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REVIEW 4 major objections 5 minor 82 references

A Physics-Flavored Transformer Network for Parametrizing Contraction Dynamics of Engineered Skeletal Muscle Tissues

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims a physics-flavored neural network can turn raw force-time curves of engineered muscle into a small set of biologically meaningful stretched-exponential parameters, and that it does so more accurately than conventional itera

desk verdict A competent engineering pipeline for automated kinetic phenotyping, but the real-data validation is self-referential and the biological claims outrun the evidence. read the letter →

arxiv 2608.03927 v1 pith:M7W75UTI submitted 2026-08-04 cs.LG

Mattias Luber , Timo Betz This is my paper
classification cs.LG
keywords physics-informedneuralnetworkstretchedexponentialskeletalmuscletissueengineeringtetanuscontractionkineticphenotypingself-supervisedlearningDuchennemusculardystrophyforce-timeanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Engineered skeletal muscle tissues are usually characterized by a single number, peak force, which throws away the shape of the contraction. This paper tries to establish that a hybrid CNN-Transformer can instead read the entire force-time profile and output kinetic parameters — effective timescales, heterogeneity exponents, fatigue onset and slope — that capture the biology of contraction. The key move is to use the same stretched-exponential physical model twice: first to generate hundreds of thousands of synthetic labeled curves for pre-training, then as a differentiable reconstruction loss that lets unlabeled real recordings be absorbed into training. If the claim holds, contraction phenotyping becomes automated, scalable, and rich enough to distinguish disease and aging states from curve shape alone.

What carries the argument

The central object is a piecewise stretched-exponential force model: a rise phase F_rise(t)=A(1−exp(−((t−t0)/τ_r)^β_r)), a smoothed fatigue factor Φ(t), and a decay phase F_decay(t)=C+(A_c−C)exp(−((t−t1)/τ_d)^β_d). This model serves double duty as the generator of synthetic training data and as the physics-based reconstruction loss. The network architecture is a CNN-Transformer in which ten learnable parameter tokens, prepended to a convolutionally extracted embedding, query the whole signal via self-attention and are mapped to the ten relaxed physical parameters. The physics loss forces the tokens to align with the kinetic constants of the model, filtering out non-physical noise while prese

What would settle it

Simultaneously record calcium transients and pillar force in the same engineered muscle contraction, compute the network's predicted τ_d and β_d on the force curve, and compare them with the independently measured decay of cytosolic calcium; if the predicted relaxation kinetics do not track the calcium signal, the parameters are curve-shape descriptors rather than biological kinetics.

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Extended reading notes

Core claim

The paper claims that a CNN-Transformer network, trained with a hybrid objective, can take raw force-time recordings of engineered skeletal muscle tetani and output the parameters of a piecewise stretched-exponential contraction model — the aggregate timescale τ, the stretching exponents β for rise and decay, the fatigue onset and slope, and the absolute force scale — with enough fidelity that the reconstructed curves match held-out experimental recordings better than a standard trust-region fitter. The authors further claim that because the physics model is used both to generate a large synthetic training corpus and as a reconstruction loss on unlabeled real data, the network self-aligns to

Load-bearing premise

The piecewise stretched-exponential equation is assumed to describe real tetanus contraction dynamics so faithfully that minimizing reconstruction error within that same model yields biologically meaningful kinetic parameters.

Editorial extensions

If this is right

  • Kinetic phenotyping no longer requires hand-picked initial guesses: the network supplies a learned prior that lands near the optimum, and post-hoc iterative refinement adds only marginal gains.
  • Unlabeled contraction data from new experiments can be pulled into training automatically through the reconstruction loss, so the pipeline improves as more measurements accumulate without manual labeling.
  • The extracted parameter vectors, rather than peak force alone, become the comparison space for drug screens and disease models, with cell lines separating cleanly in principal-component space.
  • Because the architecture handles variable-length signals, recordings of different durations can be analyzed without cropping or interpolation, easing integration across experimental setups.
  • Generalization to three independent human donor lines, including a Duchenne muscular dystrophy line, indicates the parameterization transfers across biological contexts without retraining.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The self-training recipe is not muscle-specific: any biosignal with a trusted phenomenological model could be pre-trained synthetically and self-aligned on unlabeled recordings, so the strategy plausibly transfers to cardiac twitch or calcium-transient analysis.
  • A direct test of biological validity would compare the network's relaxation parameters against an orthogonal measurement, such as simultaneous calcium imaging or stiffness measurements, because no ground-truth parameters exist for real tissues.
  • The β_d shift between healthy and Duchenne lines suggests a quantitative marker of relaxation heterogeneity, but with only three cell lines this is a hypothesis that needs a larger cohort to confirm.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents a Physics-Flavored Neural Network (PFNN) that combines a CNN-Transformer architecture with a stretched-exponential model of engineered skeletal muscle (ESM) tetanus contractions. The model is trained on 300,000 synthetic curves generated from the same stretched-exponential function, plus unlabeled real contraction curves via an unsupervised reconstruction loss, and outputs ten physical parameters. The authors report that the network accurately predicts synthetic ground-truth parameters, improves reconstruction error on real data compared to a single-initialization L-BFGS baseline, generalizes to three independent donor cell lines, and yields biological insights such as slowed kinetics in Duchenne Muscular Dystrophy and aged tissues. The core methodological contribution is the hybrid synthetic-plus-unsupervised training paradigm, which is intended to overcome the absence of labeled biological data.

