Multiscale Computer Modeling in Biomechanics and Biomedical Engineering

Fatigue damage in bone in the form of microcracks results from the repetitive loading of daily activities. It is well known that the resistance of bone at the organ level to fatigue fracture is a function of its resistance to the initiation and propagation of local microcracks at a mesoscopic scale which can lead to macrocrack growth at the organ level. Multiscale investigation of the relationship between the effect of the fatigue microcrack growth at microscopic scales and the whole bone behaviour is a subject of great interest in the research field of the biomechanics of human bone. Several finite element models (FE) have been developed in recent years in order to provide better insight and description regarding bone fatigue microcrack growth. Despite the progress in this field, there is still a lack of models integrating multiscale approaches to assess the accumulation of apparent fatigue microcracks in relation with trabecular architecture into practical FE simulations. In this chapter, a trabecular bone multiscale model based on FE simulation and neural network (NN) computation is presented to simulate the accumulation of trabecular bone crack density and crack length at a given trabecular bone site during cyclic loading. The FE calculation is performed at macroscopic level and a trained NN incorporated into a FE code is employed as a numerical device to perform the local mesoscopic computation (the behaviour law needed to compute the outputs at mesoscale is substituted by the trained NN). The input data for the NN are some trabecular morphological and material factors, the applied stress and cycle frequency. The output data are the average crack density and length computed at a given trabecular bone site.

The ability of bone tissue to adapt itself to its physical environment is the research focus of several teams all over the world. If the physical stimuli playing a role in bone remodelling are often identified, how they act and are converted into a cellular response is still an open question. The aim of this paper is, in a first part, to propose an overview on the physical factors participating in the bone remodelling process. In a second part, we present some recent developments concerning the implications of hydro–electro-chemical couplings that could modify the bone adaptation process. Since the phenomena that are involved in this mechanism are related both to the mechanical solicitations of the tissue and the physical phenomena in the vicinity of bone cells, different scales, from the organ to the cell, should be considered to go deeper in its understanding. That is why a multiscale strategy based on periodic homogenization has been carried out to propagate the multiphysics description at the cellular scale toward the macroscopic scale of the tissue. This multi-level approach is so adapted to connect macroscopic physical information to microscopic phenomena, et vice versa. Thus, using convenient simulations, we have brought a new light on classical interrogations dealing with bone adaptation. These five questions are: i. Can the sole hydro-mechanical coupling describe the poro-mechanical behaviour of bone or should we consider a modified Biot model including electro-chemical effects?; ii. Similarly, is the classical Darcy law sufficient to describe the bone interstitial fluid flow?; iii. What is the nature of the stress-induced electric potentials that can be measured in vivo?; iv. What are the consequences of the electro-chemical couplings on the shear sensitivity of the osteocytes?; v. What are the consequences of the microscopic physico-chemical properties of the bone microstructure on the mass transport within the lacuno-canalicular system? Finally, from these simple model-driven observations, we propose a new perspective to alter the current bone adaptation paradigm.

Mechanics of collagen bio-structures at different scales (nano, micro, and macro) is addressed, aiming to describe multiscale mechanisms affecting the constitutive response of soft collagen-rich tissues. Single-scale elastic models of collagen molecules, fibrils, and crimped fibers are presented and integrated by means of consistent inter-scale relationships and homogenization arguments. In this way, a unique modeling framework based on a structural multiscale approach is obtained, which allows to analyze the macroscale mechanical behavior of soft collagenous tissues. It accounts for the dominant mechanisms at lower scales without introducing phenomenological descriptions. Comparisons between numerical results obtained via present model and the available experimental data in the case of tendons and aortic walls prove present multiscale approach to be effective in capturing the deep link between histology and mechanics, opening to the possibility of developing patient-specific diagnostic and clinical tools.

