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Real-Time Musculoskeletal Surrogates for Pediatric Cerebral Palsy: a Credibility Pilot
Mohammad Arif Ul Alam
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Abstract
Abstract:Real-time musculoskeletal (MSK) surrogates could support personalized rehabilitation for children with cerebral palsy (CP), but their credibility depends on subject-wise evaluation, low inference latency, and calibrated uncertainty. We develop a subject-conditioned causal neural surrogate using OpenSim-derived static parameters, temporal joint kinematics, true muscle capacities, and training-only perturbations. On a real pediatric CP gait dataset comprising nine children, we use leave-one-subject-out validation on six development subjects and evaluate a frozen configuration once on three locked test subjects. The surrogate accurately reproduces musculotendon lengths (R-square = 0.92 in development validation and approximately 0.95 on locked subjects; nRMSE < 8%) while requiring only sub-millisecond to few-millisecond neural inference, well below a 100 ms interactive-rehabilitation target. In contrast, direct muscle-force estimation remains unstable at this small, heterogeneous scale: pooled metrics can overstate within-subject, per-muscle accuracy. A Monte Carlo credibility pilot further shows that propagating only +/-5% anthropometry and muscle-capacity variation produces severely overconfident nominal 90% intervals (approximately 4% force coverage and below 1% MT-length coverage). These results establish a leakage-free evaluation and credibility framework for pediatric MSK surrogates, while identifying force modeling and epistemic uncertainty as the central next challenges for clinically credible digital twins.
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- Source: https://arxiv.org/abs/2608.28371v1
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Real-Time Musculoskeletal Surrogates for Pediatric Cerebral Palsy: a Credibility Pilot Mohammad Arif Ul Alam Affiliation: College of Science and Technology, North Carolina A & T State University, Greensboro, NC, USA Abstract Real-time musculoskeletal (MSK) surrogates could support personalized rehabilitation for children with cerebral palsy (CP), but their credibility depends on subject-wise evaluation, low inference latency, and calibrated uncertainty. We develop a subject-conditioned causal neural surrogate using OpenSim-derived static parameters, temporal joint kinematics, true muscle capacities, and training-only perturbations. On a real pediatric CP gait dataset comprising nine children, we use leave-one-subject-out validation on six development subjects and evaluate a frozen configuration once on three locked test subjects. The surrogate accurately reproduces musculotendon lengths (R2≈0.92R^2≈ 0.92 in development validation and ≈0.95≈ 0.95 on locked subjects; nRMSE <8%<8\%) while requiring only sub-millisecond to few-millisecond neural inference, well below a 100 ms interactive-rehabilitation target. In contrast, direct muscle-force estimation remains unstable at this small, heterogeneous scale: pooled metrics can overstate within-subject, per-muscle accuracy. A Monte Carlo credibility pilot further shows that propagating only ±5%± 5\% anthropometry and muscle-capacity variation produces severely overconfident nominal 90% intervals (approximately 4% force coverage and below 1% MT-length coverage). These results establish a leakage-free evaluation and credibility framework for pediatric MSK surrogates, while identifying force modeling and epistemic uncertainty as the central next challenges for clinically credible digital twins. Keywords: Pediatric cerebral palsy, Musculoskeletal modeling, Neural surrogates, Real-time rehabilitation, Credibility assessment 1 Introduction Cerebral palsy (CP) is the leading cause of childhood physical disability and is associated with highly heterogeneous movement impairments (2007). In particular, crouch gait changes joint loading, muscle recruitment, and the mechanical demands of walking, making patient-specific analysis important for clinical decision-making Steele et al. (2010). Musculoskeletal (MSK) simulation provides a noninvasive way to estimate quantities that cannot be measured directly during gait, including muscle forces, activations, and joint moments. OpenSim and subject-scaled full-body models have made these analyses broadly reproducible and clinically relevant Seth et al. (2018); Rajagopal et al. (2016). However, conventional muscle-driven simulations remain too computationally expensive for