Research in Continuum Biomechanics and Mechanobiology

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Mathematical models and computer simulations have the great potential to substantially increase our understanding of the biophysical behaviour of the neuromuscular system. This requires detailed multiscale and multiphysics models, which, once validated, allow systematic in silico investigations that are not necessarily feasible within experiments. The aim of our group is the development of a detailed biophysical model of the entire neuromuscular system. We consider the neuromuscular system as an integrated physiological system and thus, we combine models of motor neuron pools, the sensory system and multiscale models describing the mechanical behavior of skeletal muscles. By employing sub-models that are based on strictly biophysical modeling approaches, the resulting model closely represents the underlying physiological system and thus has the potential to provide valuable insights into the complex interrelations within the healthy system and pathological conditions.

Active research focus areas:

• Analyzing the influence of sensory signals on the output of motor neuron pools.
• Developing detailed biophysical models of muscle spindles.
• Investigating the impact of the three-dimensional motor unit distribution within skeletal muscles on the force output, strain and pressure distribution. 

Review on the neuromuscular system:

Röhrle O, Yavuz US, Klotz T, Negro F, Heidlauf T. Multiscale modeling of the neuromuscular system: Coupling neurophysiology and skeletal muscle mechanics. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2019

Skeletal muscles can be voluntarily controlled by the somatic nervous system and enable motion and active force production. From a physiological point of view, the smallest functional unit of the neuromuscular system is the motor unit. A Motor unit consists of a motor neuron and all the muscle fibers it innervates. The fact that a muscle consists of many (up to several hundreds) of motor units allows to finely modulate the force output and thus enables complex motion. Specifically considering the biophysical function of skeletal muscles, the active contractile behavior of the tissue is dictated by the microstructure of the tissue and the cellular working principles of the muscle fibers. For example, after stimulating the muscle fiber membrane with an action potential, calcium release from the sarcoplasmic reticulum into the cell is triggered. Serving as a second messenger calcium enables enables cross-bridge cycling, i.e. controlling skeletal muscles molecular motor. 

Our chemo-electro-mechanical skeletal muscle models resolve all key aspects of skeletal muscles anatomy and physiology within an organ-scale continuum model. In detail, by employing a multi-scale approach the model predicts the electrical potential fields within the tissue, the biochemical behavior of the muscle cells and the deformations of the tissue induced by either physiological or other external inputs.

Thus, the chemo-electro-mechanical skeletal muscle model is e.g. applied to

• systematically analyze the influence of individual structures or phenomena on the whole system, 
• generate virtual EMG signals, which are for example useful to evaluate the goodness of automatic decomposition algorithms, 
• predict the outcome of external electrical stimulation protocols. 

We further aim to improve the physiological accuracy of the model by 

• incorporating additional structures and phenomena such as muscle spindles or the metabolic behavior of the muscle cells,
• developing new homogenization methods, which are essential for consistent and efficient multi-scale models.

Since the chemo-electro-mechanical skeletal muscle model is computationally expensive we also constantly optimize the efficiency of the simulations by 

• develop or apply suitable numerical solution methods,
• employ problem specific parallelization strategies.
  

Modelling skeletal muscle tissue within the framework of continuum mechanics demands suitable constitutive material models on the macroscopic (organ) scale. This is also the fundamental requirement for reliable three-dimensional muscle simulations using the finite-element method. Commonly used phenomenological constitutive approaches, which have to be calibrated to macroscopic experimental material tests, suffer from the high inter- and intra-tissue variability of skeletal muscle tissue and the lack of individual experiments for the specific type of muscle to be modelled. The high variations in the tissue properties are hereby originated in the highly heterogeneous microstructure of the tissue.

Hence, our research group focuses on the development and consistent formulation of novel multi-scale material models for skeletal muscle tissue. These models directly take into account the significant microstructural features, for which a lot of experimental data is available. For instance, the model directly incorporates a comprehensive formulation for the collagenous structures of the extracellular matrix, including the description of collagen fibre stiffness and the dispersion of collagen fibres. Suitable homogenisation methods are developed and applied to get the wanted macroscopic constitutive behaviour. In contrast to phenomenological approaches, the proposed multi-scale model and its underlying idea of a rigorous bottom-up framework is able to answer questions like, for instance:

• How do microstructural changes (for example, due to diseases) influence the overall behaviour of skeletal muscles?
• What kind of direction-dependent behaviour on the organ scale results from the arrangement of structures on the microscale?
• How does the stiffness of single collagen fibres influence the stress distribution of the whole muscle?

