Computational Modeling in Tissue Engineering

This chapter aims to provide an introduction to how engineering tools in general and computational models in particular can contribute to advancing the tissue engineering (TE) field. After a description of the current state of the art of TE, the developmental engineering paradigm is briefly discussed. Subsequently an overview is provided of different model categories that focus on different aspects of TE. These categories consists of the models that focus on either the TE product, the TE process or the in vivo results obtained after implantation. Generally, in all these models the aim is firstly to understand the biological process at hand and secondly to design strategies in silico to enhance the desired in vitro or in vivo behaviour. Finally, the need for quantification and parameter determination is discussed along with the computational tools and models that can be used to design the thereto required experiments in the most intelligent way.

One aspect of tissue engineering represents modelling of the extracellular matrix of connective tissue as the fiber network arrangement of the matrix determines its tensile strength. In order to define the correct position of the e.g. collagen in a structure, an optimized tertiary structure must be characterized. Existing approaches of protein models consider random packing of rigid spheres. We propose an alternative strategy to model protein structure by focusing on the folding. Our model considers (a) segments of amino-acid peptides or beads, (b) hydrogen bond distances, and (c) the distance geometry as functional components rather than minimizing distances between the centers of atoms. We reduced the molecular volume by using concepts from low dimensional topology, such as braids and surfaces, via differential geometry. A braid group maintains the continuity of a sequence while the spatial minimization is performed, and guarantees the continuity during the process. We have applied this approach to different examples of known protein sequences using ab initio protocols of ProteoRubix Systems™. Sequence files of three different proteins types were tested and modeled by ProteoRubix™ and compared to models derived by other methods. ProteoRubix™ created near-identical models with minimal computational load. This model can be expanded to large, multi-molecular network structures.

Engineered tissues are often designed to serve a mechanical role. The design and evaluation of such tissues requires a mechanical model. An important component of such models is often viscoelasticity, or the dependence of mechanical response on loading rate and loading history. In a great number of biological and bio-artificial tissues the passive tissue force (or stress) relates to changes in tissue length (or strain) in a nonlinear viscoelastic manner. Choosing and fitting nonlinear viscoelastic models to data for a specific tissue can be a computational challenge. This chapter describes the range of such models, criteria for selecting amongst them, and computational and experimental techniques needed to fit these to uniaxial data. The chapter begins with Fung’s quasi-linear viscoelastic (QLV) model, which is nearly a standard first model to try for nonlinear viscoelastic tissues. The chapter then describes the two major limitations of the Fung QLV model, and presents approaches for overcoming these. The first limitation is accuracy: the Fung QLV model imposes a severe set of restrictions on constitutive behavior, and a generalized form of the Fung QLV model is needed in many cases. The second limitation is that the Fung QLV model is cumbersome computationally, especially for calibration experiments. The Adaptive QLV model is far simpler to calibrate and provides greater flexibility than the Fung QLV model. The Adaptive QLV model extends linear viscoelastic models to incorporate nonlinearity using a principle different from that of the Fung QLV model: it adapts nonlinearity according to the instantaneous level of strain. The Adaptive QLV model can be used in simple or generalized form. The chapter concludes with a series of test protocols for calibrating QLV models along with the associated calibration procedures, using the nonlinear viscoelastic behavior of reconstituted collagen tissue as an example. The Adaptive QLV model is not only simpler to calibrate but also more accurate in predicting the mechanical response of the reconstituted collagen tissue.

Effective recapitulation of extracellular matrix properties into a Tissue Engineering strategy is strongly involved with the need for a proper transport environment. Consumption and production of soluble medium components gives rise to gradients which influence cell behavior in various ways. Understanding how transport related phenomena can shape these gradients is targeted in this chapter by the combined use of experiments and mathematical modeling. An overview of different models is given that describe solute transport and its relation to specific cell behavior. From the simulation results important information can be extracted which help to unravel mechanisms that drive solute transport. Finally we describe the genuine efforts that have been taken to translate this information into real tissue engineering setups (e.g., optimization of culture conditions and controlled-release of growth factors).