Ligaments and tendons are composed primarily of water and fibrillar type I collagen, which is hierarchically organized into complex structures that span multiple physical scales. Forces at the macroscopic joint level are transmitted via interactions at the mesoscale, microscale and nanoscale. Tissue repair and growth is mediated by fibroblasts and tenocytes, which are subjected to a unique microscale mechanical environment. The burgeoning field of multiscale modeling holds promise in filling the gaps in our understanding of structure–function relationships and mechanotransduction in these tissues, and these questions are difficult or impossible to address using experimental techniques alone. This article reviews the state of the art in multiscale modeling of ligaments and tendons, while providing sufficient background on the structure and function of these tissues to allow a reader who is new to the area to proceed without substantial outside reading. The multiscale structure of ligaments and tendons is described in detail. The available data on material characterization at different physical scales is reviewed as well. The final section of the chapter summarizes the efforts at developing and validating multiscale models that are relevant to ligament and tendon mechanics, and identifies future directions for research. Multiscale modeling of tendon and ligament holds considerable promise in advancing our understanding regarding the complex mechanisms of multiscale force transfer within these tissues.

In this chapter we will describe the latest developments in the area of lymphatic modelling. The lymphatic system is one of the key elements of the human circulation, serving the dual functions of draining interstitial fluid and returning this to the general blood circulation, together with processing this lymph fluid which is a key component of the body’s immune response system. Compared to the main cardiovascular system however, remarkably little modelling has been attempted. At the same time, the distribution of pumping activity (contractile lymphangions coupled with simple valves) throughout the system, passive primary lymphatics and complex lymph nodes combining to form an active network, makes the system a prime candidate for multiscale modelling.

The motility of the intestines is partly governed by a bioelectrical activity termed intestinal slow wave activity; however, the dynamics of the electromechanical relationship have remained poorly defined. With the recent advances in continuum-based multi-scale modeling techniques, we present a modeling framework to investigate the electromechanical coupling in a segment of small intestine. The overall modeling framework included three parts: (i) an anatomical model describing the geometry and makeup of the smooth muscle fibers; (ii) an electrical model describing the slow wave propagation; and (iii) a mechanical model describing the active and passive tension laws during contraction. The resultant intraluminal pressure was approximated using Lamé’s thick-walled cylinder equation. This modeling framework demonstrates the potential to be used in investigating the effects of intestinal slow wave dysrhythmias on the motility of the small intestine, and may be extended in the future to incorporate additional regulatory factors and pathways.

Blood vessels exhibit a remarkable ability to adapt in response to sustained alterations in hemodynamic loads and diverse disease processes. Although such adaptations typically manifest at the tissue level, underlying mechanisms exist at cellular and molecular levels. Dramatic technological advances in recent years, including sophisticated theoretical and computational modeling, have enabled significantly increased understanding at tissue, cellular, and molecular levels, yet there has been little attempt to integrate the associated models across these length and time scales. In this chapter, we suggest a new paradigm for identifying strengths and weaknesses of models at different scales and for establishing congruent models that more completely predict vascular adaptations. Specifically, we show the importance of linking intracellular with cellular models and cellular models with tissue level models. In this way, we propose a new approach for incorporating events across these three levels, thus providing a means to predict phenomena that can only emerge from a system of integrated interactions.

The human body, and hence the vascular system, is by its very nature a dynamic multiscale hierarchial system. This multiscale nature encompasses different length scales, from molecular and cellular levels to the tissue and organ level, as well as different physical phenomena, such as mechanical, biological and chemical processes. In arteries, vascular cells alter their growth, phenotype and extracellular matrix production in response to macro mechanical changes. These cell level events can in turn accumulate and emerge at the tissue level as pathological conditions such as atherosclerosis and intimal hyperplasia. These cardiovascular diseases evolve through adaptation of cells and tissues over days to months also demonstrating the multiscale nature of vascular diseases with respect to time. The challenge in vascular multiscale modelling is to create a framework which can incorporate the key mechanical, biological and chemical characteristics of this complex system at these various space and time scales to successfully capture the long-term behaviour of the system. Such a framework can then be used to gain additional insights with regards to pathological conditions within the vascular system and to improve the design of medical devices used to treat such pathologies. In the following chapter, a review will be presented of some relevant studies reported in literature which have used multiscale modelling approaches to elucidate the growth and remodelling mechanisms underlying vascular diseases, such as atherosclerosis, in-stent restenosis and intimal hyperplasia.

