Adaptive surrogate modeling for efficient coupling of musculoskeletal control and tissue deformation models

FDM data driven U-Net as a 2D Laplace PINN solver

Efficient solution of partial differential equations (PDEs) of physical laws is of interest for manifold applications in computer science and image analysis. However, conventional domain discretization techniques for numerical solving PDEs such as Finite Difference (FDM), Finite Element (FEM) methods are unsuitable for real-time applications and are also quite laborious in adaptation to new applications, especially for non-experts in numerical mathematics and computational modeling. More recently, alternative approaches to solving PDEs using the so-called Physically Informed Neural Networks (PINNs) received increasing attention because of their straightforward application to new data and potentially more efficient performance. In this work, we present a novel data-driven approach to solve 2D Laplace PDE with arbitrary boundary conditions using deep learning models trained on a large set of reference FDM solutions. Our experimental results show that both forward and inverse 2D Laplace problems can efficiently be solved using the proposed PINN approach with nearly real-time performance and average accuracy of 94% for different types of boundary value problems compared to FDM. In summary, our deep learning based PINN PDE solver provides an efficient tool with various applications in image analysis and computational simulation of image-based physical boundary value problems.

Inter-strides variability affects internal foot tissue loadings during running

Running overuse injuries result from an imbalance between repetitive loadings on the anatomical structures and their ability to adapt to these loadings. Unfortunately, the measure of these in-vivo loadings is not easily accessible. An optimal amount of movement variability is thought to decrease the running overuse injury risk, but the influence of movement variability on local tissue loading is still not known. A 3D dynamic finite element foot model driven by extrinsic muscle forces was developed to estimate the stress undergone by the different internal foot structures during the stance phase. The boundary conditions of different trials with similar running speed were used as input. Variability in bone stress (10%) and cartilage pressure (16%) can be expected while keeping the overall running speed constant. Bone and cartilage stress were mainly influenced by the muscle force profiles rather than by ground reaction force. These findings suggest, first, that the analysis of a single trial only is not representative of the internal tissue loadings distribution in the foot and second, that muscle forces must be considered when estimating bone and cartilage loadings at the foot level. This model could be applied to an optimal clinical management of the overuse injury.

Integration of neural architecture within a finite element framework for improved neuromusculoskeletal modeling

Neuromusculoskeletal (NMS) models can aid in studying the impacts of the nervous and musculoskeletal systems on one another. These computational models facilitate studies investigating mechanisms and treatment of musculoskeletal and neurodegenerative conditions. In this study, we present a predictive NMS model that uses an embedded neural architecture within a finite element (FE) framework to simulate muscle activation. A previously developed neuromuscular model of a motor neuron was embedded into a simple FE musculoskeletal model. Input stimulation profiles from literature were simulated in the FE NMS model to verify effective integration of the software platforms. Motor unit recruitment and rate coding capabilities of the model were evaluated. The integrated model reproduced previously published output muscle forces with an average error of 0.0435 N. The integrated model effectively demonstrated motor unit recruitment and rate coding in the physiological range based upon motor unit discharge rates and muscle force output. The combined capability of a predictive NMS model within a FE framework can aid in improving our understanding of how the nervous and musculoskeletal systems work together. While this study focused on a simple FE application, the framework presented here easily accommodates increased complexity in the neuromuscular model, the FE simulation, or both.

A Conceptual Blueprint for Making Neuromusculoskeletal Models Clinically Useful

The ultimate goal of most neuromusculoskeletal modeling research is to improve the treatment of movement impairments. However, even though neuromusculoskeletal models have become more realistic anatomically, physiologically, and neurologically over the past 25 years, they have yet to make a positive impact on the design of clinical treatments for movement impairments. Such impairments are caused by common conditions such as stroke, osteoarthritis, Parkinson’s disease, spinal cord injury, cerebral palsy, limb amputation, and even cancer. The lack of clinical impact is somewhat surprising given that comparable computational technology has transformed the design of airplanes, automobiles, and other commercial products over the same time period. This paper provides the author’s personal perspective for how neuromusculoskeletal models can become clinically useful. First, the paper motivates the potential value of neuromusculoskeletal models for clinical treatment design. Next, it highlights five challenges to achieving clinical utility and provides suggestions for how to overcome them. After that, it describes clinical, technical, collaboration, and practical needs that must be addressed for neuromusculoskeletal models to fulfill their clinical potential, along with recommendations for meeting them. Finally, it discusses how more complex modeling and experimental methods could enhance neuromusculoskeletal model fidelity, personalization, and utilization. The author hopes that these ideas will provide a conceptual blueprint that will help the neuromusculoskeletal modeling research community work toward clinical utility.