Significance. If the extracted parameters were shown to correspond to true biological kinetics, the PFNN would be a valuable tool for high-throughput functional phenotyping of engineered muscle, replacing simple peak-force metrics. The architecture is sensible, and the synthetic-to-real adaptation strategy is a pragmatic response to label scarcity. The integrated-gradients attribution analysis is a nice attempt to show that the network attends to physiologically relevant time windows. However, the current real-data evaluation is self-referential: the metric used to validate the model on real data, MSE(y_real, F(p_hat)), is the same physics-reconstruction loss used to train the model on unlabeled real data. Low reconstruction error therefore does not establish that the predicted parameters are biologically meaningful, especially since the paper explicitly states that no experimental ground truth exists. The biological conclusions in Section 3.5 rest on this unvalidated link.

major comments (4)
  1. [Section 3.2, 3.4; Eq. (3)] The real-data evaluation is circular. In Eq. (3), the model is trained on unlabeled real data by minimizing L_fit(B_real) = MSE(y_real, F(p_hat_real)). The validation metrics in Tables 2 and 3 are exactly this same reconstruction MSE. A low value only shows that the network has internalized the stretched-exponential model F, not that the predicted tau, beta, alpha, etc. correspond to true biological kinetics. Section 2.1 confirms that no ground-truth parameters exist. To support the abstract's claim of 'high-fidelity parameterization' and the biological insights in Section 3.5, the authors need external validation: e.g., use the known pharmacological interventions from the primary dataset (ref. 26) to check that predicted parameters shift in expected directions, compare against an independent biophysical assay, or at minimum report a formal identifiability analysis on synthetic data with
  2. [Section 3.1, Fig. 4] The synthetic evaluation only tests whether the network can invert its own generator. Both training and test sets are generated from the same stretched-exponential model F, so high R² values (0.85–1.00) demonstrate identifiability within that family, but not that F adequately represents real ESM dynamics. The 'visual indistinguishability' claim in Fig. 3 is qualitative, and Fig. 4B already shows systematic deviations in the decay regime on real data. The authors should quantify the domain shift, e.g., by comparing the distribution of residuals between synthetic and real curves, or by testing on synthetic data generated from perturbed/out-of-distribution parameters. Without this, the statement that 'synthetic data is a strong basis' is not fully supported.
  3. [Section 3.4, Table 2, Fig. 7] The L-BFGS baseline is not a strong comparator. The initialization is a single 'average curve' (Fig. 7), and the claim that L-BFGS 'became trapped in local minima' is not established because only one starting point was tried. Classical curve fitting routinely uses multi-start or random restarts, especially for non-convex problems. The authors should compare against multi-start L-BFGS (e.g., 50–100 random initializations, reporting best-of-N) or a global optimizer. Beating a single-initialization solver shows only that the network provides a better warm start, not that the network is more accurate than classical fitting. This directly affects the headline result 'surpasses L-BFGS in terms of accuracy'.
  4. [Section 3.5, Fig. 9] The biological parameter comparisons across cell lines are based on point estimates without uncertainty quantification or statistical significance testing. The PCA clustering in Fig. 9 could arise from differences in noise, curve shape outside the model family, or non-identifiable parameter combinations, rather than from true physiological differences. The authors should provide confidence intervals for the predicted parameters (e.g., via ensembling or dropout) and perform proper statistical tests (e.g., mixed-effects models accounting for multiple curves per tissue) before claiming that DMD and aged tissues show 'significantly decayed' kinetics or that beta_d heterogeneity differs. The manuscript's own caveat that only three cell lines were studied does not address this more fundamental issue.
minor comments (5)
  1. [Eq. (2), Section 2.3.2] Equation (2) contains a typo: the last term should be MSE(y_real, F(p_hat_real)), not MSE(y_real, p_hat_real). Equation (3) is correct, but the inconsistency is confusing.
  2. [Figure 6] The integrated-gradient attribution scores are described qualitatively. Please specify how scores were normalized across samples, how the average was computed, and whether any statistical threshold was used to define 'sharp' peaks.
  3. [Section 5, Limitations] The Limitations section discusses only computational complexity and sampling-rate flexibility; it does not mention the absence of ground-truth validation for real-data parameters or the potential non-identifiability of the stretched-exponential parameters. Given that this is the central risk to the biological interpretation, it should be explicitly acknowledged.
  4. [General] There are several typographical errors: 'intragration' (Introduction), 'unsuperwised' (Fig. 2), 'paramters' (Fig. 9), 'appraoch' (Conclusion). A careful proofread is needed.
  5. [Table 1] A, t0, and C are listed as 'Unconstrained' although the physical model likely expects A and C to be non-negative. Please clarify or add appropriate transformations.

Circularity Check

2 steps flagged · score 6.0 of 10

Real-data validation is self-referential: the evaluation metric is the same physics reconstruction loss used in training, and the visual 'confirmation' of the model is produced by fitting the model to the data.

