Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

A Modeling Framework for Computational Physiology

  • Peter HunterEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_29-1

Definitions

Physics based – the use of physical principles such as the laws of conservation of mass and conservation of energy.

Computational physiology – the use of computer models containing equations that represent physiological processes and the solution of these equations with numerical methods designed for digital computers (such as the finite element methods discussed in this entry).

Multiscale modeling – the use of mathematical models that incorporate physical processes operating at more than one spatial scale (e.g., at the level of cells as well as a whole organ).

Standards-based computational frameworks – this refers to the use of community-agreed standards for the encoding of models to help ensure that they are reproducible and reusable.

Introduction

Anatomy and physiology are the disciplines that underpin the practice of medicine. Discoveries from molecular biology, including the sequencing of the human genome in 2000 (https://en.wikipedia.org/wiki/Human_genome), are having an increasing impact, particularly for so-called precision medicine (https://en.wikipedia.org/wiki/Precision_medicine) where treatment strategies take into account SNP (single nucleotide polymorphism) profiles from an individual patient, but these are still minor contributors to current medical practice. Medical imaging technologies, on the other hand, have had a huge impact and can be coupled with continuum modeling approaches to enhance scientific understanding and, in a clinical context, to extract useful diagnostic information or to aid surgical planning. Finite element methods provide the means to solve the equations of continuum mechanics on complex anatomical domains, but to be useful for clinical diagnostics, the constitutive parameters of these continuum models must be linked through multiscale modeling to the molecular scale where diseases and drugs operate. In this brief summary of computational physiology, some of the particular features of physiological systems and biological materials are first highlighted, and then the standards-based computational frameworks needed to deal with these features in the context of multiscale modeling are discussed. A particular focus of the entry is the use of a consistent mathematical framework based on the exchange of energy between different biophysical systems. This combines field concepts and Galerkin finite element methods using high-order basis functions for solving the partial differential equations of continuum mechanics at the three-dimensional spatial scale, with bond graph methods at the zero-dimensional scale of cellular processes. Multiscale models of the heart are used to illustrate these principles. The focus of the entry is on the principles and approaches used for multiscale computational physiology. Details of the continuum equations and finite element methods are not presented as these are well documented elsewhere in the encyclopedia and in the references given in the entry.

Progress in physiology (and biomedical science generally) depends on three fundamental pillars: firstly, the development of new instrumentation (including medical devices) to observe and measure new phenomena; secondly, the experimental work that with these instruments yields the material properties of cells, tissue, and organs (and is used to validate bioengineering models); and, thirdly, the development of mathematical models based on biophysical principles to interpret the complex data provided by these measurements and to capture the discovered mechanisms in a quantitative form that respects the underlying laws of physics. Although the two are complementary, engineering is different from science in one important aspect. Whereas scientific enquiry typically uses reductive methods to tease apart complex mechanisms, engineering uses integrative approaches to put the pieces back together again – it builds systems approaches in order to understand complex processes through predictive models (often supplemented these days with machine learning approaches to uncover patterns in the data). The sheer complexity of biological systems, however, requires a highly systematic approach to organizing models and data, and this, in turn, requires the development and implementation of standards, databases, and open-source software built on those standards.

Characteristics of Physiological Systems and Biological Materials

There are several notable characteristics of mammalian physiological systems that are highly relevant to continuum modeling. These are:
  1. 1.

    All physiological mechanisms depend on multiple coupled physical processes – mechanics (solid and fluid), electrical processes (sometimes electromagnetic), heat flow, and biochemical fluxes and reactions. The units and modeling framework below needed to capture energy flow between these physical processes are discussed below.

     
  2. 2.

    All biological materials are anisotropic and usually orthotropic (different mechanical, electrical, and biochemical properties in three orthogonal directions). This is especially true for soft tissues but is generally also the case for hard tissue such as bone and cartilage.

     
  3. 3.

    All biological tissues are inhomogeneous in their material properties and in the local orientation of the material axes used to express the anisotropy.