The novel multi-scale model is an important step towards subject-specific material models.

This is done in close collaboration with the University of Pennsylvania, USA.

Skeletal muscles play an important role in overall health; from movement range, resistance and strength, to protein reservoir, muscles have the ability to adapt to any specific intensity of physical activity. A particular adaptation feature can be targeted by means of exercise. In principle, there are two kinds of training protocols: training for resistance and training for endurance, but a combination of these two protocols lead to intermediate outcomes.

Exercise, mechanical, or electrical stimuli trigger biochemical responses known as signaling pathways. These pathways control the adaptation processes at the cell level. Muscle cells are also known as muscle fibers, and there are many fiber types that share characteristics with one of two categories: some muscle cells (known as Type I fibers) adapt to sustained activities such as walking, or running a marathon, and are fatigue resistant; other cells (known as Type II fibers) adapt to activities that require force or power, such as sprinting, but are easily fatigable. As any muscle has a different proportion of fiber types, an adaptation process involves the transition between fiber types.

The aim of this research is to model the hypertrophy, atrophy, and fiber shifting processes of muscle tissue as the outcome of different training protocols. We assume that the training input stimulates a biochemical system that drives the protein content and fiber shifting of the tissue, and as a consequence, the model predicts force and fatigue adaptation. This is in collaboration with the National University of Colombia (Bogotá).

Real-world human movement generally arises from an interaction of multiple muscles. Experimental techniques may be used to measure outputs of the human movement system, e.g., via electromyography, sonomyography and/or optical techniques and while this data may be used to infer the underlying mechanisms behind human movement, e.g., using decomposition techniques to predict motor-neuron firings for a particular movement, there still remains a very large amount of uncertainty since the exact inputs to the system are not known. 

Therefore, our research focuses on the use of physiologically realistic multi-muscle simulation models to better understand the underlying mechanisms of healthy and pathological human movement.

Active research focus areas:

• Upper arm modelling, consisting of six muscles capable of flexion/extension and supination/pronation to investigate muscle and motor-unit synergies for applications in ergonomics, human-machine -interaction and sports and rehabilitation
• Pervasive simulation to enable resource-poor systems like laptops or tablets to perform complex simulation tasks. For example by using cloud computing to distribute the work-load, or machine learning methods to benefit from a (offline) learning processes.
• Surrogate modelling techniques to create a computationally highly efficient version of a model of the human upper arm for real-time simulation and visualization and interaction in virtual and/or augmented reality. 
• Use of surrogate models in our investigations of different activation strategies to generate desired movements and possibly even feedback mechanisms.

This is in close collaboration with the IPVS and VISUS at the University of Stuttgart and the Fraunhofer IPA.

Modelling the musculoskeletal system using a continuum-mechanical approach based on the theory of finite hyperelasticity, together with fine finite element discretisations is not only a challenging, but also a computationally expensive task. Especially in the many query context, which for example can arise when patient specific data is needed, or if parameter studies for healthy and pathological conditions shall be conducted, the simulations become prohibitively expensive or even unfeasible. For this reason, one needs to find ways to reduce the computational costs and this is where the mathematical field of model order reduction (MOR) suggests itself.

One possibility of reducing the computational effort of simulating a high dimensional system of differential equations, in the context of MOR referred to as full order model (FOM), is the projection based MOR. Essentially, this approach aims at reducing the number of degrees of freedom of the system by projecting it onto a lower dimensional subspace of the original high dimensional solution space. That way, a reduced order model (ROM) of lower dimension is obtained. For the determination of a suitable low dimensional subspace, the proper orthogonal decomposition (POD) is a well established and widely applied method. The idea of the POD is based on a split of necessary computations into an offline and an online phase. Therein, it is assumed, that during the offline phase, time and resources are unlimited, i.e. computations involving high dimensional operations can be executed, while aiming at computing only operations of low dimensional complexity during the online phase.

The challenge of constructing a fast and stable skeletal muscle model lies in considering and preserving the structural properties of the FOM when trying to set up a suitable method to obtain the ROM. Properties we focus on are

• dynamics
• incompressibility
• nonlinearity

which means we obtain a nonlinear differential algebraic equation system (DAE) with three fields to be solved for and a saddle point problem in the linear solve.

A summary of the research activities within this field will be published soon.

A summary of the research activities within this field will be published soon.