Adherent cells in vivo often reside on basement membranes which are thin, sheet-like specialized extracellular matrices that function as cellular anchorage sites, physical barriers, and signaling hubs. An important physical attribute of basement membranes is their three-dimensional topography due to the complex microstructural organization of their constituent proteins. This topography takes the form of an intricate network of fibers and pores with characteristic dimensions on the order of 100 nm on top of which lies a larger, micron-scale topography in the form of anisotropic undulations. Various types of engineered microstructured substrates have been developed to study the impact of topographical cues on cellular structure and function in vitro. One such system that we have been using is a substrate that consists of arrays of microgrooves that are intended to mimic the anisotropic organization of the basement membrane and on which cells can be directly cultured.

Our studies of the effect of substrate topography on cellular structure and function have focused principally on the vascular endothelium, the monolayer of cells lining the inner surfaces of all blood vessels. In medium and large arteries, chronic endothelial inflammation is a trigger for atherosclerosis, the disease that leads to heart attacks and strokes. Interestingly, atherosclerotic lesions develop preferentially in arterial regions where endothelial cells are cuboidal and randomly oriented, whereas arterial zones that are characterized by highly elongated and aligned endothelial cells remain largely spared from the disease. Therefore, understanding the relationships between endothelial cell shape/alignment and function is of fundamental interest, a question that we are tackling using topographic microgroove substrates.

In this presentation, I will discuss four aspects of endothelial cell responsiveness to microgroove substrates and highlight opportunities for computational modeling in each of these aspects. First, I will show how microgroove substrates can be used to noninvasively control endothelial cell shape and alignment and will describe our understanding of the mechanisms that underlie cell shape regulation by microgrooves. Second, I will describe dynamic live-cell recordings that demonstrate that microgrooves can orient the direction of migration of endothelial cells within monolayers and can lead to a unique pattern of collective cell migration that takes the form of antiparallel streams. Modeling the endothelial monolayer as an active fluid with the effect of the microgrooves considered as an energetic constraint on cell orientation predicts the emergence of the antiparallel streams as well as the dimensions of these streams. Third, I will show how microgrooves lead to extensive deformation of endothelial cells and their nuclei and will evoke the interesting notion of using these deformations to diagnose certain diseases that involve abnormalities in cellular and nuclear mechanical properties. Finally, I will describe the competition between microgroove-derived contact stresses on the cells’ basal surface with flow forces on the cells’ apical surface and how this competition is a key determinant of endothelial cell shape and alignment.

Osteoporosis (OP) is a chronic progressive bone disease which affects a large portion of the elderly population worldwide. OP is characterized by a slow reduction of bone matrix and changes in the bone matrix properties which ultimately leads to whole (organ) bone fractures [1].

Novel drug treatments are developed to more effectively reduce the risk of bone fractures. Assessing the effects of novel and existing treatments on OP is challenging due to the complexity of the bone remodeling process, its effects on the bone matrix and the different spatial and temporal scales involved. Identification and characterization of various bone biomarkers has significantly improved our understanding of OP pathophysiology. The bone matrix and its constituents are specific bone biomarkers measured at a particular bone site. On the other hand, biochemical ligands released during bone remodeling and measured in blood or urine are non-specific bone biomarkers. These biomarkers can be used to characterize the underlying bone mechanobiological system and drug treatment effects [1].

Recently, disease system analysis (DSA) has been proposed as a novel approach to quantitatively characterize drug effects on disease progression [1]. DSA integrates physiology, disease progression and drug treatment in a comprehensive mechanism-based modelling framework using a large amount of complementary biomarker data. DSA applied to population based structural models of whole bony organs (e.g. femur and vertebra) can be used to perform in silico trials of drug efficacy for osteoporosis treatment. In this presentation, I will present latest mechanistic pharmacokinetic-pharmacodynamic (PK/PD) models of osteoporosis treatments. Examples of currently used drug interventions including denosumab [2,3] romozosumab [4], and PTH [5] treatments will serve as discussion points on which mechanisms are essential for accurate bone remodeling simulations. Bone matrix mineralization turns out to be an essential model feature that is required to predict BV/TV changes for the case of anti-catabolic drug treatments of OP [3]. Finally, I will elucidate on how PK/PD models, typically applied at the tissue scale, can be adapted to the whole organ scale and coupled with structural finite element simulations of bone in order to predict effects of drug treatments on bone strength and fracture risk.

Acknowledgments: Dr Pivonka acknowledges support from the Australian Research Council (IC190100020) and (DP230101404).

References: ”[1] S. Trichilo and P. Pivonka, Disease systems analysis in osteoporosis and mechanobiology, in Multiscale mechanobiology of bone remodelling and adaptation, Editor P. Pivonka, CISM Courses and Lectures No. 1406, Springer, 2017; [2] S. Scheiner et al.. Mathematical modeling of postmenopausal osteoporosis and its treatment by the anti-catabolic denosumab, Int. Journal for Numerical Methods in Biomedical Engineering, 30(1), pp1-27, 2014; [3] J. Martinez-Reina and P. Pivonka, Effects of long-term treatment of denosumab on bone mineral density: insights from an in-sillico model of bone mineralization, Bone, 125, pp87-95, 2019; [4] M. Martin et al., Assessment of Romozosumab efficacy in the treatment of postmenopausal osteoporosis: results from a mechanistic PK-PD mechanostat model of bone remodeling, Bone, 133, pp1-16, 2020; [5] M. Lavaill et al., Effects of PTH treatment in osteoporosis, BMMB, pp1-16, 2020.”

As the key organ for metabolic processes in the human body, the liver is responsible for essential processes like fat storage or detoxification. Liver diseases can trigger growth processes in the liver, disrupting important hepatic function-perfusion processes[1]. To better understand the interplay between hepatic perfusion, metabolism and tissue in the hierarchically organized liver structure, we developed a multicomponent, poro-elastic multiphasic and multiscale function-perfusion model [2,3], using a multicomponent mixture theory based on the Theory of Porous Media (TPM). The multiscale approach considers the different functional units of the liver, so-called liver lobules, with an anisotropic blood flow via the sinusoids (slender capillaries between periportal field and central vein), and the hepatocytes, where the biochemical metabolic reactions take place. On the lobular scale, we consider a tetra-phasic body, composed of a porous solid structure representing healthy tissue, a liquid phase describing the blood, and two solid phases with the ability of growth and depletion representing the fat tissue and the tumor tissue. The phases consist of a carrier phase, called solvent, and solutes, representing microscopic components, e.g. nutrients, dissolved in the solvent. To describe the influences of the resulting tissue growth, the model is enhanced by a kinematic growth approach using a multiplicative split of the deformation gradient into an elastic and a growth part, dependent on the fat accumulation and tumor development. To describe the metabolic processes as well as the production, utilization and storage of the metabolites on the cellular scale, a bi-scale PDE-ODE approach with embedded coupled ordinary differential equations is used. In order to represent realistic conditions of the liver, experimentally or clinically obtained data such as changes in perfusion, material parameters or tissue morphology and geometry are integrated as initial boundary conditions or used for parametrization and validation [4]. Data integration approaches like machine learning are developed for the identification, processing and integration of data. A workflow is designed that directly prepares the model for clinical application by (semi-)automatically processing the data, considering uncertainties, and reducing computation time.

Acknowledgements: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the following projects: 312860381 (Priority Programme SPP 1886, Subproject 12); 390740016 (Germany’s Excellence StrategyEXC 2075/1, Project PN 2-2A); 436883643 (Research Unit Programme FOR5151, Project P7); 465194077 (Priority Programme SPP 2311, Project Sim-LivA); 463296570 (Priority Programme SPP 1158, Antarctica)

REFERENCES
[1] Christ, B., ..., Ricken, T. et al. [2017], Front. Physiol., doi: 10.3389/fphys.2017.00906 .
[2] Ricken, T., et al. [2015], Biomech. Model. Mechanobiol., doi: 10.1007/s10237-014-0619-z .
[3] Ricken, T., and Lambers, L. [2019], GAMM-Mitteilungen, doi: 10.1002/gamm.201900016 .
[4] Seyedpour, S. M, ..., and Ricken, T. [2021], Front. Physiol., doi: 10.3389/fphys.2021.733393

Physiologically-based pharmacokinetic (PBPK) models are powerful tools for studying drug metabolism and its effect on the human body. Here, we present our work on PBPK models for metabolic phenotyping and liver function evaluation [1-5]. To develop and validate our models, we established the first open pharmacokinetics database, PK-DB, which contains curated data from over 600 clinical studies. Our models can be individualized and stratified, allowing us to simulate the effect of lifestyle factors and co-administration on drug metabolism.

We have applied our models to various clinical questions, such as simulating individual outcomes after hepatectomy using an indocyanine green model and studying the effect of CYP2D6 gene variants using a model of dextromethorphan coupled with drug-gene interactions. These models are constructed hierarchically, describing metabolism and other biological processes of organs such as liver and kidney coupled to whole-body physiology. All models and data are available for reuse in a reproducible manner encoded in the Systems Biology Markup Language (SBML).

One of the major challenges in PBPK modeling is the integration of these models with cellular and continuum-biomechanical models. By coupling PBPK models with these models, it becomes possible to provide drug concentrations at the site of action as boundary conditions and time-varying inputs to continuum-biomechanical models. This approach enables us to study the impact of systemic chemotherapy on tumor growth or to investigate changes in perfusion and pressure at the whole-body level on transport and biomechanical properties at the tissue scale.

To demonstrate the effectiveness of this approach, we present an example of PBPK modeling coupled with a multiscale and multiphase continuum-biomechanical model for the liver. Our model integrates partial differential equations (PDE) on the lobular scale with ordinary differential equations (ODE) on the cellular and whole-body scale, resulting in a PDE-ODE coupling. In our showcase, we investigate the drug metabolization process in the liver, considering the possible damage to the liver tissue due to necrosis caused by a toxic side product. The substrates and products of this process can be transported via the systemic circulation and be removed by the kidneys. By coupling PBPK models with continuum-biomechanical models, we can study the transport and biomechanical properties of these substances at both the tissue and whole-body levels. This coupling approach provides a promising method for studying drug metabolism and its effect on the human body at multiple scales, ranging from cellular to whole-body.

Our approach allows us to investigate the impact of this drug metabolization process on the liver tissue and how it affects the overall functioning of the body. Additionally, we can study the impact of various factors, such as different drug dosages, on this process and determine the optimal drug dosage for a given patient. This integrated approach provides a comprehensive understanding of the drug metabolization process and its impact on the human body, paving the way for more effective drug development and personalized medicine.