     
  4. 4.

    Soft biological tissues are almost incompressible, since they are composed largely of water. Sometimes an assumption of incompressibility is justified, but sometimes the movement of fluid within the soft tissue must also be modeled.

     
  5. 5.

    Soft biological materials are highly nonlinear in their mechanical properties and usually exhibit strain-softening behavior – i.e., become stiffer as the strain increases (i.e., J-shaped stress-strain curves). An assumption of elastic behavior is often justified, but sometimes the viscous properties of tissues must also be taken into account.

     

Another feature of biological systems is that attempts to understand them usually need to take account of their physical properties at multiple spatial and temporal scales – and hence the term “multiscale modeling.” One illustration of this is in use of the equations of finite elasticity to understand ventricular mechanics. The constitutive equations that characterize the nonlinear, orthotropic properties of the myocardial tissue can only be obtained from experiments on very limited tissue samples (Nielsen et al. 1991), and often a more useful approach is to relate the macroscopic properties to observations and models of the tissue microstructure (Hooks et al. 2002). As these multiscale frameworks encompass more and more anatomical and physiological detail, it becomes essential to develop a modular approach to modeling in which the details of a particular subsystem are encapsulated in a module based on model-encoding standards (Hunter and Borg 2003). This is discussed further below, where the CellML and FieldML modeling standards for computational physiology are introduced. However, imaging technologies are presented first, followed by the very important topic of choosing an appropriate set of units for modeling multiple coupled physical processes.

Imaging at Multiple Scales

An ability to image the structure and function of biological tissues at various spatial scales from protein and subcellular levels to tissue, organ, and organ system levels is fundamental to progress in modeling. Some of these imaging techniques are illustrated in Fig. 1. Starting at the smallest scale, X-ray diffraction techniques yield protein structure at about 0.5 nm resolution (Fig. 1a). Electron tomography has the capacity to image 3D subcellular structure at the 5 nm level (Fig. 1b). With image registration across multiple scanned tissue samples, confocal imaging can provide 3D reconstruction of complete transmural cross-sections of, for example, the ventricular wall (see Fig. 1c). Multiphoton and confocal imaging routinely gives 500 nm resolution on soft tissues, although with the use of single-molecule fluorescent markers such as GFPs coupled to quenching techniques that eliminate background illumination, optical imaging techniques are now achieving 50 nm resolution. Micro-CT, in which a specimen is imaged with a collimated X-ray source from multiple incident angles to build a 3D tomographic reconstruction, provides 0.5–5 μm resolution on suitable contrast-enhanced tissue (Fig. 1d). Magnetic resonance imaging (MRI) has the capacity to image intact organs at about 0.1–1 mm resolution (see Fig. 1e) and when used with tagging can provide direct information on tissue movement in a clinical setting. A number of other imaging technologies, such as helical scan CT, positron emission tomography (PET), and ultrasound (US), are used clinically for noninvasive imaging of structure and function at about 1 mm resolution.
Fig. 1

Some of the imaging techniques used to measure anatomical structure at different spatial scales

These imaging technologies provide the data to model anatomical structure at the multiple spatial scales on which the biophysical processes underlying physiological function can be defined, as illustrated below for the heart. But first, consistent physical units and the use of bond graphs are needed to ensure that the mechanical, electrical, and biochemical processes underpinning physiological function are based on a consistent physics framework.

Units Associated with Physical Processes

The seven units (ampere, kelvin, second, meter, kilogram, candela, mole) of the Système international d’unités (SI) are the starting points for any discussion of units. The way that these are measured, however, has recently been revised such that four of the SI base units – namely, the kilogram, the ampere, the kelvin, and the mole – will be redefined in terms of fixed numerical values of the Planck constant (h, defines the relationship between the energy of a photon and its frequency), the elementary charge (e, the charge on an electron), the Boltzmann constant (kB, defines the relationship between the average kinetic energy of particles in a gas and the temperature of the gas), and the Avogadro constant (NA, the number of particles contained in one mole of substance), respectively (www.bipm.org/en/measurement-units/rev-si).

