Neuromuscular modelling

Neuromuscular modelling is a method for creating computational models of neuromuscular functions derived from algorithms and their resulting representation of cohesion between anatomy, muscular mechanics, and the neural control that commands the system. This field of neuromechanics is used to build computational systems that hypothesize locomotion and other neuromuscular purposes in order to gain insight and make additional hypotheses.^[1] The use of the word hypothesis is integral in this situation because each of the methods in this field of modelling makes use of assumptions and predictions, whether it be in the types of models used for joints (such as articulating surfaces), or the solution method used (such as inverse).^[2]^[3]

Methods and Data Capture

Electromyography

Electromyography (EMG) signals are documented by placing electrodes on muscle groups or individual muscles.^[4]^[5] When muscles contract, signals are produced which are controlled by the nervous system. These signals represent both the physiological and anatomical properties of muscles. Common muscles used in data capture for locomotion include the gluteus (maximus and medius), rectus femoris, vastus medialis and laterialis, biceps femoris, tibialis anterior, peroneus longus and brevis, soleus, and gastrocnemius medialis and lateralis.^[6] There are two main types of EMG data, surface and intramuscular. Surface EMG data is recorded by non-invasive electrodes placed in the main areas of the muscles previously mentioned. The only difference between surface and intramuscular is that of invasiveness. Intramuscular EMG data is captured from needles or wires inserted into the muscle.^[7]^[4] The signals captured are then transformed into ingestible data which is used in the neuromuscular model.

Mathematical Analysis

Since neuromuscular models are hypothesis and assumption driven, there needs to exist some form of statistical and mathematical analysis to determine model efficacy.

Non-negative Matrix Factorization Algorithms

The Non-negative matrix factorization is a method for dimensional reduction and feature extraction of non-negative data and is used in many fields such as artificial intelligence, signal processing, and bioinformatics. ^[8] NNMF uses a series of algorithms in multivariate analysis to remove negative factors of a matrix resulting in matrices with no negative elements. The data is numerically approximated, which is essential in for applications such as muscular activity (inherently non-negative) and makes calculations much more simple.

$X \approx W H$

Where $X$ is $n \times p$ , $W$ is $n \times r$ , $H$ is $r \times p$ , $r \leq p$ . We assume $X_{i j}, W_{i j}, H_{i j} \geq 0$ .

VAF Index

$V A F = 1 - (\frac{S S E}{T S S})$ where $S S E$ is the sum of squared errors between the experimental and represents the unexplained variation, given by $v a r (y_{i} - {\hat{y}}_{i})$ , and $T S S$ is the sum of total squares which quantifies the total variation, given by $v a r (y_{i})$ .^[9]

The higher the VAF, the more similar the two models are. The VAF index is used in neuromuscular modelling to assess the accuracy of reconstructed EMG data. The extracted NNMF matrix and experimental data are put through the VAF process in a series of iterations to produce factorizations with randomized factors and weights. The dimensionality is increased until a minimum threshold is met and the resulting factorized components are ingested into the model.

Monte-Carlo Approaches

The Monte-Carlo methods involves a broad spectrum of computational algorithms that use repeated random sampling to obtain deterministic numerical results and estimations of unknown parameters.^[10]^[11] They are widely used when modelling any sort of neural system because they allow for the modelling of complex situations with many random variables. This is why they are widely favored when the analytical or numerical solutions do not exist or are too complex to implement. The method generally follows 5 main steps:

Determine statistical properties of possible inputs and create parametric model
Generate many sets of possible inputs and parameters
Perform deterministic calculation
Evaluate the model
Statistically analyze

Modelling Techniques

Musculoskeletal Modelling

Components commonly included in multijoint model of movement adapted from Pandy (2001)

Forward dynamics model adapted by Gibet et. al.

Musculoskeletal modelling typically involves numerically representing the mechanics of the muscular system and understanding the overall mechanical dynamics combined with neural control models. As with the theme of any anatomical modelling, experimental data serves as the most reliable source of information but can be difficult to acquire. This is where the need for computer models arose as they provide access to parameters that cannot be easily gained through experimental data. Musculoskeletal modelling typically includes the components in a multijoint model of movement [see figure]. Musculoskeletal computational models are favorable because muscle forces cannot be measured without invasive procedures. The skeleton itself can be computationally modeled in two or three dimensions, but regardless of dimensionality, the relationships between the forces applied to the skeleton and resulting segments can always be expressed as

$M (\underline{q}) \underline{\ddot{q}} + C (\underline{q}) \underline{{\dot{q}}^{2}} + \underline{G} (\underline{q}) + R \underline{q} F^{M T} + E (\underline{q}, \underline{\dot{q}}) = \underline{0}$ ,

Where $\underline{q}, \underline{\dot{q}}, \underline{\ddot{q}}$ are respectively vectors of generalized coordinates, velocities, and accelerations; $M (\underline{\dot{q}})$ is the system mass matrix and $M (\underline{q}) \underline{\ddot{q}}$ a vector of inertial forces and torques; $C (\underline{q}) \underline{{\dot{q}}^{2}}$ is a vector of centrifugal and Coriolis forces and torques; $\underline{G} (\underline{q})$ is a vector of gravitational forces and torques; $R (\underline{q})$ is the matrix of muscle moment arms; ${\underline{F}}^{M T}$ is a vector of musculotendon torques; and $E (\underline{q}, \underline{\dot{q}})$ is a vector of external forces and torques applied to the body by the environment.^[12]