  1. fitted input called prediction [Section 2.6 (Eq. 3, Objective) and Section 3.4 / Tables 2-3]
    "The physics-fit loss (Lf it) also used an MSE loss, but between the input force-time signal and the curve reconstructed by passing the predicted parameters through the physics flavored model F(ˆp). ... Objective= 1 N NX i=1 MSE(yreal,i, F(ˆpreal,i))"

    The reported real-data 'accuracy' is exactly the Lfit term used to train on unlabeled real data (Eq. 3), computed through the same stretched-exponential F that generated the synthetic labels. Section 2.1 states 'no experimental ground truth exists for direct model supervision.' Therefore low MSE(y_real, F(p_hat)) only shows that the network reconstructs curves through the model it was trained to use; it cannot validate that the predicted tau, beta, alpha are biologically meaningful. The parameter 'predictions' are scored by the very objective they were trained to minimize.

  2. self definitional [Section 3.1 and Figure 3 caption]
    "For direct comparison, the synthetic cohorts are centered around the respective experimental curves: the optimal parameter fit is highlighted in dark red ... The synthetic profiles are visually near-indistinguishable from the real measurements ... This high degree of morphological overlap confirms that the stretched exponential model provides a sufficient and robust descriptor of the complex, multi-modal dynamics inherent in engineered skeletal muscle contractions."

    The synthetic curves are 'centered around the respective experimental curves' via the 'optimal parameter fit' of the very same model F. The visual near-indistinguishability is therefore the fitting residual, not an independent test of whether the stretched-exponential family truly generates the real signals. This overlap is used as the 'cornerstone' justification that F represents the true biological signal, making the model-adequacy evidence circular and load-bearing for the later claim that extracted parameters are physically meaningful.

full rationale

The PFNN architecture and its synthetic-data inversion are not themselves circular: Figure 4 is an internal consistency check of learning the inverse of F, and generalizing reconstruction to held-out real curves is a nontrivial empirical result. However, the paper's central claim of extracting 'physically meaningful parameters' rests on two circular validation moves. First, real-data performance (Tables 2-3) is measured by the same physics reconstruction loss Lfit used in training on unlabeled real data, with the same model F; because no ground truth exists for the kinetic parameters, low MSE is a self-consistency check within the assumed model family, not evidence of biological correctness. Second, the visual evidence that F adequately describes real contractions is produced by fitting F to each real curve, so the match is manufactured by construction. The remaining external support for model adequacy is the authors' own prior work [26], which is load-bearing but not independently verified in this manuscript. The L-BFGS comparison only shows escape from one local basin, not parameter correctness. These issues undercut the biological-interpretation claims while leaving the curve-reconstruction capability intact, so a score of 6 is appropriate rather than a higher score.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The work introduces no new physical entities. The central assumption burden is that the stretched-exponential model, inherited from the authors' prior paper, is both sufficient and identifiable for real ESM contractions, and that minimizing self-reconstruction error yields biologically meaningful parameters despite the absence of any external parameter ground truth.

free parameters (4)
  • Training weighting coefficients lambda, gamma, and real-data sampling probability p_real = not reported; selected via Optuna
    Equation 3; these are tuned to the validation set, so the headline MSE partly reflects hyperparameter selection on the evaluation set.
  • Synthetic data parameter ranges and noise scales = not reported
    Section 2.5 states parameters are varied over physiologically relevant ranges with Gaussian noise, but exact ranges and noise amplitudes are not given, blocking exact replication.
  • CNN-Transformer architecture hyperparameters = not reported fully
    Sections 2.4 and 2.6 describe the architecture conceptually and say Optuna was used, but exact layer counts, dimensions, heads, and learning rates are omitted.
  • Physical model parameters (A, tau_r, beta_r, tau_d, beta_d, t0, dur, t_delta, alpha, sigma_smooth, C) = per-curve network predictions, no independent ground truth
    These are the output variables of the claimed parameterization; they are fitted to each curve and never validated against an external biological measurement, so the correctness of the central extraction claim rests on fitted values.
assumptions (3)
  • domain assumption KWW stretched-exponential piecewise model (Eq. 1 and Section 2.3.1) faithfully represents tetanus contraction dynamics, including fatigue.
    The model is taken from the authors' prior work [26]; the paper's synthetic data, training loss, and evaluation all use this same model, so the biological validity of the extracted parameters depends entirely on this form.
  • domain assumption Synthetic parameter ranges and noise model cover the empirical distribution of real ESM recordings.
    Section 2.5 says parameters are varied to cover empirical distribution, but exact ranges are not reported; if the synthetic prior misses real phenotypes, the zero-shot and self-alignment results do not transfer.
  • ad hoc to paper Minimizing reconstruction error through the model yields biologically meaningful parameter values.
    Section 2.1 explicitly notes no ground truth exists for the parameters; the paper treats the physics-loss optimum as the correct parameterization without an external check.

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Cite this review

Pith. "Pith review of A Physics-Flavored Transformer Network for Parametrizing Contraction Dynamics of Engineered Skeletal Muscle Tissues." pith.science (2026). https://pith.science/paper/M7W75UTI

@misc{pith2026260803927,
  author       = {Pith},
  title        = {Pith review of: A Physics-Flavored Transformer Network for Parametrizing Contraction Dynamics of Engineered Skeletal Muscle Tissues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7W75UTI}},
  note         = {Machine review of arXiv:2608.03927}
}
read the original abstract

Engineered Skeletal Muscle Tissues (ESMs) have become a key structure for biomedical disease modeling and pharmacological screening, yet their functional characterization often relies on simplistic metrics like peak force, discarding critical kinetic information. This is partially due to the high level of mathematical complexity which mechanistic models introduce to capture these dynamics. Hence, exactly the complexity prevents scalable application and widespread adaptation in the field. Here we present a Physics-Flavored Neural Network (PFNN) that automates the kinetic phenotyping of ESMs. Our architecture integrates a stretched-exponential physical model into a CNN-Transformer, enabling the extraction of physically meaningful parameters directly from force-time profiles. To address the scarcity of labeled biological data, we employ a hybrid training paradigm: the model develops a "physical intuition" on synthetic data before undergoing unsupervised self-alignment on unlabeled real-world measurements. Our results demonstrate that this physics-flavored approach achieves high-fidelity parameterization across diverse contractile phenotypes and cell lines, including Duchenne Muscular Dystrophy models. Our scalable, self-improving pipeline bridges the gap between idealized biophysics and noisy \emph{in vitro} data, providing a robust tool for high-throughput biophysical research.