From a continuum modeling perspective, it is useful to have a slightly different set of units derived from these seven SI base units, since there is a fundamental distinction between the units that establish the space-time dimensions of the universe (meter, second); the unit of energy (joule) – bearing in mind that energy is the only quantity (apart from meters and seconds) that is common to all domains of physics; and the remaining four units that deal with everything else. These last four are most usefully based on countable quantities: coulomb (number of elementary charges), candela (number of photons), mole (number of atoms), and a unit of entropy (number of probability states). Thus the appropriate base units for continuum mechanics, with their abbreviations, are joule (J), second (s), meter (m), coulomb (C), candela (cd), mole (mol), and entropy (e).

Note that the three fundamental domains of physics relevant to physiology, namely, mechanics, electromagnetics, and biochemistry and heat flow, all require the units joule, second, and meter, but are supplemented with, respectively, meter (mechanics), coulomb and candela (electromagnetics), and mole and entropy (biochemistry and heat flow). Within a three-dimensional spatial context, these give rise to, respectively, Euler’s equations (and hence the equations of finite elasticity theory for solids and the Navier-Stokes equations for fluids), Maxwell’s equations, and reaction-diffusion equations. Since physiology involves all of these domains, there is a need to provide a consistent set of units for energy exchange between the domains and a consistent approach to formulating the relevant equations. Variational principles and Galerkin finite element methods provide the consistent approach for three-dimensional spatial models, and bond graphs, discussed next, provide this consistency for zero-dimensional lumped parameter models (based on ordinary differential and algebraic equations).

Physical Processes Defined with Bond Graphs

Many physical phenomena can be described by a potential (μ) in units of Joules/some_quantity where the quantity q could be meters (for mechanics), moles and entropy (for biochemistry and heat flow), or coulombs and candela (for electromagnetism). The potential drives a flow (\( {\boldsymbol v} {=}\frac{d\boldsymbol{q}}{dt} \)) in units of quantity. s−1, such that their product μ (J.quantity−1) × v (quantity.s−1) = Power (J.s−1). A bond with covariablesμ and v (see Fig. 2) is therefore used to represent the transmission of energy and power, with the arrowhead indicating the assumed direction of positive power flow as shown in Fig. 2.
Fig. 2

A “bond,” shown as an arrow, is associated with a potential μ and a flow v

The flow v and potential μ must satisfy conservation laws (e.g., mass or charge conservation for v and force balance, Kirchhoff’s voltage law, or Gibbs free energy relations for μ).

The quantity (q), whose rate of change is v (i.e., \( \frac{d\boldsymbol{q}}{dt}=\boldsymbol{\upsilon} \)), can be stored statically in a “capacitor” with a dependence on potential μ given empirically by μ=Eq or dynamically in an “inductor” with μ=La\( \boldsymbol{a}=\dot{\boldsymbol{\upsilon}}=\ddot{\boldsymbol{q}} \), where \( \boldsymbol{a}=\dot{\boldsymbol{\upsilon}}=\ddot{\boldsymbol{q}} \). Energy can also be dissipated by a “resistor” in proportion to v with an empirical relation μ = Rv. The domain-specific units for the capacitance E, inductance L, and resistance R are given in Table 1.
Table 1

The units of capacitance, resistance, and inductance for electrical, mechanical, thermal, and chemical systems

Component type

Electrical

Solid mechanics

Fluid mechanics

Heat flow

Biochemistry

Capacitance E

J.C−2

J.m−2

J.m−6

J.e−2

J.Mol−2

Resistance R

J.s.C−2

J.s.m.−2

J.s.m.−6

J.s.e−2

J.s.Mol−2

Inductance L

J.s2.C−2

J.s2.m−2

J.s2.m−6

J.s2.e−2

J.s2.Mol−2

Bond graphs use the concept of a 0-node and a 1-node. The 0-node defines a common potential μ and imposes a conservation constraint based on v – this is volume or mass conservation if q is volume or mass, charge conservation if q is charge, or stoichiometric relations for chemical species in a biochemical reaction. The1-node defines a common flow v and imposes a conservation constraint based on μ – this is Kirchhoff’s voltage law for electrical circuits, force balance for mechanical systems, heat flux balance for thermal systems, or Gibbs free energy changes (the second law of thermodynamics) for biochemical systems.