ReadySim: A computational framework for building explicit finite element musculoskeletal simulations directly from motion laboratory data

Musculoskeletal modeling allows researchers insight into joint mechanics which might not otherwise be obtainable through in vivo or in vitro studies. Common musculoskeletal modeling techniques involve rigid body dynamics software which often employ simplified joint representations. These representations have proven useful but are limited in performing single-framework deformable analyzes in structures of interest. Musculoskeletal finite element (MSFE) analysis allows for representation of structures in sufficient detail to obtain accurate solutions of the internal stresses and strains including complex contact conditions and material representations. Studies which performed muscle force optimization directly in a finite element framework were often limited in complexity to minimize computational time. Recent advances in computational efficiency and control schemes for muscle force prediction have made these solutions more practical. Yet, the formulation of subject-specific simulations remains a challenging problem. The objectives of this work were to develop an open-source computational framework to build and run simulations which (a) scale the size of MSFE models and efficiently estimate (b) joint kinematics and (c) muscle forces from human motion data collected in a typical gait laboratory. A computational framework was built using MATLAB and Python to interface with model input and output files. The software uses laboratory marker data to scale model segment lengths and estimate joint kinematics. Concurrent muscle force and tissue strain estimations are performed based on the estimated kinematics and ground reaction forces. This software will improve the usability and consistency of single-framework MSFE simulations. Both software and template model are made freely available on SimTK.Novelty Statement Single framework musculoskeletal modeling directly in a finite element environment for muscle force estimation and tissue strain analysis. Open dissemination of unilateral musculoskeletal finite element model and software used in manuscript.

A lower extremity model for muscle-driven simulation of activity using explicit finite element modeling

A key strength of computational modeling is that it can provide estimates of muscle, ligament, and joint loads, stresses, and strains through non-invasive means. However, simulations that can predict the forces in the muscles during activity while maintaining sufficient complexity to realistically represent the muscles and joint structures can be computationally challenging. For this reason, the current state of the art is to apply separate rigid-body dynamic and finite-element (FE) analyses in series. However, the use of two or more disconnected models often fails to capture key interactions between the joint-level and whole-body scales. Single framework MSFE models have the potential to overcome the limitations associated with disconnected models in series. The objectives of the current study were to create a multi-scale FE model of the human lower extremity that combines optimization, dynamic muscle modeling, and structural FE analysis in a single framework and to apply this framework to evaluate the mechanics of healthy knee specimens during two activities. Two subject-specific FE models (Model 1, Model 2) of the lower extremity were developed in ABAQUS/Explicit including detailed representations of the muscles. Muscle forces, knee joint loading, and articular contact were calculated for two activities using an inverse dynamics approach and static optimization. Quadriceps muscle forces peaked at the onset of chair rise (2174 N, 1962 N) and in early stance phase (510 N, 525 N), while gait saw peak forces in the hamstrings (851 N, 868 N) in midstance. Joint forces were similar in magnitude to available telemetric patient data. This study demonstrates the feasibility of detailed quasi-static, muscle-driven simulations in an FE framework.

A computationally efficient strategy to estimate muscle forces in a finite element musculoskeletal model of the lower limb

Concurrent multiscale simulation strategies are required in computational biomechanics to study the interdependence between body scales. However, detailed finite element models rarely include muscle recruitment due to the computational burden of both the finite element method and the optimization strategies widely used to estimate muscle forces. The aim of this study was twofold: first, to develop a computationally efficient muscle force prediction strategy based on proportional-integral-derivative (PID) controllers to track gait and chair rise experimental joint motion with a finite element musculoskeletal model of the lower limb, including a deformable knee representation with 12 degrees of freedom; and, second, to demonstrate that the inclusion of joint-level deformability affects muscle force estimation by using two different knee models and comparing muscle forces between the two solutions. The PID control strategy tracked experimental hip, knee, and ankle flexion/extension with root mean square errors below 1°, and estimated muscle, contact and ligament forces in good agreement with previous results and electromyography signals. Differences up to 11% and 20% in the vasti and biceps femoris forces, respectively, were observed between the two knee models, which might be attributed to a combination of differing joint contact geometry, ligament behavior, joint kinematics, and muscle moment arms. The tracking strategy developed in this study addressed the inevitable tradeoff between computational cost and model detail in musculoskeletal simulations and can be used with finite element musculoskeletal models to efficiently estimate the interdependence between muscle forces and tissue deformation.