Figures

Figures reproduced from arXiv: 2608.03927 by the authors.

Figure 1
Figure 1. Sketch of the experimental data derivation for the force-time profiles of engineered [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Overview of the physics-flavored CNN-Transformer Architecture. The model itself [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The upper row displays experimental force-time profiles (blue), illustrating a wide spec [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Quantitative and qualitative evaluation of the CNN-Transformer network trained ex [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Qualitative zero-shot evaluation of the semi-supervised model, trained on synthetic [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Attribution scores for the different parameter predictions. The plots illustrate the tem [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Visualization of the initial parametrization for the L-BFGS-based parameter estimation. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Visualization of the fit derived by the model (black) and the measurement data (colored) [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: A) Comparison of the predicted contractile parameters for different cell lines. The healthy cell line AB1167 shows a much higher peak force and faster contraction/relaxation dy￾namics compared to the diseased (K571) or aged (KM670) counterparts. Whereas the hetero￾geni…

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Works this paper leans on

82 extracted references · 62 canonical work pages

  1. [1]

    Afshar Bakooshli, M., Lippmann, E.S., Mulcahy, B., Iyer, N., Nguyen, C.T., Tung, K., Stew- art, B.A., van den Dorpel, H., Fuehrmann, T., Shoichet, M., Bigot, A., Pegoraro, E., Ahn, H., Ginsberg, H., Zhen, M., Ashton, R.S., and Gilbert, P .M. (2019). A 3d culture model of inner- vated human skeletal muscle enables studies of the adult neuromuscular junctio...

    work page doi:10.7554/elife.44530 2019

  2. [2]

    Shahriyari, M., Islam, M.R., Sakib, S.M., Rinn, M., Rika, A., Kr¨uger, D., Kaurani, L., Gisa, V., Winterhoff, M., Anandakumar, H., Shomroni, O., Schmidt, M., Salinas, G., Unger, A., Linke, W.A., Zsch¨untzsch, J., Schmidt, J., Bassel-Duby, R., Olson, E.N., Fischer, A., Zimmermann, W.H., and Tiburcy, M. (2022). Engineered skeletal muscle recapitulates human...

    work page doi:10.1002/jcsm.13094 2022

  3. [3]

    Y oshida, A., Baba, K., Takahashi, H., Nagese, K., and Shimizu, T. (2025). One-step fab- rication of 3d-aligned human skeletal muscle tissue and measurement of contractile force for preclinical drug testing. Materials today. Bio31, 101456. doi:10.1016/j.mtbio.2025. 101456

    work page doi:10.1016/j.mtbio.2025 2025

  4. [4]

    Rajabian, N., Shahini, A., Asmani, M., Vydiam, K., Choudhury, D., Nguyen, T., Ikhapoh, I., Zhao, R., Lei, P ., and Andreadis, S.T. (2021). Bioengineered skeletal muscle as a model of muscle aging and regeneration. Tissue engineering. Part A27, 74–86. doi:10.1089/ten. TEA.2020.0005

    work page arXiv 2021

  5. [5]

    Vesga-Castro, C., Aldazabal, J., Vallejo-Illarramendi, A., and Paredes, J. (2022). Contractile force assessment methods for in vitro skeletal muscle tissues. eLife11. doi:10.7554/ eLife.77204

    2022

  6. [6]

    Moyle, L.A., Jacques, E., and Gilbert, P .M. (2020). Engineering the next generation of hu- man skeletal muscle models: From cellular complexity to disease modeling. Current Opinion in Biomedical Engineering16, 9–18. doi:10.1016/j.cobme.2020.05.006

    work page doi:10.1016/j.cobme.2020.05.006 2020

  7. [7]

    van der Wal, E., Iuliano, A., in ’t Groen, S.L.M., Bholasing, A.P ., Priesmann, D., Sharma, P ., den Hamer, B., Saggiomo, V., Kr ¨uger, M., Pijnappel, W.W.M.P ., and de Greef, J.C. (2023). Highly contractile 3d tissue engineered skeletal muscles from human ipscs reveal similarities with primary myoblast-derived tissues. Stem cell reports18, 1954–1971. doi...

    work page doi:10.1016/j.stemcr.2023.08.014 2023

  8. [8]

    R ¨ohrle, O., Y avuz, U.S ¸ ., Klotz, T., Negro, F ., and Heidlauf, T. (2019). Multiscale modeling of the neuromuscular system: Coupling neurophysiology and skeletal muscle mechanics. Wiley interdisciplinary reviews. Systems biology and medicine11, e1457. doi:10.1002/ wsbm.1457

    2019

Show all 82 references
  1. [9]