The theory of bond graphs was pioneered by Henry Paynter at MIT in the 1960s (Paynter 1961) and further developed by Karnopp et al. (2012) in a series of textbooks aimed at mechanical engineers. When Breedveld (1984) added the theory of network thermodynamics, pioneered by Aharon Katchalsky (Oster et al. 1971), bond graphs evolved to become a more general systems theory. It provides a biophysically and thermodynamically consistent framework on which to base CellML models. The modern application of bond graphs to biological systems has been pioneered by Gawthrop and Crampin (Gawthrop and Crampin 2014, 2016, Gawthrop et al. 2015).

Figure 3 illustrates a generic model construction. In the middle layer, a 0-node (defining the potentials on all incident bonds) is followed by a 1-node (defining the flows on all incident bonds), since otherwise the potential and flow on a bond would not be unique. Since energy is conserved through these nodes, ∑μv = 0 (summed over the incident bonds). For a 0-node μ is constant and hence ∑v = 0 (flow balance), whereas for a 1-node, v is constant and hence ∑μ = 0 (potential balance). Note that these balance equations can be generated automatically from the bond graph diagram (see bottom layer of Fig.~3). The static and dynamic storage components are usually associated with 0-nodes since they enter the flow balance (and require the integration of v to obtain q or the differentiation of v to obtain a). Dissipation (the resistance component), on the other hand, is associated with an energy balance 1-node. The constitutive relations for the capacitor, inductor, and resistor (shown in the top layer) must be stated explicitly as they are not associated with the physical equations. Note that there is no requirement that these constitutive relations should be linear.
Fig. 3

The generic layout for a bond graph model. The middle layer (“network topology”) is the sequence of 0-nodes and 1-nodes that establish the topology of the model and generate the flow or energy balance equations shown in the bottom layer. The top layer (“constitutive relations”) provides the empirically defined energy storage and dissipation mechanisms

The equations generated by bond graph models are encoded in CellML (https://www.cellml.org), archived in the Physiome Model Repository (PMR; https://models.physiomeproject.org/cellml), and solved with freely available open-source tools such as OpenCOR (http://www.opencor.ws). CellML is discussed further below. These models are then incorporated into tissue-/organ-level models described below.

Multiscale Modeling of the Heart

In this section the general principles involved in multiscale modeling are illustrated by focusing on the coupled physical processes that underpin the function of the heart. Models of torso anatomy and whole heart structure based on continuum models and finite element techniques (described below) are shown at the top in Fig. 4. These models incorporate tissue structure and tissue-level properties such as the mechanical stress-strain characteristics and the electrical conductivity. The continuum models link to the cell level models of electrophysiology, calcium transport, myofilament mechanics, and metabolic and signaling pathways shown at the lower left. At the cellular scale (illustrated in the lower center), these processes are, in turn, driven by subcellular components with their own complex spatial organization. Finally the link to protein structure and gene regulation is depicted at the bottom right. Note that this “Cardiac Physiome” project is one example from the more general Physiome Project (Hunter and Borg 2003) which now includes all 12 organ systems of the body. All of these models use biophysically based modeling, since the only way to interpret data from multiple spatial scales is via the biophysical mechanisms that operate at each scale and, since modeling biophysical processes generally involves spatial gradients and information about three-dimensional structure, the biophysical models need to be based on anatomically defined geometric models.
Fig. 4

The multiscale cardiac physiome modeling hierarchy from genes to the whole organism

Parameters used in a model at one scale can often be derived from a more detailed model at a lower spatial scale. The cell models shown at the bottom left of Fig. 4 define multiple processes within the cardiac myocyte and are based on the bond graph approach to ensure consistency. The finite element continuum models shown on the top right deal with the spatially varying fibrous-sheet tissue structure and include the vascular bed.