Tissue engineering uses porous biomaterial scaffolds to support the complex tissue healing process to fulfill two main functions: (1) to support mechanical loading and (2) to allow mass transport. Computational methods have been extensively applied to characterize scaffold morphology and to simulate different biological processes of tissue engineering. In addition, phenomena such a cell seeding, cell migration, cell proliferation, cell differentiation, vascularisation, oxygen consumption, mass transport or scaffold degradation can be simulated using computational methods. A review of the different methods used to model scaffolds in tissue engineering is described in this chapter.

In this chapter, we outline the use of computational modeling and novel experimental methods to develop tissue engineering scaffolds as delivery devices for exogenous and endogenous cues, including biochemical and mechanical signals, to drive the fate of mesenchymal stem cells (MSCs) seeded within. Tissue regeneration in mature organisms recapitulates de novo tissue generation during organismal development. This gave us the impetus to develop tissue engineering scaffolds that deliver mechanical and chemical cues intrinsic to the environment of cells during mesenchymal condensation, which marks the initiation of skeletogenesis during development. Cell seeding density and mode of achieving density (protocol) have been shown to effect dilatational (volume changing) stresses on stem cells and deviatoric (shape changing) stresses on their nuclei. Shear flow provides a practical means to deliver mechanical forces within scaffolds, resulting in both dilatational and deviatoric stresses on cell surfaces. Both spatiotemporal mechanical cue delivery and mechanically modulated biochemical gradients can be further honed through optimization of scaffold geometry and mechanical properties. We use computational fluid dynamics (CFD) coupled with finite element analysis (FEA) modeling to predict flow regimes within the scaffolds and optimize flow rates to simulate seeded cells. This chapter outlines to major advantages of using computational modeling to design and optimize tissue engineering scaffold geometry, material behavior, and tissue ingrowth over time.

Cryopreservation of biological material is a crucial step of tissue engineering, but biological material can be damaged by the cryopreservation process itself. Depending on some bio-physical properties that change from cell to cell lineages, an optimum cryopreservation protocol needs to be identified for any cell type to maximise post-thaw cell viability. Since a prohibitively large set of operating conditions has to be determined to avoid the principal origins of cell damage (i.e., ice formation and solution injuries), mathematical modelling represents a valuable alternative to experimental optimisation. The theoretical analysis traditionally adopted for the cryopreservation of a cell suspension addresses only a single, average cell size and ascribes the experimental evidence of different ice formation temperatures to statistical variations. In this chapter our efforts to develop a novel mathematical model based on the population balance approach that comprehensively takes into account the size distribution of a cell population are reviewed. According to this novel approach, a sound explanation for the experimental evidence of different ice formation temperatures may now be given by adopting a fully deterministic criterion based on the size distribution of the cell population. In this regard, the proposed model represents a clear novelty for the cryopreservation field and provides an original perspective to interpret system behaviour as experimentally measured so far. First our efforts to successfully validate the proposed model by comparison with suitable experimental data taken from the literature are reported. Then, in absence of suitable experimental data, the model is used to theoretically investigate system behaviour at various operating conditions. This is done both in absence or presence of a cryo-protectant agent, as well as when the extra-cellular ice is assumed to form under thermodynamic equilibrium or its dynamics is taken into account consistently by means of an additional population balance. More specifically, the effect of the cell size distribution on system behaviour when varying cooling rate and cryo-protectant content within practicable values for a standard cryopreservation protocol is investigated. It is demonstrated that, cell survival due to intra-cellular ice formation depends on the initial cell size distribution and its osmotic parameters. At practicable operating conditions in terms of cooling rate and cryo-protectant concentration, intra-cellular ice formation may be lethal for the fraction of larger size classes of the cell population whilst it may not reach a dangerous level for the intermediate size class cells and it will not even take place for the smaller ones.