    Lieber, R.L., and Ward, S.R. (2013). Cellular mechanisms of tissue fibrosis. 4. structural and functional consequences of skeletal muscle fibrosis. American journal of physiology. Cell physiology305, C241–52. doi:10.1152/ajpcell.00173.2013

    2013

  2. [10]

    Hill, C., Brunello, E., Fusi, L., Ovejero, J.G., and Irving, M. (2021). Myosin-based regulation of twitch and tetanic contractions in mammalian skeletal muscle. eLife10. doi:10.7554/ eLife.68211. 22

    2021

  3. [11]

    Wang, D., Wu, J., Xu, Z., Jia, J., Lai, Y ., and He, Z. (2025). Increased matrix stiffness promotes slow muscle fibre regeneration after skeletal muscle injury. Journal of cellular and molecular medicine29, e70423. doi:10.1111/jcmm.70423

    2025 doi

  4. [12]

    Wakeling, J.M., Febrer-Nafr ´ıa, M., and de Groote, F . (2023). A review of the efforts to de- velop muscle and musculoskeletal models for biomechanics in the last 50 years. Journal of biomechanics155, 111657. doi:10.1016/j.jbiomech.2023.111657

    2023

  5. [13]

    HUXLEY , A.F ., and NIEDERGERKE, R. (1954). Structural changes in muscle during con- traction; interference microscopy of living muscle fibres. Nature173, 971–973. doi: 10.1038/173971a0

    1954 doi

  6. [14]

    Oomens, C.W.J., Maenhout, M., van Oijen, C.H., Drost, M.R., and Baaijens, F .P . (2003). Finite element modelling of contracting skeletal muscle. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences358, 1453–1460. doi:10.1098/ rstb.2003.1345

    2003

  7. [15]

    Mili ´cevi´c, B., Ivanovi´c, M., Stojanovi ´c, B., Milo ˇsevi´c, M., Koji´c, M., and Filipovi ´c, N. (2022). Huxley muscle model surrogates for high-speed multi-scale simulations of cardiac contrac- tion. Computers in biology and medicine149, 105963. doi:10.1016/j.compbiomed.2...

    2022 doi

  8. [16]

    Villota-Narvaez, Y ., Garz´on-Alvarado, D.A., R¨ohrle, O., and Ram´ırez-Mart´ınez, A.M. (2022). Multi-scale mechanobiological model for skeletal muscle hypertrophy. Frontiers in physiol- ogy13, 899784. doi:10.3389/fphys.2022.899784

    2022

  9. [17]

    Zeng, W., Hume, D.R., Lu, Y ., Fitzpatrick, C.K., Babcock, C., Myers, C.A., Rullkoetter, P .J., and Shelburne, K.B. (2023). Modeling of active skeletal muscles: a 3d continuum approach incorporating multiple muscle interactions. Frontiers in bioengineering and biotechnology 11...

    2023

  10. [18]

    Zahalak, G.I. (1981). A distribution-moment approximation for kinetic theories of muscular contraction. Mathematical Biosciences55, 89–114. doi:10.1016/0025-5564(81)90014-6

    1981 doi

  11. [19]

    van Soest, A.J.K., Casius, L.J.R., and Lemaire, K.K. (2019). Huxley-type cross-bridge mod- els in largeish-scale musculoskeletal models; an evaluation of computational cost. Journal of biomechanics83, 43–48. doi:10.1016/j.jbiomech.2018.11.021

    2019 doi

  12. [20]

    Wang, M., Jin, D., Wang, H., Xu, X., and Zheng, S. (2025). Multi-scale modeling and simu- lation of skeletal muscles with different fatigue degrees based on microphysiology. Scientific reports15, 11020. doi:10.1038/s41598-025-87443-4

    2025 doi

  13. [21]

    Heidlauf, T., Klotz, T., Rode, C., Altan, E., Bleiler, C., Siebert, T., and R ¨ohrle, O. (2016). A multi-scale continuum model of skeletal muscle mechanics predicting force enhancement based on actin-titin interaction. Biomechanics and modeling in mechanobiology15, 1423–

    2016

  14. [22]

    Hern ´andez-Gasc´on, B., Grasa, J., Calvo, B., and Rodr ´ıguez, J.F . (2013). A 3d electro- mechanical continuum model for simulating skeletal muscle contraction. Journal of theoret- ical biology335, 108–118. doi:10.1016/j.jtbi.2013.06.029

    2013 doi

  15. [23]

    Karami, M., Zohoor, H., Calvo, B., and Grasa, J. (2023). A 3d multi-scale skeletal mus- cle model to predict active and passive responses. application to intra-abdominal pressure prediction. Computer Methods in Applied Mechanics and Engineering415, 116222. doi: 10.1016/j.cma.2...

    2023

  16. [24]

    Lemaire, K.K., Baan, G.C., Jaspers, R.T., and van Soest, A.J.K. (2016). Comparison of the validity of hill and huxley muscle-tendon complex models using experimental data obtained from rat m. soleus in situ. The Journal of experimental biology219, 977–987. doi:10.1242/ jeb.128280

    2016

  17. [25]

    Dennis, R.G., Kosnik, P .E., Gilbert, M.E., and Faulkner, J.A. (2001). Excitability and contrac- tility of skeletal muscle engineered from primary cultures and cell lines. American journal of physiology. Cell physiology280, C288–95. doi:10.1152/ajpcell.2001.280.2.C288

    2001 doi

  18. [26]

    Luber, M., Schmelz, B., Lenz, C., and Betz, T. (2026). Stretched Exponential Modeling Reveals Drug-Specific Kinetics in Human Engineered Skeletal Muscle. . doi:10.64898/ 2026.06.08.730797

    2026

  19. [27]

    Raissi, M., Perdikaris, P ., and Karniadakis, G.E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics378, 686–707. doi:10. 1016/j.jcp....