[1] https://doi.org/10.1093/nar/gkaa990
[2] https://doi.org/10.3389/fphar.2021.752826
[3] https://doi.org/10.3389/fphys.2021.730418
[4] https://doi.org/10.3389/fphys.2021.757293
[5] https://doi.org/10.1101/2022.08.23.504981

Liver transplantation is the only curative treatment option for acute and chronic end-stage liver disease. As a result of demographic change and a more western way of life, the number of elderly multi-morbid prospective recipients and donors grows. Liver grafts from such donors, so-called marginal liver grafts, are often affected by hepatic steatosis compromising the quality of the donor organ. One reason for this is the alteration of the tissue structure, resulting in an impaired perfusion, in turn affecting hepatic metabolism and organ function. In case of a marginal graft, the surgeon is faced with the clinical decision to accept or reject the organ, increasing either the recipient’s postoperative risk or the risk of death on the waiting list significantly. Two major challenges for marginal grafts are the storage between procurement of the organ and transplantation (cold ischemia time) alongside damage after reperfusion, the so-called ischemia reperfusion injury (IRI).

For this purpose, we use computational multiphase and multiscale continuum-biomechanical modeling of biological tissue to simulate the hepatic deformation-perfusion-function relationship (cf. [1, 2]). Based on the theory of porous media (TPM, cf. [3]), we describe the functional liver units, the liver lobules, as a homogenized porous medium. A poroelastic multiphase and multiscale function-perfusion model is obtained by considering the porous liver tissue saturated with an anisotropic blood flow and coupling the metabolic processes at the cellular level within a bi-scale approach (cf. [4]). This approach combines partial differential equations (PDE) on the lobular scale with ordinary differential equations (ODE) on the cellular scale resulting in a PDE-ODE coupling. At the lobular scale, we consider healthy liver, necrotic, and fat tissue as solid phases, while blood is given as a fluid phase. Based on this, systems biology models can be used to model the energy balance, cell death, and functionality of each hepatocyte at the cellular level. Thus, it is possible to describe the ischemic damage caused by nutrient depletion, in turn influencing the perfusion at the lobular scale.

To make patient-specific predictions, we enhance the model through the integration of experimental and clinical data for validation and parameterization. This involves using machine-learning-based image analysis to read the geometry of liver lobules and zonation patterns of steatosis from histopathological images, in addition to laboratory data to provide initial and boundary values for solutes and the ODE models as well as information on the transplantation process, e.g., cold ischemia time. The simulation yields spatial evolutions of the considered phases and solutes within the liver lobules over time based on the given data. Such an in-silico model, which connects structure, perfusion, and function of the liver via the interaction between mechanical properties of the graft, hepatic perfusion, and the affected metabolism could facilitate clinical decision-making cf. [2].

[1] B. Christ et al. Frontiers in Physiology 8:00906 (2017)

[2] B. Christ et al. Frontiers in Physiology 12:733868 (2021)

[3] W. Ehlers. „Foundations of multiphasic and porous materials” in Porous Media: Theory, Experiments and Numerical Applications (2002).

[4] T. Ricken et al. BMMB 14:515-536 (2015)

Various research works from the biomechanics community focus on the inflation of pressurized tubular materials as this problem can be directly linked to the modeling of arterial tissue (see, e.g. Merodio & Ogden). In the modeling, the material is often taken to be hyperelastic, and the relation between loading deformation is given in terms of a strain energy density function. This function is often decomposed to account for the contribution of the different types of constitutes. The pressurization of the tubular structure is accompanied by various types of responses in both the stable and unstable parts of the inflation (see, e.g. Topol et al. 2022).

Biological soft tissue consists of cellular and non-cellular components. The relationship between stress and strain in the material evolves, and is determined by viscoelastic and tissue remodeling processes (see, e.g. Topol et al. (2021), Wineman & Pence (2021)). Due to the interplay of countless biological and chemical processes, the mechanical properties of tissue will be time-dependent. These processes may be initiated, stimulated, and mediated by different physical processes, for example in the form of mechanical stimuli in static or dynamic form (Stoffel et al., 2017).

This work on the inflation behavior of pressurized tubes under a time-dependent behavior of the material. It can be shown that the inflation of the material shows a large variety of responses that depend on various factors, that include the geometry of the considered problem (H. Topol et al. 2019)). A certain focus is given to the influence of the fiber natural configuration of the inflation behavior, which may differ from that of the embedding ground substance.

References
J. Merodio and R. W. Ogden. Extension, inflation and torsion of a residually stressed circular cylindrical tube. Continuum Mech. Thermodyn. 28:157–174 (2016)

M. Stoffel, W. Willenber, M. Azarnoosh, N. Fuhrmann-Nelles, B. Zhou, B. Markert. Towards bioreactor development with physiological motion control and its applications. Med. Eng. Phys 39: 106–112 (2017)

H. Topol, H. Demirkoparan, T. J. Pence. Morphoelastic fiber remodeling in pressurized thick-walled cylinders with application to soft tissue collagenous tubes. Eur. J. Mech. A/Solids 77: 103800 (2019)

H. Topol, H. Demirkopararn, T. J. Pence. Fibrillar Collagen: A Review of the Mechanical Modeling of Strain Mediated Enzymatic Turnover. Appl. Mech. Rev. 73: 050802, 2021

H. Topol, N. K. Jha, H. Demirkoparan, M. Stoffel, and J. Merodio. Bulging of inflated membranes made of fiber reinforced materials with different natural configurations. Eur. J. Mech. A/Solids 94: 104670 (2022)

A. Wineman and T. J. Pence. Fiber-reinforced composites: nonlinear elasticity and beyond. J. Eng. Math. 127: 30 (2021)

Chiral molecules form a plethora of liquid-crystalline phases, even in equilibrium. Liquid-crystalline phases formed by chiral and active agents have features that are even more counterintuitive. For instance, while in passive matter, chirality tends to cloak itself from the long-wavelength elastic and hydrodynamic properties, activity reveals its effect in cholesteric and chiral columnar phases, leading to unique forms of odd elastic behaviours. Specifically, cholesteric phases display a unique odd elastic force density tangential to the contours of the constant mean curvature of layer undulation. This ultimately leads to the formation of an antiferromagnetically organised, columnar, vortex-lattice array in extensile active cholesteric fluids. Columnar materials which are both polar and chiral surprisingly display two-dimensional odd elasticity, even though the system is three-dimensional. The interplay of this odd elasticity with three-dimensional Stokesian hydrodynamics leads to an oscillatory optical mode.
The interplay of chirality and activity also leads to liquid-crystalline phases without a passive analogue, such as time liquid crystals. In this phase, planar-chiral active elements — spinners — that tend to align with each other trade in rotation symmetry breaking for time-translation symmetry breaking. That is, in these spontaneously rotating aligned states, rotation symmetry is restored in a time-averaged sense. Therefore, while such states are aligned, they escape the celebrated Simha-Ramaswamy instability that plagues uniaxial active suspensions by spinning out of unstable configurations. In my talk, I will discuss both new phases of chiral active liquid crystals and their properties and the effect of activity on classical chiral liquid crystalline phases. I will specifically highlight how the interplay of chirality and activity allows the former to affect dynamic and static properties of ordered states to a much greater degree than in passive soft materials, as summarised above.
Given that most biomaterials are chiral, biological systems ranging in scales from subcellular to all the way to organisms are the natural domain for detecting chiral active liquid crystalline organisations and their effect. Indeed, some of the phases that I will describe in my talk were subsequently observed in in vitro experiments with properties consistent with my predictions. Furthermore, given chiral active liquid crystals have material properties that can, in principle, make them useful for engineering applications, a further important avenue of research will involve synthesising them artificially.

Whether one considers swarming insects, flocking birds, or bacterial colonies, collective motion arises from the coordination of individuals and entails the adjustment of their respective velocities. In particular, in close confinement, such as those encountered by dense cell populations during development or regeneration, collective migration can only arise coordinately. Yet, how individuals unify their velocities is often not understood. Focusing on a finite number of cells in circular confinements, we identify waves of polymerizing actin that function as a pacemaker governing the speed of individual cells. We show that the onset of collective motion coincides with the synchronization of the wave nucleation frequencies across the population. Employing a simpler and more readily accessible mechanical model system of active spheres, we identify the synchronization of the individuals' internal oscillators as one of the essential requirements to reach the corresponding collective state. The mechanical 'toy' experiment illustrates that the global synchronous state is achieved by nearest neighbor coupling. We suggest by analogy that local coupling and the synchronization of actin waves are essential for emergent, self-organized motion of cell collectives.

In this talk, I will describe two biologically inspired systems that can be described using the same geometrical Hamiltonian formalism. The first is ATP synthase proteins which rotate in a biological membrane. The second is swimming micro-organisms such as bacteria or algae confined to a 2D film. I will show that in both cases, the active systems self-assemble into distinct structural states - the rotating proteins rearrange into a hexagonal lattice, whereas the micro-swimmers evolve into sharp lines with a particular tilt. While the two systems differ both on the microscopic, local interaction, as well as the emerging, global structure, I will show that their dynamics originate from similar geometrical conservation laws dictated by a Hamiltonian formalism applicable to a broad class of fluid flows.

Many active biological particles, such as swimming microorganisms or motor-proteins, do work on their environment by going though a periodic sequence of shapes. Interactions between particles can lead to the phase-synchronization of their duty cycles. We consider collective dynamics in a suspension of
such active particles coupled through hydrodynamics. We demonstrate that the emergent non-equilibrium states feature stationary patterned flows and robust unidirectional pumping states under confinement. Moreover the phase-synchronized state of the suspension exhibits spatially robust chimera patterns in which synchronized and phase-isotropic regions coexist within the same system. These findings demonstrate a new route to pattern formation and could guide the design of new active materials. An extension of the same theory for treating ciliated surfaces quantitatively captures the instabilities and flow pupmping behaviour of ciliated carpets and metachronal waves.

Axolotl salamanders regrow entire limbs throughout life using a molecular machinery similar to that used in the embryonic development of our limbs. A series of genetically-regulated molecular markers, known as morphogens, coordinate key mechanisms of this process. But exactly how morphogen patterning drives skeletal joint formation is not fully understood yet. Computational models provide a means to explore the factors influencing this process. In addition, we have access to three-dimensional (3D) microscopy images of morphogen expression in regenerating axolotl forelimbs obtained using a novel technique [1], which can help inform such models. But to effectively abstract key concepts and test hypotheses about how biological processes work using computational models, we must have a thorough grasp on how model parameters and conditions influence predicted outcomes.

One of the prevailing theories to explain the formation of morphogen patterns was proposed by the mathematician Alan Turing [2]. Starting from a nearly uniform initial state, and through suitable nonlinear interactions between reacting and diffusing morphogens, stable spatial patterns, known as Turing patterns, are obtained. Turing patterns have been used to investigate the formation of skeletal limb structures, but previous studies only used one or two-dimensional models, which have limited applicability due to the joint’s asymmetrical structure. Extending to 3D is challenging due to the wider variety of patterns and increased complexity.

We use linear stability analysis and finite element modeling to predict Turing pattern emergence in a 3D generic domain using the Schnakenberg activator-substrate model [3]. We provide a framework to identify the critical factors necessary for specific morphogen pattern emergence, and explore the role of initial conditions, model parameters and domain size on the predictions. We have observed that initial conditions on the activator have a stronger impact on the final pattern than initial conditions on the substrate, which has important implications for modeling morphogen expression in joint formation.