interactive rehabilitation or continuously updated digital twins. Neural surrogates offer a practical route to real-time MSK analysis. Recent studies have used temporal neural networks and physics-informed objectives to estimate muscle forces or joint kinematics from motion and electromyography signals Zhang et al. (2022b); Liu et al. (2024); Zhang et al. (2022a). These results are promising, but most evidence is based on healthy adults, a small number of muscles, or task-specific datasets. Pediatric CP creates a more difficult setting because gait patterns, body dimensions, strength, and neuromuscular control vary substantially between children. Further, muscle force is not directly observed: it is an underdetermined model estimate whose value depends on the musculoskeletal model, optimization method, and physiological assumptions Roelker et al. (2020); Falisse et al. (2018). Consequently, a surrogate should be evaluated not only by a favorable frame-pooled score, but also by its ability to generalize across held-out children. Credibility is equally important for clinical digital twins. ASME V&V 40 and recent FDA guidance emphasize that the evidence supporting a computational model should be appropriate for its intended context of use U.S. Food and Drug Administration (2023). This requires transparent data partitioning, validation on independent subjects, and explicit assessment of uncertainty rather than relying on point predictions alone. In this preliminary study, we develop a real-time, subject-conditioned MSK surrogate for pediatric CP gait and contribute: (i) a leakage-free evaluation protocol with leave-one-subject-out development validation and a locked held-out test set; (i) a causal surrogate conditioned on OpenSim-derived static parameters and true muscle capacities; (i) an honest characterization of the limitations of small-cohort muscle-force estimation; and (iv) a Monte Carlo credibility pilot showing that input-parameter propagation alone does not yield calibrated uncertainty. The subject-level cohort and evaluation partition are summarized in Table 1. Together, these results establish a rigorous experimental foundation for future credible pediatric MSK digital twins. Table 1: Cohort and subject-level evaluation partition. Frame counts denote aligned kinematic/CMC ground-truth frames. Dev: development LOSO; Test: locked held-out evaluation. Subject Severity Frames Split MI01 Mild 674 Dev MI02 Mild 893 Dev MO02 Moderate 799 Dev MO03 Moderate 749 Dev SE01 Severe 838 Dev SE02 Severe 889 Dev MI03 Mild 736 Test MO04 Moderate 1,166 Test SE05 Severe 804 Test Total — 7,548 6 Dev / 3 Test 2 Method 2.1 Subject-Conditioned Causal Surrogate We developed a compact causal temporal convolutional network (TCN) with approximately 0.9 million trainable parameters. TCNs provide a computationally efficient mechanism for using a finite history of motion while preserving causality, making them appropriate for real-time sequence inference Bai et al. (2018). At time t, the model receives a 251-frame (≈ 250 ms) history of 23 joint coordinates and their speeds (46 dynamic features), together with subject-specific static parameters extracted from the scaled OpenSim model. These parameters include muscle-specific maximum isometric forces (FmaxF ), body-scale factors, model mass, and muscle geometric and contractile properties. OpenSim provides a transparent basis for these subject-scaled mechanistic quantities Seth et al. (2018); Rajagopal et al. (2016). The dynamic sequence is encoded by the causal TCN, while a small multilayer perceptron encodes static parameters. Their fused representation produces muscle force, activation, musculotendon (MT) length, and moment-arm outputs: (^t,^t,ℓ^t,^t) ( f_t, a_t, _t, R_t) =gθ(t−W+1:t,), =g_θ(X_t-W+1:t,s), (1) ^t τ_t =^t⊤^t,^t=max⊙^t⊙σ(t), = R_t f_t, f_t=F a_t σ(z_t), where W=251W=251, X denotes kinematics, s denotes static subject parameters, R contains muscle moment arms, and τ is the resulting net muscle joint moment. The final expression enforces a muscle-specific force-capacity bound using true OpenSim FmaxF values. The model predicts 92 muscle forces, activations, and MT lengths, as well as moment arms for 23 independent coordinates. 