Rice, Winslow, and Hunter developed three further tension development models to evaluate cooperativity mechanisms [325]. The model of Peterson-Hunter-Berman and Landesberg-Sideman were named model 1 and model 2, respectively. The three new models of Rice-Winslow-Hunter were named model 3, 4, and 5. These models include the dependency of tension development by [Ca2+]i and the sarcomere length SL. Moreover, they include in each model a different number of cooperativity mechanisms. Model 3 considers the binding of calcium to troponin and the subsequent shifting of tropomyosin. Model 4 and 5 extend this feature by adding more cross-bridge binding states per troponin-tropomyosin complex.
7.5.1 Model 3
This model including the Tm–Tm cooperativity mechanism describes the actin myosin inter-action with six states divided into two groups: T and TCa as well as N0, N1, P0, P1 (fig. 7.3).
T and TCa express the binding of calcium to troponin, the other four states are the same as the ones from the Peterson-Hunter-Berman model. TCa represents the binding between tro-ponin and calcium for the trotro-ponin group, whereas the symbol T describes unbound trotro-ponin.
The influence of XBs on the affinity of calcium for troponin is neglected in this model. The Tm–Tm cooperativity mechanism, which considers the overlapping of tropomyosin strands, is incorporated by changing the dependency rate of troponin to the shifting of tropomyosin.
The mathematical equation describing the relation between the states are:
T + T Ca = 1 N0 + N1 + P 0 + P 1 = 1
The transition rates from unbound troponin T to bound troponin T Ca and vice versa are represented by the coefficient kon and kof f, respectively. The transition is described by a first order differential equation, considering [Ca2+]i:
7.5. Rice-Winslow-Hunter Models 97
1 XB
Ca2+ N1
k−1 1
k k
N0
P0
k−1
on off
T
TCa
k (TCa,SL) k (TCa,SL)1
g (SL)10
P1
g (SL)10
f01 f
g (SL) f
P2 P3
12
g (SL)21
23
32
0 XBs 2 XBs 3 XBs
Fig. 7.4. State diagram of the 4th Rice-Winslow-Hunter model with Tm–Tm and XB–XB cooperativity mechanisms. The diagram shows the extension of the former model with two additional states, which represents the number of cross-bridges. Iterative coupling between XBs indicates the support of an existing cross-bridge on the occurrence of further XBs.
d dt
T T Ca
= −kon([Ca2+]i) kof f
kon([Ca2+]i) −kof f
T T Ca
The transition rates between N0 and P0 as well as between N1 and P1 are described with the coefficients k1 and k−1, respectively, which depend on sarcomere length and T Ca as the integration of the Tm–Tm cooperativity mechanism (fig. 7.3). For the transition from P0 to P1 and vice versa, the coefficients f and g, respectively, are used, whereas for the transiton from N1 to N0 solely g is present. The coupling between tropomyosin states can be described by the first order differential equation:
d dt
N0 P 0 P 1 N1
= M
N0 P 0 P 1 N1
with the 4 × 4 transition matrix M :
M =
−k1 k−1 0 g
k1 −k−1− f g 0 0 f −g − k−1 k1
0 0 k−1 −k1− g
k1 is a function of the sarcomere length and bound troponin. The parameters are detailed in tab. 7.3. The resulting normalized tension Tn is calculated by eqs. 7.3 and 7.4.
7.5.2 Model 4
This model is an extension to model 3 considering the Tm–Tm and the XB–XB cooperativity mechanism. The XB–XB cooperativity mechanism is integrated into the model by adding two further XB states P2 and P3, which describe the number of possible interacted XBs per functional troponin-tropomyosin complex. The second and the third XB depends iteratively on the first XB with shifted tropomyosin (fig. 7.4).
The changes of the maximum tension during fixed calcium concentration but varying sarcom-ere lengths is reproducible using model 4 compared to model 3. A sarcomsarcom-ere length dependent unbinding rate of the cross-bridges g(SL) is incorporated to consider these findings.
The slowing of the relaxation due to several XB bindings (1–3) is prevented by an increasing unbinding rate (g10, g21, g32). Another effect of the addition of two further states is the possibility that tropomyosin can be held in the opened state by several cross-bridges. This is implemented in the model by preventing any transitions from P 2 or P 3 directly to the N states.
The rising probability of building cross-bridges due to cross-bridges, i.e. the XB-XB coopera-tivity mechanism, is realized in this model by a rising building rate (f01 – f23) in dependance of the amount of already existing and possible XBs.
The first order differential equation for this model is:
d dt
N0
P 0 P 1 N1 P 2 P 3
= M
N0 P 0 P 1 N1 P 2 P 3
with the 6 × 6 transition matrix M , which is dependent on the bound troponin and the sarcomere length:
M =
−k1 k−1 0 g10 0 0
k1 −k−1− f01 g10 0 0 0
0 f01 −g10− k−1− f12 k1 g12 0
0 0 k−1 −k1− g01 0 0
0 0 f12 0 −g21− f23 g32
0 0 0 0 f23 −g32
The factors of the transition matrix are depicted in tab. 7.3. The binding of calcium to troponin is described equivalent to model 3. The developed normalized tension of this model is weighted with the additional cross-bridge states:
Tn= α(P 1 + N1 + 2P 2 + 3P 3) Tmax
with the overlap function α of the Peterson-Hunter-Berman model (eq. 7.4).
n
N1
k−1 1
N0
P0
k−1 k (TCa,SL)
k (TCa,SL)1
g (SL)10
P1
g (SL)10
f01 f
g (SL) f
P2 P3
12
g (SL)21
23
32
0 XBs 1 XB 2 XBs 3 XBs
kon off
T
TCa
Ca2+
k (T )
Fig. 7.5. The state diagram of the 5th Rice-Winslow-Hunter model with Tm–Tm, XB–XB, and XB–Tn cooperativity mechanism includes the influence of cross-bridge on the troponin interaction with calcium, as the XB–Tn cooperativity mechanism.