Our findings establish the groundwork for upcoming research, where we aim to employ the model developed here to predict the morphogen expression resulting from the axolotl joint formation experiments. Only through a comprehensive understanding of the factors influencing pattern emergence will we be able to successfully use Turing patterns as a model for morphogen patterning in joint formation.

References

[1] Lovely AM et al. HCR-FISH in Ambystoma mexicanum Tissue. In: Salamanders Methods in Molecular Biology. New York, 2023. p.109–22, doi:10.1007/978-1-0716-2659-7_6.

[2] Turing A. The chemical basis of morphogenesis. Philos Trans R Soc London, 1952; 237(641):37-72, doi:10.1007/BF02459572.

[3] 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.

Cells are constantly interacting with their environment, adapting their behavior in response to different stimuli and environmental conditions. This cellular adaptation occurs via changes in gene expression derived from changes in the physiological environment, giving rise to phenotypic plasticity. Phenotypic plasticity plays a key role in many steps of cancer progression, to the point that it has been included among the Hallmarks of Cancer. Indeed, it partly explains some of the most characteristic features of cancer, such as metastasis or drug resistance, probably the main challenges for improving cancer prognosis. Hence, understanding the mechanisms that trigger cellular adaptation is crucial for advancing in the study of this disease and its eventual treatments

In this line, mathematical models and simulation have great potential for gaining insight into complex cell process such as adaptation, and testing hypotheses regarding the effects of different environmental conditions in the adaptive response of tumors. In this work, we present a novel mathematical framework to model cellular adaptation and phenotypic heterogeneity in cell populations interacting with their environment. The model is based on the concept of internal variables, which are used to model cell state and regulate cell behaviour, in addition to Partial Differential Equations (PDEs) to describe the evolution of cell populations and the spatial concentration of chemical species from a continuum point of view. The proposed approach allows to consider not only cell response to environmental changes, but also reversibility and inheritance typical of phenotypic changes.

After presentation of the model, we particularize it to the case of glioblastoma (GBM) adaptation to hypoxia. GBM is the most common and lethal primary brain tumor. It has a dismal prognosis with a 5-year survival rate of 5%. The poor response of GBM to treatment is a consequence of its intrinsic and acquired drug resistance. This resistance is enhanced by hypoxia, a defining feature of GBM. Therefore, it is important to study how hypoxia governs cellular adaptation in GBM to improve our understanding on treatment response and, eventually, GBM prognosis.

The objective of this study is analyzing the potential of the derived model for capturing important biological trends in GBM evolution and adaptation under variable oxygen concentrations. After an extensive parametric analysis, different oxygenation conditions are tested. The results show the flexibility of the model for capturing the variability existing among GBM tumors. The model is also able to capture some observed experimental trends, such as the increased aggressiveness and resilience of tumors undergoing cyclic hypoxia.

In short, the developed framework presents an alternative to modelling cellular adaptation and may, with suitable validation, help in designing predictive tools as well as in silico clinical trials.

Last few years have brought into broader scientific audience examples of very robust and prominent periodic spatio-temporal patterns in biological systems [1,2]. Until recently, patterns such as spirals and planar wave trains were observed almost exclusively in pure chemical nonlinear systems. Reaction-diffusion equations are commonly used for understanding such phenomena. In these systems, onset of periodic wave trains is described by the so called wave instability (WI). Here we present generic three component activator-depleted substrate-inhibitor model which is a minimal model for patterns created by small GTPase signalling molecule RhoA. The model reproduces patterns which are observed experimentally [1] and explains some quantitative relationships between amplitude, temporal period and concentrations of some components.

Activity of RhoA may also induce contraction of cell surface (cortex) by activation of a motor protein myosin, which may result in advection of chemical components. Hence, our system becomes a reaction-diffusion-advection system with temporally and spatially dependent velocity field. We are interested in modification of chemical instabilities by biomechanical modes and interactions of those modes in simple reaction-diffusion-active gel framework. In this approach cell cortex is treated as a visco-elastic fluid in which active stress is controlled locally by the chemical subsystem [3]. We found that biomechanical coupling may increase parameter range in which spatially homogenous solution is unstable and periodic spatial patterns are possible. We show that interactions between chemical and biomechanical modes may lead to the change of spatiotemporal characteristics of patterns created by the chemical subsystem, their destabilization as well as formation of new patterns. Moreover, we discuss the role of the delay between RhoA activity and active stress. 2D simulations on the plane and stability analysis of 1D homogenous and nonhomogeneous solutions are presented.

[1] A. Michaud, M. Leda, Z. Swider, S. Kim, J. He, J. Valley, J. Huisken, J. Landino, A. Goryachev, G. von Dassow, W. Bement, A versatile cortical pattern-forming circuit based on Rho, F-actin, Ect2, and RGA-3/4. J. Cell Biol., 221(8):e202203017, (2022)

[2] W. Bement, M. Leda, A. Moe, A. Kita, M. Larson, A. Golding, C. Pfeuti, K-C. Su, A. Miller, A. Goryachev, G. von Dassow, Activator-inhibitor coupling between Rho signalling and actin assembly makes the cell cortex an excitable medium, Nature Cell Biol., 17(11), 1471 – 1483 (2015).

[3] Frank Jülicher et al, Hydrodynamic theory of active matter, Rep. Prog. Phys. 81, 076601 (2018).

Red blood cells (RBCs) are known as important formed elements in the blood. The functions of RBCs are to transport oxygen from the lungs to the tissues and metabolic waste, carbon dioxide, from the tissues to the lungs and to maintain systemic acid/base equilibrium. Along with that, RBC is known to release adenosine triphosphate (ATP) when it is subjected to shear stress. Subsequently, ATP molecules bind to purinergic receptors, to activate a cascade of biochemical reactions in endothelial cells (ECs) to release sequestrated ubiquitous calcium ion from endoplasmic reticulum (ER). The response of EC to ATP is in the form of transient calcium. Calcium is well known to regulate the activity of many enzymes in order to maintain the cellular homeostasis. Nevertheless, unregulated free calcium concentration could lead to serious pathological conditions such as cell death. In order to avoid such consequences, EC manages to maintain its physiological calcium concentration in the presence of ATP using ATPdriven pumps present in the plasma/ER membrane and the desensitization of the purinergic receptors. In order to understand how ATP released from RBCs affect the intracellular homeostasis in a vascular wall, we firstly developed a minimal calcium model, which guarantees the intracellular calcium homeostasis. Secondly, we couple it to RBC flow in a two-dimensional channel for a given imposed parabolic flow. In simulation, we use immersed boundary-lattice Boltzmann method (IB-LBM) to solve the fluid flow and the ATP release from RBC. We carried out several simulations varying flow strength, channel width, and concentrations of RBC (hematocrit) in order to emulate the blood flow in microcirculation. The endothelium helps maintaining a steady ATP concentration, avoiding the abnormal rise in the ATP concentration released from RBCs. With varying flow strength and hematocrit for a given channel width, we found that the ATP concentration and the cytoplasmic transient concentration increase with increase in the flow strength as well as hematocrit, and this leads to the cytoplasmic transient time decrease. Due to the relatively small peak times and amplitudes of cytoplasmic calcium at high flow strength as compared to that at low flow strength for all hematocrit, there is a possibility of calcium propagation from the high flow strength region to the low flow strength region for the coordination of cellular functions. Similarly, we observed a possibility of calcium propagation from low confined channels to the medium or highly confined channels for all hematocrit for a given flow strength. It would be interesting to carry out a further study in a vascular network in order to get more insights on the calcium propagation as each branch of vascular network may have unequal concentrations of RBC and the flow strength.

Spider silk is an extraordinary protein material that exhibits counterintuitive mechanical behaviors such as a reduction in stiffness of several orders of magnitude, supercontraction (i.e. a shortening of up to ~60% in length), and twist upon exposure to high humidity. These non-trivial responses originate from a unique polymeric structure made of crystalline domains that are embedded in a highly aligned amorphous matrix. Broadly, high humidity leads to water uptake by the silk, which motivates the dissociation of intermolecular hydrogen cross-linking bonds. In this talk, I will present energetically motivated models that explain the origin of supercontraction and twist in spider silk. Using tools from statistical mechanics, I will show that the dissociation of intermolecular bonds gives rise to a transition from a glassy to a rubbery phase, an increase in entropy, and a decrease in free energy. These factors shed light on the underlying mechanisms that govern supercontraction and agree with experimental findings. In addition, I will employ a continuum-based framework to show that the twist behavior originates from helical features that exist in a glassy spider silk fiber. The merit of these works is two-fold: (1) they account for the microstructural evolution of spider silk in response to water uptake and (2) they provide a method to characterize the microstructural evolution of hydrogen-bond dominated networks. The insights from the presented models pave the way to the design of novel biomimetic fibers with non-trivial properties.

1) N. Cohen, “The underlying mechanisms behind the hydration-induced and mechanical response of spider silk”, Journal of the Mechanics and Physics of Solids, 172:105141, 2023.

2) N. Cohen and C.D. Eisenbach, “Humidity-Driven Supercontraction and Twist in Spider Silk”, Physical Review Letters, 128:098101, 2022.

3) N. Cohen, M. Levin, and C.D. Eisenbach, “On the origin of supercontraction in spider silk”, Biomacromolecules, 22:993-1000, 2021.

The study of the mechanical properties of viscoelastic composites has been of great interest due to their unique characteristics. Following the methodology proposed in [1,2], we study the effective properties of nonlinear viscoelastic heterogeneous materials. With this goal, we employ the asymptotic homogenisation technique to decouple the equilibrium equation into a cell and a homogenised problem. The theory developed in this work is specialised to the case of a strain energy density of Saint-Venant type, with the second Piola-Kirchhoff stress tensor also featuring a viscous contribution. Within this setting, we frame the general theory in the case of infinitesimal displacements to make use of the correspondence principle which results from the employment of the Laplace transform. This choice is also advantageous to avoid the numerical complications arising in a finite theory. Furthermore, it permits obtaining the classical cell and homogenised problems in linear viscoelasticity as a special case. We frame our analysis by considering the case of uniaxially fibre-reinforced composites and, taking inspiration from [3], we write short formulae for the effective coefficients associated with the antiplane problem. Finally, after selecting different constitutive models for the terms associated with the memory of the constituents, our results evidence that the approximations of the semi-analytical method converge rapidly and comparisons with data available in the literature show a good agreement. Further developments of this work aim to generalise the model using the covariant formulation of continuum mechanics and to include a broader analysis of different microstructural geometric arrangements of nonlinear viscoelastic composites. A further scope is to frame the general theory in biological scenarios of interest. These include but are not limited to, biological fibrous tissues, such as muscles and connective tissue. It is expected that further research in this area will lead to new research questions in materials science and biomathematics.

[1] Pruchnicki, E. Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. Acta Mechanica 1998, 129, 139–162. https://doi.org/10.1007/bf01176742.