Efficacy and efficiency of multivariate linear regression for rapid prediction of femoral strain fields during activity

Multivariate Linear Regression-based (MLR) surrogate models were explored to reduce the computational cost of predicting femoral strains during normal activity in comparison with finite element analysis. The musculoskeletal model of one individual, the finite-element model of the right femur, and experimental force and motion data for normal walking, fast walking, stair ascent, stair descent, and rising from a chair were obtained from a previous study. Equivalent Von Mises strain was calculated for 1000 frames uniformly distributed across activities. MLR surrogate models were generated using training sets of 50, 100, 200 and 300 samples. The finite-element and MLR analyses were compared using linear regression. The Root Mean Square Error (RMSE) and the 95th percentile of the strain error distribution were used as indicators of average and peak error. The MLR model trained using 200 samples (RMSE < 108 µε; peak error < 228 µε) was used as a reference. The finite-element method required 66 s per frame on a standard desktop computer. The MLR model required 0.1 s per frame plus 1848 s of training time. RMSE ranged from 1.2% to 1.3% while peak error ranged from 2.2% to 3.6% of the maximum micro-strain (5020 µε). Performance within an activity was lower during early and late stance, with RMSE of 4.1% and peak error of 8.6% of the maximum computed micro-strain. These results show that MLR surrogate models may be used to rapidly and accurately estimate strain fields in long bones during daily physical activity.

A subject-specific integrative biomechanical framework of the pelvis for gait analysis

Analysis of the human locomotor system using rigid-body musculoskeletal models has increased in the biomechanical community with the objective of studying muscle activations of different movements. Simultaneously, the finite element method has emerged as a complementary approach for analyzing the mechanical behavior of tissues. This study presents an integrative biomechanical framework for gait analysis by linking a musculoskeletal model and a subject-specific finite element model of the pelvis. To investigate its performance, a convergence study was performed and its sensitivity to the use of non-subject-specific material properties was studied. The total hip joint force estimated by the rigid musculoskeletal model and by the finite element model showed good agreement, suggesting that the integrative approach estimates adequately (in shape and magnitude) the hip total contact force. Previous studies found movements of up to 1.4 mm in the anterior-posterior direction, for single leg stance. These results are comparable with the displacement values found in this study: 0-0.5 mm in the sagittal axis. Maximum von Mises stress values of approximately 17 MPa were found in the pelvic bone. Comparing this results with a previous study of our group, the new findings show that the introduction of muscular boundary conditions and the flexion-extension movement of the hip reduce the regions of high stress and distributes more uniformly the stress across the pelvic bone. Thus, it is thought that muscle force has a relevant impact in reducing stresses in pelvic bone during walking of the finite element model proposed in this study. Future work will focus on including other deformable structures, such as the femur and the tibia, and subject-specific material properties.

Synthesis of Subject-Specific Human Balance Responses Using a Task-Level Neuromuscular Control Platform

Many activities of daily living require a high level of neuromuscular coordination and balance control to avoid falls. Complex musculoskeletal models paired with detailed neuromuscular simulations complement experimental studies and uncover principles of coordinated and uncoordinated movements. Here, we created a closed-loop forward dynamic simulation framework that utilizes a detailed musculoskeletal model (19 degrees of freedom, and 92 muscles) to synthesize human balance responses after support-surface perturbation. In addition, surrogate response models of task-level experimental kinematics from two healthy subjects were provided as inputs to our closed-loop simulations to inform the design of the task-level controller. The predicted muscle activations and the resulting synthesized subject joint angles showed good conformity with the average of experimental trials. The simulated whole-body center of mass displacements, generated from a single kinematics trial per perturbation direction, were on average, within 7 mm (anterior perturbations) and 13 mm (posterior perturbations) of experimental displacements. Our results confirmed how a complex subject-specific movement can be reconstructed by sequencing and prioritizing multiple task-level commands to achieve desired movements. By combining the multidisciplinary approaches of robotics and biomechanics, the platform demonstrated here offers great potential for studying human movement control and subject-specific outcome prediction.

Evaluation of a Surrogate Contact Model in Force-Dependent Kinematic Simulations of Total Knee Replacement

Knowing the forces in the human body is of great clinical interest and musculoskeletal (MS) models are the most commonly used tool to estimate them in vivo. Unfortunately, the process of computing muscle, joint contact, and ligament forces simultaneously is computationally highly demanding. The goal of this study was to develop a fast surrogate model of the tibiofemoral (TF) contact in a total knee replacement (TKR) model and apply it to force-dependent kinematic (FDK) simulations of activities of daily living (ADLs). Multiple domains were populated with sample points from the reference TKR contact model, based on reference simulations and design-of-experiments. Artificial neural networks (ANN) learned the relationship between TF pose and loads from the medial and lateral sides of the TKR implant. Normal and right-turn gait, rising-from-a-chair, and a squat were simulated using both surrogate and reference contact models. Compared to the reference contact model, the surrogate contact model predicted TF forces with a root-mean-square error (RMSE) lower than 10 N and TF moments lower than 0.3 N·m over all simulated activities. Secondary knee kinematics were predicted with RMSE lower than 0.2 mm and 0.2 deg. Simulations that used the surrogate contact model ran on average three times faster than those using the reference model, allowing the simulation of a full gait cycle in 4.5 min. This modeling approach proved fast and accurate enough to perform extensive parametric analyses, such as simulating subject-specific variations and surgical-related factors in TKR.