The multiple biophysical processes underlying the models are identified in Fig. 5. Models of the tissue structure and material properties are considered in the next section
Fig. 5

Types of model used in the multiscale modeling hierarchy. Models based on systems of ODEs and algebraic equations are shown in blue (so-called “lumped parameter” models), and these are encoded in CellML. Models that require the solution of partial differential equations are shown in pink and are encoded in FieldML. The FieldML models link to CellML models at material points in the tissue. The arrows above are shown as unidirectional, but, in fact, information flows both ways

Cell-Level Modeling

The process of modeling myocardial activation (and hence cardiac arrhythmias) starts with models of the excitability of cardiomyocytes – the cardiac muscle cells that both support electrical propagation in heart muscle and generate force to cause heart contraction. The ion channels – primarily sodium (for generating the rapid upstroke of the action potential), calcium (for generating the internal rise in calcium concentration that initiates contraction), and potassium (for repolarizing the cell to its resting membrane potential) – have been studied for over 50 years using voltage clamp and patch clamp techniques to define their current/voltage/time characteristics. Models of the whole cell are built from the individual ion channel models (Nickerson and Hunter 2006) and incorporated into tissue scale models. Other membrane proteins acting as ion exchangers (such as the Na-Ca exchanger that uses the Na gradient to remove Ca from the cell) and ATP-dependent pumps (such as the Na-K pump) that maintain ion concentrations are also included in these models. Note that these “lumped parameter” cell models do not incorporate the three-dimensional structure of the myocyte.

The CellML Standard

Many models of biological processes at the subcellular level ignore the detailed three-dimensional structure of the cell and model the cell excitability, mechanics, calcium (or other second messenger) transients, motility, signaling, metabolism or gene regulation, etc., in a “lumped parameter” system of ordinary differential equations (ODEs). Sometimes the models also include nonlinear algebraic equations or need to solve constrained optimization problems as part of the solution strategy, but they do not require the solution of partial differential equations (PDEs). The markup language CellML (www.cellml.org) has been developed to provide an unambiguous definition of these lumped parameter models. The language is designed to support the definition and sharing of models of biological processes by including information about model structure (how the parts of a model are organizationally related to one another), mathematics (equations describing the underlying biological processes), and metadata (additional information about the model that allows scientists to search for specific models or model components in a database or other repository).

CellML has a simple structure based upon connected components. These components abstract concepts by providing well-defined interfaces to other components and encapsulate concepts by hiding details from other components. Connections provide the means for sharing information by associating variables visible in the interface of one component with those in the interface of another component. Consistency is enforced by requiring that all variables be assigned appropriate physical units which must match when variables are connected. Public and private interfaces enable encapsulation hierarchies, providing further mechanisms for information hiding and abstraction. Model reuse is facilitated by the import element, enabling new models to be constructed by combining existing models into model hierarchies. The CellML 1.1 standard is available at (www.cellml.org/specifications/cellml_1.1). Note that another markup language (SBML www.sbml.org) is in widespread use for describing biochemical reaction networks.

Modeling Organ Anatomy and Tissue Structure

Finite element methods are used to solve the partial differential equations (see below) that express the physical laws underlying the physiological function of organs. High-order finite element basis functions can be used to capture the heart’s anatomy (geometry and tissue structure) in an efficient manner (Hunter and Smaill 1989). For example, a fitted tricubic Hermite finite element model is shown with a translucent epicardial surface in Fig. 6a, and the changing orientations of the muscle fibers throughout the myocardium are shown in Fig. 6b–d – see further details below.
Fig. 6

(a) The tricubic Hermite finite element mesh with a translucent epicardial surface (bd). The orientation of muscle fibers is shown in red, at the endocardial surface (b), mid-wall (c), and epicardial surface (d)