[2] Ramírez-Torres, A.; Di Stefano, S.; Grillo, A.; Rodríguez-Ramos, R.; Merodio, J.; Penta, R. An asymptotic homogenization approach to the microstructural evolution of heterogeneous media. International Journal of Non-Linear Mechanics 2018, 106, 245–257. https://doi.org/10.1016/j.ijnonlinmec.2018.06.012

[3] Rodríguez-Ramos, R.; Otero, J.; Cruz-González, O.; Guinovart-Díaz, R.; Bravo-Castillero, J.; Sabina, F.; Padilla, P.; Lebon, F.; Sevostianov, I. Computation of the relaxation effective moduli for fibrous viscoelastic composites using the asymptotic homogenization method. International Journal of Solids and Structures 2020, 190, 281–290. https://doi.org/10.1016/j.ijsolstr.2019.11.014.

Stress shielding minimization is one of the major issues in implant design. Currently, aseptic loosening brought on by stress shielding is one of the primary causes of revision surgery. As bone regeneration is triggered by a stress stimulus, poor load transmission to the bone can result in a low stimulus, which can cause bone decay and other issues.

The development and evolution of additive manufacturing techniques have made it possible to significantly reduce stiffness by introducing porosity into implant materials as virtually any shape can be manufactured through those processes, regardless of shape complexity. In order to encourage cell adhesion and proliferation, porous geometries are frequently used in the construction of scaffolds. Additionally, porous geometries allow for the fine tuning of mechanical properties through changes to its topological design.

Feed-forward neural networks are able to do complex non-linear mapping between the input and output data. It has been shown that a neural network with one hidden layer and a number n of neurons in capable of representing any function. When a neural network presents several hidden layers, it is usually considered a deep learning approach.

The objective of this study is to create an optimal design by training a neural network to produce the ideal unit cell topology for a given constitutive elastic matrix. As a result, the network has the ability to reverse the homogenization process my mapping the relationship between the constitutive elastic properties and the unit cell geometry. A feed-forward neural network was created and trained in MATLAB with data generated from a set of several different geometries.

To each of these geometries, homogenization with periodic boundary conditions was performed. The lattice was modelled as a biphasic material where the solid phase was modeled with the properties of the material and the remainder area of the representative volume element (RVE) was considered to be a void phase with 1e-06 of the Young’s modulus of the solid material in order to reduce the influence of these elements to the homogenized constitutive matrix. A uniform mesh of square 2D elements allows to directly impose the periodic boundary conditions. The original geometry is therefore simplified to fit the uniform mesh where each element, a structured square, is either attributed solid or void properties.

The constitutive matrix used as the input of the network was obtained by applying a deformation gradient leading to a strain state where all components, but one is null, and thus, the macro-stress tensor of the RVE is equal to one line in the constitutive matrix. The linear-elastic analysis is run using ABAQUS as the solver.

Composite or microstructured materials have been long since considered as important means to engineer and optimize mechanical properties for specific applications. With the advent of additive manufacturing (also known as 3D-printing), production of artificial constructs conceived to possess specific optimal properties is now becoming possible, with applications ranging from construction to biomimetic materials. The architecture of such composites is typically based on designing features at a small scale, that lead to the desired large-scale behavior of structural elements. In light of this, theoretical studies of composites often involve asymptotic homogenization or alternative upscaling techniques. Most often, a periodic assembly of basic units is a practical way to build metamaterials.

In the design and analysis of composite materials based on periodic arrangements of sub-units it is of paramount importance to control the emergent material symmetry in relation to the elastic response. In numerous applications it would be useful to obtain effectively isotropic materials. While these typically emerge from a random microstructure, it is not obvious how to achieve isotropy with a periodic order. I has been long since recognized that inclusions distributed on a two-dimensional hexagonal lattice orthogonally extruded in the third dimension can give rise to transversely isotropic materials. Nevertheless, in spite of some computational evidence emerged in recent years, a generalization of this result to full three-dimensional isotropy has so far remained elusive.

We present a rigorous and yet simple proof of the fact that a periodic arrangement on a face-centered cubic lattice of spherical inclusions of an isotropic solid within an isotropic matrix gives rise to a large-scale isotropic response [Mech. Res. Commun. 126 (2022) 104001]. In doing so, we also show that any rhomboidal computational cell that generates such a lattice can be used to successfully design homogenized solids in which the material symmetry is not affected by the periodicity of the construction, since the latter would preserve even the largest possible symmetry group. It is significant to observe that the geometric symmetry group of such rhomboidal cells is strictly smaller than the symmetry group of the lattice they generate, but the lattice and not the cell is the geometrically relevant structure when analyzing large-scale properties.

We frame our discussion in the context of linear elasticity by introducing a normalized Voigt representation of the elasticity tensor which is very convenient for the identification of material parameters and symmetries for inclusion lattices. Such a representation hinges on the definition of a basis for the space of symmetric tensors that is linked to the generators of the periodic lattice. In this way, the coefficients of the 6 by 6 elasticity matrix that describes the linear stress-strain relation acquire a material meaning which is independent of the coordinate basis chosen to represent the tensors.

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.

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.

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.

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.

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.

References:

[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.

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.

The expression "Hierarchical fibre bundles" indicates structures with one principal direction. The structure is composed of multiple fibrillar elements, i.e. with one principal direction. Interfaces may or may not be present between the fibrillar elements. Finally, each fibrillar element is itself defined as a fibre bundle, made of fibrils with different mechanical and geometrical properties. The properties of each scale arise from the properties of the lower scale.
Hierarchical fibre bundles are abundant in nature across a variety of anisotropic biological materials, such as tendons, muscles and silk. These structures often exhibit exceptional fracture properties with outstanding toughness, thanks to the presence of structural proteins and/or to bio-mineralisation and their arrangement across different scales.
Analysing the fracture behaviour of complex fibre-based structures is still a challenging task, even when using advanced computational tools. Current numerical models often require a set of assumptions and (over-)simplifications. In particular, many models require the knowledge a priori of the fracture path. Biological materials, however, are characterised by complex internal structures with random features; moreover, for nano- and micro-bundles, it is often impossible to capture the fracture phenomenon empirically. The fracture path is thus often unknown, and it is one of the open questions which needs to be addressed using mechanical models. Here, we propose an alternative solution to model the fracture mechanics of fibre-based materials, which combines the mathematical Fibre Bundle Model (FBM) and Finite Element (FE) methods.
The Fibre Bundle model is a general theory of failure based on the statistical distribution of the critical elongations of the fibres forming the bundle. FBM assumes the bundle as a system of parallel one-dimensional elements, being clamped at their extremities and undergoing external forces along their main direction. Being a mathematical tool, the FBM is based on analytical formulations and thus it can be computed at practically instantaneous speed using computational solvers. In this work, we use the FBM to obtain general properties of the fracture behaviour of a fibre bundle undergoing elongation along its main axis, and to describe the stress-strain curves characterising the fibrils in the structure. The non-linear stress-strain curve obtained is utilised as an input for an FE approach, which aids us in comprehending the deformation and fracture behaviour of the fibrils and the interaction between different fibrils and the eventual surrounding interfaces.
We demonstrate the utility of the approach by applying it to spider-silk threads and bio-polymeric fibre bundles that are known for their non-linear properties. The enhanced numerical approach to the fracture mechanics of mixed fibre bundles will improve our understanding of the function of biological structures and bio-inspired materials across scales.

Constant demand for improvement of cardiovascular treatments and necessity for better understanding of soft tissue behaviour is frequently tackled with finite element (FE) modelling. The accuracy of FE simulations considerably depends on precise and reliable material parameters. They are obtained through the process of material characterization that consists of experimental testing and subsequent data analysis. When it comes to soft tissue, planar biaxial tensile tests are widely applied because they resemble in vivo conditions fairly well and material is assumed as anisotropic. In spite of planar biaxial tensile tests being the go-to type of experiment, they are still not standardized. The possible variations are numerous, from specimen size and different gripping mechanisms to the choice of preconditioning stretches, experimental protocols, and optimization procedures during data analysis. Limited availability of soft tissue requires simple and tiny square specimens. Before the start of experiment, specimens are mounted with hooks, rakes, or sutures and slightly preloaded to avoid the specimen hanging and to ensure that its orientation is perpendicular to loading axes. Application of the initial loads reflects in a different stress state compared to the reference configuration. This pre-stretching of the specimen cannot be measured due to the nature of experiment. Thus, measured initial force results in zero stretches during the material parameter estimation. However, these occurrences are mostly neglected. Either existing pre-stretches are disregarded or initial forces are annulled manually. Obviously, neither of the cases corresponds to real stress state.

In this work we developed the procedure to calculate mounting pre-stretches and consequently material parameters are determined more accurately. Standard optimization procedure is based on minimizing the least squares difference between experimental and model Cauchy stresses. Errors induced while calculating experimental Cauchy stresses lead to inaccurate material parameters. To avoid such errors, additional deformation gradient that represents mounted specimen in a state prior to experiment, has to be introduced. Since mounting pre-stretches are not measured, a new data fitting process is established with supplementary nested loop that will iteratively correct pre-stretches to satisfy equilibrium between initial experimental and model stresses. The procedures were first applied on the series of simulated virtual planar biaxial experiments where the exact material parameters can be set and compared to the obtained ones. Holzapfel-Gasser-Ogden strain energy function was used in virtual experiments while material parameters were based on the literature. Furthermore, data fitting procedures were applied to data gathered from biaxial experiments on aortic tissue samples. The analysis has shown significant difference between obtained material parameters, especially in cases where HGO structural parameters such as fibre family angle and in-plane dispersion are considered as fitting parameters. The rate of error increases with the amount of applied pre-stretches and also decreases with increase of maximum achieved experimental stretches. Finally, annulling initial force proved to be better approach if only standard optimization is to be applied.

The dura mater is the outermost layer of the meninges, a layered membrane that covers and protects the brain and spinal cord. It is a dense and fibrous connective tissue that is composed of collagen and elastin fibers, proteoglycans, and other extracellular matrix components [1]. The dura mater plays an important role in maintaining the stability and integrity of the brain and spinal cord, as well as in regulating the cerebrospinal fluid dynamics and protecting the neural tissue from mechanical stress and injury [2]. Despite its important role in, e.g., traumatic brain injury pathology it is frequently neglected in computational and physical human head models. Until now, the biomechanical failure tests have not yielded information about the anisotropic mechanical behavior of various locations of the human dura mater under physiological conditions. Additionally, limited data exists on the orientation of collagen fibers, which has not been thoroughly quantified.

Therefore, this study aimed to investigate the mechanical and structural anisotropy of different locations of the human dura mater and to correlate mechanical data with the quantification of the orientation of collagen fibers. To achieve this objective, sixty samples from six donors were subjected to quasi-static, uniaxial extension tests until failure in a heated tissue bath, utilizing a Z020 torsion multi-axis testing system (ZwickRoell AG, Ulm, Germany) together with an Aramis image correlation system (GOM, Braunschweig, Germany). Additionally, the collagen microstructure of samples from four donors was analyzed using second-harmonic generation imaging.

