2O2 and mediator hydroquinone (HQ). (B) Varying recombinant DSG3 protein concentrations were used to generate a calibration plot. The sensitivity of DSG3 sensor using recombinant protein was 5646 nA mL [fg protein] 1 cm 2. (C) Protein extracts of primary human oral squamous carcinomas (T14) made with RIPA buffer were processed for detection of DSG3. High DSG3 levels were found to be present in all the samples, and this was conrmed by Western blot analysis of the same extracts for DSG3 (D). Tubulin was used as loading control. 1000 tumor cells, while the remaining had between 13,000 and 16,000 cells (Table 1). Using the nanosensors, we evaluated total protein extracts from lymph nodes for levels of DSG3. As seen in Fig. 5A, DSG3 was essentially not detected in the normal lymph nodes, with marginal values likely a reection of very limited residual non-specic binding of Ab2-MB-HRP complex, giving rise to minimal amperometric signal. In contrast, all of the positive, metastatic lymph nodes showed high levels of DSG3 protein expression. To validate these measurements, the same protein extracts were also analyzed by Western blotting (Fig. 5B). No DSG3 was detected in the negative lymph nodes, while in all the positive lymph nodes, bands for DSG3 and its multiple glycosylated forms were readily seen, and the intensity of the corresponding bands correlated with DSG3 levels quantied by the nanosensors. When total DSG3 was normalized by the number of tumor cells in each metastatic lesion, positive lymph nodes expressed approximately 150 fg DSG3 per tumor cell (Table 1), well above the threshold for DSG3 detection. Together, this indicates that the nanosensor-based detection of DSG3 could be a highly sensitive and specic method for the identication of squamous carcinoma metastases in clinical practice. This rapid method was capable of measuring DSG3 levels even from as little as a single cell, suggesting that this technique may represent a potent tool for the ultrasensitive detection of the presence or absence of lymph node invasion in human oral cancer patients. Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 99 V. Patel et al. / Oral Oncology 49 (2013) 93101 Lymphnodes(+) 1 2 2 H&E Lymphnodes(-) 1 50m 100m 100m 50m 100m 50m 100m 50m 100m 50m 100m 50m 100m 50m IF 100m 50m DAPI/Vimentin/DSG3 Lymph node (-) Lymph node (+) 1000 [DSG3], pg mL-1 1 2 3 1 2 3 100 10 DSG3 1 0.1 0.01 1 2 3 1 2 3 Tubulin Lymph node - Lymph node + Figure 5 Detection of DSG3 in metastatic human cervical lymph nodes. H&E stained cryosections of representative non-metastatic ( ) and metastatic (+) human cervical lymph nodes were scanned and the total number of tumor cells per section was quantied (Table 1). Serial sections of these lymph nodes were evaluated by immunouorescence for DSG3 and detected only in metastatic lymph nodes (green). Vimentin (red) was used to identify stromal tissue, and nuclei of all cells were stained blue with DAPI (Fig. 5A). Protein extracts made from single cryosections of lymph nodes were used for the detection of DSG3 by Western blot analysis and DSG3 quantication using nanosensors. DSG3 levels were similar to background for all non-metastatic samples, while DSG3 levels in all metastatic cases were proportional to the number of invading HNSCC cells (Fig. 5B). Table 1 DSG3 detection in metastastic lymph nodes. Cryosections of non-metastatic (N ) (n = 3) and metastatic (N+) (n = 3) human cervical lymph nodes were collected and analyzed by nanosensor detection. The total number of tumor cells per cryosection was evaluated, and used to estimate the total amount of DSG3 per tumor cell. Samples Detected DSG3 (pg/mL) Tumor cells per section Detected DSG3 (fg/tumor cell) N 1 N 2 N 3 N+1 N+2 N+3 0.01 0.03 0.02 427 6274 4512 697 13,843 16,576 202 150 90 Discussion The spread of primary HNSCC lesions to locoregional lymph nodes has often already occurred at the time of diagnosis, thus compromising the prognosis and long-term survival of HNSCC patients.3 Accurate diagnosis of lymph node metastases remains difcult, and many patients that do not present cancer dissemination to the lymph nodes (N0) may be subjected to unnecessary elective surgery. On the other hand, small lesions may be difcult to identify within the lymph nodes in cryosections when histophatologic evaluation is performed while the patient is in the operating room. Hence, some patients may miss a therapeutic opportunity due to false negative diagnosis of lymph node metastasis. Here, we demonstrate that DSG3 is expressed in normal oral squamous mucosa, and in all HNSCC lesions and their metastatic cervical lymph nodes. Indeed, the presence of DSG3 in lymph nodes can be exploited to detect Downloaded for Anonymous User (n/a) at Marian University from ClinicalKey.com by Elsevier on March 08, 2021. For personal use only. No other uses without permission. Copyright 2021. Elsevier Inc. All rights reserved. 100 V. Patel et al. / Oral Oncology 49 (2013) 93101 micrometastatic lesions, which can serve as a sensitive marker of HNSCC progression. We also show the feasibility of using a rapid, low-cost nanostructured immunoarray device for the detection of DSG3 protein in metastatic lymph nodes in newly diagnosed HNSCC patients, which can improve diagnosis and guide the most effective therapeutic options. Most current technologies for cancer detection and diagnostics are not suitable for the differentiation of normal versus metastatic lymph nodes at early stages of cancer progression, and efforts to address this gap have been met with mixed results. Currently, the gold standard for identication of metastasis is the serial sectioning and histopathological analysis of tissue specimens by H&E staining and immunohistochemistry.30 This provides key information needed for TNM (tumor-node-metastasis criteria) classication of HNSCC patients. However, a risk remains that micrometastases may go undetected in otherwise negative lymph nodes. IHC performed on serial sections for cytokeratin may help in detecting metastases, but unfortunately this low-throughput method requires signicant investment of time and expense, and it is often performed after surgery. Considering the false-negative rate and the sampling error that are encountered by H&E examination alone, a reliable and rapid predictive