Another important component of heart muscle structure is the way that muscle “fibers” (cardiac myocytes) are coupled into layered sheetlike structures separated by “cleavage planes” (LeGrice et al. 1995). The cells are bound into layers three to five cells thick, as shown in Fig. 7.
Fig. 7

The fibrous-sheet structure of ventricular myocardium. The schematic on the right shows how the cells are bound into layers three to five cells thick that form the myocardial “sheets” (a). A macroscopic view of the branching sheet structures is shown in (b), and an electron microscopy image shows the slightly distended sheets, bound together by collagen in (c). Note how capillaries are embedded within the sheets (at a one-to-one relationship with myocytes) in (c)

The schematic on the right (Fig. 7a) shows a segment of wall removed from the left ventricle. The muscle fibers lie substantially parallel to the wall tangent plane at any point between the inner (endocardial) and outer (epicardial) surfaces. The changing fiber orientation is shown by the thin white rods, and in the lower schematic, the myocytes are shown organized into layers. The sheet orientation can also be seen in a macroscopic surface image of a transmural slice in Fig. 7b. When the slightly expanded muscle structure is examined under electron microscopy (Fig. 7c), the branching nature of the sheets can be seen, as can the perimysial collagen joining the sheets together. At every point in the tissue, therefore, there are three “material axes” (axes effectively tethered to the tissue material). One is the “fiber axis,” the second is the “sheet axis” (orthogonal to the fibers in the plane of the sheet), and the third is the “sheet normal” orthogonal to the sheet plane. Collagen surrounding the sheets and binding the cells together is called “endomysial collagen,” and collagen between the sheets, and also often forming large axially aligned thick bundles that appear to limit stretch in the muscle fiber direction, is called “perimysial collagen” (LeGrice et al. 1997).

The spatial variation of the fibrous-sheet architecture of the ventricular myocardium is shown in Fig. 8. A material axis aligned with the fiber direction at any point is given a coordinate η1. The fibers are tethered by collagen into sheets, three to five cells thick, that branch continually but at any point can be defined by another material axis with coordinate η2 that is orthogonal to the fiber axis in the unloaded reference state of the tissue. A third material axis, orthogonal to the fiber-sheet plane, is defined with coordinate η3. On the epicardial surface of the ventricle, the fiber-sheet plane is tangent to this surface so that η3 points out from the wall as shown in Fig. 8b. The angle that the fiber axis makes with the circumferential direction of the ventricle is called the “fiber angle,” and this varies from about −60° on the epicardial surface to +80° on the endocardial surface (as can be seen in Fig.~6). The sheet axis rotates through the wall from being tangential to the wall at the epicardium to approximately transmural in the center of the wall to again tangential to the wall at the endocardium. The spatial variations in fiber and sheet angles throughout the myocardium are defined by finite element fields.
Fig. 8

The fibrous-sheet architecture of the ventricular myocardium. (a) The schematic of the organization of fibers and sheets reproduced from Fig. 7a. (b) An illustration of the fiber-sheet material axes defined by the coordinates (η1, η2, η3) in relation to the heart model (see text). The changing orientation of these axes across the ventricular wall is shown at close-up (c) and across the whole heart (d)

Modeling Whole Heart Function

The Cardiac Physiome project is modeling the heart as an integrated system of coupled physical processes (large deformation tissue mechanics, ventricular blood flow, myocardial excitation). It also addresses multiscale issues, such as the coupling of ion channel models into myocardial reaction-diffusion equations. Figure 9 shows the major systems of equations which include the equations for large deformation elasticity theory, the equations that couple wall mechanics to fluid mechanics (both ventricular and coronary flow), and the reaction-diffusion equations that describe myocardial excitation in terms of membrane ion channels.
Fig. 9

The major biophysical processes operating at the whole heart level. Large deformation soft tissue mechanics is coupled with myocardial excitation and ventricular fluid mechanics. The processes within the oval shapes are continuum models, and the models shown in boxes are encoded in zero-dimensional CellML models