A Computational Efficient Method to Assess the Sensitivity of Finite-Element Models: An Illustration With the Hemipelvis

Assessing the sensitivity of a finite-element (FE) model to uncertainties in geometric parameters and material properties is a fundamental step in understanding the reliability of model predictions. However, the computational cost of individual simulations and the large number of required models limits comprehensive quantification of model sensitivity. To quickly assess the sensitivity of an FE model, we built linear and Kriging surrogate models of an FE model of the intact hemipelvis. The percentage of the total sum of squares (%TSS) was used to determine the most influential input parameters and their possible interactions on the median, 95th percentile and maximum equivalent strains. We assessed the surrogate models by comparing their predictions to those of a full factorial design of FE simulations. The Kriging surrogate model accurately predicted all output metrics based on a training set of 30 analyses (R2 = 0.99). There was good agreement between the Kriging surrogate model and the full factorial design in determining the most influential input parameters and interactions. For the median, 95th percentile and maximum equivalent strain, the bone geometry (60%, 52%, and 76%, respectively) was the most influential input parameter. The interactions between bone geometry and cancellous bone modulus (13%) and bone geometry and cortical bone thickness (7%) were also influential terms on the output metrics. This study demonstrates a method with a low time and computational cost to quantify the sensitivity of an FE model. It can be applied to FE models in computational orthopaedic biomechanics in order to understand the reliability of predictions.

Development of an open-source cosimulation method of the knee

Rigid body dynamics and soft tissue loads are solved simultaneously in a cosimulation framework to couple musculoskeletal dynamics and tissue mechanics. The goal of this work was to implement a validated, open-source cosimulation framework of the knee to determine how this coupling affects computed cartilage loads. The kinematic knee joint of a generic whole body model in the open-source software OpenSim was replaced by a previously developed discrete element knee model that consisted of a six degree of freedom (dof) tibiofemoral joint and one dof patellofemoral joint. A serial approach was initially used to estimate muscle forces and cartilage contact loads for a simple flexion movement. Then, a cosimulation framework was implemented for a simple knee flexion movement in which neuromusculoskeletal dynamics and knee mechanics were simultaneously solved using a computed muscle control (CMC) algorithm. This work highlights that the choice of computation method and the precise acquisition of all the dofs of the knee are important factors to consider when estimating soft tissue loads.

Efficient Computation of Cartilage Contact Pressures within Dynamic Simulations of Movement

The objective of this study was to assess the use of an advanced collision detection algorithm to simulate cartilage contact pressure patterns within dynamic musculoskeletal simulations of movement. We created a knee model that included articular cartilage contact for the tibiofemoral and patellofemoral joints. Knee mechanics were then predicted within the context of a dynamic gait simulation. At each time step of a simulation, ray-casting was used in conjunction with hierarchical oriented bounding boxes (OBB) to rapidly identify regions of overlap between articulating cartilage surfaces. Local cartilage contact pressure was then computed using an elastic foundation model. Collision detection implemented in parallel on a GPU provided up to a 10× speed increase when using high resolution mesh densities that had >10 triangles/mm2. However, pressure magnitudes converged at considerably lower mesh densities (2.6 triangles/mm2) where CPU and GPU implementations of collision detection exhibited equivalent performance. Simulated tibiofemoral contact locations were comparable to prior experimental measurements, while pressure magnitudes were similar to those predicted by finite element models. We conclude the use of ray-casting with hierarchical OBB for collision detection is a viable method for simulating joint contact mechanics in human movement.

Surrogate modeling of deformable joint contact using artificial neural networks

Deformable joint contact models can be used to estimate loading conditions for cartilage-cartilage, implant-implant, human-orthotic, and foot-ground interactions. However, contact evaluations are often so expensive computationally that they can be prohibitive for simulations or optimizations requiring thousands or even millions of contact evaluations. To overcome this limitation, we developed a novel surrogate contact modeling method based on artificial neural networks (ANNs). The method uses special sampling techniques to gather input-output data points from an original (slow) contact model in multiple domains of input space, where each domain represents a different physical situation likely to be encountered. For each contact force and torque output by the original contact model, a multi-layer feed-forward ANN is defined, trained, and incorporated into a surrogate contact model. As an evaluation problem, we created an ANN-based surrogate contact model of an artificial tibiofemoral joint using over 75,000 evaluations of a fine-grid elastic foundation (EF) contact model. The surrogate contact model computed contact forces and torques about 1000 times faster than a less accurate coarse grid EF contact model. Furthermore, the surrogate contact model was seven times more accurate than the coarse grid EF contact model within the input domain of a walking motion. For larger input domains, the surrogate contact model showed the expected trend of increasing error with increasing domain size. In addition, the surrogate contact model was able to identify out-of-contact situations with high accuracy. Computational contact models created using our proposed ANN approach may remove an important computational bottleneck from musculoskeletal simulations or optimizations incorporating deformable joint contact models.