The pulmonary circulation is a unique low resistance system that carries almost the entire cardiac output, and is responsible for the essential role of providing oxygenated blood to the body. As the pulmonary circulation differs from the systemic circulation in its development, structure, and function, it is often most appropriate to study the mechanisms that contribute toward pulmonary vascular disease separately from those of systemic vascular disease at the genetic, cellular, tissue and organ level. Here we review the development of multi-scale, anatomically based models of the pulmonary circulation. These models aim to describe the interaction of structural and functional aspects of the pulmonary circulation that are the most important in determining the effective uptake of oxygen to the blood. We describe how these models have been used to understand normal lung physiology and to explain outcomes in pulmonary disease. Finally, we consider the future of multi-scale modeling in the pulmonary circulation and discuss what can be learned from well-developed multi-scale models of the pulmonary airspaces that interact closely with the lung’s circulatory system.

A pressure ulcer is a form of tissue degeneration as a result of sustained mechanical loading. In the last 3 decades a lot of research has been done to understand the aetiology of pressure ulcers. It has become clear that the initial signs of tissue damage are found at the cell scale. That is where the damage process starts that eventually leads to severe wounds. In order to define damage thresholds or to understand what cells “feel” it is necessary to have information on the mechanical status of cells at a scale in the order of micrometres. How the external loading, that is gravitational body forces and reaction forces at supporting surfaces on patients in a bed or a wheel chair, is transferred to a local mechanical state within tissues depends on tissue morphology, mechanical properties and other boundary conditions and requires an analysis at the scale of the order of centimetres to a meter. This cannot be done in one single analysis covering the entire range of scales. This chapter summarizes some work that our group has done in the last 10 years on multi-scale modelling of soft tissues that was aimed at understanding some of the phenomena that play a role in pressure ulcer development. The work has shown the potential of multi-scale modelling to gain insight in the very complex interactions at cell level. It was shown that the heterogeneity in the microstructure has a profound impact on the way cells deform as well as the mechanical property changes of the cell after they become damaged.

The interacting effects operating on subcellular (gene regulatory processes), cellular (interactions between neighbouring cells) and population (signalling molecule transport) scales are exemplified and explored through simple multiscale models. Specific attention is focused on how the upregulation (or downregulation) of small numbers of discrete cells can influence the behaviour of the population as a whole, by investigating toy models for positive autoregulation and by the simulation of a much more detailed model for quorum sensing within a Gram-positive population of bacteria. The implications for delays associated with gene expression are also investigated in a spatio-temporal context through the analysis of blow-up behaviour, as a mathematical symptom of upregulation through positive feedback, in some model reaction-diffusion delay equations.

This chapter is meant as an overview of our already published work that we carry out on modeling wound healing on the cellular, colony and tissue scale, though we detail the description of some stochastic principles that appear in our models. The relation between the scales is described in terms of the underlying biological and mathematical concepts. We also present the implications and applicability of the mathematical models studied.

Tissue necrosis and calcification significantly affect cancer progression and clinical treatment decisions. Necrosis and calcification are inherently multiscale processes, operating at molecular to tissue scales with time scales ranging from hours to months. This chapter details key insights we have gained through mechanistic continuum and discrete multiscale models, including the first modeling of necrotic cell swelling, lysis, and calcification. Among our key findings: necrotic volume loss contributes to steady tumor sizes but can destabilize tumor morphology; steady necrotic fractions can emerge even during unstable growth; necrotic volume loss is responsible for linear ductal carcinoma in situ (DCIS) growth; fast necrotic cell swelling creates mechanical tears at the perinecrotic boundary; multiscale interactions give rise to an age-structured, stratified necrotic core; and mechanistic, patient-calibrated DCIS modeling allows us to assess our working biological assumptions and better interpret pathology and mammography. We finish by outlining our integrative computational oncology approach to developing computational tools that we hope will one day assist clinicians and patients in their treatment decisions.

Multiscale modeling has now been well-accepted as a powerful tool to quantitatively represent, simulate, understand, and predict cancer progression and development across multiple biological scales. In this chapter, we focus on a specific type of multiscale cancer models where molecular signaling profiles are explicitly linked to the determination of cellular phenotypic changes. These models are particularly suitable for exploring the relationship between signaling dynamics within each individual cancer cell and the emergent cancer behavior on the multicellular level. We also discuss current challenges and future directions of this molecular signaling-incorporated multiscale cancer modeling approach.