7.5. Rice-Winslow-Hunter Models 99 7.5.3 Model 5
The fifth model is a modification of model 4. No changes in the amount of states is found (fig. 7.5). This model including the Tm–Tm and XB–XB cooperativity mechanisms integrates the XB–Tn cooperativity mechanism by including a feedback of the tension generating states N1, P 1, P 2, and P 3 to the unbinding rate of calcium from troponin. With this feature the new calcium release coefficient k0of f is defined as:
kof f0 = kof f(1 − Tn/2)
This causes a strong feedback effect of tension on the binding rate between troponin and calcium.
The system of equations describing the model is the same as in model 4 (eqs. 7.5.2 – 7.5.2) using the parameters listed in tab. 7.3. Two slightly different versions of the 5th model are described.
Table 7.3. Transition coefficients of the three Rice-Winslow-Hunter models.
Rice 3 Rice 4 Rice 5
kon 40 µM−1s−1 20 µM−1s−1 20 µM−1s−1
kof f 20 s−1 40 s−1 40 s−1
1. vers.: kof f’=kof f(1–T/2) 2. vers.: kof f’=kof f(1–3T/4) k1 k−1(T Ca/K1/2)N k−1(T Ca/K1/2)N k−1(T Ca/K1/2)N
k−1 45 s−1 45 s−1 45 s−1
f 10 s−1 10 s−1 10 s−1
1 XB: f01=2f f01=3f,
2 XB: f01=2f, f12=7f f12=10f, 3 XB: f01=3f, f12=10f, f23=7f f23=7f
g 20 s−1 g01(SL)=g(SL) g01(SL)=g(SL)
g12(SL)=2g(SL) g12(SL)=3g(SL) g23(SL)=2g(SL) g23(SL)=3g(SL)
g(SL) g*[1+(1–SLnorm)1,6] g*[1+(1–SLnorm)1,6]
1 XB: g*=25 s−1 1 XB: g*=25 s−1 2 XB: g*=27,5 s−1 2 XB: g*=27,5 s−1
3 XB: g*=30 s−1 3 XB: g*=30 s−1 F α(P1+N1)/Fmax α(P1+N1+2P2+3P3)/Fmax α(P1+N1+2P2+3P3)/Fmax
Fmax f/(f+g) P1max+2P2max+3P3max P1max+2P2max+3P3max
P1max=f01g2,1’g3,2’/P
P1max=f01g2,1’g3,2’/P P2max=f01f12’g3,2’/P
P2max=f01f12’g3,2’/P P3max=f01f12’f23’/P
P2max=f01f12’g3,2’/P P g*1,0g*2,1g*3,2+f01g*2,1g*3,2+ g*1,0g*2,1g*3,2+f01g*2,1g*3,2+
f01f12g*3,2+ f01f12f23 f01f12g*3,2+f01f12f23
g*i,j gi,jg* gi,jg*
K1/2 (1+KCa/(1,5 µM – (1+KCa/(1,5 µM – (1+KCa/(1,8 µM – SLnorm1,0 µM ))−1 SLnorm1,0 µM ))−1 SLnorm1,0 µM ))−1
N 7+3 SLnorm 5+3 SLnorm 1. vers.: 3,4+1,4 SLnorm
2. vers.: : 2,6+SLnorm
SLnorm (SL–1,7 µM ) (SL–1,7 µM ) (SL–1,7 µM )
/(23 µM –1,7 µM ) /(23 µM –1,7 µM ) /(23 µM –1,7 µM ) KCa kof f/kon=0,5 µM kof f/kon=2 µM kof f/kon=2 µM
8
Mathematical and Physical Basics
Modeling of the electromechanical behavior of the human heart requires knowledge of several mathematical and physical basics as well as numerical methods. They are detailed in this chapter as far as they are used to model the anatomical, electrophysiological, and tension development properties of the heart, or since they are necessary for understanding the results achieved using the simulations.
Most of the biophysical models describing the electrophysiological behavior as well as the tension development are based on so-called Markov models, which describe the process of transition between discrete states of a system. These processes are time dependent and thus necessitate the numerical solution of ordinary differential equations describing the properties of the cells. Coupling of several cells into a net describing the tissue as an electrically coupled system requires knowledge of the theory of electrical fields and the electrical properties of the tissue. The bidomain model, which is used in this work to describe the electrical coupling of cells, is based on the generalized Poisson’s equation for stationary electrical fields. This kind of approach delivers partial differential equations for the excitation conduction. The partial differential equations describing the electrical activity of the cells in combination with the conduction of excitation are discretized with the finite difference or the finite element method and a system of linear algebraic equations is achieved. This linear equation system is solved directly or iteratively. The mathematics behind this modeling is explained in the first part of this chapter.
Recently published measurement data has to be considered in order to generate new or to adapt available electrophysiological and tension development models. The finding of the correct parameters of the model is achieved by comparing the simulated behavior of e.g. an ionic channel to the measured equivalent. In order to optimize the parameters of the model, minimization techniques are utilized as explained in this chapter.
The last section of this chapter deals with the matrix diagonalization method. It is used in this work for determining the first principal axis of anatomical structures whose fibers are arranged in bundles. These structures, e.g. fast conducting pathways in the atrium, have strong anisotropic features. The matrix diagonalization method is used to define the local longitudinal direction of the fibers in a region of interest.