An Open-Source Toolbox for Surrogate Modeling of Joint Contact Mechanics

GOAL

Incorporation of elastic joint contact models into simulations of human movement could facilitate studying the interactions between muscles, ligaments, and bones. Unfortunately, elastic joint contact models are often too expensive computationally to be used within iterative simulation frameworks. This limitation can be overcome by using fast and accurate surrogate contact models that fit or interpolate input-output data sampled from existing elastic contact models. However, construction of surrogate contact models remains an arduous task. The aim of this paper is to introduce an open-source program called Surrogate Contact Modeling Toolbox (SCMT) that facilitates surrogate contact model creation, evaluation, and use.

METHODS

SCMT interacts with the third-party software FEBio to perform elastic contact analyses of finite-element models and uses MATLAB to train neural networks that fit the input-output contact data. SCMT features sample point generation for multiple domains, automated sampling, sample point filtering, and surrogate model training and testing.

RESULTS

An overview of the software is presented along with two example applications. The first example demonstrates creation of surrogate contact models of artificial tibiofemoral and patellofemoral joints and evaluates their computational speed and accuracy, while the second demonstrates the use of surrogate contact models in a forward dynamic simulation of an open-chain leg extension-flexion motion.

CONCLUSION

SCMT facilitates the creation of computationally fast and accurate surrogate contact models. Additionally, it serves as a bridge between FEBio and OpenSim musculoskeletal modeling software.

SIGNIFICANCE

Researchers may now create and deploy surrogate models of elastic joint contact with minimal effort.

Deconstructing the Anterior Cruciate Ligament: What We Know and Do Not Know About Function, Material Properties, and Injury Mechanics

Anterior cruciate ligament (ACL) injury is a common and potentially catastrophic knee joint injury, afflicting a large number of males and particularly females annually. Apart from the obvious acute injury events, it also presents with significant long-term morbidities, in which osteoarthritis (OA) is a frequent and debilitative outcome. With these facts in mind, a vast amount of research has been undertaken over the past five decades geared toward characterizing the structural and mechanical behaviors of the native ACL tissue under various external load applications. While these efforts have afforded important insights, both in terms of understanding treating and rehabilitating ACL injuries; injury rates, their well-established sex-based disparity, and long-term sequelae have endured. In reviewing the expanse of literature conducted to date in this area, this paper identifies important knowledge gaps that contribute directly to this long-standing clinical dilemma. In particular, the following limitations remain. First, minimal data exist that accurately describe native ACL mechanics under the extreme loading rates synonymous with actual injury. Second, current ACL mechanical data are typically derived from isolated and oversimplified strain estimates that fail to adequately capture the true 3D mechanical response of this anatomically complex structure. Third, graft tissues commonly chosen to reconstruct the ruptured ACL are mechanically suboptimal, being overdesigned for stiffness compared to the native tissue. The net result is an increased risk of rerupture and a modified and potentially hazardous habitual joint contact profile. These major limitations appear to warrant explicit research attention moving forward in order to successfully maintain/restore optimal knee joint function and long-term life quality in a large number of otherwise healthy individuals.

Co-simulation of neuromuscular dynamics and knee mechanics during human walking

This study introduces a framework for co-simulating neuromuscular dynamics and knee joint mechanics during gait. A knee model was developed that included 17 ligament bundles and a representation of the distributed contact between a femoral component and tibial insert surface. The knee was incorporated into a forward dynamics musculoskeletal model of the lower extremity. A computed muscle control algorithm was then used to modulate the muscle excitations to drive the model to closely track measured hip, knee, and ankle angle trajectories of a subject walking overground with an instrumented knee replacement. The resulting simulations predicted the muscle forces, ligament forces, secondary knee kinematics, and tibiofemoral contact loads. Model-predicted tibiofemoral contact forces were of comparable magnitudes to experimental measurements, with peak medial (1.95 body weight (BW)) and total (2.76 BW) contact forces within 4-17% of measured values. Average root-mean-square errors over a gait cycle were 0.26, 0.42, and 0.51 BW for the medial, lateral, and total contact forces, respectively. The model was subsequently used to predict variations in joint contact pressure that could arise by altering the frontal plane joint alignment. Small variations (±2 deg) in the alignment of the femoral component and tibial insert did not substantially affect the location of contact pressure, but did alter the medio-lateral distribution of load and internal tibia rotation in swing. Thus, the computational framework can be used to virtually assess the coupled influence of both physiological and design factors on in vivo joint mechanics and performance.

