Conference Agenda

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Session Overview
MS22-3: Continuum biomechanics of active biological systems
Friday, 22/Sept/2023:
1:30pm - 3:50pm

Session Chair: Martin Schanz
Session Chair: Tim Ricken
Location: Cupola Hall

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1:30pm - 1:50pm

Development and simulation of a detailed modelling human leg – application in in silico orthopedics

O. Avci, A. Ranjan, A. C. Yildiz

Fraunhofer IPA, Germany

There is a shortage of physics-based tools, available to orthopaedic surgeons, to quantify their everyday decision-making measures towards resolving their patients’ orthopaedic disorders. Such decisions rely purely on static medical imaging and surgical experience gathered over the years. As such, there is general consensus about the lack of understanding surrounding the consequences of performed surgical interventions on the resulting patients’ biomechanics. Such problems are prevalent across orthopaedics, esp. with implants, where the influence of implantation on individual biomechanics is unknown. Hence, at this juncture, the subject-specific interaction of the implant with the patient’s locomotor system cannot be determined. Our motivation is to overcome these issues using high-fidelity, physiologically realistic, in-silico analysis of the patient’s musculoskeletal disorder at a given biomechanical joint, thereby enabling implant testing for an individual patient or even performing large scale clinical trials for a cohort of virtual patients. In this regard, I will be present our current work on Fraunhofer IPA’s In-silico Human Modelling platform (ISHM), ranging from medical imaging to complex 3d biomechanical simulations of the musculoskeletal system simulations with very detailed physiologically human models.

The biomechanics of every single joint in the human body is extremely complex. Their physiological motions within the framework of stable joint mechanics are balanced by a complex system of pre-stretched muscles, tendons, ligaments and various other connective tissues and controlled by targeted muscle activation. Such a biomechanical joint system reacts very sensitively to changes in the properties of its components. Therefore, forced joint models commonly used in simulations cannot represent the real joint biomechanics as they end up generating unphysical coercive forces. As a result, we get inaccurate muscle forces and joint kinematics, far from absolute physiological reality. In the current study, I will present some physiological joint FE analyzes of the foot, knee, and hip without joint constraints. The muscle-driven forward analysis, together with the physiological musculoskeletal model, allows us to understand joint mechanics much better, especially when it comes to the biomechanical analysis of subjects with musculoskeletal disorders. Patient-specific model generation, the biomechanics of soft- and hard-tissues and accurate representation of joint biomechanics are of significant importance for realistic FEA. Such analysis would pave the way for in silico engineering of medical products in orthopaedics such as implants, prosthetics and orthoses. Improved functional fitting of the products will enhance their performance, which shall have a positive impact on the patient’s biomechanics and their overall quality of life.

1:50pm - 2:10pm

Exploring bone remodelling via a novel micromorphic approach

A. Titlbach1, A. Papastavrou1, A. McBride2, P. Steinmann2,3

1Nuremberg Institute of Technology, Germany; 2University of Glasgow, United Kingdom; 3Universität Erlangen-Nürnberg, Germany

Bone tissue possesses a remarkable capacity for adaptation to external loads, allowing it to modify its structure and density accordingly. Cancellous bone, the spongy network of trabeculae that constitutes the inner part of bones, undergoes microstructural changes in response to under- or overloading, which may result in the reinforcement or narrowing of its constituent trabeculae and alteration of the microstructural pattern. Due to the microstructure, size-dependent effects may play a role. Multi-scale approaches consider bone as continuum matter and resolve the trabecular structure directly at the subscale. This is, however, algorithmically cumbersome to implement and, above all, involves additional computational effort.

Here we present a micromorphic approach that accounts for both the heterogeneous substructure of the material, without resolving it explicitly, and the nonlocality of the bone remodelling process, which is physiologically motivated by spatially correlated mechanosensing and regulation. Our approach has the advantage of being able to dispense with laborious neighborhood sampling, as is the case with integral approaches, and higher continuity requirements, as is the case with higher gradient approaches. Our approach is implemented in the open source finite element environment deal.II.

Our methodology employs nominal bone density as a macroscopic quantity for the bone mass to volume ratio in the underlying trabecular microstructure. This approach allows us to account for the heterogeneous nature of bone while avoiding the need to resolve individual trabeculae. In this model, we use the theory of open-system thermodynamics, which assumes a mass source proportional to the change in nominal density over time. This mass source is equated with a mechanical stimulus, and the stored energy is compared to an attractor, which represents a biological stimulus that drives remodelling. In the local case, the stored energy is a local quantity that depends on macroscopic deformation. In our novel non-local approach, we extend this to include a micromorphic and a scale-bridging component, allowing us to incorporate non-locality with a characteristic length scale for the heterogeneous microstructure and a scale-bridging parameter that couples the micro and macro deformation. This approach enables us to account for the interaction between continuum points and to model how points in the material that are not directly loaded react to the loading of their neighbors.