The equations governing mechanical and electrical function of the heart are derived from the physical conservation laws – conservation of mass, conservation of momentum, and conservation of current. These equations are formulated in a so-called weak form that uses spatial integrals over the myocardial domain. The equations also require the specification of “constitutive” relations that characterize tissue properties – the elastic or stress-strain properties in the case of the equations representing conservation of momentum (Nash and Hunter 2001) – and the relationship between current and voltage gradient (the tissue conductivity) in the case of the equations governing conservation of current flow in the myocardium (Hunter et al. 2003). These constitutive laws are defined relative to the material fibrous-sheet coordinate axes described above. Note that the concept of a “constitutive relation” is a crucial step in multiscale modeling because it summarizes the electrical and mechanical behavior of a complex three-dimensional network of muscle cells and extracellular matrix. Constitutive relations can be measured experimentally or can be derived from microstructurally detailed models in order to establish a better understanding of the multiscale relationships (see Nash and Hunter 2001). Wang et al. (2017) discuss the solution of ventricular mechanics using the finite elasticity equations with orthotropic constitutive laws. Chen et al. (2017) and Neic et al. (2017) discuss clinically oriented studies of the electrical activation of myocardial tissue. Lee et al. (2016) consider multiphysics coupling issues.

Discussion

A computational physiology modeling framework has been described that incorporates the anisotropic, inhomogeneous, and nonlinear properties of tissues within anatomically accurate finite element models of body organs. The continuum equations representing mechanical, electrical, and biochemical processes are linked to cellular processes defined with bond graph CellML models in order to link the physiological function of whole organs to underlying molecular processes. Note that the use of bond graphs at the lumped parameter subcellular level and Galerkin finite element techniques at the tissue/organ level ensures that mass balance and energy balance equations are satisfied. The heart has been used here to illustrate the physics-based modeling principles, but the methods are equally applicable to any organ. For similar applications to the lungs, see Tawhai and Bates (2011) and Lin et al. (2013), for the digestive tract, see Du et al. (2016, 2017), and for the musculoskeletal system, see Fernandez et al. (2004, Fernandez and Pandy 2006).