Anthropomorphic Models

Computerized anthropomorphic modelling typically exists in one of three forms: equation-based mathematical functions, digital volume arrays, or hybrid equation-voxel models that mathematically describe boundaries from extracted definitions and data. The equation-based, or stylized, models use combinations of simple surface equations, such as spheres, discs, cylinders. Voxel-based models are chiefly derived from voxels, or value on a grid in a 3D space, taken from anatomical images obtained from live subjects or cadavers using a mode of medical imaging.^[13]^[14] One limitation of the equation-based models is that they are mostly suited for applications regarding anatomical variability or temporal changes. Voxel-based models bring discretization errors which requires refined data sets to reduce these errors. However, with the rise of high-performing computing, dynamic 4D stylized models are continuously being improved. Current research is in robust computer models that can accurately portray a population of patients. These models often use a 4 dimensional Monte Carlo simulation. However, when modelling anthropomorphically, faithful representations are very difficult to obtain as most neuromuscular systems are highly non-linear with an inane amount of degrees of freedom. Because of this, the system is described using functional blocks which represent the different stages of motor control [see figure]. These functional blocks are then characterized by a function of $f$ which establishes a relationship between the variables $y (t)$ and the command function $u (t)$ such that

Block Definition Diagram of Neuromuscular System.

$\dot{y} (t) = f (t, y (t), u (t))$ which is then optimized by $J (u) \int_{t_{i}}^{t_{f}} η (t, y (t), u (t)) d t + C (y (t_{f}))$

System-focused Models

Another method of neuromuscular modelling involves looking at the neuromuscular system as a whole and applying a methodological-based approach through Model-Based Systems Engineering (MBSE) and capturing views using the general modelling language, SysML. Applying the methodologies of MBSE allows practitioners to manage the complexity that arises from the high degrees of freedom and dimensionality found in the neuromuscular system.^[15] It does this by breaking the system as a whole into different views which can be captured using SysML. The most common modelling software is Cameo Systems Modeler, which is an object-oriented (OO) program. This allows it to use other OO languages such as Java, Python, etc to interface with other applications such as MATLAB and AutoCAD for kinematic analysis and optimization methods, such as the genetic algorithm. SysML offers a large repository of modelling semantics so the user can create views that capture the behavior and structure of the system as well as the requirements. The behavior can be shown as an activity, sequence, state machine, or use case diagram. The structure can be represented through block definitions, internal blocks (also parametric), and top-level package diagrams.^[16]^[17] SysML is useful for neuromuscular modelling because it can provide cross-sections of different views through use of the many types of models available. Each model serves as a "slice" of the whole model which forms the entire picture when pieced together. The figure to the right shows a basic block definition diagram of the neuromuscular system to provide a reference.

Muscle Excitation Profile-Driven Modelling

Muscle excitation is the stimulation of the muscle that results in a profile of the excitation/contraction for that specified muscle. Human locomotion is best described as an impulsive excitation of groups of muscles and musculotendon units. The interesting aspect is that the timings depends on the goal of the task. Since there are exist an infinite number of possibilities, these excitations are best captured in a model by approximating the experimental non-negative factors for each motor component. The approximation function found in literature uses single impulse Gaussian curves. The curves are used to explain and approximate the temporal modulation of the non-negative profiles as a function of the percentage of the gait cycle.^[18] This is given by

$g (t) = h \cdot e^{- \frac{(t - b)^{2}}{2 \cdot c^{2}}} - s$

Where $t$ is the time frame; $h$ is the function parameters of the curve peak height; $b$ is the position of the center peak; $c$ is the width of the curve bell; and $s$ is the vertical shift.^[19]^[20]^[21]^[22]

Hill-Type Modelling

Hill-type modelling refers to the 3-element biomechanical muscle model derived by A. V. Hill, a famous British physiologist. The model elements typically involves a contracting component (C) in series with an elastic component (E). It is widely used to explain biomechanics and human movement. Notably, hill-type modelling is often used in conjunction with EMG data. The core component of this model is the contractile component which is the parts of the muscle that produce force as a response to excitation.^[23]

As this is both a kinetic and kinematic model, these forces for the contracting component are shown in equilibrium as -

$L_{m} = L_{C} + L_{E}$

$F_{C} = F_{E}$

Where $L_{m}$ is the muscle origin-to-insertion length; $L_{C}$ and $L_{E}$ are the contracting and elastic component lengths; and $F_{C}$ and $F_{E}$ are the forces.

The pennation angle (which is the angle between the longitudinal axis of the entire muscle and its fibers) denoted by $θ$ ,

$L_{m} = L_{C} \cos θ + L_{E}$

$F_{C} \cos θ = F_{E}$

$L_{C C} \sin θ = L_{o} \sin θ_{o}$

Where $L_{o}$ is the optimal length of $C$ when considering the force-length relationship and $θ_{o}$ is the pennation angle when $L_{C} = L_{o}$ .

