Computational cardiology. Modeling of anatomy, electrophysiology, and mechanics.

*(English)*Zbl 1051.92025
Lecture Notes in Computer Science 2966. Berlin: Springer (ISBN 3-540-21907-2/pbk). xvii, 322 p. (2004).

The topic of this work is computer-based, mathematical modeling of the heart. The application of modeling as a research, development and clinical tool is stressed as a promising attempt to address many problems in cardiology, heart surgery and biomedical engineering. The modeling includes the areas of anatomy, electrophysiology, excitation propagation, force development and mechanics as well as the coupling of these areas. The work shows knowledge at molecular, cellular and macroscopic level, how these areas can be studied with measurements, and how these areas can be modeled with mathematical methods. Detailed information of phenomena on each of these levels is necessary to understand the ideas behind the modeling.

Special attention is given to macroscopic and integrative modeling with the aim of reconstruction of the whole heart behavior. At macroscopic level anatomical models provide a basis for electrophysiological and mechanical models. Of particular interest is the assessment of influence of cardiac deformation to initiation and propagation of electrical excitation and to the force development. Various simulation studies are demonstrated, which provide information about this assessment.

Detailed modeling of the macroscopic cardiac anatomy is commonly performed on base of medical imaging systems, which are used in clinical routine and research. The resulting data is transformed with methods of digital image processing to obtain a representation of anatomy, which is suitable for the target application. Different levels of spatial description can be distinguished, ranging from analytical, comprehensive approaches to detailed descriptions on base of millions of volume elements. Commonly, ultrasonic (US), magnetic resonance (MRT), and computed tomography (CT) are used for imaging of the heart. These imaging systems use tissue dependent variations of the acoustic impedance leading to reflection of ultrasonic waves and of their scattering, of the absorption of X-rays, and of the resonance behavior of nuclei to get information of the tissue distribution inside a body. An alternative data source for macroscopic modeling of human anatomy provides the Visible Human Project of the National Library of Medicine, Bethesda, Maryland (USA).

In the sections of the book, modeling of the orientation and lamination of myocytes is restricted to a macroscopic, averaged perspective onto the cellular geometry. This perspective is often taken in modeling of complex, inhomogeneous structures, e.g., continuum modeling of electrophysiology and structure mechanics, and allows a simplified treatment, especially, if the microscopic inhomogeneity of attributes can be neglected. In the macroscopic perspective the orientation and lamination of myocytes is defined by averaging and interpolation. The local attributes can be viewed as averaged macroscopic quantities, e.g., the spatial averaged principal axis of the myocytes and orientations of the lamination.

The strategies for modeling of the orientation and lamination of myocytes can be divided into two groups. The first group includes methods, which base on the measurement of the attributes, i.e., histological studies of surfaces of tissue sections, and recently developed imaging techniques. The second group consists of rule-based methods, which apply rules constructed from anatomical studies.

Knowledge concerning the electrophysiology of the heart is necessary for the understanding of many aspects of physiological and pathophysiological cardiac behavior. The electrophysiology is tightly coupled with the mechanic deformation and the pump function of the heart by controlling the development of tension. Furthermore, various mechano-electrical feedback mechanisms influence the cardiac electrophysiology. Different electrophysiological experiments and modeling approaches are described. The description starts with the measuring and modeling approaches of phenomena of cellular components, followed by the whole cell electrophysiological behavior and mechanisms of the intercellular excitation propagation. The description of the electrophysiology of cellular components concerns primarily the phenomena resulting from the behavior of the cell. In this context, the classical work of Hodgkin and Huxley is presented, who delivered quantitative data of the electrophysiology of a squid axon and constructed a mathematical model. Most electrophysiological models of nerve and muscle cells base on their mathematical formulation.

In the last years a large number of models of myocytes was constructed, with increasing abilities to describe the different electrophysiological mechanisms. Primarily, the models are produced from animal experiments using the mathematical formulations of Hodgkin and Huxley. The modeling approaches of the cell membrane, ionic channels and pumps, exchanger and intracellular components are combined to describe the behavior of a whole cell.