2.2 Supervision and Physics-Informed Objective Ground-truth forces and activations were obtained from the existing OpenSim computed muscle control (CMC) workflow, and MT lengths, moment arms, and inverse-dynamics joint moments were generated from each subject-scaled model. This use of OpenSim-derived labels follows established musculoskeletal simulation practice Thelen et al. (2003); Seth et al. (2018). We optimized a multi-task objective: ℒ=λfℒf+λaℒa+λℓℒℓ+λRℒR+λτℒτ,L= _fL_f+ _aL_a+ _ L_ + _RL_R+ _τL_τ, (2) where each term is a normalized mean-squared error for force, activation, MT length, moment arm, or joint moment, respectively. The torque term is evaluated only on available lower-limb coordinates: ℒτ=1||∑j∈(τ^t,j−τt,jID)2.L_τ= 1|J| _j ( τ_t,j-τ^ID_t,j )^2. (3) Thus, predicted muscle forces are coupled to joint-level mechanics through the moment-arm relation in Eq. 1. Physics-informed learning has shown promise for musculoskeletal inference, but its benefit depends on label quality, modeling assumptions, and the available data regime Zhang et al. (2022b). Accordingly, we report a direct ablation in Section 3, rather than presuming that this term improves force estimation. 2.3 Training-Only Perturbations To improve robustness without contaminating evaluation, we applied synthetic perturbations exclusively to training windows. These included independent ±5%± 5\% perturbations of FmaxF and anthropometry-related static features, small additive kinematic sensor noise, and temporary occlusion of up to two input channels. Validation and locked-test windows remained unperturbed. This separation is important because augmentation must not create an implicit pathway from evaluation subjects into model selection. 2.4 Leakage-Free Evaluation Protocol Our primary methodological contribution is a subject-level evaluation protocol. The six development subjects were evaluated with leave-one-subject-out (LOSO) validation: five subjects were used for training and the remaining subject for validation in each fold. Dynamic and static normalizers, missing-value imputations, augmentation, and early-stopping decisions were fitted using only the corresponding training subjects. Three additional children, one per crouch-severity group, were retained as a locked held-out test set. The final architecture and training duration were frozen from the development folds before this test set was evaluated once. We report within-subject performance as the primary measure, including subject-level and mean-per-muscle R2R^2 values. Frame-pooled/global R2R^2 is reported only alongside these measures because it can be inflated by between-subject force-scale differences. The complete surrogate, data-labeling, test-firewall, and credibility-pilot workflow is shown in Figure 1. The protocol and uncertainty screening pilot are intended to support a risk-informed credibility workflow consistent with the principles of ASME V&V 40 and current FDA guidance for computational modeling and simulation U.S. Food and Drug Administration (2023). Figure 1: Overview of the subject-conditioned musculoskeletal surrogate and leakage-free evaluation protocol. Pediatric CP gait data are processed with subject-scaled OpenSim models and computed muscle control to generate force, activation, musculotendon-length, moment-arm, and inverse-dynamics labels. Windowed kinematics and static model parameters are provided to a causal TCN. Development subjects are used exclusively for leave-one-subject-out model selection and training-only augmentation. A firewall separates the locked held-out evaluation from development. The frozen model is subsequently used for a Monte Carlo credibility pilot with ±5%± 5\% input variation and nominal 90% prediction-interval coverage assessment. 3 Results 3.1 Musculotendon Geometry Surrogate The causal subject-conditioned surrogate reproduced musculotendon (MT) lengths accurately. Development leave-one-subject-out (LOSO) validation achieved pooled MT-length R2=0.9245R^2=0.9245, while the frozen model achieved R2=0.9477R^2=0.9477 on the locked held-out subjects (Table 2; Figure 2). Locked-test MT-length nRMSE was 7.69% overall, with values of 6.51%, 6.38%, and 9.51% for the mild, moderate, and severe held-out subjects, respectively. This result establishes that the pipeline can replace the OpenSim geometry mapping with high accuracy in a causal setting. However, MT length is a smooth, largely deterministic function of joint configuration in the scaled musculoskeletal model Seth et al. (2018); Thelen et al. (2003). Thus, this result should be interpreted as validation of the data alignment, subject conditioning, and real-time geometry surrogate, rather than as evidence that MT-length prediction is the most difficult scientific component of the problem. 