test to determine lymph node metastases is needed.31 Application of new technologies such as real-time quantitative PCR (qPCR), to look at mRNA levels of molecules expressed by oral squamous tissues have shown encouraging results.32 While this improves upon some of the limitations of IHC detection of cytokeratins, independent studies have found some inconsistencies in the precise cytokeratin to be analyzed. For example, from a three-marker analysis, only cytokeratin 14 was reliably detected by qPCR in several oral cancer cell lines and tissues, and sensitive enough to detect down to a single cancer cell in a background of Jurkat cells, essentially representing a model of lymph node metastasis.33 In another study, cytokeratin 17 was demonstrated to be far superior at discriminating positive lymph nodes while cytokeratin 14 was less informative, although in parallel histological analysis, this was only achieved if metastasis had exceeded 450 lm, leaving a high probability of micrometastasis going undetected.34 While the sensitivity of qPCR for detecting cytokeratins is unquestionable, its ability to reliably and consistently detect these molecules in a single tumor cell embedded within normal lymphatic tissues still remains a challenge. The met-receptor is over-expressed in several metastatic carcinomas including HNSCC,35,36 and with minimal to none in lymphatic cells, its presence in lymph nodes may be exploited for predicting metastasis. Indeed, qPCR analysis detected met expression in 40% of invaded lymph nodes, and interestingly exceeding the sensitivity of cytokeratins, which were tested in the same sample cohort.37 Use of multiple markers may improve detection of metastatic lymph nodes, and in this regard, mRNA for DSG3 (referred to as pemphigus vulgaris antigen, PVA) and TACSTD1 (tumor-associated calcium signal transducer 1), have been previously reported to be highly expressed in HNSCC, and successfully integrated into a multiplex qRT-PCR assay for metastatic prediction, achieving a remarkable accuracy.38,39 DSG3 mRNA levels in lymph nodes have been also touted as potential predictors of HNSCC progression.38,40 Generally, the use of qPCR greatly improves the sensitivity of detection of target genes, but the need of high quality RNA extracted from tissues remains a signicant technical hurdle, such as the presence of contaminants and RNA degradation that can severely interfere with data interpretation. It is noteworthy that mRNA levels may not accurately reect protein expression, as many post translational regulatory processes may allow or prevent the accumulation of translated products, and for predictive biomarkers, the presence of the target protein may be better suited. In this regard, our proteomics analysis of HNSCC and normal oral epithelial tissues suggested that DSG3 is preferentially expressed in squamous tissues.21 By examining hundreds of cancer lesions representing some of the most prevalent human malignancies we now show that DSG3 is highly expressed in all tumors derived from cells of squamous epithelial origin, such as lung SCC and HNSCC, with more variable expression in adenocarcinomas of the colon, prostate, breast and lung, likely reecting their glandular epithelial cell of origin. For HNSCC, we noticed a lower expression of DSG3 in poorly differentiated lesions, aligned with prior reports.41 However, all HNSCC cases analyzed expressed DSG3, albeit in some lesions not all tumor cells expressed this marker. Thus, although the possibility exists that in some invaded lymph nodes the level of DSG3 may be below our detection limit, our collective ndings indicate that DSG3 is highly expressed in all human SCCs but not expressed in normal lymph nodes, and that DSG3 can be detected even in small clusters of malignant cells invading the lymph nodes, thus serving as a marker for the metastatic spread of the cancerous lesions. We now exploited this information, the availability of highly specic monoclonal antibodies detecting different epitopes in DSG3, and our recently established biomarker detection platform using microuidic immunoarray devices featuring nanostructured electrodes,26 to develop an assay system enabling the rapid and ultrasensitive detection of DSG3 protein in complex tissue extracts, with minimal non-specic binding. The method was sensitive enough to detect isolated tumor cells, and certainly small groups of cells in a single cryosection of a positive lymph node. Taken together, we can conclude that the ability to quantitate femtogram levels of DSG3 can be used for the intraoperational detection of the presence of even few invasive human squamous epithelial cells in cryosections of lymph nodes of HNSCC patients, hence aiding the pathologists and surgeons to make informed decisions about appropriate treatment options. We expect that similar approaches can be used to optimize the detection of additional cancer biomarkers in lymph node sections, thus increasing the basis for successful clinical prediction. Collectively, combined with a simple work-ow and a short assay time, these features described in this study, hold promise for the development of point-of-care clinical screening techniques to identify HNSCC patients with metastatic disease. Indeed, the encouraging results described in this proof of principle study may provide the rationale for future validation of this diagnostic strategy in larger multicenter studies. Conict of interest statement None declared. Acknowledgements This work was supported by the Intramural Program of the National Institute of Dental and Craniofacial Research, National Institutes of Health, and through grant R01EB014586 from the National Institute of Biomedical Imaging and Bioengineering (JRF). Appendix A. 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- Creador:
- Singh, Bhuvanesh, Marsh, Christina A., Sinha,Uttam K., Rusling, James F., Malhotra, Ruchika, Veenstra, Timothy D., Doci, Colleen L., Gutkind, J. Silvio, Patel, Vyomesh, Nathan, Cherie-Ann O., Martin, Daniel, and Molinolo, Alfredo A.
- Descripción:
- OBJECTIVES: The diagnosis of cervical lymph node metastasis in head and neck squamous cell carcinoma (HNSCC) patients constitutes an essential requirement for clinical staging and treatment selection. However, clinical...
- Tipo de recurso:
- Article