TISSUE MODELLING

In the Cell Modelling section a model that describes the changes of the membrane voltage of the cardiomyocyte or the transmembrane potential was given. If we have modelled the behaviour of the heart cell, how can we model the behaviour of the heart tissue?

At the example of the study by Crampin, E. J., M. Halstead, et al. (2004), Exp Physiol 89(1): 1-26, let's consider a one-dimensional row of myocytes as shown infigure 5 of the publication or figure 1 below. Each cardiomyocyte has a slightly different resting transmembrane potential (because of the difference in the expression levels of proteins involved in the generation of ionic currents) and these differences become larger upon activation. However, as adjacent cells communicate with each other via their gap junctions, ionic currents pass from one cell to the next one: cells are considered to be electrically coupled. As a result, the difference in transmembrane potential between adjacent cells is small. In figure 5a of the publication or figure 1 below, the transmembrane potential of each cell is represented with a dot below the cell. By "linking the dots" in a "mathematical manner" we obtain a curve which represents the mean of the potential variation in the tissue. The spatial variation of transmembrane potential is termed a "field" and it can be represented mathematically by dividing the spatial domain into subdomains called "finite elements". This is shown in figure 5b of the paper (figure 1 below), where three finite elements are defined. Nodes are chosen at the boundary of the two elements, and the potential of these nodes is used to calculate the potential field within an element. As one node belongs to two elements, the potential is continuous accross element boundaries. This is the basis for the approximation of the cell-to-cell domain with a continuous field representation (continuum) and the implementation of continuum modelling.

Figure 1: Figure 5 from the study by Crampin, E. J., M. Halstead, et al. (2004), Exp Physiol 89(1): 1-26 and legend of the publication below.

"Finite element representation of cellular potentials. A, a one-dimensional row of myocytes, each with transmembrane potential shown below by the dots. B, the mean potential distribution is approximated by a smooth curve constructed from piecewise polynomials. The three sections separated by dashed lines are called finite elements.C, the field inside each element is given by an interpolation of potentials defined at the nodes (shown by the red dots). "

In the same physical domain, more than one fields could exist. In this approach, only one domain, the cell-to cell domain, was considered and this was linked to the cardiomyocyte transmembrane potential field. Similarly, it is possible to also add a second domain, the extracellular domain, and link it to the extracellular potential field. This would constitute a bidomain model. An example of a bidomain model is given below.

In the study by Hooks, D. A., K. A. Tomlinson, et al. (2002), Circ Res 91(4): 331-338, the authors initiated their modelling efforts by performing a detailed morphologic characterization of a myocardial sample. They took a small piece block of rat myocardium with the dimensions 0.8μM * 0.8μM * 3.7mm, as shown in the first figure of the publication or the first supplemental data movie of the paper.

Using a confocal microscope they did "X-rays" or more correctly "laser-rays" of slices/layers and by joining these together they reconstructed the myocardial tissue. The movie represents the three-dimensional reconstruction of rat left ventricular myocardium (upper panel) from layers of 80μM thickness.

Figure 2Screen capture from the  first supplemental data movie of the study by Hooks, D. A., K. A. Tomlinson, et al. (2002), Circ Res 91(4): 331-338  representing a three-dimensional reconstruction of rat myocardium from layers of 80μM thickness.

The link below is relevant to this process.
http://www.physiome.ox.ac.uk/projects_cardiac_tissue_structure.html

The scientists identified the areas with cells (green) and the areas that are devoid of cells (extracellular space) and which actually run as clefts or «cleavage planes» discontinuities among cardiomyocytes (lower panel). We can appreciate that cardiomyocytes are arranged in branching layers and the orientation of the layers can be measured (e.g . angle of 75°). The authors used the finite element technique to create a detailed geometrical or three-dimensional profile of the cleavage planes. In total, 1540 elements and 2803 nodes were used for accurate geometrical description of the clefts.

The authors then proceeded in modelling the propagation of the electric signal in this block. From cell modelling we know how the signal is propagated within a cardiomyocyte and from one cardiomyocyte to another. Interlaminar clefts must be viewed as discontinuities in the intracellular domain, across which there is zero intracellular flux, in other words, these regions are considered to be insulated.

A fundamental law of physics dictates that current flow is conserved in the tissue. Therefore, the equations from ion channels electrophysiological cell models will be coupled to reaction-diffusion equations representing the conservation of flow in the tissue and will be solved to determine the propagation of electrical current in the myocardium.

Mathematically, this is implemented with partial differential equations (PDEs) (integral equations) which balance fluxes, such as electrical current. As mentioned in the study by Crampin, E. J., M. Halstead, et al. (2004), Exp Physiol 89(1): 1-26, in order to solve these balance equations for a specific material, as is in this case the myocardial tissue, we need another equation, called material law or constitutive equation, which is derived by experimental observation of the material. In order to model propagation of an electric signal, we need to know the "resistance" or "conductivity" of the material. For conservation of current, the material law will link current to voltage, via conductivity for the material.

Empirically, we can understand that the propagation of the electic signal will vary in different dimensions. It will be faster for instance along the myofiber axis, if it travels inside a cell or from one cell via its gap junctions to the next one, but slower in another axis where clefts are present.

For this purpose, scientists create conductivity "maps", called "conductivity tensors" which define conductivities in different dimensions. A conductivity tensor is assigned at each point in space.

The equations of the bidomain model are described for instance in the study by Linge, S., J. Sundnes, et al. (2009), Philos Transact A Math Phys Eng Sci367(1895): 1931-1950. For a brief reference we can consult equations (1.1) to (1.5) and the descriptive paragraph that follows them (note: the symbol denotes a gradient).

Hooks, D. A., K. A. Tomlinson, et al. (2002), proceeded to a finite element solution of the bidomain equations and generated the "discontinuous model". They then followed another modelling approach:  they created a "continuous model" by replacing the detailed 3D structure with a structure approximation of the average behaviour of the tissue to provide a continuous description of structural variation (tissue considered as a continuum). To this purpose they chose their parameters in such a way as to minimize the differences between the solutions of the two models. Obtaining a continuous model is very important as it can be used in further studies for organ modelling. Its value consists in the fact that it actually predicts how a specific area will behave even if we do not take into account the detailed measurements we have made. This means that we could apply this model "safely" for regions that have not been structurally characterized.

In the second supplemental movie of the above publication and also in the third figure of the paper, the results of simulations for the discontinuous (upper panel) and the continuous model (lower panel) are shown. The reconstructed tissue block was stimulated at twice diastolic threshold by applying transmembrane current for 2 ms at its center.

The propagation of the electric signal in the block is shown. Transmembrane potentials are plotted through time (0-22ms) on 7 equally spaced planes in the tissue block (the color code is mentioned in the third figure). The site of stimulation is marked with a black dot at the center of the block.

Note: As mentioned in the study, the computational time required to obtain the solutions of the model was approximatively 85 hours of CPU time with 8 processors of an SGI Origin 2000 (250 MHz R10000 CPU units).

References

Crampin, E. J., M. Halstead, et al. (2004). "Computational physiology and the Physiome Project." Exp Physiol 89(1): 1-26.
Hooks, D. A., K. A. Tomlinson, et al. (2002). "Cardiac microstructure: implications for electrical propagation and defibrillation in the heart." Circ Res 91(4): 331-338.
Linge, S., J. Sundnes, et al. (2009). "Numerical solution of the bidomain equations." Philos Transact A Math Phys Eng Sci 367(1895): 1931-1950.