The data obtained were used to determine the failure stress and strain, E-moduli, and a microstructurally motivated material model was employed to examine local differences in both structure and mechanics. Among others, significant differences in both collagen fiber dispersion and main fiber orientation were found. The structural parameters obtained from only four out of six donors yielded good fitting results to the structurally motivated mechanical model for the remaining two donors in all directions. This study establishes a foundation for further research on the microstructure and mechanical properties of the human dura mater, enabling realistic modeling and prediction of tissue failure.

References

[1] J. T. Maikos, R. A. Elias, and D. I. Shreiber, "Mechanical properties of dura mater from the rat brain and spinal cord," J Neurotrauma, vol. 25, no. 1, pp. 38-51, Jan 2008, doi: 10.1089/neu.2007.0348.

[2] D. J. Patin, E. C. Eckstein, K. Harum, and V. S. Pallares, "Anatomic and biomechanical properties of human lumbar dura mater," Anesth Analg, vol. 76, no. 3, pp. 535-40, Mar 1993, doi: 10.1213/00000539-199303000-00014

Cardiovascular diseases are still one of the major cause of death worldwide. They are responsible for significant morbidity and therefore both noninvasive and especially invasive treatments are still being developed. Our research group is working in the field of new materials for cardiovascular surgery.

There are several procedures in the field of cardiovascular surgery which are based on the mechanical interaction of the arterial wall and the external support. A typical example is Pulmonary Artery Banding (PAB) in infant patients. PAB is a palliative procedure that reduces pulmonary over-circulation in neonates suffering from certain congenital heart diseases and constitutes the first stage of intervention prior to the complete repair of cardiac defects. The principle of PAB consists in the implantation of a band around the pulmonary artery, which results in the reduction of blood flow.

Our goal is to develop resorbable fabric based on composite material consisting of a polylactide and polycaprolactone (PLA/PCL) copolymer nanofibrous reinforcement combined with a collagen matrix for use in cardiovascular surgery as the pulmonary artery banding material. The resulting composite composed of these polymers prepared by electrospinning followed by the impregnation of the layers with collagen (type I) dispersion is then collagen-poly[L-lactide/caprolactone] (COLL-PLCL). This compound is bioresorbable and gives the advantage of PAB as a one-step procedure.

In our work, we present the results of the comparison of the mechanical properties of our COLL-PLCL material with two kinds of commercially available Gore material that can be used in pulmonary artery banding. They are the Gore Pericardial Membrane and the Gore Cardiovascular Patch, and both materials are chemically based on the form of polytetrafluoroethylene (PTFE).

The strips of all three materials underwent uniaxial tensile tests. In the case of the COLL-PLCL material, it was a monotonic tensile test, whereas the Gore materials were tested cyclically. The COLL-PLCL materials were tested in the hydrated state.

The results of our experiments show that all three materials exhibit an approximately linear mechanical response under primary loading. Cyclic loading led to so called preconditioned state, in which the Gore materials exhibited some non-linearities in their response. The preconditioned Gore-Tex materials were also considerably stiffer than those in the pristine state. Gore materials showed higher stress at failure than COLL-PLCL material whereas strain at failure was approximately the same. What is significant is that the COLL-PLCL material was much more compliant than Gore. We hypothesize that it will therefore cause a less mechanobiological response to compliance mismatch during banding.

Although the biomechanics of the circulatory system is a very dynamically developing field, the study of venous aneurysms is somewhat outside the mainstream. Although aneurysm, a bulge of the blood vessel wall, is mainly known as a disease of the arteries, it can also be encountered in veins. It occurs mainly at the site of an arterio-venous shunt, which is artificially created as a so-called vascular access for hemodialysis. Other venous aneurysms are quite rare. However, the number of patients for dialysis is increasing, so the need to study tissue biomechanics in vascular access is a current scientific problem.

The exact mechanism of arterio-venous shunt aneurysm formation, apart from the rather vague claim that it is a consequence of altered hemodynamic conditions, is still unknown. The same applies to the constitutive models and their parameters found for such affected vein walls. However, it should be emphasized that not only the change in hemodynamics due to arterialization of the vein, but also the repeated needling of the tissue during the hemodialysis procedure are involved in the formation of the arteriovenous shunt aneurysm. These lead to significant scarring of the vessel wall, which results in a change in mechanical properties compared to healthy vein. As with any scar, we should expect a significant increase in connective tissue content.

The long-term goal of our study is to experimentally identify material models for pathologically affected veins and compare them with the behavior of healthy tissue. We then intend to use the constitutive models for the arterio-venous shunt aneurysm tissue in computational fluid dynamics simulations with a deformable wall to determine the changes that occur in the hemodynamics of the aneurysm at the vascular access.

As a first step in our study, we perform tensile experiments with vein wall samples, both healthy ones and veins with a created arteriovenous shunt. We chose to model the tissue behavior as elastic after preconditioning. The constitutive model used for the vein wall is adopted from the work of Gasser Holzapfel and Ogden (2006), or we model the vein wall as an incompressible, anisotropic, hyperelastic material that exhibits significant material nonlinearity. Because aneurysmal vein wall samples are collected during surgery from living patients, it is not possible to compare them with other donor veins, as they are not naturally available. Comparisons will be made against the characteristics of veins taken to create an aorto-coronary bypass graft. The number of experimentally measured samples in both groups is not yet large enough to achieve significant population differences. Therefore, our present results should be considered as preliminary.

Tissue fibrosis, classically associated with excessive accumulation of extracellular matrix (ECM) by activated fibroblasts, can occur in several pathological conditions. Fibroblasts are mechanosensitive cells and increased stiffness of the ECM activates fibrotic pathways [1]. To mimic the three-dimensional cell microenvironment in vitro, fibroblasts are cultured in 3D spheroids. Mechanical assessment of cell spheroids is accomplished via parallel plate compression and mechanical properties in a continuum sense are determined from force vs displacement (F-δ) curves. Current analysis methods are often limited to Hertzian theory and its modifications, that only account for small deformations [2]. However, spheroids exhibit low stiffness, thus undergoing large deformations at small external forces, and therefore current contact mechanics models fail to describe such behavior [3,4]. Here, we use the Tatara model, which considers both nonlinear elasticity and large deformations to describe the mechanics of cell spheroids under parallel-plate compression.

Cell spheroids consisting of primary human fibroblasts cultured for four days were subjected to parallel-plate compression testing (MicroSquisher, CellScale) fitted with a round tungsten cantilever and accompanying SquisherJoy V5.23 software (CellScale, Ontario, Canada). The fluid bath test chamber was filled with sterile phosphate buffered saline (pH=7.4). Stage and optics were calibrated according to manufacturer’s instructions. Samples were compressed up to 50% apparent linear strain at different displacement rates. F–δ data was fitted using linear least squares regression on the Hertz and Tatara model and its extended version (custom MATLAB code) with fully constrained contact points (F =0, δ =0). Images were captured via two digital cameras during compression and static states to determine contact radius and lateral expansion of cell spheroids over a range of deformations.

The dependence of compression modulus on the displacement rate was examined. The Hertzian (and extended) theory was successfully applied to the small strain regime of F-δ curves to extract the compression modulus. In contrast to the Hertzian theory, for compressive displacements over 10-25% (depending on displacement rate), where the force follows the third and fifth power of the displacement, the Tatara numerical analysis theory better described the non-linear behavior of cell spheroids. The predicted contact radius of cell spheroids was found to be in good agreement with the data obtained from image analysis. However, the lateral expansion was underestimated by the Tatara model, a difference which may arise from the adopted constant-volume assumption.

The Hertzian theory and its modifications can be applied for strain in the range of 10%-25% while, for larger deformations (above 25% strain), the extended Tatara model better describes the deformation behavior of cell spheroids also providing their lateral expansion. While Hertzian contact theory and its modifications is applicable for small deformations, hyperelastic and nonlinear elasticity theory should be employed to better describe the data at larger deformations.

1. M. Jones et al, eLife, 7 :1-24, 2018

2. Y. Efremov et al, Scientific Reports, 7:1-14, 2017.

3. Y Tatara et al, JSME INT A-SOLID M, 36:190-196, 1993.

4. K. Liu et al, J. Phys. D, 31 :294-303, 1998

Background. Despite biomechanical factors playing a pivotal role in the onset and development of disease [1], little is known concerning the fracture behaviour of most biological tissues. Most experimental testing techniques fail to acquire data for the mechanical fracture characterisation of soft tissues. Therefore, we propose the novel symmetry-constraint Compact Tension (symconCT) test for aortic tissue fracture testing. It allows for stable crack propagation and the in-depth exploration of fracture processes. Finite Element Method (FEM) modelling was combined with Digital Image Correlation (DIC) to identify the specimen-specific constitutive properties.

Material and Methods. The classical CT test was augmented with an elastic metal beam connecting the two clamps [2]. It pre-strained the 30x35 mm² specimens and forced the fracture to develop in the middle of the specimen. Specimens with a 10 mm pre-notch were obtained from the intima-media compound of the porcine abdominal aorta. Tensile loading, either in the axial or circumferential direction, was applied orthogonal to the pre-notch at 3 mm min^-1 (ADMET eXpert 4000 Universal Testing System). 2D FEM models were developed (Abaqus 2019) for six specimens (representing mean ± standard deviation of recorded symconCT data sets), incorporating a viscoelastic Yeoh model and an isotropic bilinear cohesive zone model. Whilst the viscous properties were fixed, the two elastic Yeoh coefficients, c₁ and c₂, were calibrated against the clamp force and DIC-based Green-Lagrange strains. The cohesive strength and the fracture energy were also calibrated against the force.

Results. Given the load aligned with the main collagen fibre direction in the media, circumferentially loaded specimens reached higher forces, with a mean of 5.76 ± 0.69 N against 3.89 ± 0.57 N of the axially loaded specimens (t-test, p<0.05). Whilst circumferential loading resulted in a zig-zag crack, axial loading provoked a straight fracture. The DIC showed the maximum principal strain being orthogonal to the crack tip. The identified elastic properties were c₁ = 34.70 ± 1.52 kPa and c₂ = 98.70 ± 5.63 kPa under circumferential loading and c₁ = 14.54 ± 5.24 kPa and c₂ = 67.34 ± 6.65 kPa under axial loading. The cohesive strength resulted in 235.62 ± 24.04 kPa and 242.40±72.48 kPa, the fracture energy in 1.57 ± 0.82 kJ m² and 0.96 ± 0.34 kJ m² under circumferential and axial loading, respectively. Given the calibrated FEM models, the FEM crack extensions closely matched the experimental ones.

Conclusions. The symconCT test allowed for stable crack propagation, permitting the exploration of the fracture behaviour ahead of the crack tip. The DIC-based principal strains revealed mode-I failure dominating the fracture in both loading conditions. The FEM models accurately simulated the tissue fracture. Although inertial effects were overaccelerated (mass-scaling), good agreement with the experimental recordings was observed. Additionally, the vessel wall shows anisotropic mechanical properties [3], characteristics which could have further improved our results.