Mesenchymal Stem Cell (MSC)-based therapies have been suggested as a particular promising strategy in tissue regeneration. These cells can easily be obtained from the patient and are able to produce a large number of progeny that can be induced to form connective tissue. However, rapid amplification of the isolated cells is required for their therapeutic application. While already the isolated populations are heterogeneous regarding various functional and molecular aspects, this heterogeneity further evolves during amplification. Understanding the origin and development of MSC heterogeneity will help to improve MSC culture conditions and thus facilitate their clinical use. We here review recent results on MSC heterogeneity and introduce a mathematical framework that approaches MSC heterogeneity on the single cell level. This approach bases on the concept of noise-driven MSC differentiation and allows describing MSC heterogeneity with respect to their differentiation state and age. It is capable of describing the impact of MSC heterogeneity on in vitro expansion and differentiation. We present new results on the formation of an age structure in MSC populations in vitro and the age-dependent differentiation structure of MSC populations. Moreover, we discuss open questions regarding MSC adaptation to changing environments and the cell intrinsic control of state fluctuations.

In clinical tissue engineering, both safety and effectiveness are definite requirements that should be satisfied. Conventional cell biology techniques are facing limitations in the quality assurance step of cell production for clinical therapy. Image-based cell quality assessment offers a great potential, because it is the only way to non-destructively and repeatedly assess cellular phenotypes and irregularities. To effectively assess cell quality using the multiple parameters derived from time course cell imaging, machine learning models, which have been effectively used to connect biological phenomena with biological measurements in the field of bioinformatics, are promising approaches for achieving high accuracy. Here, we present the recent results of our successful cell quality modeling and discuss its possibility and considerations on further application in clinical cell therapy.

By providing replacements for damaged tissues and organs, in vitro tissue engineering has the potential to become a viable alternative to donor-provided organ transplant, which is increasingly hampered by a shortage of available tissue. The complexity of the myriad biophysical and biochemical processes that together regulate tissue growth renders almost impossible understanding by experimental investigation alone. Mathematical modelling applied to tissue engineering represents a powerful tool with which to investigate how the different underlying processes interact to produce functional tissues for implantation. The aim of this review is to demonstrate how a combination of mathematical modelling, analysis and in silico computation, undertaken in collaboration with experimental studies, may lead to significant advances in our understanding of the fundamental processes that regulate biological tissue growth and the optimal design of in vitro methods for generating replacement tissues that are fully functional. With this in mind, we review the state-of-the-art in theoretical research in the field of in vitro tissue engineering, concentrating on continuum modelling of cell culture in bioreactor systems and with particular emphasis on the generation of new tissues from cells seeded on porous scaffolds. We highlight the advantages and limitations of different mathematical modelling approaches that can be used to study aspects of cell population growth. We also discuss future mathematical and computational challenges and interesting open questions.

A common technique for in vitro cartilage regeneration is to seed a porous matrix with cartilage cells and to culture the construct in static conditions or under medium perfusion in a bioreactor. An essential step toward the development of functional cartilage is to understand and control the tissue growth phenomenon in such systems. The growth process depends on various space- and time-varying biophysical variables of the environment surrounding the cartilage cells, primarily mass transport and mechanical variables, all involved in the cell biological response. Moreover, the growth process is inherently multiscale, since cell size (10 μm), scaffold pore size (100 μm), and cellular construct size (10 mm) pertain to three separate spatial scales. To obtain a quantitative understanding of cartilage growth in this complex multiphysics and multiscale system, advanced mathematical models and efficient scientific computing techniques have been developed. In this chapter, we discuss the existing knowledge in this field and we present the most recent advancements for the numerical simulation of cartilage tissue engineering.

Tissue engineered vascular grafts cultured in vitro are often done so under static conditions, which forces a diffusion-only mass transport regime for nutrient delivery and metabolite removal. Some bioreactor culture methods employ mechanical stimulation to improve material strength and stiffness; however, even with mechanical stimulation, engineered tissues are likely to operate in a diffusional transport regime for nutrient delivery and metabolite removal. In this study, we present an analysis of dissolved oxygen (DO) transport limitations that can arise in statically cultured vascular grafts and highlight bioreactor designs that improve transport, particularly by perfusion of medium through the interstitial space by transmural flow. A computational analysis is provided in conjunction with empirical data to support the models. Our goal was to investigate designs that would eliminate nutrient gradients that are evident using static culture methods in order to develop more uniform engineered vascular tissues, which could potentially improve mechanical strength and stiffness.

