# A Modeling Framework for Computational Physiology

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**DOI:**https://doi.org/10.1007/978-3-662-53605-6_29-2

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## 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

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

- 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.
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

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 (k_{B}, defines the relationship between the average kinetic energy of particles in a gas and the temperature of the gas), and the Avogadro constant (N_{A}, 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

*) in units of Joules/some_quantity where the quantity*

**μ***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*

**q***quantity*.

*s*

^{−1}, such that their product

*(*

**μ***J.quantity*

^{−1}) ×

*(*

**v***quantity.s*

^{−1}) = Power (J.s

^{−1}). A

*bond*with

*covariables*

*and*

**μ***(see Fig. 2) is therefore used to represent the*

**v***transmission of energy and power*, with the arrowhead indicating the assumed direction of positive power flow as shown in Fig. 2.

The flow * v* and potential

*must satisfy conservation laws (e.g., mass or charge conservation for*

**μ***and force balance, Kirchhoff’s voltage law, or Gibbs free energy relations for*

**v***).*

**μ***), whose rate of change is*

**q***(i.e., \( \frac{d\boldsymbol{q}}{dt}=\boldsymbol{\upsilon} \)), can be stored*

**v***statically*in a “capacitor” with a dependence on potential

*given empirically by*

**μ***=E*

**μ***or*

**q***dynamically*in an “inductor” with

*=L*

**μ***\( \boldsymbol{a}=\dot{\boldsymbol{\upsilon}}=\ddot{\boldsymbol{q}} \), where \( \boldsymbol{a}=\dot{\boldsymbol{\upsilon}}=\ddot{\boldsymbol{q}} \). Energy can also be*

**a***dissipated*by a “resistor” in proportion to

*with an empirical relation*

**v***= R*

**μ***. The domain-specific units for the capacitance E, inductance L, and resistance R are given in Table 1.*

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

Component type | Electrical | Solid mechanics | Fluid mechanics | Heat flow | Biochemistry |
---|---|---|---|---|---|

| J.C | J.m | J.m | J.e | J.Mol |

| J.s.C | J.s.m. | J.s.m. | J.s.e | J.s.Mol |

| J.s | J.s | J.s | J.s | J.s |

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

*– this is volume or mass conservation if*

**v***is volume or mass, charge conservation if*

**q***is charge, or stoichiometric relations for chemical species in a biochemical reaction. The*

**q****1-node**defines a common flow

*and imposes a conservation constraint based on*

**v***– 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.*

**μ**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).

**μ***= 0 (summed over the incident bonds). For a 0-node*

**v***is constant and hence ∑*

**μ***= 0 (flow balance), whereas for a 1-node,*

**v***is constant and hence ∑*

**v***= 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*

**μ***to obtain*

**v***or the differentiation of*

**q***to obtain*

**v***). 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.*

**a**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

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.

## 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

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).

*η*

_{1}. 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

*η*

_{2}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

*η*

_{3}. On the epicardial surface of the ventricle, the fiber-sheet plane is tangent to this surface so that

*η*

_{3}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.

## Modeling Whole Heart Function

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).

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