References.

[1] N. Baeyens, M. A. Schwartz, Mol. Biol. Cell, vol. 27, pp. 7-11, 2016.

[2] M. Alloisio, M. Chatziefraimidou, J. Roy, T. Gasser, Acta Biomaterialia (under review).

[3] T.C. Gasser, Vascular Biomechanics, ISBN: 978-3-030-70965-5. Springer 2021.

The ability of bone to sense and react to its mechanical environment is well-known and undisputed. In qualitative terms, the prolonged exposure of bone tissue to increased mechanical loading (with respect to a "normal" physiological load level) leads to a corresponding increase in bone mass; usually associated with the filling of the bone tissue-surrounding pore spaces by additional bone tissue. A decreased mechanical loading, on the other hand, causes a corresponding depletion of bone tissue, leading to enlarged pore spaces and a decreased bone mass.

Several stimuli have been proposed as potentially relevant for the mechanobiological regulation of bone tissue, including direct cell stretching, hydrostatic pressure, and fluid flow-induced shear stresses. Furthermore, it is well known that bone tissue exhibits piezoelectric properties, and it has been suggested that piezoelectric excitation may be a contributing factor to the mechanobiological regulation of the activities of cells residing in the bone pore spaces as well. In this contribution, the focus is on the latter proposition. Due to the very small length scales at which the related processes occur, direct experimental validation or falsification of this hypothesis has turned out to be out of reach. Here, we investigate this potentially important mechanism by means of continuum micromechanics-inspired multiscale modeling.

In particular, the concept of continuum micromechanics, originally developed for elasticity upscaling (or, homogenization), was utilized for deriving and homogenizing a so-called electromechanical tensor, comprising stiffness as well as electrical quantities, and for up- and down-scaling mechanical and piezoelectric properties. This modeling concept was applied to bone tissue, spanning thereby the hierarchical organization from the molecular level of bone (where the so-called elementary constituents of bone can be identified) to macroscopic bone tissue. Applying this model to the multiscale micromechanical representation of bone tissue has revealed that the electrical stimuli arriving at the level of bone cells, in response to physiological macroscopic loading, is much too small for effectively stimulating bone cell activities. This result suggests that piezoelectric effects may be a contributing, but not a major factor for bone cell excitation. This contribution is concluded by an interpretation of the obtained results in the context of other mechanical stimuli - i.e., hydrostatic pressure and fluid flow-induced shear stresses - previously investigated also by means of continuum micromechanics-inspired multiscale methods, and by comparing their respective importance for the bone remodeling process (as suggested based on the computational results), revealing that the pressurization of bone pore spaces may be of bigger relevance than the other stimuli.

For accurate quantification of mineralised tissues using polychromatic lab x-ray sources, one needs to do two things: 1) to integrate for a very long time to get a good signal to noise ratio and 2) to calibrate the x-ray source and the detector response. At Queen Mary we build our own X-ray Micro-Tomography (XMT, synonym µCT) systems specifically designed for accurate quantification of mineral concentration. We build on the pioneering work of Professor James Elliott, the co-inventor of x-ray micro-tomography. We use CCD cameras and time delay integration to allow long integration times, and calibrate every scan with a multi-metal carousel, enabling us to correct beam hardening effects, model the x-ray emission spectrum and model our detector response which allow us to derive the sample Linear Attenuation Coefficient (LAC) and hence mineralisation.

Getting improved contrast [i.e., mineral concentration] resolution is crucial to all applications of XMT to studying skeletal tissues and especially where there are mixed tissue types with different degrees of mineralisation. The common clinically important fractures of the cortices of vertebral bodies and the femoral neck involve variable amounts of calcified ligament: this, and calcified cartilage in the end plates, reach higher levels of mineralisation than bone. They also have different histology, ‘grain’ and fracture properties. With our own systems, we can distinguish these tissue phases, but this is not possible in commercial systems. Mistakes will therefore be made is measuring cortical thickness – never mind that the tissue is not bone.

Context. Bone disorders like osteoporosis as well as active lifestyles increase the occurrence of bone fractures and joints damage. In orthopaedic and dental surgery, the implantation of biomaterials within bone tissue to restore the integrity of the treated organ has become a standard procedure. Their long-term stability relies on osseointegration phenomena, where bone grows onto and around metallic implants, creating a bone-implant interface. The bone tissue is a highly hierarchical material which evolves spatially and temporally during this healing phase. A deeper understanding of its biomechanical characteristics is needed, as they are determinant for the surgical success. In this context, we propose a multiscale homogenization model to compute the effective elastic properties of bone tissue (macroscale) as a function of the distance from the implant, based on the structure and composition at lower scales.

Methods. The model considers three scales: mineral foam, ultrastructure, and bone tissue. The elastic properties and the volume fraction of the elementary constituents of bone matrix (mineral, collagen and water), the orientation of the collagen fiber relatively to the implant surface and the mesoscale porosity constitute the input data of the model. At each scale, the continuum micromechanics theory based on the famous Eshelby’s representation of the uniform elastic field inside the ellipsoidal inclusion is applied. Experimental data were obtained from Ti6Al4V coin-shaped implants that were osseointegrated into cortical rabbit bones. The mineral platelet orientation -assumed to be parallel to the collagen fibers- and the mesoscale porosity were retrieved using small-angle X-ray scattering (SAXS) and using light microscopy, respectively (Le Cann et al., Acta Biomater, 116:391–399, 2020). The effect of their spatial variation on the bone anisotropic properties in the proximity of the implant were investigated.

Results and discussion. The findings revealed a strong variation of the components of the effective elasticity tensor of the bone as a function of the distance from the implant. The effective elasticity is primarily sensitive to the porosity (mesoscale) rather than to the collagen fibers orientation (submicroscale). However, the orientation of the fibers has a significant influence on the isotropy of the bone tissue, leading to a high degree of anisotropy when the fibers are oriented with an angle close to 45° with respect to the implant surface. When analyzing the symmetry properties of the effective elasticity tensor, the ratio between the isotropic and hexagonal components is determined by a combination of the porosity and the fibers orientation. A decrease in the porosity leads to a decrease of the bone isotropy and to an increase of the impact of the fibers’ orientation.

Conclusions. These results demonstrate that the collagen fiber orientation affects the effective elastic properties of the bone throughout the remodeling process in the proximity of an implant. Collagen fiber orientation should be taken into account to properly describe the effective elastic anisotropy of bone at the organ scale.

Magnesium implants appear as promising technology for load-carrying, bioresorbable bone regeneration tools (Hofstetter 2014 [1], Kraus 2012 [2]). In this context, quantitative exploration of the biomechanics and mechano-biology of the bone-implant compound system is of great interest. As a corresponding contribution, we here present an analytical beam theory representation (Pircher 2021 [3]) of a cylindrical magnesium implant located roughly in the middle of a rat femur and approximately orthogonal to the axis of the femoral shaft, which we map onto CT and SAXS-tomographic images of the pre-clinical animal test setting (Liebi 2021 [4], Grünewald [5] ). This reveals the correspondence between mechanical stress state distributions and micro-texture (collagen fiber orientation) across the bony organ. In more detail, the femur is first represented as a polygon with two straight lines, one associated with the femoral neck, and the other one associated with the femoral shaft. This polygon is the basis of a beam representation, where the shaft line is fixed at the knee, and the femoral head (the end of the neck) is loaded by external forces arising from standing. For the stress analysis, the implant is represented by the geometrical object "solid circular cylinder“, such that the cylinder’s volume, its radial normal component of the inertia tensor, its center of gravity, and its principal axes are identical to the corresponding quantities describing all the voxels showing the implant in the CT image. In the same way, the shaft region in proximity of the implant is represented by (i) a full cylinder representing the entire shaft, and (ii) a hollow cylinder representing the cortical shaft compartment only. Then, classical Bernoulli-Euler beam theory is applied to the bony portions of the implant-bone compound structure, with the beam cross-sections being dictated by the interaction of the roughly orthogonally positioned geometrical objects. As a measure for correspondence between stress states and texture directions, we evaluated the dot products between principcal stress directions and directions of maximum shear stress. As a result, somewhat surprisingly, not the principal stress directions, but the loading directions associated with maximum shear stress appear as the key mechano-biological drivers. Accordingly, the shear tractions of largest magnitude act orthogonal as well as parallel to the main texture direction, i.e. the collagen fiber direction.

[1] Hofstetter et al. (2014) JOM 66.4, pp. 566–572, 10.1007/s11837-014-0875-5.

[2] Kraus et al. (2012) Acta Biomaterialia 8.3, pp. 1230–1238, 10.1016/j.actbio.2011.11.008

[3] Pircher (2021) Thesis Wien, 20.500.12708/79654

[4] Liebi et al (2021) Acta Biomaterialia, 134, 804-817, 10.1016/j.actbio.2021.07.060

[5] Grünewald et al (2016) Biomaterials 76, pp. 250–260, 10.1016/j.biomaterials.2015.10.054.

Bone remodeling is a very complex and fine-tuned process, which is necessary to ensure a healthy bone structure. If this process gets out of balance – e.g., because of hormonal disbalance or the impact of bone metastases – pathologies like osteoporosis can appear. In this contribution we introduce a novel computational approach to investigate this balance by connecting the bone remodeling process with its microenvironment. Our goal is to better understand the well-balanced and complex dynamic of the subprocesses involved in healthy bone remodeling.

We implement a 3D stochastic cellular automaton (SCA), where voxels interact only with their nearest neighbors in a scaffold representing bone tissue. At each time point, each voxel can take one of four different states that stand for the different phases of bone remodeling: formation, quiescent bone, resorption, and environment. To create a compact representation of the frequency-dependent interaction of those voxel states we make use of methods borrowed from evolutionary game theory for the update rule of the cellular automaton [1]. This representation encodes knowledge about the mutual impact the different actors of bone remodeling (osteocytes, osteoclasts and osteoblasts) have on each other. Each parameter in the model has therefore a direct connection to the biological processes.

First, we set up simulations of the model with either only resorption or only formation. This choice reduced the model complexity and allowed us to determine parameter spaces for a self-regulating behavior for each of them. The self-regulating behavior is defined by resorption or formation starting and ending without further parameter tuning. Parameters outside the range of self-regulation will lead to either osteolytic lesions (resorption) or heterotopic ossification (formation). Further analyses supported the approach of a spatial model with a small neighborhood to simulate the local phenomena observed in bone remodeling.

Next, we coupled the two processes of resorption and formation. In the limit of separation of time scales, our model showed that self-regulating resorption followed by self-regulating formation reproduces the physiological bone remodeling behavior. Further analysis will create a more fluid coupling of the two processes while involving more parameters.

The model has the potential to use the role of the microenvironment to evaluate the impact of additional factors, such as drugs or bone metastases. We are planning on using experimental in vivo data from a breast cancer bone metastasis mouse model [2], which includes spatial and temporal dynamic of early osteolytic lesions, to fit additional parameters. Hopefully, these findings will add to the discussion, how pathological behavior might be controlled, if not even reversed.