Different modeling approaches of the excitation propagation in the myocardium can be distinguished depending on the representation of the microscopic and macroscopic anatomy as well as depending on the approximation of the cellular electrophysiology. Modeling approaches that use only macroscopic information allow combining of cells and their common treatment. In contrast, models using microscopic anatomical information at a cellular level split cells into components, which are separately treated.

In the last years different approaches for the macroscopic excitation propagation were developed: 1) celular automata, 2) reaction diffusion systems. Microscopic and macroscopic models allow the inclusion of anisotropic effects resulting from the orientation of myocytes, e.g., by using conductivity tensors. Commonly, the methods are applied on a one-, two- or three-dimensional lattice, whereby each cell’s intracellular space is coupled directly only with the intracellular space of adjacent cells. A biophysically well-founded approach consists of combining detailed electrophysiological models of single myocardial cells with models of the electrical current flow through the intra- and extracellular space as well as the gap junctions. The electrophysiological cell models describe the concentration and flow of ions as well as the conductivity of cellular structures and the transmembrane voltage by a set of coupled differential equations. The results of many experimental studies were used to model the relationship between strain and stress by assumption of a convenient template strain energy function and parameter fitting procedures. The parameters were determined by numerical experiments reconstructing the measurements.

In the next sections some modeling approaches are described. The development of models and the inclusion of different properties are illustrated. The properties, are e.g., inhomogeneity, anisotropy, and multiple domains. In multiple domain approaches different strain energy functions, each defined on a domain, contribute to the summary strain energy. Commonly, the energy density function does not comprise plasto- and viscoelastic effects. A biophysically, microscopic anatomically based mathematical description, i.e., the theory of sliding filaments, was proposed in 1953 by Huxley and Niedergerke as well as by Huxley and Janson. This description of force development in striated muscle and derived models is introduced in the following sections.

Of special interest for biophysically motivated modeling are descriptions of cellular force developments, which base on electrophysiological quantities delivered, e.g., by electrophysiological cell models. The concentration of intracellular calcium \([Ca^{2+}]_{i}\) is used to define rate coefficients, which depict the interaction between states of actin and myosin. The states describe, e.g., the binding of intracellular \(Ca^{2+}\) to the troponin complex and the cross bridge cycling. Further parameters influencing the rate coefficients are the sarcomere length and the state variables. Efficient working necessitates an adaptation to suddenly emerging demands and loads. The adaptation is performed by a well ordered interplay between cellular electrophysiology, intercellular excitation propagation and cellular force development. Different feedback mechanisms, i.e., mechanoelectrophysiological and mechano-force development feedbacks, support the interplay. A reconstruction of this interplay can be achieved by coupled models of different type, particularly logical, force development and deformation models.

Electro-mechanic models at cellular, macroscopic and whole heart level are presented, which result from a coupling of diverse models. The electro-mechanical models are illustrated by simulations. Electro-mechanic models of different levels and applied techniques are shown: 1) Models of electrophysiology and force development at the level of single cells. The models are described by systems of coupled ordinary differential equations. The models include geometric descriptions. The resultant stretch is derived. 2) Excitation propagation and cardiac force development at whole heart level. A rule-based method, a cellular automaton, is working in the spatial domain of a realistic anatomical heart model. 3) Electro-mechanics in areas of myocardium. Cellular models of electrophysiology and force development are coupled with excitation propagation and deformation models. The excitation propagation is reconstructed with the bidomain model. For simulations of deformation the mechanical material properties are described as hyperelastic.