A dynamic finite element analysis of human foot complex in the sagittal plane during level walking

The objective of this study is to develop a computational framework for investigating the dynamic behavior and the internal loading conditions of the human foot complex during locomotion. A subject-specific dynamic finite element model in the sagittal plane was constructed based on anatomical structures segmented from medical CT scan images. Three-dimensional gait measurements were conducted to support and validate the model. Ankle joint forces and moment derived from gait measurements were used to drive the model. Explicit finite element simulations were conducted, covering the entire stance phase from heel-strike impact to toe-off. The predicted ground reaction forces, center of pressure, foot bone motions and plantar surface pressure showed reasonably good agreement with the gait measurement data over most of the stance phase. The prediction discrepancies can be explained by the assumptions and limitations of the model. Our analysis showed that a dynamic FE simulation can improve the prediction accuracy in the peak plantar pressures at some parts of the foot complex by 10%–33% compared to a quasi-static FE simulation. However, to simplify the costly explicit FE simulation, the proposed model is confined only to the sagittal plane and has a simplified representation of foot structure. The dynamic finite element foot model proposed in this study would provide a useful tool for future extension to a fully muscle-driven dynamic three-dimensional model with detailed representation of all major anatomical structures, in order to investigate the structural dynamics of the human foot musculoskeletal system during normal or even pathological functioning.

3D finite element models of shoulder muscles for computing lines of actions and moment arms

Accurate representation of musculoskeletal geometry is needed to characterise the function of shoulder muscles. Previous models of shoulder muscles have represented muscle geometry as a collection of line segments, making it difficult to account for the large attachment areas, muscle-muscle interactions and complex muscle fibre trajectories typical of shoulder muscles. To better represent shoulder muscle geometry, we developed 3D finite element models of the deltoid and rotator cuff muscles and used the models to examine muscle function. Muscle fibre paths within the muscles were approximated, and moment arms were calculated for two motions: thoracohumeral abduction and internal/external rotation. We found that muscle fibre moment arms varied substantially across each muscle. For example, supraspinatus is considered a weak external rotator, but the 3D model of supraspinatus showed that the anterior fibres provide substantial internal rotation while the posterior fibres act as external rotators. Including the effects of large attachment regions and 3D mechanical interactions of muscle fibres constrains muscle motion, generates more realistic muscle paths and allows deeper analysis of shoulder muscle function.

Multiscale mechanics of articular cartilage: potentials and challenges of coupling musculoskeletal, joint, and microscale computational models

Articular cartilage experiences significant mechanical loads during daily activities. Healthy cartilage provides the capacity for load bearing and regulates the mechanobiological processes for tissue development, maintenance, and repair. Experimental studies at multiple scales have provided a fundamental understanding of macroscopic mechanical function, evaluation of the micromechanical environment of chondrocytes, and the foundations for mechanobiological response. In addition, computational models of cartilage have offered a concise description of experimental data at many spatial levels under healthy and diseased conditions, and have served to generate hypotheses for the mechanical and biological function. Further, modeling and simulation provides a platform for predictive risk assessment, management of dysfunction, as well as a means to relate multiple spatial scales. Simulation-based investigation of cartilage comes with many challenges including both the computational burden and often insufficient availability of data for model development and validation. This review outlines recent modeling and simulation approaches to understand cartilage function from a mechanical systems perspective, and illustrates pathways to associate mechanics with biological function. Computational representations at single scales are provided from the body down to the microstructure, along with attempts to explore multiscale mechanisms of load sharing that dictate the mechanical environment of the cartilage and chondrocytes.

Chondrocyte deformations as a function of tibiofemoral joint loading predicted by a generalized high-throughput pipeline of multi-scale simulations

Cells of the musculoskeletal system are known to respond to mechanical loading and chondrocytes within the cartilage are not an exception. However, understanding how joint level loads relate to cell level deformations, e.g. in the cartilage, is not a straightforward task. In this study, a multi-scale analysis pipeline was implemented to post-process the results of a macro-scale finite element (FE) tibiofemoral joint model to provide joint mechanics based displacement boundary conditions to micro-scale cellular FE models of the cartilage, for the purpose of characterizing chondrocyte deformations in relation to tibiofemoral joint loading. It was possible to identify the load distribution within the knee among its tissue structures and ultimately within the cartilage among its extracellular matrix, pericellular environment and resident chondrocytes. Various cellular deformation metrics (aspect ratio change, volumetric strain, cellular effective strain and maximum shear strain) were calculated. To illustrate further utility of this multi-scale modeling pipeline, two micro-scale cartilage constructs were considered: an idealized single cell at the centroid of a 100×100×100 μm block commonly used in past research studies, and an anatomically based (11 cell model of the same volume) representation of the middle zone of tibiofemoral cartilage. In both cases, chondrocytes experienced amplified deformations compared to those at the macro-scale, predicted by simulating one body weight compressive loading on the tibiofemoral joint. In the 11 cell case, all cells experienced less deformation than the single cell case, and also exhibited a larger variance in deformation compared to other cells residing in the same block. The coupling method proved to be highly scalable due to micro-scale model independence that allowed for exploitation of distributed memory computing architecture. The method's generalized nature also allows for substitution of any macro-scale and/or micro-scale model providing application for other multi-scale continuum mechanics problems.