[1] M. D. Ryser and K.A. Murgas, Bone remodeling as a spatial evolutionary game, Journal of Theoretical Biology, 2017

[2] S. A. E. Young, A.-D. Heller et al., From breast cancer cell homing to the onset of early bone metastasis: dynamic bone (re)modeling as a driver of osteolytic disease, bioRxiv preprint

The mechanical quality of trabecular bone is influenced by its inhomogeneous mineral content and spatial distribution at a microscopic scale. The bone remodelling process, which is the concerted action of osteoclastic bone resorption followed by osteoblastic bone formation, controls bone turnover. During the bone formation process (FP) the deposited organic collagenous matrix (i.e., osteoid) becomes mineralised. The latter process is regulated by the mineralisation kinetics which exhibits two distinct phases: a fast primary mineralisation phase lasting for several days to a few weeks and a secondary mineralisation phase that can last from several months to years. Variations in bone turnover and mineralisation kinetics can be observed in the bone mineral density distribution (BMDD), which can be used to distinguish healthy from pathological bone tissue at the scale of the bone matrix. Here, we propose a discrete statistical spatio-temporal bone remodelling model to study the effects of activation frequency (Ac.f) and mineralisation kinematics on the BMD distribution. In this model individual basic multicellular units (BMUs) are activated discretely on trabecular surfaces which then undergo typical bone remodelling sequence, i.e., resorption, reversal, osteoid formation, and mineralisation, with the latter process following a double exponential law. Our simulation results highlight that trabecular BMDD is strongly regulated by Ac.f and kinetics of secondary mineralization (t₂) in a coupled way. Indeed, Ca wt% increases with lower Ac.f and fast secondary mineralisation, while lower Ca wt% values are obtained for higher Ac.f and slower secondary mineralisation. Accordingly, the dynamic equilibrium can be achieved for different combinations of Ac.f and t₂. For example, a mean Ca wt% of 25, which has been reported in the literature based on qBEI experiments, can be obtained with Ac.f = 4 BMU/year/mm³ and t₂ = 8 years or with Ac.f = 6 BMU/year/mm³ and t₂ = 6 years. This close link between Ac.f and t₂ on BMDD results shows the importance of taking both characteristics into account in order to draw meaningful conclusions about bone quality. Indeed, pathological conditions such as osteoporosis demonstrate a similar pattern. We investigated post-menopausal and senile osteoporosis (type I and type II, respectively) and hypothesised that high Ac.f or very long formation period (FP) would result in type I osteoporosis, whereas underfilling and no filling would result in type II osteoporosis. Our results show that the apparent density and bone mineral fraction were similar for all osteoporosis hypotheses except for the non-filling one. However, when examining BMDD, significant variations were observed, especially in type I simulations. High Ac.f resulted in low Ca wt%, consistent with post-menopausal osteoporosis, while large FP led to high Ca wt%, which is only seen in type II osteoporosis.

The unique and somewhat enigmatic properties of biological apatites have made them an attractive area of research for many decades, with studies aiming to provide a comprehensive understanding of apatite physiochemistry and biological properties. Our work, however, initially began unconventionally at the end of the story, through investigating how the unique properties of biological apatites could be applied to the fields of forensic science, medicine and archaeology and finally ending with understanding the fundamental features of these properties. Our work with Professor John Clement began in the late 1990’s, with a collaboration between Professor Keith Rogers and Dr Sophie Beckett, on species differentiation of heated bone. The results of the study clearly demonstrated that apatite characteristics of unheated and heated bone exhibit significant inter-species variation, quantifiable using X-ray diffraction analysis. The work also highlighted the potential capability of distinguishing human from non-human bone based on apatite physicochemistry. Discussions with John throughout this project centred around ‘apatites not being a fixed piece of chemistry’ and there being ‘a need to understand the fundamental features of apatites to truly appreciate the applications of the work’.

Dr Charlene Greenwood’s PhD project followed, which aimed to provide a new insight into the fundamental mechanisms and processes associated with physicochemical changes to bone during heat treatment. The work, which studied both unheated and heated bone mineral chemistry, considered a controversial in vivo crystallite size control mechanism for biological apatites and mathematically (through Arrenhius and Johnson-Mehl-Avrami equations) described crystallisation kinetics of apatite during heating. Of importance from this work, was the role of carbonate in biological apatite formation and crystal growth, with Dr Emily Arnold’s PhD furthering our understanding of carbonates within the apatite lattice through the application of X-ray pair distribution functions. The work allowed the carbonate within the apatite lattice to be examined on a local scale during heating, with the results suggesting that carbonate does affect intermediate (and therefore likely average) order, but not local order. This insight is significant when considering apatite physiochemistry and disease, and area of research our team is particularly focused on.

Our work in the field of apatites and disease focuses on understanding apatite physicochemisty and disease initiation and progression. Our work on osteoporosis, which was heavily supported by John and the Melbourne Femur Research Collection, identified new fracture risk biomarkers based on tissue features, with differences in apatite crystal chemistry, nanostructure and microstructure identified between fracture and non-fracture groups. John’s legacy has continued with our work on ectopic calcification chemistry and its association with cancer. The team are currently exploring the view that calcification physicochemical characteristics could contain additional or independent source of diagnostic information, and we are applying this hypothesis to both breast and prostate microcalcifications to identify new chemical biomarkers for early diagnostics and prognostics.

Our presentation aims to discuss our fascinating journey with apatites and John, from the applications to their fundamental properties, with the work and John’s legacy continuing, even if we did take ‘the long way round’.

Introduction: Quantifying the effect of a new drug in the presence of natural variability between individuals is a major challenge of drug development. For bone diseases, where mechanics is an important outcome, one approach is to apply a non-intrusive uncertainty quantification method such as Monte Carlo on finite element (FE) models of bones, but with the drawback that a large number of simulations are usually necessary [1]. In this study, a stochastic FE (sFE) solver was developed that uses an intrusive method to predict the variation in bone mechanics given the variability in bone tissue elastic properties.

Method: The sFE solver is developed based on ParOSol, an open-source matrix-free FE solver specialized in bone mechanics. ParOSol solves the system of equations without storing the calculated matrix, resulting in a minimum memory footprint. Developing the sFE solver necessitates significant development in different modules from pre-processing to post-processing using C++ and Python programming languages. The core of the sFE solver is written by C++ codes, and Python is used to facilitate pre-processing and post-processing steps and to carry out the stochastic analysis. Here, the polynomial chaos expansions (PCE) method is implemented to quantify the uncertainty of bone mechanics efficiently, and the stiffness matrix is calculated using Hermite polynomials [2]. The new system of equations is numerically solved using Jacobi’s smoother and preconditioned conjugate gradients method. In this study, multigrid and parallelization features are disabled. Young’s modulus and von Mises stress are considered uncertain input and output variables, respectively. Propagated uncertainty through von Mises stress is quantified using statistical analysis of PCE coefficients. The implementation was used to analyse a column with a square cross-section considered a simplified bone to evaluate the correctness of the implementation process. The column meshes using hexahedral elements with 3 elements in width and 10 elements in height. Here, Young’s modulus is homogeneous and normally distributed with mean value and coefficient of variation equal to 1000 MPa and 10%, respectively. Young’s modulus is estimated using a linear combination of two orders of Hermite polynomials. Poisson’s ratio is deterministic and homogeneous equal to 0.3. The bottom face of the column is fixed in the height direction, and the upper face is under displacement-controlled compression (10% of height). Also, rigid body motions are fixed.

Results: Since Young’s modulus is homogeneous and has identical variability through all the elements, it is expected that all stress components will vary identically (i.e. normally distributed with a 10% coefficient of variation). The propagated uncertainty through the predicted von Mises stress is quantified and found to be less than 0.1% different from the analytical result (with respect to mean and variance).

Conclusion: The present evaluation provides an initial demonstration of the correctness of sFE implementation.

Acknowledgements: This study was funded by the EPSRC New Investigator Award (Grant Reference Numbers: EP/V050346/1).

Reference:

[1] La Mattina AA. et al. Ann Biomed Eng 2023; 51:117-124.

[2] Xiu D, Princeton, NJ, USA:Princeton Univ. Press, (2010).

Our research presents a multidisciplinary bio-optical-mechanical approach for describing the three-dimensional (3D) motion of healthy and pathological sperm cells during a free swim in the human female reproductive system. Our novel method is based on a multidisciplinary mechanical-numerical-modeling and experimental-optical approach that combines prior knowledge of the sperm cell's internal structure, and a description of its flagellar beating patterns.

The interaction of human sperm cells with the female reproductive system significantly impacts fertilization. On average, a few dozen to hundreds of sperm cells reach the fertilization site, where only a single sperm cell fertilizes an oocyte. This may be one of the strictest selection processes created by evolution, but its design and role are not fully understood. The prospective correlation between sperm morphology, motility, and uterine interaction should affect the diagnosis and prognosis of male fertility, especially in pathological cases of abnormal sperm morphology and motility. In addition, such analysis may help obtain a more informed sperm selection for in vitro fertilization (IVF), where a male sperm cell fertilizes a female oocyte in a dish. Because the physiological mechanisms of the natural selection of sperm cells in the female body are bypassed in IVF, it is not possible to predict which individual sperm cell would be the one that is most likely to fertilize the oocyte naturally and result in a healthy child, especially in pathological cases. Moreover, the relationship between morphology, motility, cell–surface interaction, and fertilization potential is still not completely clear. Understanding sperm 3D motility while swimming inside the female reproductive tract may lead to improved sperm cell classification and sperm selection for IVF.

We developed a dynamic 3D mechanical finite-element numerical model of sperm cell swimming inside the human female reproductive system and scored the pathological cells according to their chances of reaching the oocyte site. The sperm-cell models, including the full 3D sperm geometry, were preliminary constructed based on experimentally acquired dynamic 3D refractive index profiles of sperm cells swimming freely in vitro as imaged by high-resolution optical diffraction tomography, and then further developed. We numerically stimulated hundreds of cells swimming in a dish and in the female body, with normal and abnormal morphology. We verified that the number of normal sperm cells that succeeded in reaching the fallopian tube sites is greater than the number of abnormal sperm cells. Hence, besides the cell morphology influence, swimming under the female body’s environmental conditions significantly affects the behavior of sperm cells, especially abnormal sperm cells. Moreover, both cell 3D morphology and motility in the complicated geometry of the uterus are significant factors for filtering pathological sperm cells until they reach the oocyte site. Our method may lead to changes in sperm cell classification and evaluation and to new biophysical analysis tools to fill in the gaps from previous studies. Specifically, understanding the migration of normal and pathological human sperm cells inside the female reproductive system can improve the strategies of human sperm cell selection in IVF and fertility evaluation.

Connective tissue is one of the basic tissue types of the body. One of these connective tissues is fascia which is surrounding and interpenetrating skeletal muscle, joints, organs, nerves, and vascular beds. It serves several important functions including transmission of mechanical forces between muscles, tendons, ligaments and bones, protection of organs forming a barrier, regulation of mechanical stress by adsorbing, storing and releasing kinetic energy.