3.2 Real-Time Inference For one causal input window, mean neural inference time was 7.41 ms on a single-thread CPU and 3.16 ms on an NVIDIA A100 GPU. Both measurements are substantially below the 100 ms interactive-rehabilitation criterion. The result therefore establishes real-time feasibility for the neural surrogate component. We do not report a numerical speedup relative to an OpenSim CMC or static-optimization solve because an equivalent end-to-end OpenSim latency measurement was not collected in the current notebook. Such a comparison should be added in a follow-up experiment; otherwise, an explicit orders-of-magnitude speedup claim would not be supported by the present evidence. 3.3 Direct Muscle-Force Estimation Remains Difficult Force prediction was substantially less stable than MT-length prediction. Across development LOSO folds, pooled force R2R^2 was 0.2165 with nRMSE of 34.55%. Subject-level force R2R^2 ranged from −0.635-0.635 to 0.486. On the locked test set, the pooled force R2R^2 increased to 0.6056, but this pooled value masked pronounced heterogeneity: force R2R^2 was −1.052-1.052 for MI03, 0.239 for MO04, and 0.733 for SE05. A negative value means that the model performed worse than a subject-specific mean-force predictor. The complete subject-wise force and MT-length results are shown in Table 2 and Figure 2. This discrepancy illustrates why pooled force metrics must not be used as the primary claim in a small, heterogeneous pediatric cohort. Frame-pooled/global R2R^2 can partly reflect between-subject differences in force scale rather than within-subject temporal tracking. We therefore emphasize subject-wise results in Figure 2 and recommend reporting normalized force, f/Fmaxf/F , in future experiments as a scale-robust secondary outcome. This normalized metric was not computed in the current notebook and is not retrospectively claimed here. Overall, the kinematic and latency components of the feasibility hypothesis are supported, whereas the direct muscle-force accuracy target is not yet met. 3.4 Torque-Consistency Diagnostic The physics-informed torque pathway did not yield meaningful joint-moment accuracy in the present configuration. Development LOSO pooled joint-moment R2R^2 was −0.0055-0.0055, and subject-wise values ranged from −0.024-0.024 to 0.000 (Table 2). This indicates that the masked torque-consistency term was empirically inert at the available sample size and label quality. This finding does not invalidate physics-informed musculoskeletal learning; rather, it shows that including a mechanically motivated loss does not guarantee improved generalization Zhang et al. (2022b). A future formal ablation should compare otherwise identical models trained with and without the torque term, and should evaluate normalized force targets, Hill-type force generation, and improved joint-moment labels. Table 2: Subject-wise global R2R^2 results. Each subject-level value is computed over that subject’s evaluation windows and output dimensions. Joint-moment labels were generated only for development LOSO evaluation. Set Subject Force MT length Joint moment Dev. MI01 0.380 0.974 -0.019 Dev. MI02 -0.050 0.512 -0.024 Dev. MO02 0.441 0.951 0.000 Dev. MO03 0.486 0.987 -0.014 Dev. SE01 -0.635 0.986 -0.013 Dev. SE02 0.374 0.919 -0.007 Dev. Pooled 0.217 0.925 -0.006 Locked MI03 -1.052 0.962 — Locked MO04 0.239 0.961 — Locked SE05 0.733 0.919 — Locked Pooled 0.606 0.948 — Figure 2: Subject-wise global R2R^2 for muscle force and MT length. Development subjects are evaluated by leave-one-subject-out validation; locked subjects are evaluated once after model configuration freezing. MT-length prediction is consistently strong except for one development subject, whereas force prediction varies sharply across children. This variation demonstrates why pooled force R2R^2 must be interpreted alongside subject-wise metrics. 4 A Credibility (VVUQ) Pilot We conducted a screening-level uncertainty-propagation experiment using the frozen locked-test model. For each held-out subject, we drew 100 Monte Carlo samples with independent ±5%± 5\% perturbations of muscle-specific FmaxF values and anthropometry-related static inputs. We evaluated nominal 90% prediction intervals over randomly sampled causal windows and measured empirical coverage and mean interval width. This experiment is a sensitivity and calibration diagnostic, not a complete credibility assessment. The resulting intervals were severely overconfident (Figure 3). Across the locked cohort, nominal 90% intervals achieved only 4.35% empirical coverage for muscle force and 0.70% coverage for MT length. Mean interval widths were 8.68 N for force and 0.00062 m for MT length. The same pattern appeared for every held-out subject: force coverage ranged from 3.94% to 4.72%, and