A data-driven surrogate model to connect scales between multi-domain biomechanics simulations

A data driven surrogate was developed to bridge the gap between finite element and multibody modeling and to expand the information available from a rigid multibody cartilage simulation. An indentation experiment performed on canine stifle cartilage was modeled in both paradigms with acceptable accuracy and the data were used to create the surrogate. Neural networks were found to adequately approximate the von Mises stress calculated by the finite element model based on force values provided from the multibody model with a correlation coefficient over 0.96.

Adaptive surrogate modeling for expedited estimation of nonlinear tissue properties through inverse finite element analysis

Simulation-based prediction of specimen-specific biomechanical behavior commonly requires inverse analysis using geometrically consistent finite element (FE) models. Optimization drives such analyses but previous studies have highlighted a large computational cost dictated by iterative use of nonlinear FE models. The goal of this study was to evaluate the performance of a local regression-based adaptive surrogate modeling approach to decrease computational cost for both global and local optimization approaches using an inverse FE application. Nonlinear elastic material parameters for patient-specific heel-pad tissue were found, both with and without the surrogate model. Surrogate prediction replaced a FE simulation using local regression of previous simulations when the corresponding error estimate was less than a given tolerance. Performance depended on optimization type and tolerance value. The surrogate reduced local optimization expense up to 68%, but achieved accurate results for only 1 of 20 initial conditions. Conversely, up to a tolerance value of 20 N(2), global optimization with the surrogate yielded consistent parameter predictions with a concurrent decrease in computational cost (up to 77%). However, the local optimization method without the surrogate, although sensitive to the initial conditions, was still on average seven times faster than the global approach. Our results help establish guidelines for setting acceptable tolerance values while using an adaptive surrogate model for inverse FE analysis. Most important, the study demonstrates the benefits of a surrogate modeling approach for intensive FE-based iterative analysis.

Complex integrative morphological and mechanical contributions to ACL injury risk

By failing to consider the integrative impact of key morphological and neuromechanical factors within the anterior cruciate ligament injury mechanism, we consider the current injury prevention model to be flawed. Critical links between these factors continue to be identified, suggesting that a successful prevention model should entrench neuromuscular control strategies that can successfully cater to individual morphological vulnerabilities.

Concurrent musculoskeletal dynamics and finite element analysis predicts altered gait patterns to reduce foot tissue loading

Current computational methods for simulating locomotion have primarily used muscle-driven multibody dynamics, in which neuromuscular control is optimized. Such simulations generally represent joints and soft tissue as simple kinematic or elastic elements for computational efficiency. These assumptions limit application in studies such as ligament injury or osteoarthritis, where local tissue loading must be predicted. Conversely, tissue can be simulated using the finite element method with assumed or measured boundary conditions, but this does not represent the effects of whole body dynamics and neuromuscular control. Coupling the two domains would overcome these limitations and allow prediction of movement strategies guided by tissue stresses. Here we demonstrate this concept in a gait simulation where a musculoskeletal model is coupled to a finite element representation of the foot. Predictive simulations incorporated peak plantar tissue deformation into the objective of the movement optimization, as well as terms to track normative gait data and minimize fatigue. Two optimizations were performed, first without the strain minimization term and second with the term. Convergence to realistic gait patterns was achieved with the second optimization realizing a 44% reduction in peak tissue strain energy density. The study demonstrated that it is possible to alter computationally predicted neuromuscular control to minimize tissue strain while including desired kinematic and muscular behavior. Future work should include experimental validation before application of the methodology to patient care.

Surrogate articular contact models for computationally efficient multibody dynamic simulations

Contact occurs in a wide variety of multibody dynamic systems, including the human musculoskeletal system. However, sensitivity and optimization studies of such systems have been limited by the high computational cost of repeated contact analyses. This study presents a novel surrogate modeling approach for performing computationally efficient three-dimensional elastic contact analyses within multibody dynamic simulations. The approach fits a computationally cheap surrogate contact model to data points sampled from a computationally expensive elastic contact model (e.g., a finite element or elastic foundation model) and resolves several unique challenges involved in applying surrogate modeling techniques to elastic contact problems. As an example application, we performed multibody dynamic simulations of a Stanmore wear simulator machine using surrogate and elastic foundation (EF) contact models of a total knee replacement. Accuracy was assessed by performing eleven dynamic simulations with both types of contact models utilizing large variations in motion and load inputs to the machine. Wear volumes predicted with the surrogate contact models were within 1.5% of those predicted with the EF contact models. Computational speed was assessed by performing five Monte Carlo analyses (over 1000 dynamic simulations each) with surrogate contact models utilizing realistic variations in motion and load inputs. Computation time was reduced from an estimated 284 h per analysis with the EF contact models to 1.4 h with the surrogate contact models (i.e., 17 min vs. 5 s per simulation), with higher wear sensitivity observed for motion variations than for load variations. The proposed surrogate modeling approach can significantly improve the computational speed of multibody dynamic simulations incorporating three-dimensional elastic contact models with general surface geometry.