The models showed to be challenging concerning computing resources. Particularly the modeling of deformation demands significant parts of calculation time. Exploiting of new numeric methods, e.g., meshless techniques for solving of differential equations, and software development, e.g., grid technologies for computing, seems to be a chance to satisfy these demands. Despite the limitations of the models characteristic micro- and macro-scopic phenomena of cardiac electro-mechanics were reconstructed. The presented methods show a strategy, which can be adapted, e.g., for improvement of biomedical instrumentation and pharmaceuticals as well as for clinical cardiologic diagnosis and planning of therapies. The inclusion of patient specific data, ranging from genetic scans to medical images, is possible and offers the adaptation of models to patient specific characteristics

Special attention is given to macroscopic and integrative modeling with the aim of reconstruction of the whole heart behavior. At macroscopic level anatomical models provide a basis for electrophysiological and mechanical models. Of particular interest is the assessment of influence of cardiac deformation to initiation and propagation of electrical excitation and to the force development. Various simulation studies are demonstrated, which provide information about this assessment.

Detailed modeling of the macroscopic cardiac anatomy is commonly performed on base of medical imaging systems, which are used in clinical routine and research. The resulting data is transformed with methods of digital image processing to obtain a representation of anatomy, which is suitable for the target application. Different levels of spatial description can be distinguished, ranging from analytical, comprehensive approaches to detailed descriptions on base of millions of volume elements. Commonly, ultrasonic (US), magnetic resonance (MRT), and computed tomography (CT) are used for imaging of the heart. These imaging systems use tissue dependent variations of the acoustic impedance leading to reflection of ultrasonic waves and of their scattering, of the absorption of X-rays, and of the resonance behavior of nuclei to get information of the tissue distribution inside a body. An alternative data source for macroscopic modeling of human anatomy provides the Visible Human Project of the National Library of Medicine, Bethesda, Maryland (USA).

In the sections of the book, modeling of the orientation and lamination of myocytes is restricted to a macroscopic, averaged perspective onto the cellular geometry. This perspective is often taken in modeling of complex, inhomogeneous structures, e.g., continuum modeling of electrophysiology and structure mechanics, and allows a simplified treatment, especially, if the microscopic inhomogeneity of attributes can be neglected. In the macroscopic perspective the orientation and lamination of myocytes is defined by averaging and interpolation. The local attributes can be viewed as averaged macroscopic quantities, e.g., the spatial averaged principal axis of the myocytes and orientations of the lamination.

The strategies for modeling of the orientation and lamination of myocytes can be divided into two groups. The first group includes methods, which base on the measurement of the attributes, i.e., histological studies of surfaces of tissue sections, and recently developed imaging techniques. The second group consists of rule-based methods, which apply rules constructed from anatomical studies.

Knowledge concerning the electrophysiology of the heart is necessary for the understanding of many aspects of physiological and pathophysiological cardiac behavior. The electrophysiology is tightly coupled with the mechanic deformation and the pump function of the heart by controlling the development of tension. Furthermore, various mechano-electrical feedback mechanisms influence the cardiac electrophysiology. Different electrophysiological experiments and modeling approaches are described. The description starts with the measuring and modeling approaches of phenomena of cellular components, followed by the whole cell electrophysiological behavior and mechanisms of the intercellular excitation propagation. The description of the electrophysiology of cellular components concerns primarily the phenomena resulting from the behavior of the cell. In this context, the classical work of Hodgkin and Huxley is presented, who delivered quantitative data of the electrophysiology of a squid axon and constructed a mathematical model. Most electrophysiological models of nerve and muscle cells base on their mathematical formulation.

In the last years a large number of models of myocytes was constructed, with increasing abilities to describe the different electrophysiological mechanisms. Primarily, the models are produced from animal experiments using the mathematical formulations of Hodgkin and Huxley. The modeling approaches of the cell membrane, ionic channels and pumps, exchanger and intracellular components are combined to describe the behavior of a whole cell.

Different modeling approaches of the excitation propagation in the myocardium can be distinguished depending on the representation of the microscopic and macroscopic anatomy as well as depending on the approximation of the cellular electrophysiology. Modeling approaches that use only macroscopic information allow combining of cells and their common treatment. In contrast, models using microscopic anatomical information at a cellular level split cells into components, which are separately treated.

