Model cardiac muscle

Weiner and Rosenbluth had used a discrete diffusion model with excitable kinetics to study phenomenologically observed spatial phenomena in cardiac muscle tissue. 

The major drawback of these models, however, is their lack of a clear reference between model components and constituent parts of the biological system (e.g. structures like ion channels, transporter proteins, receptors, etc.). These models, therefore, do not permit the simulation of patho-physiological detail, such as the series of events that follows a reduction in oxygen supply to the cardiac muscle and, ultimately, causes serious disturbances in heart rhythm. 

These detailed cell models can be used to study the development in time of processes like myocardial ischaemia (a reduction in coronary blood flow that causes under-supply of oxygen to the cardiac muscle), or effects of genetic mutations on cellular electrophysiology. They allow to predict the outcome of changes in the cell s environment, and may even be used to assess drug actions. 

The principle of thermal recycling is also used in reactors with a boiling layer, in which the heat from the hot region of the reactor is transported to the cold region by circulating solid particles suspended in the gas flow.15 Methods of the theory of chemical reactor regulation have been successfully used in other sciences as well. We note the model of Belousov-Zhabotinskii, proposed for the description of heart disease, of spasmatic contractions of the cardiac muscle. 

Fig. 13. The titin Z-repeats and a Z-band assembly model. The N-termini of titin filaments from adjoining sarcomeres overlap in the Z-band. This part of titin comprises a modular region of so-called Z-repeats, each about 45 residues long, the number of which is related to fiber type 24 repeats occur in fast muscles, 57 occur in slow and cardiac muscles. This correlates with the Z-band appearance since fast and slow fibers have narrow and wide Z-bands, respectively, as shown in Fig. 12. The measured axial spacing between a-actinin bridges is about 19.2 nm (Luther and Squire, 2002), a distance that is too long for a single Z-repeat (A) to stretch to. Perhaps the bridge separation is related to two levels of Z-repeats (C).

The long-lasting cardiac action potential is a consequence of such an evolutionary compromise in the development of potassium currents in the heart. Early work on these channels [5] showed that they could be divided into two classes channels that close on depolarization, Iki, and channels that open during depolarization, including the various components of Ik and the transient outward current, it0. At rest, Iki is switched on and holds the resting potential at a very negative level, where the other K+ currents are switched off. On depolarization iki rapidly switches off, while the other currents take time to activate and cause repolarization. This analysis of the potassium channels in cardiac muscle formed the basis of the first biophysically detailed model [6] and remains the basis of all subsequent models [7-9]. 

Then x variable plays in Zeeman s model the role of length of a fibre of the cardiac muscle while the b variable corresponds to the electrochemical control (contraction of the cardiac muscle is triggered by a biochemically generated electric impulse). A stable stationary point E may occur near the point B which is infinitely sensitive to perturbations. To transfer the system from the stable stationary point E to B, a perturbation of the system is required if E is located close to B the perturbation can be small. The mechanism of switching the heart from the state of equilibrium E (lack of heartbeat) to the state of action involves removing the system from the state E to B by way of stimulation, for example by an electric impulse. On reaching the state B the model system imitates the heartbeat — this is the trajectory BB CC E. A subsequent cycle requires the repeated stimulation at the point E. 

PaUadino, J.L. 1990. Models of Cardiac Muscle Contraction and Relaxation. Ph.D. dissertation, University of Pennsylvania, Philadelphia. Univ. Microforms, Ann Arbor, Ml. 

Among the most important developments of the past 50 years has been the creation of a series of mathematical and numerical models of the time course and voltage dependence of the change in membrane permeabilities. These membrane models originally described the behavior of nerve membrane and has now been extended to many other tissues, such as skeletal muscle and cardiac muscle. 

FIGURE 54.6 Cardiac muscle isometric twitch tension generated by a model of rat cardiac contraction (courtesy Dr. Julius Guccione) (a) Developed twitch tension as a function of time and sarcomere length (b) Peak isometric twitch tension vs. sarcomere length for low and high calcium concentration. 

The above relationships which describe the distributed properties of the mechanical parameters, as well as coronary perfusion and energy demand laws, are the physiological cornerstones for the development of a global 3-D geometrical model. The local properties of the cardiac muscle can be approximated by symmetrical models which utilize distributed properties as a function of y, the radial distance from the endocard. This results in the evaluation of the constants which relate the local phenomena of the cardiac muscle to local structure, stress, strain and strain rate of the muscle. 

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