    2019

  20. [28]

    Cai, S., Mao, Z., Wang, Z., Yin, M., and Karniadakis, G.E. (2021). Physics-informed neural networks (pinns) for fluid mechanics: a review. Acta Mechanica Sinica37, 1727–1738. doi: 10.1007/s10409-021-01148-1

    2021 doi

  21. [29]

    Janssen, M., Chan, C.k., Davelaar, J., Natarajan, I., Olivares, H., Ripperda, B., R ¨oder, J., Rynge, M., and Wielgus, M. (2025). Deep learning inference with the event horizon telescope. Astronomy & Astrophysics698, A60. doi:10.1051/0004-6361/202553784

    2025 doi

  22. [30]

    Ahmadi, N., Cao, Q., Humphrey, J.D., and Karniadakis, G.E. (2026). Physics-informed ma- chine learning in biomedical science and engineering. Annual review of biomedical engi- neering28, 309–336. doi:10.1146/annurev-bioeng-110824-124907

    2026 doi

  23. [31]

    Jeong, Y .J., and Moon, T. (2025). Physics-informed fine-tuning for physics discovery from random and sparse data. Engineering Applications of Artificial Intelligence162, 112132. doi:10.1016/j.engappai.2025.112132

    2025

  24. [32]

    Guerrero-Viu, J., Izquierdo, S., Schr ¨oppel, P ., and Brox, T. (). Semi-supervised disparity estimation with deep feature reconstruction. arXiv. doi:10.48550/ARXIV.2106.00318

    doi arXiv

  25. [33]

    Wang, Z., Yu, X., Wang, C., Chen, W., Wang, J., Chu, Y .H., Sun, H., Li, R., Li, P ., Y ang, F ., Han, H., Kang, T., Lin, J., Y ang, C., Chang, S., Shi, Z., Hua, S., Li, Y ., Hu, J., Zhu, L., Zhou, J., Lin, M., Guo, J., Cai, C., Chen, Z., Di Guo, Y ang, G., and Qu, X. (2025). ...

    2025

  26. [34]

    Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., De- hghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., and Houlsby, N. (). An image is worth 16x16 words: Transformers for image recognition at scale. arXiv. doi: 10.48550/ARX...

    2010 doi arXiv

  27. [35]

    Devlin, J., Chang, M.W., Lee, K., and Toutanova, K. (). Bert: Pre-training of deep bidirec- tional transformers for language understanding. arXiv. doi:10.48550/ARXIV.1810.04805

    doi arXiv

  28. [36]

    Hofemeier, A.D., Limon, T., Muenker, T.M., Wallmeyer, B., Jurado, A., Afshar, M.E., Ebrahimi, M., Tsukanov, R., Oleksiievets, N., Enderlein, J., Gilbert, P .M., and Betz, T. (2021). Global and local tension measurements in biomimetic skeletal muscle tissues re- veals early mec...

    2021 doi

  29. [37]

    Madden, L., Juhas, M., Kraus, W.E., Truskey, G.A., and Bursac, N. (2015). Bioengineered human myobundles mimic clinical responses of skeletal muscle to drugs. eLife4, e04885. doi:10.7554/eLife.04885

    2015 doi

  30. [38]

    Smith, A.S., Luttrell, S.M., Dupont, J.B., Gray, K., Lih, D., Fleming, J.W., Cunningham, N.J., Jepson, S., Hesson, J., Mathieu, J., Maves, L., Berry, B.J., Fisher, E.C., Sniadecki, N.J., Geisse, N.A., and Mack, D.L. (2022). High-throughput, real-time monitoring of engi- neered...

    2022 doi

  31. [39]

    Tollitt, B.R., Jones, S.W., Ohana, J., Henstock, J.R., Jackson, M.J., and McArdle, A. (2025). A comparison of human skeletal muscle cell maturation in 2d versus 3d culture: A quantita- tive proteomic study. Physiological reports13, e70420. doi:10.14814/phy2.70420

    2025 doi

  32. [40]

    Tiper, Y ., Xie, Z., Hofemeier, A., Lad, H., Luber, M., Krawetz, R., Betz, T., Zimmermann, W.H., Morton, A.B., Segal, S.S., and Gilbert, P .M. (2025). Optimizing electrical field stim- ulation parameters reveals the maximum contractile function of human skeletal muscle microti...

    2025

  33. [41]

    Hofemeier, A.D., Ristau, M., Luber, M., Muenker, T.M., Herkenrath, F ., Schmelz, B., Schar- fenstein, L.M., Brandt, M., Malova, P ., Shahriyari, M., Rinn, M., Haertter, D., Tiburcy, M., Zimmermann, W.H., Lenz, C., Mamchaoui, K., Bigot, A., Nguyen, J., Gilbert, P .M., Li `evre,...