In the last years different approaches for the macroscopic excitation propagation were developed: 1) celular automata, 2) reaction diffusion systems. Microscopic and macroscopic models allow the inclusion of anisotropic effects resulting from the orientation of myocytes, e.g., by using conductivity tensors. Commonly, the methods are applied on a one-, two- or three-dimensional lattice, whereby each cell’s intracellular space is coupled directly only with the intracellular space of adjacent cells. A biophysically well-founded approach consists of combining detailed electrophysiological models of single myocardial cells with models of the electrical current flow through the intra- and extracellular space as well as the gap junctions. The electrophysiological cell models describe the concentration and flow of ions as well as the conductivity of cellular structures and the transmembrane voltage by a set of coupled differential equations. The results of many experimental studies were used to model the relationship between strain and stress by assumption of a convenient template strain energy function and parameter fitting procedures. The parameters were determined by numerical experiments reconstructing the measurements.

In the next sections some modeling approaches are described. The development of models and the inclusion of different properties are illustrated. The properties, are e.g., inhomogeneity, anisotropy, and multiple domains. In multiple domain approaches different strain energy functions, each defined on a domain, contribute to the summary strain energy. Commonly, the energy density function does not comprise plasto- and viscoelastic effects. A biophysically, microscopic anatomically based mathematical description, i.e., the theory of sliding filaments, was proposed in 1953 by Huxley and Niedergerke as well as by Huxley and Janson. This description of force development in striated muscle and derived models is introduced in the following sections.

Of special interest for biophysically motivated modeling are descriptions of cellular force developments, which base on electrophysiological quantities delivered, e.g., by electrophysiological cell models. The concentration of intracellular calcium \([Ca^{2+}]_{i}\) is used to define rate coefficients, which depict the interaction between states of actin and myosin. The states describe, e.g., the binding of intracellular \(Ca^{2+}\) to the troponin complex and the cross bridge cycling. Further parameters influencing the rate coefficients are the sarcomere length and the state variables. Efficient working necessitates an adaptation to suddenly emerging demands and loads. The adaptation is performed by a well ordered interplay between cellular electrophysiology, intercellular excitation propagation and cellular force development. Different feedback mechanisms, i.e., mechanoelectrophysiological and mechano-force development feedbacks, support the interplay. A reconstruction of this interplay can be achieved by coupled models of different type, particularly logical, force development and deformation models.

Electro-mechanic models at cellular, macroscopic and whole heart level are presented, which result from a coupling of diverse models. The electro-mechanical models are illustrated by simulations. Electro-mechanic models of different levels and applied techniques are shown: 1) Models of electrophysiology and force development at the level of single cells. The models are described by systems of coupled ordinary differential equations. The models include geometric descriptions. The resultant stretch is derived. 2) Excitation propagation and cardiac force development at whole heart level. A rule-based method, a cellular automaton, is working in the spatial domain of a realistic anatomical heart model. 3) Electro-mechanics in areas of myocardium. Cellular models of electrophysiology and force development are coupled with excitation propagation and deformation models. The excitation propagation is reconstructed with the bidomain model. For simulations of deformation the mechanical material properties are described as hyperelastic.

The models showed to be challenging concerning computing resources. Particularly the modeling of deformation demands significant parts of calculation time. Exploiting of new numeric methods, e.g., meshless techniques for solving of differential equations, and software development, e.g., grid technologies for computing, seems to be a chance to satisfy these demands. Despite the limitations of the models characteristic micro- and macro-scopic phenomena of cardiac electro-mechanics were reconstructed. The presented methods show a strategy, which can be adapted, e.g., for improvement of biomedical instrumentation and pharmaceuticals as well as for clinical cardiologic diagnosis and planning of therapies. The inclusion of patient specific data, ranging from genetic scans to medical images, is possible and offers the adaptation of models to patient specific characteristics

Reviewer: Fatima T. Adylova (Tashkent)

##### MSC:

92C50 | Medical applications (general) |

92-08 | Computational methods for problems pertaining to biology |

92C30 | Physiology (general) |

92-02 | Research exposition (monographs, survey articles) pertaining to biology |

92C55 | Biomedical imaging and signal processing |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

92C05 | Biophysics |