We present this approach in detail and demonstrate its efficacy using benchmark examples. We then apply this approach to long tubular bones and compare it with a series of CT images of femoral heads.

2:10pm - 2:30pm

Electro-mechanical modeling and simulation of the human heart

T. Gerach, J. Krauß, C. Wieners, A. Loewe

Karlsruhe Institute of Technology (KIT), Germany

Mathematical models of the heart have evolved from single-physics representations on simplified geometries to coupled multi-physics representations of the whole heart with high anatomical fidelity. Here we present a fully-coupled electromechanical model of the human heart including all four chambers. Electrophysiology, active continuum biomechanics, and closed-loop circulation are modeled in a multi-scale approach. State-of-the-art models based on human physiology are used to describe membrane kinetics, excitation-contraction coupling and active tension generation in the atria and the ventricles. The validity of the model is demonstrated through simulations on a personalized whole heart geometry based on magnetic resonance imaging data of a healthy volunteer.

The proposed framework for the fully coupled cardiac electro-mechanical problem comprises a detailed description of appropriate boundary conditions such as a lumped parameter model of the human circulatory system and a contact handling that replicates the effects of the tissue surrounding the heart. To solve the coupled electro-mechanical problem, we apply a staggered scheme in time where the monodomain equation describing cardiac electrophysiology and the non-linear deformation resulting from the balance of active tension, passive material forces, chamber pressure and contact to the surrounding tissue are solved sequentially. Additionally, the proposed electro-mechanical whole-heart model is coupled to a 0D closed-loop model of the cardiovascular system. Here, we use a quasi Newton method to update the pressure values in all four chambers and reach convergence in fewer iterations compared to standard Newton methods.

We provide parameterizations for the fully-coupled excitation-contraction model for cells of the atrial and ventricular myocardium. Both the intracellular calcium transient and the tension development match data of human tissue preparations from literature. Coupling the 0D lumped parameter model of human circulation to all four chambers of the 3D electromechanical model enables a faithful reproduction of the major phases of the cardiac cycle as well as the characteristic figure of eight shape in the atrial pressure volume loops and flow patterns observed in clinical practice.
After introducing the coupled model, we provide application examples how such models can be used to generate mechanistic insight for clinical challenges. Selected examples include the in silico study of (i) the side effects of atrial ablation to treat electrical arrhythmia on whole-heart mechanical function and (ii) the electro-mechanical pathomechanisms underlying the breakdown of ventricular wringing rotation in heart failure. For (i), we provide biomechanical evidence that atrial ablation has acute effects not only on atrial contraction but also on ventricular performance. Therefore, the position and extent of ablation scars is not only important for the termination of arrhythmias but is also determining long-term pumping efficiency. For (ii), we show that isolated changes of the electromechanical activation sequence in the left ventricle are not sufficient to reproduce the rotation pattern changes observed in vivo and suggest that further patho-mechanisms are involved.

2:30pm - 2:50pm

Numerical treatment and sensitivity analysis of a cell-based mathematical model of tissue regeneration

E. Grosjean, B. Simeon

Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Germany

The crescent-shaped fibro-cartilaginous menisci located between the articulating surfaces of the femur and tibia are essential for the structural integrity of a healthy knee. Until recently, partial meniscectomy was considered the gold standard for treating meniscal lesions. However, due to the poor mid- and long-term outcomes of meniscus material dissection, surgical meniscus treatment paradigms have shifted in the last decade to promote healing through repair or regeneration. For cell colonization and meniscus replacement, various scaffolds have been proposed, including synthetic polymers, hydrogels, ECM components, and tissue-derived materials. So far, however, the outcomes of these studies seem inconsistent and indicate the need for a more fundamental understanding of the basic control mechanisms in cell-scaffold interactions under different environmental parameters.
In this context, we are working on simulations of a PDE-ODE system that model the dynamics of two cell populations involved in cartilage tissue production (adipose derived stem cells and chondrocytes). They are expected to migrate, proliferate, and differentiate within a perfusion chamber containing an artificial scaffold (poro-elastic medium) impregnated with a chemoattractant. To simulate this problem, we have used a first order discontinuous Galerkin scheme in space and an implicit Euler scheme in time.

This system is linked to a fluid problem, representing a mechanical stimulus. It is modeled by a Biot-Darcy system coupled to unsteady Stokes equations. To take into account all the coupled interface conditions, we have employed Nitsche's method.

In this macroscopic problem, several factors are important, most notably the stem cell differentiation that is expected to be mostly induced by mechanical stress inside the porous medium of the scaffold. Our goal is to identify all of the parameters of interest and investigate their impact. There are several methods available in this framework.
When dealing with discretized solutions of parameter-dependent PDEs, the sensitivities with respect to certain parameters of interest can be computed directly from the original problem, but each parameter of interest requires the solution of a new system: it is the "direct method". As a result, when dealing with a large number of parameters, the "adjoint method" may be a viable option.