We present a fluid-solid-growth (FSG) computational framework to simulate the mechanobiology of the arterial wall. The model utilises a realistic constitutive model that accounts for the structural arrangement of collagen fibres in the medial and adventitial layers, the natural reference configurations in which the collagen fibres are recruited to load bearing and the (normalised) mass-density of the elastinous and collagenous constituents. Growth and remodelling (G&R) of constituents is explicitly linked to mechanical stimuli: computational fluid dynamic analysis produces snapshots of the frictional forces acting on the endothelial cells; a quasi-static structural analysis is employed to quantify the cyclic deformation of the vascular cells. We apply the computational framework to simulate the evolution of a specific vascular pathology: abdominal aortic aneurysm (AAA). Two illustrative models of AAA evolution are presented. Firstly, the degradation of elastin (that is observed to accompany AAA evolution) is prescribed, and secondly, it is linked to low levels of wall shear stress (WSS). In the first example, we predict the development of tortuosity that accompanies AAA enlargement, whilst in the latter, we illustrate that linking elastin degradation to low WSS leads to enlarging fusiform AAAs. We conclude that this computational framework provides the basis for further investigating and elucidating the aetiology of AAA and other vascular diseases. Moreover, it has immediate application to tissue engineering, e.g., aiding the design and optimisation of tissue engineered vascular constructs.

This chapter gives a short overview of computational models in cartilage and bone tissue engineering with a focus on how mechanical cues can regulate tissue regeneration on multiple levels, from cell phenotype to tissue architecture. The chapter begins with a brief review of single cell models with a focus on cell-substrate interactions and cytoskeletal remodelling. After summarising a number of current theories for mechanoregulated tissue differentiation, we explain how such hypotheses can either be corroborated or rejected by attempting to simulate in vivo regenerative events. We then outline a recently introduced model for MSC differentiation based on substrate stiffness and oxygen tension as well as how tissue phenotype and organisation can be explored simultaneously within a computational model. The application of computational models to aid in the design of scaffolds for bone and cartilage repair is demonstrated. We also outline how such models can be used in the design and analysis of bioreactors, demonstrating how changes in tissue structure in response to mechanical loading during bioreactor culture can potentially impact the mechanical properties of the final engineered constructs. The chapter closes with a short overview of multiscale models with relevance to tissue engineering.

Angiogenesis is essential for complex biological phenomena such as tissue engineering and bone repair. The ability to heal in these processes strongly depends on the ability of new blood vessels to grow. Capillary growth and its impact on human health has been focus of intense research from an in vivo, in vitro and in silico perspective. In fact, over the last decade many mathematical models have been proposed to understand and simulate the vascular network. This review addresses the role of the vascular network in well defined and controlled processes such as wound healing or distraction osteogenesis and covers the connection between vascularization and bone, starting with the biology of vascular ingrowth, moving through its impact on tissue engineering and bone regeneration, and ending with repair. Furthermore, we also describe the most recent in-silico models proposed to simulate the vascular network within bone constructs. Finally, discrete as well as continuum approaches are analyzed from a computational perspective and applied to three distinct phenomena: wound healing, distraction osteogenesis and individual cell migration in 3D.

One of the most promising recent achievements in the field of regenerative medicine was the first successful transplantation of a tissue-engineered trachea (Macchiarini et al. The Lancet 372(9655), 2023–2030). This land-mark operation has paved the way for developing a host of successful stem cell-based therapies for treating disease, and the exciting possibility of the tissue engineering of whole organs. It has also provided the opportunity for new directions in mathematical and computational modelling for tissue engineering. By way of describing an approach to modelling the regeneration of a tissue-engineered trachea seeded with cells in situ this chapter will highlight some of the opportunities and challenges involved in applying mathematical models to these new therapies.