MT-length coverage ranged from 0.37% to 0.88%. Thus, the propagated input variation represents only a small fraction of the observed prediction error. This failure is itself an informative result. The dominant uncertainty source is likely model-form and epistemic uncertainty associated with limited, heterogeneous pediatric training data, rather than the narrow input-parameter variation considered here. Consequently, input-only propagation is insufficient for clinically consequential or regulatory-grade credibility claims. Future work will add epistemic uncertainty through deep ensembles Lakshminarayanan et al. (2017), heteroscedastic prediction heads, and calibration methods such as conformal prediction Angelopoulos and Bates (2023). In the spirit of ASME V&V 40, this pilot establishes a measurable credibility gap and identifies the additional validation evidence needed before the surrogate could support a higher-consequence decision U.S. Food and Drug Administration (2023). Figure 3: Monte Carlo credibility pilot on locked held-out subjects. The dashed line denotes nominal 90% coverage. Propagating only ±5%± 5\% variation in muscle capacity and anthropometry-related static inputs yields severely overconfident intervals: pooled empirical coverage is 4.35% for force and 0.70% for MT length. Narrow interval widths show that input-parameter variation captures only a small component of total predictive uncertainty. 5 Discussion and Limitations This preliminary study establishes a rigorous evaluation substrate for pediatric MSK surrogates rather than a claim that all components of a clinically deployable digital twin have been solved. The leakage-free subject protocol, locked-test firewall, train-only preprocessing, and fast MT-length predictions demonstrate that real pediatric CP data can support reproducible, real-time surrogate evaluation. As summarized in Table 2 and Figure 2, however, the heterogeneous force results show that direct regression of raw muscle force from kinematics remains insufficient at this exploratory scale. In particular, pooled force metrics should not substitute for within-subject, per-muscle performance because body size and force scale can inflate frame-pooled agreement. Several limitations define the next research steps. First, the cohort contains only nine children and one lower-limb gait trial per subject, as summarized in Table 1; larger and more diverse pediatric datasets are required to model inter-subject heterogeneity and to support reliable subgroup analyses. Second, CMC-derived force and activation labels are physics-based model estimates rather than in-vivo measurements. Future work should therefore predict normalized force (f/Fmaxf/F_ ), incorporate Hill-type muscle dynamics, and evaluate sensitivity to alternative optimization and muscle-model assumptions Thelen et al. (2003); Seth et al. (2018). Third, the present torque-consistency pathway did not provide measurable improvement, indicating that moment supervision, moment-arm estimation, loss weighting, or the force parameterization requires redesign rather than merely stronger regularization. Finally, the credibility pilot in Figure 3 demonstrates that propagating plausible input uncertainty is not equivalent to calibrated predictive uncertainty. Regulatory-grade credibility requires explicit treatment of model-form and epistemic uncertainty, for example through ensembles, heteroscedastic prediction, and conformal calibration Lakshminarayanan et al. (2017); Angelopoulos and Bates (2023); U.S. Food and Drug Administration (2023). These findings directly motivate the next phase of the program: expansion to additional pediatric subjects, improved physics-consistent force modeling, calibrated uncertainty quantification, and extension from lower-limb gait to upper-limb rehabilitation tasks. 6 Conclusion We developed a leakage-free, subject-conditioned MSK surrogate pipeline for real pediatric CP gait data. The approach meets real-time latency and MT-length accuracy targets, while quantifying the unresolved difficulty of small-cohort muscle-force estimation and showing that input-only uncertainty propagation is severely overconfident. Together, these results, summarized in Table 2 and Figures 2–3, provide a credible preliminary foundation and a focused technical agenda for in-silico pediatric neurorehabilitation. References [1] (2007) A report: the definition and classification of cerebral palsy april 2006. Developmental Medicine & Child Neurology 49 (s109), p. 8–14. External Links: Document Cited by: §1. 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