The forces for the elastic component are given by the following nonlinear model which assumes the force-extension relationship is quadratic -

$f_{C} = {\begin{cases} K_{1} (L_{E} - L_{u})^{2}, & if L_{E} > L_{u} \\ 0, & if L_{E} \leq L_{u} \end{cases}$

$K_{1} = F_{o} / (U_{o} L_{u})^{2}$

Where $F_{o}$ is the maximum isometric force and $U_{o}$ is the strain of the elastic component when loaded with $F_{o}$ .

Hill-type models also receive time-varying neural excitation signals $u (t)$ that represents the total motor unit action potentials and ranges. The response, given by $a$ , is a nondimensional activation parameter. These activation dynamics are represented by -

$\dot{α} = (u - α) (c_{1} u + c_{2})$

$c_{2} = 1 / τ$

$c_{1} = T / τ_{1} - c_{2}$

Where $τ_{1}$ and $τ_{2}$ are the time constants for rising and falling activation.

Validation

Model validation^[24] is an important effort in modelling. It ensures the accuracy and reliability of the created models and increases model fidelity. Normalized root mean squared deviation and logistic regression are two methods of validation but there are many more used to check the accuracy and performance of of the models.

Normalized Root Mean Squared Deviation

For neuromuscular modeling, validation is paramount to authenticity. Many researchers use normalized root mean squared deviation (NRMSD) as a measure of model success. The formula is derived from root-mean-square deviation or root-mean-square error which is a popular method of statistically measuring the differences between values predicted by a model and the values observed. This method is mostly used for validation in EMG data-driven modelling as it allows the researchers to compare the differences between the captured EMG data and the outputs of their model. The NRMSD is the same method except it normalizes the data, or rather, allows comparison between differently scaled sets of data. It measures the difference of the predicted and actual models by percentages, where lower values indicate less residual variance.^[25]^[26]^[27]

$N R M S D = \frac{\sqrt{\frac{1}{N} \sum_{i = 1}^{N} (\hat{X_{i}} - X_{i})^{2}}}{m a x (\hat{X}, X) - m i n (\hat{X}, X)}$

Logistic Regression^[28]

Logistic regression is a statistical method which uses regression models to predict a dependent variable from a set of predictors. These models can be used as validation because they provide a way to compare numerical results and determine whether or not they are acceptable as description of the data.^[29]^[30] See regression validation. This method reevaluates the accuracy of computed results by dynamically comparing the regression model with the predicted model. The regression model is often created in tandem with the neuromuscular model and used as an in situ performance predictor. Since regression can help determine the fit of data, it is used to compare the differences between things such as muscle weightings with their absolute value in the baseline condition. It should be noted that most regression modelling of the neuromuscular system is non-linear, in that the observation data is a non-linear combination of the determined parameters and often uses more than one independent variable.^[18]

Applications

Robotics

Neuromuscular models can help to shape the underlying processes in robotics which can improve the anthropomorphic features of these machines. These models originally were developed to help understand human motor control but the advances in computing allowed predictive models to be created to complement the experimental data and advance our knowledge of human locomotion. As the human body is a complex system, it can be viewed from an engineering perspective. The engineered model of human locomotion integrates three layers: the skeletal system, the muscle-tendon actuators, and controller architectures which simulates neural circuitry. These, combined with the concept of impedance control, form the current paradigm of robotic anthropomorphic locomotion. These neuromuscular models also form the predictive backbone for generalizing robotic behavior. They allow testing and fine-tuning of the robotic locomotive function and can facilitate integrating these models with the model-based controllers of humanoid robots.^[31]

Physical Therapy

As neuromuscular model fidelity improves, these models can serve as a predictor for normalized locomotive behavior which, in turns, provides a baseline.^[32]^[1] This baseline can be used to compare the locomotion of rehabilitating patients to measure progress and compare with past results to visually show improvement or success. The baseline can also serve as a measure from which to measure the severity of the issue that is impeding neuromuscular function.^[33]

Human-Centered Assistive Devices

The design of human-centered assistive devices can be driven by neuromuscular modelling as these models can provide insight and relevant unique characteristics of the rehabilitating subject. These predictive models are used to simulate motion prediction, neural control, central pattern generators and more in order to develop new methods and approaches in the field of rehabilitation and device design. They also are used to simulate the effects of neurological of physical conditions such as stroke, amputation, and various injuries on neural control and muscle properties. These can then be used in the design of assistance devices such as prosthesis, orthoses, and exoskeletons.^[32]^[34]

Medicine

These neuromuscular models can also be used to simulate the effects of various medical procedures on the system such as tendon transfer, joint replacement, biomedical implants, and electric stimulation controllers.^[1] Simulating the effects of these can assist medical practitioners in demonstrating the viability of a proposed solution, ensuring correct size and placing of a solution (if necessary), and limit invasiveness (procedure permitting). These predictive models and simulations will also allow future medical work and research to be conducted without the need for invasive procedures but ensuring fidelity.

References

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