    2022 doi

  34. [42]

    Kohlrausch, R. (1854). Theorie des elektrischen r ¨uckstandes in der leidener flasche. An- nalen der Physik167, 56–82. doi:10.1002/andp.18541670103

    doi

  35. [43]

    Rausch, M., B ¨ohringer, D., Steinmann, M., Schubert, D.W., Schr ¨ufer, S., Mark, C., and Fabry, B. (2020). Measurement of skeletal muscle fiber contractility with high-speed traction microscopy. Biophysical journal118, 657–666. doi:10.1016/j.bpj.2019.12.014

    2020 doi

  36. [44]

    Moreno-Justicia, R., van der Stede, T., Stocks, B., Laitila, J., Seaborne, R.A., van de Loock, A., Lievens, E., Samodova, D., Mar ´ın-Arraiza, L., Dmytriyeva, O., Browaeys, R., van Vossel, K., Moesgaard, L., Yigit, N., Anckaert, J., Weyns, A., van Thienen, R., Sahl, R.E., Zano...

    2025

  37. [45]

    Juhas, M., Y e, J., and Bursac, N. (2015). Design, evaluation, and application of engineered skeletal muscle. Methods (San Diego, Calif.)99, 81–90. doi:10.1016/j.ymeth.2015.10. 002

    2015 doi

  38. [46]

    Cheng, C.S., Davis, B.N.J., Madden, L., Bursac, N., and Truskey, G.A. (2014). Physiology and metabolism of tissue-engineered skeletal muscle. Experimental biology and medicine (Maywood, N.J.)239, 1203–1214. doi:10.1177/1535370214538589

    2014 doi

  39. [47]

    Istratov, A.A., and Vyvenko, O.F . (1999). Exponential analysis in physical phenomena. Re- view of Scientific Instruments70, 1233–1257. doi:10.1063/1.1149581

    1999 doi

  40. [48]

    Spencer, R.G., and Bi, C. (2020). A tutorial introduction to inverse problems in magnetic resonance. NMR in biomedicine33, e4315. doi:10.1002/nbm.4315. 25

    2020 doi

  41. [49]

    Sabett, C., Hafftka, A., Sexton, K., and Spencer, R.G. (2017). L 1 , l p , l 2 , and elastic net penalties for regularization of gaussian component distributions in magnetic resonance relaxometry. Concepts in Magnetic Resonance Part A46A. doi:10.1002/cmr.a.21427

    2017 doi

  42. [50]

    Elton, D.C. (). Stretched exponential relaxation. arXiv. doi:10.48550/ARXIV.1808.00881

    doi arXiv

  43. [51]

    Berberan-Santos, M.N., Bodunov, E.N., and Valeur, B. (2005). Mathematical functions for the analysis of luminescence decays with underlying distributions 1. kohlrausch decay func- tion (stretched exponential). Chemical Physics315, 171–182. doi:10.1016/j.chemphys. 2005.04.006

    2005 doi

  44. [52]

    Lindsey, C.P ., and Patterson, G.D. (1980). Detailed comparison of the williams–watts and cole–davidson functions. The Journal of Chemical Physics73, 3348–3357. doi:10.1063/ 1.440530

    1980

  45. [53]

    Johnston, D.C. (2006). Stretched exponential relaxation arising from a continuous sum of exponential decays. Physical Review B74. doi:10.1103/PhysRevB.74.184430

    2006 doi

  46. [54]

    Lukichev, A. (2019). Physical meaning of the stretched exponential kohlrausch function. Physics Letters A383, 2983–2987. doi:10.1016/j.physleta.2019.06.029

    2019 doi

  47. [55]

    Lee, K.C., Siegel, J., Webb, S.E., L ´evˆeque-Fort, S., Cole, M.J., Jones, R., Dowling, K., Lever, M.J., and French, P .M. (2001). Application of the stretched exponential func- tion to fluorescence lifetime imaging. Biophysical journal81, 1265–1274. doi:10.1016/ S0006-3495(01)75784-0

    2001

  48. [56]

    Flomenbom, O., Velonia, K., Loos, D., Masuo, S., Cotlet, M., Engelborghs, Y ., Hofkens, J., Rowan, A.E., Nolte, R.J.M., van der Auweraer, M., de Schryver, F .C., and Klafter, J. (2005). Stretched exponential decay and correlations in the catalytic activity of fluctuating singl...

    2005 doi

  49. [57]

    Song, J., Holten-Andersen, N., and McKinley, G.H. (2023). Non-maxwellian viscoelastic stress relaxations in soft matter. Soft matter19, 7885–7906. doi:10.1039/d3sm00736g

    2023 doi

  50. [58]

    Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., and Polosukhin, I. (). Attention is all you need. arXiv. doi:10.48550/ARXIV.1706.03762

    doi arXiv

  51. [59]

    Akiba, T., Sano, S., Y anase, T., Ohta, T., and Koyama, M. (). Optuna: A next-generation hyperparameter optimization framework. arXiv. doi:10.48550/ARXIV.1907.10902

    1907 doi arXiv

  52. [60]

    Sundararajan, M., Taly, A., and Y an, Q. (). Axiomatic attribution for deep networks. arXiv. doi:10.48550/ARXIV.1703.01365

    doi arXiv

  53. [61]

    Kokhlikyan, N., Miglani, V., Martin, M., Wang, E., Alsallakh, B., Reynolds, J., Melnikov, A., Kliushkina, N., Araya, C., Y an, S., and Reblitz-Richardson, O. (). Captum: A unified and generic model interpretability library for pytorch. arXiv. doi:10.48550/ARXIV.2009.07896

    2009 doi arXiv

  54. [62]

    Liu, D.C., and Nocedal, J. (1989). On the limited memory bfgs method for large scale opti- mization. Mathematical Programming45, 503–528. doi:10.1007/BF01589116

    1989 doi

  55. [63]

    Schmidt, M. (2005). minfunc: unconstrained differentiable multivariate optimization in mat- lab. Software available at http://www. cs. ubc. ca/ schmidtm/Software/minFunc. htm. 26

    2005

  56. [64]

    Mestre, R., Garc ´ıa, N., Pati ˜no, T., Guix, M., Fuentes, J., Valerio-Santiago, M., Almi ˜nana, N., and S´anchez, S. (2021). 3d-bioengineered model of human skeletal muscle tissue with phenotypic features of aging for drug testing purposes. Biofabrication13. doi:10.1088/ 1758...