Cross-References

References

  1. Breedveld PC (1984) Physical systems theory in terms of bond graphs. PhD thesis University of TwenteGoogle Scholar
  2. Chen Z, Niederer SA, Shanmugam N, Sermesant M, Rinaldi CA (2017) Cardiac computational modeling of ventricular tachycardia and cardiac resynchronization therapy: a clinical perspective. Minerva Cardioangiol 65(4):380–397Google Scholar
  3. Du P, Paskaranandavadivel N, Angeli TR, Cheng LK, O’Grady G (2016) The virtual intestine: in silico modeling of small intestinal electrophysiology and motility and the applications. Wiley Interdiscip Rev Syst Biol Med 8(1):69–85CrossRefGoogle Scholar
  4. Du P, O’Grady G, Cheng LK (2017) A theoretical analysis of anatomical and functional intestinal slow wave re-entry. J Theor Biol 21(425):72–79CrossRefGoogle Scholar
  5. Fernandez JW, Pandy MG (2006) Integrating modelling and experiments to assess dynamic musculoskeletal function in humans. Exp Physiol 91(2):371–382CrossRefGoogle Scholar
  6. Fernandez JW, Mithraratne P, Thrupp SF, Tawhai MH, Hunter PJ (2004) Anatomically based geometric modelling of the musculo-skeletal system and other organs. Biomech Model Mechanobiol 2(3):139–155CrossRefGoogle Scholar
  7. Gawthrop PJ, Crampin EJ (2014) Energy based analysis of biochemical cycles using bond graphs. Proc R Soc A 470(2171):20140459. https://doi.org/10.1098/rspa.2014.0459CrossRefGoogle Scholar
  8. Gawthrop PJ, Crampin EJ (2016) Modular bond-graph modelling and analysis of biomolecular systems. IET Syst Biol. https://doi.org/10.1049/iet-syb.2015.0083CrossRefGoogle Scholar
  9. Gawthrop PJ, Cursons J, Crampin EJ (2015) Hierarchical bond graph modelling of biochemical networks. Proc R Soc A 471(2184):20150642. https://doi.org/10.1098/rspa.2015.0642MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hooks DA, Tomlinson KA, Marsden SG, Le Grice IJ, Smaill BH, Pullan AJ, Hunter PJ (2002) Cardiac microstructure: implications for electrical propagation and defibrillation in the heart. Circ Res 91(4):331–338CrossRefGoogle Scholar
  11. Hunter PJ, Borg TK (2003) Integration from proteins to organs: the Physiome project. Nat Rev Mol Cell Biol 4(3):237–243CrossRefGoogle Scholar
  12. Hunter PJ, Smaill BH (1989) The analysis of cardiac function: a continuum approach. Prog Biophys Mol Biol 52:101–164CrossRefGoogle Scholar
  13. Hunter PJ, Pullan AJ, Smaill BH (2003) Modeling total heart function. Annu Rev Biomed Eng 5:147–177CrossRefGoogle Scholar
  14. Karnopp DC, Margolis DL, Rosenberg RC (2012) System dynamics, 5th edn. WileyGoogle Scholar
  15. Lee J, Cookson A, Roy I, Kerfoot E, Asner L, Vigueras G, Sochi T, Michler C, Smith N, Nordsletten D (2016) CHeart: Multiphysics computational modelling in CHeart. SIAM J Sci Comput 38:C150–C178CrossRefGoogle Scholar
  16. LeGrice IJ, Smaill BH, Chai LZ, Edgar SG, Gavin JB, Hunter PJ (1995) Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Am J Phys 269:H571–H582Google Scholar
  17. LeGrice IJ, Hunter PJ, Smaill BH (1997) Laminar structure of the heart: a mathematical model. Am J Phys 272:H2466–H2476Google Scholar
  18. Lin CL, Tawhai MH, Hoffman EA (2013) Multiscale image-based modeling and simulation of gas flow and particle transport in the human lungs. WIREs Syst Biol Med 5(5):643–655CrossRefGoogle Scholar
  19. Nash MP, Hunter PJ (2001) Computational mechanics of the heart. J Elast 61(1–3):113–141zbMATHGoogle Scholar
  20. Neic A, Campos FO, Prassl AJ, Niederer SA, Bishop MJ, Vigmond EJ, Plank G (2017) Efficient computation of electrograms and ECGs in human whole heart simulations using a reaction-eikonal model. J Comp Phy 346:191–211MathSciNetCrossRefGoogle Scholar
  21. Nickerson DP, Hunter PJ (2006) The Noble cardiac ventricular electrophysiology models in CellML. Prog Biophys Molec Biol 90:346–359CrossRefGoogle Scholar
  22. Nielsen PMF, Hunter PJ, Smaill BH (1991) Biaxial testing of membrane biomaterials: testing equipment and procedures. ASME J Biomech Eng 113(3):295–300CrossRefGoogle Scholar
  23. Oster G, Perelson A, Katchalsky A (1971) Network thermodynamics. Nature (Lond) 234:393CrossRefGoogle Scholar
  24. Paynter H (1961) Analysis and Design of Engineering Systems. MIT, CambridgeGoogle Scholar
  25. Tawhai MH, Bates JH (2011) Multi-scale lung modeling. J Appl Physiol 110(5):1466–1472CrossRefGoogle Scholar
  26. Wang VY, Hussan JR, Yousefi H, Bradley CP, Hunter PJ, Nash MP (2017) Modelling cardiac tissue growth and remodelling. J Elast 129(1–2):283–305MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Auckland Bioengineering InstituteUniversity of AucklandAucklandNew Zealand

Section editors and affiliations

  • Daniel Balzani
    • 1
  1. 1.Ruhr-Universität BochumBochumGermany