Concerning the mechanical behavior, fascia, is an incompressible, hyperelastic, non-linear and anisotropic material. Anisotropic behavior is given by the spatial orientation of the collagen fibers, which changes along the sheet to ensure a properly response to mechanical demands. As soft tissue, fascia also exhibits viscoelastic properties.

Today, computational simulation is a very powerful tool for the study and analysis of pathologies, treatments and surgeries. To do it correctly, it is necessary an exhaustive characterization and the use of an adequate constitutive model with the ability to predict the behavior of tissues, such as fascia. The present work aims to investigate in depth the mechanical behavior of deep fascia by means of a multidimensional characterization which includes uniaxial (UT), biaxial (BxT) and planar tension (PT) tests. Besides, a constitutive model is proposed to fit tests and obtain material parameters.

Tests were carried out with fascia from sheep. Different shapes were used for testing: dog bone for UT with a central region of interest of 25x5 mm (5:1 aspect ratio), being 25 mm the distance between clamps, rectangular for PT 35x15 mm (width x large), and cruciform shape for BxT with a central region of interest of 15x15 mm. Two directions were defined based on punch orientation with respect to the fibers: longitudinal which has the sample’s axial direction parallel to fibers, and transverse, which has it perpendicular. Testing protocols were defined based on literature and on our own previous tests. For UT and PT three strain levels (2,5%, 5% y 7,5%) are considered performing five cycles in each of them with a strain rate of 10%/min, for BxT case, a 10% strain level is defined undergoing the sample to five cycles and five ratios of one direction to the other, strain rate of 20%/min. Strain measurements and displacements were obtained using GOM Correlate, a digital image correlation software.

To obtain the material parameters by test fitting, a coupled and uncoupled exponential type energy function (SEF) are proposed that consider two perpendicular fiber directions following Stecco 2009.

Results show stress values like to those found in literature that used animal model for testing. For UT, in longitudinal direction, mean stress value was 3,96 MPa, and in transverse 0,6; PT shows for longitudinal and transverse 0,43 and 0,11 MPa respectively; finally, for BxT in case of 1:1 ratio, mean stresses were 3,16 and 1,2 MPa longitudinal to the transverse.

The fitting results show that an uncoupled exponential type function is be able to fit the UT or the equibiaxial experimental data, however fails to predict using the fitting parameters other experiments. On the contrary, the coupled exponential SEF shows good results during the fitting and prediction processes.

Introduction: Microdamage accumulated by cyclic loading or single overloading events contributes to bone fragility through a reduction in stiffness and strength [1]. QCT-based computational modelling cannot incorporate existing in vivo microdamage due to limited resolution. MR imaging on the other hand, is sensitive to pathophysiological changes to adjacent bone marrow that is ‘hidden’ to clinical CT imaging. In the case of repetitive trauma, signal hyperintensity in fluid sensitive sequences is indicative of a stress response where edema, haemorrhage and hyperaemia are present alongside microdamage [2]. Here, we aim to quantify this signal hyperintensity and use it to derive a pre-existing damage variable that represents the underlying tissue damage prior to overloading. We incorporate this variable into a constitutive model to investigate its influence on material and whole bone stiffness and strength.

Materials and Methods: We use the equine athlete as a model for microdamage induced stress fracture where high-speed exercise induces subchondral microdamage. Distal metacarpals (MC3) from n=5 Thoroughbred racehorses were scanned by clinical QCT (0.3 mm voxel size), calibrated to bone mineral density (BMD) and converted to bone volume fraction (BV/TV). MR images (T1w, STIR) were acquired at 3T (0.3 mm voxel size) and registered to the QCT data. Regions of ‘dense’ or ‘sclerotic’ subchondral bone, where microdamage coalesces [3], were segmented from T1w images. Patch-based similarity [4] was used to generate pseudoCT (pCT) images from STIR images. A relative increase in STIR intensity in the dense subchondral bone returned a lower pCT-derived BMD than QCT. This reflects the presence of underlying porosities such as microdamage and increased vasculature [2]. We derived a pre-existing damage variable (D^pex) which affects stiffness and strength and incorporated it into an isotropic, asymmetric BV/TV-dependent elasto-viscoplastic material model (UMAT, Abaqus v6.16) [5,6,7]. Voxel based FEA was used to compress MC3 condyles in silico before (D^pex=0) and after the inclusion of accumulated ‘hidden’ porosity (D^pex>0) to investigate its influence on whole bone mechanical properties.

Results: pCT BMD in dense subchondral bone was lower in all MC3 bones. Incorporating resulted in a median reduction of material stiffness and strength of 20.3% and 20.9% in tension and compression. Inclusion of D^pex reduced whole bone stiffness and strength, and their reduction correlated with D^pex (R²=0.74, R²=0.89). Previous experimental results support our interpretation [8,9].

Discussion: We propose a methodology for incorporating MRI signal hyperintensity into QCT-based FE models to include pre-existing porosities that cannot be detected by clinical CT. Our results illustrate the value of multimodal imaging to potentially capture existing microdamage in vivo. As we use clinical imaging techniques, our results may also aid research in diseases such as osteoarthritis and bone cancer.

References: [1] Seref-Ferlengez, BoneKEy Rep., 4:1, 2015; [2] Taljanovic, Skeletal Radiol., 37:423, 2008; [3] Muir, Bone., 38:342, 2006; [4] Andreasen Med. Phys., 42:1596, 2015; [5] Schwiedrzik & Zysset. BMMB, 12:201, 2013; [6] Schwiedrzik BMMB, 12:1155, 2013; [7] Mirzaali JMBBM, 49:355, 2015. [8] Schwiedrzik Nature Materials 13:740, 2014; [9] Mirzaali Bone 93:196, 2016

Acknowledgements: HBLB VET/CS/027, Siemens Project IPA 42, Leverhulme Trust RPG-2020-215, EPSRC EP/P005756/1.

Neuromusculoskeletal computational models are useful for biomechanical simulations, allowing the analysis of human movement, development of protocols for rehabilitation, physical exercises and training, as well as the development of assistive devices, such as orthoses and prostheses. One of the several types of movements that can be simulated using neuromusculoskeletal models is the sit-to-stand, which is frequently requested during the performance of various activities of daily life of any individual. Despite the importance of simulating such a movement, whether for understanding its execution, analyzing ergonomic aspects, developing assistive devices or identifying possible causes of injury, there is a lack of neuromusculoskeletal computational models dedicated to this type of simulation and that can represent well the biomechanics of a bipedal individual. The OpenSim, an open-source and freely available environment for modeling, simulation and analyzing of the human movement, provides a neuromusculoskeletal model dedicated to squat-to-stand, but such model is not bipedal, which requires the simplification that the individual simulated performs the movement with complete homogeneity in both legs, which does not always happen, especially in cases of hemiparesis, a condition of muscle weakness on only one side of the body due to neuromotor disease (e.g. stroke). Thus, the objective of this work was to develop a bipedal neuromusculoskeletal model capable of representing the human biomechanics of the lower limbs and dedicated to sit-to-stand or squat-to-stand movement simulations. For the development of such a model, we started from the already validated and well accepted model gait10dof18musc freely available by OpenSim. Such model contains 10 degrees of freedom, being able to move in the sagittal plane, and 18 muscles responsible for the execution of the movements. However, such a model is not dedicated to the sit-to-stand movement, being initially designed only for gait. To obtain the model dedicated to the sit-to-stand movement proposed in this work, Hunt-Crossley contact force elements were added between the foot and the ground and kinematic and dynamic constraints were elaborated. In order to test the model obtained for the proposed movement, a predictive simulation was conducted, with the model starting in a sitting position, performing the sit-to-stand and standing stably. Joint movements (hips, knees and ankles) and muscle activations were analyzed. It was verified that the proposed model can perform the sit-to-stand movement well, in a stable way, generating coherent kinematic and dynamic results. The main limitation of the model is related to the foot-to-ground contact, which does not always have enough static for a stable movement, a point that will be improved in later works. In the future, it is intended to improve the model, adding contact force between the hamstrings and a platform capable of representing a seat, improving the foot-to-ground contact ratio and testing it under conditions of hemiparesis.

Breast-cancer and Prostate-cancer are among the most prevalent cancers in women and men, respectively. The World-Health-Organization estimates that about a million deaths occur due to breast and prostate-cancer worldwide due to these cancers each year. Although mostly curable when detected early at the primary site, both of these cancers are incurable when the cancer metastasizes to a distant location in the body, which for these two cancers is eventually bone. There is a scarcity of available human samples and animal models fail due to death preceding bone metastasis; hence a huge unmet need for development of robust in vitro models of bone metastasis. We have developed a novel testbed using a bone mimetic nanoclay scaffold to regenerate human bone followed by sequential seeding prostate and breast-cancer cells obtained from commercial and patient derived cell lines to generate tumors. Extensive analysis of the tumors using gene and protein expression assays and imaging confirm that the testbeds can replicate tumors during mesenchymal-to-epithelial-transition (MET). While many biomarkers exist for evaluation of cancer at the primary site, there are no known bone metastasis markers. Mechanical properties of cells and tissues can capture the complex biological phenomena of adhesion and colonization at the bone site. We also observe via imaging, significant changes to the cytoskeleton quantity and organization within the cells as cancer progresses at bone metastasis. We measured mechanical response of cells using direct-nanoindentation experiments on cancer cells from tumors generated on the testbeds to describe the evolution of cellular properties during cancer progression at the bone metastasis sites. We conducted the nanoindentation experiments under static and dynamic modes to evaluate elastic moduli, hardness, and viscoelastic properties of cancer cells over time. The force-displacement response, elastic moduli, hardness, plastic deformation, viscoelastic properties were captured, with confocal imaging of the cytoskeleton and gene and protein expressions at the same time points. Our results indicate significant reduction in elastic modulus and increased fluid-like behavior of bone metastasized breast-cancer cells (MCF-7) caused by depolymerization and reorganization of F-actin. On the other hand, bone metastasized triple negative cells (MDA-MB-231) showed insignificant changes in elastic modulus and F-actin reorganization over time. We also measured changes to nanomechanical properties of MDA PCa2b(PCa) prostate-cancer cells during the MET and cancer bone metastasis progression over time. The stiffness of PCa cells decreases with metastasis; however, the mechanical plasticity increases during the same time, suggesting that PCa cells become softer on undergoing MET and softer with metastasis progression. In all cases, the imaging and gene, protein expression studies, and computational modeling point towards the depolymerization of actin and reorganization of the cytoskeleton as key factors in the evolution of cell mechanics. In addition, we also subjected the cancer cells to physiologically relevant fluid induced shear stresses that are experimentally enabled using specially designed bioreactors as well as computationally modelled. The role of fluid-derived stresses on migratory characteristics and apoptosis potential of cells is also evaluated. These studies present the use of mechanics-based characterizations as a potential new tool for evaluation of metastasis progression.