Because the flow direction in our experiment alternates at a frequency of 1 Hz, the long-term simulation with a time span of up to 28 days is definitely a tremendous challenge. Thus, we will show how to efficiently combine reduced basis techniques and sensitivities computation with these methods to further reduce computational time. We will concentrate on non-intrusive reduced basis methods that do not require any changes to the High-Fidelity (HF) code. They only use the HF code as a "black box" solver. These adaptations will be numerically demonstrated with several results from our finite element model problem.

2:50pm - 3:10pm

Turing patterns in growing three-dimensional domains

S. Ben Tahar1, E. Comellas2, J. J Muñoz2, S. J Shefelbine1

1Northeastern University, United States of America; 2Universitat Politecnica de Catalunya (UPC), Spain

Embryonic development is a complex and fascinating process that has puzzled researchers for over a century. One of its greatest mysteries is how cells within a homogeneous mass can differentiate and organize into a wide variety of patterns. A key factor in this process is the role of morphogens. Morphogens are chemicals signals that cells use to communicate with each other, and they play a crucial role in determining cell fate and tissue specialization. Alan Turing proposed a model for how morphogens interact, known as the reaction-diffusion system [1]. This model relies on a diffusion-driven instability and nonlinear feedback between chemical species. Since Turing's time, researchers have used reaction-diffusion systems to model patterning in a variety of biological applications. In recent years, computational and experimental models have further validated the relevance and potential of this model.

The emergence of Turing patterns in a 3D growing domain has been sparsely investigated. The few studies exploring 3D Turing patterns indicate that the extension from 2D to 3D domains is not straightforward because the introduction of an additional dimension leads to a wider variety of patterns [2]. Previous studies do not focus on the biological context of embryogenesis and do not consider tissue growth. This gap in the literature highlights the need for further investigation on Turing patterns formation in 3D growing domains.

We explore the pattern evolution in 3D growing domains using finite elements analysis. As the growth process occurs on a much larger time scale than the reaction-diffusion one, we did not couple the domain growth with the pattern formation. Instead, for a given geometry, we computed the final steady-state pattern before growing the domain and relaunching the simulation. To model tissue growth two techniques were investigated: one simulating the apical tissue growth which is the cell formation at the distal or medial end and the second simulating anisotropic expansion. In the first case, new elements are added to one side of the mesh. In the second case, the whole mesh is stretched in one direction. Because these two different ways produce different pattern outcomes, pattern selection and emergence of bifurcations are affected due to the non-linearities of the system.

Overall, our study provides new insights into the role of different growth types when modelling 3D Turing patterns.


[1] Turing AM. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. doi: 10.1007/BF02459572

[2] Ben Tahar S at al. Turing pattern prediction in three-dimensional domains: the role of initial conditions and growth. Preprint on bioRxiv, 2023; doi: 10.1101/2023.03.29.534782.

3:10pm - 3:30pm

Variational modeling of biomechanical systems

M. Liero1, D. Peschka2

1Weierstrass Institute Berlin, Germany; 2Freie Universität Berlin, Germany

The consistent mathematical modeling of (active) biomechanical systems
at large strains based on partial differential equations is a challenging task.
In particular, the evolution of most biological systems is not just driven by mechanics, but is typically the result of coupling with other physical or chemical fields. Thus, the consistent coupling e.g. to reaction-diffusion equations or phase-field models becomes necessary to develop a general theory of out-of-equilibrium nonlinear thermodynamics for biological systems.

In this talk, we demonstrate that for dissipative biomaterials, coupled balance equations can be expressed as GENERIC or gradient systems. These frameworks describe the evolution of a system through thermodynamical potentials and geometric structures, encoding the reversible or irreversible nature processes using internal variables, such as concentration of biomolecules, phase-fields and growth variables, in addition to the elastic deformation. This point-of-view enables us to derive thermodynamically consistent coupled field equations systematically and reveals underlying coupling mechanisms. Moreover, the additional variational structure of the evolution equations can be exploited to establish structure-preserving numerical discretization schemes.

We discuss the applicability of the GENERIC framework and gradient systems for a biomechanical model describing the spatio-temporal evolution of brain atrophy in Alzheimer’s disease recently proposed by Schäfer, Weickenmeier, and Kuhl.
The model combines an anisotropic reaction–diffusion model for the time-dependent local concentration of misfolded tau protein with a large deformation shrinking model to predict the concentration-dependent regional tissue atrophy.
Similar models are relevant for the description of nonlinearly coupled transport and phase-transition processes in hydrogels and are therefore also highly relevant for biomedical applications.

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