    2021

  57. [65]

    Larsson, L., Degens, H., Li, M., Salviati, L., Lee, Y .I., Thompson, W., Kirkland, J.L., and San- dri, M. (2019). Sarcopenia: Aging-related loss of muscle mass and function. Physiological reviews99, 427–511. doi:10.1152/physrev.00061.2017

    2019

  58. [66]

    Korhonen, M.T., Cristea, A., Al ´en, M., H ¨akkinen, K., Sipil ¨a, S., Mero, A., Viitasalo, J.T., Larsson, L., and Suominen, H. (2006). Aging, muscle fiber type, and contractile function in sprint-trained athletes. Journal of applied physiology (Bethesda, Md. : 1985)101, 906–9...

    2006

  59. [67]

    Nicolas-Metral, V., Raddatz, E., Kucera, P ., and Ruegg, U.T. (2001). Mdx myotubes have normal excitability but show reduced contraction-relaxation dynamics. Journal of muscle research and cell motility22, 69–75. doi:10.1023/a:1010384625954

    2001 doi

  60. [68]

    Goldstein, J.A., and McNally, E.M. (2010). Mechanisms of muscle weakness in muscular dystrophy. The Journal of general physiology136, 29–34. doi:10.1085/jgp.201010436

    2010 doi

  61. [69]

    Nesmith, A.P ., Wagner, M.A., Pasqualini, F .S., O’Connor, B.B., Pincus, M.J., August, P .R., and Parker, K.K. (2016). A human in vitro model of duchenne muscular dystro- phy muscle formation and contractility. The Journal of cell biology215, 47–56. doi: 10.1083/jcb.201603111

    2016 doi

  62. [70]

    Riddell, D.O., Hildyard, J.C.W., Harron, R.C.M., Taylor-Brown, F ., Kornegay, J.N., Wells, D.J., and Piercy, R.J. (2023). Longitudinal assessment of skeletal muscle functional mechanics in the de50-md dog model of duchenne muscular dystrophy. Disease models & mechanisms

    2023

  63. [71]

    Burr, A.R., and Molkentin, J.D. (2015). Genetic evidence in the mouse solidifies the calcium hypothesis of myofiber death in muscular dystrophy. Cell death and differentiation22, 1402–

    2015

  64. [72]

    doi:10.1242/dmm.050395

    doi

  65. [73]

    Islam, M.A., Jia, S., and Bruce, N.D.B. (). How much position information do convolutional neural networks encode?. arXiv. doi:10.48550/ARXIV.2001.08248

    2001 doi arXiv

  66. [74]

    He, K., Zhang, X., Ren, S., and Sun, J. (2015). Spatial pyramid pooling in deep convolu- tional networks for visual recognition. IEEE transactions on pattern analysis and machine intelligence37, 1904–1916. doi:10.1109/TPAMI.2015.2389824

    2015

  67. [75]

    Mareedu, S., Million, E.D., Duan, D., and Babu, G.J. (2021). Abnormal calcium handling in duchenne muscular dystrophy: Mechanisms and potential therapies. Frontiers in physiology 12, 647010. doi:10.3389/fphys.2021.647010

    2021

  68. [76]

    Vennerød, C.B., Kjærran, A., and Bugge, E.S. (). Long short-term memory rnn. arXiv. doi: 10.48550/ARXIV.2105.06756

    doi arXiv

  69. [77]

    Anwar, A., Khalifa, Y ., Coyle, J.L., and Sejdic, E. (2025). Transformers in biosignal analysis: A review. Information Fusion114, 102697. doi:10.1016/j.inffus.2024.102697. 27

    2025

  70. [78]

    Chung, J., Gulcehre, C., Cho, K., and Bengio, Y . (). Empirical evaluation of gated recurrent neural networks on sequence modeling. arXiv. doi:10.48550/ARXIV.1412.3555

    doi arXiv

  71. [79]

    Gu, A., and Dao, T. (). Mamba: Linear-time sequence modeling with selective state spaces. arXiv. doi:10.48550/ARXIV.2312.00752. 28

    doi arXiv

  72. [81]

    Choromanski, K., Likhosherstov, V., Dohan, D., Song, X., Gane, A., Sarlos, T., Hawkins, P ., Davis, J., Mohiuddin, A., Kaiser, L., Belanger, D., Colwell, L., and Weller, A. (). Rethinking attention with performers. arXiv. doi:10.48550/ARXIV.2009.14794

    2009 doi arXiv

  73. [1412]

    doi:10.1038/cdd.2015.65

    2015 doi

  74. [1437]

    doi:10.1007/s10237-016-0772-7

    doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.