Modeling, Simulation, and Optimal Initiation Planning For Needle Insertion Into the Liver

Needle insertion simulation and planning systems (SPSs) will play an important role in diminishing inappropriate insertions into soft tissues and resultant complications. Difficulties in SPS development are due in large part to the computational requirements of the extensive calculations in finite element (FE) models of tissue. For clinical feasibility, the computational speed of SPSs must be improved. At the same time, a realistic model of tissue properties that reflects large and velocity-dependent deformations must be employed. The purpose of this study is to address the aforementioned difficulties by presenting a cost-effective SPS platform for needle insertions into the liver. The study was constrained to planar (2D) cases, but can be extended to 3D insertions. To accommodate large and velocity-dependent deformations, a hyperviscoelastic model was devised to produce an FE model of liver tissue. Material constants were identified by a genetic algorithm applied to the experimental results of unconfined compressions of bovine liver. The approach for SPS involves B-spline interpolations of sample data generated from the FE model of liver. Two interpolation-based models are introduced to approximate puncture times and to approximate the coordinates of FE model nodes interacting with the needle tip as a function of the needle initiation pose; the latter was also a function of postpuncture time. A real-time simulation framework is provided, and its computational benefit is highlighted by comparing its performance with the FE method. A planning algorithm for optimal needle initiation was designed, and its effectiveness was evaluated by analyzing its accuracy in reaching a random set of targets at different resolutions of sampled data using the FE model. The proposed simulation framework can easily surpass haptic rates (>500 Hz), even with a high pose resolution level (∼30). The computational time required to update the coordinates of the node at the needle tip in the provided example was reduced from 177 s to 0.8069 ms. The planning accuracy was acceptable even with moderate resolution levels: root-mean-square and maximum errors were 1 mm and 1.2 mm, respectively, for a pose and PPT resolution levels of 17 and 20, respectively. The proposed interpolation-based models significantly improve the computational speed of needle insertion simulation and planning, based on the discretized (FE) model of the liver and can be utilized to establish a cost-effective planning platform. This modeling approach can also be extended for use in other surgical simulations.

An elaborate data set characterizing the mechanical response of the foot

Mechanical properties of the foot are responsible for its normal function and play a role in various clinical problems. Specifically, we are interested in quantification of foot mechanical properties to assist the development of computational models for movement analysis and detailed simulations of tissue deformation. Current available data are specific to a foot region and the loading scenarios are limited to a single direction. A data set that incorporates regional response, to quantify individual function of foot components, as well as the overall response, to illustrate their combined operation, does not exist. Furthermore, the combined three-dimensional loading scenarios while measuring the complete three-dimensional deformation response are lacking. When combined with an anatomical image data set, development of anatomically realistic and mechanically validated models becomes possible. Therefore, the goal of this study was to record and disseminate the mechanical response of a foot specimen, supported by imaging data. Robotic testing was conducted at the rear foot, forefoot, metatarsal heads, and the foot as a whole. Complex foot deformations were induced by single mode loading, e.g., compression, and combined loading, e.g., compression and shear. Small and large indenters were used for heel and metatarsal head loading, an elevated platform was utilized to isolate the rear foot and forefoot, and a full platform compressed the whole foot. Three-dimensional tool movements and reaction loads were recorded simultaneously. Computed tomography scans of the same specimen were collected for anatomical reconstruction a priori. The three-dimensional mechanical response of the specimen was nonlinear and viscoelastic. A low stiffness region was observed starting with contact between the tool and foot regions, increasing with loading. Loading and unloading responses portrayed hysteresis. Loading range ensured capturing the toe and linear regions of the load deformation curves for the dominant loading direction, with the rates approximating those of walking. A large data set was successfully obtained to characterize the overall and the regional mechanical responses of an intact foot specimen under single and combined loads. Medical imaging complemented the mechanical testing data to establish the potential relationship between the anatomical architecture and mechanical responses and to further develop foot models that are mechanically realistic and anatomically consistent. This combined data set has been documented and disseminated in the public domain to promote future development in foot biomechanics.

Computational Models for Neuromuscular Function

Computational models of the neuromuscular system hold the potential to allow us to reach a deeper understanding of neuromuscular function and clinical rehabilitation by complementing experimentation. By serving as a means to distill and explore specific hypotheses, computational models emerge from prior experimental data and motivate future experimental work. Here we review computational tools used to understand neuromuscular function including musculoskeletal modeling, machine learning, control theory, and statistical model analysis. We conclude that these tools, when used in combination, have the potential to further our understanding of neuromuscular function by serving as a rigorous means to test scientific hypotheses in ways that complement and leverage experimental data.