Analysis of the Underlying Mechanism of Frank-Starling Law of a Constructive Hemodynamics Model

Transcription

1 Analysis of the Unerlying Mechanism of Frank-Starling Law of a Constructive Hemoynamics Moel Mitsuharu Mishima 1, Takao Shimayoshi 2, Akira Amano 3, Tetsuya Matsua 1 1 Grauate School of Informatics, Kyoto University, Japan 2 ASTEM Research Institute of Kyoto, Japan 3 Department of Bioinformatics, Ritsumeikan University, Japan Abstract Frank-Starling law, which escribes a relation between cariac contraction energy an en-iastolic volume, is of importance, but the etaile mechanism is not quantitatively unerstoo yet. In this paper, the mechanism of a linear relation between en-iastolic volume an en-systolic pressure as a part of Frank-Starling law is analyze by means of computer simulation. A hemoynamics moel, which is constructe by composing a vascular system moel, a left ventricular ynamics moel an a myocarial cell moel, reprouce a linear relation between en-iastolic volume an en-systolic pressure successfully. In this paper, the simulation results an the etaile analysis are reporte. Keywors myocarial cell moel, excitation-contraction coupling moel, hemoynamic simulation, Frank-Starling law, circulatory equilibrium 1 Introuction It is important to unerstan the etaile mechanisms of the heart, which is an organ to circulate bloo into the whole boy. A well known law on a cariac function, Frank- Starling law[1] escribes that the energy of myocarial contraction is proportional to the en-iastolic volume within the physiological range. The mechanism is qualitatively explaine as the following[2]; 1) as the bloo inflow to the heart increases, en-iastolic volume increases an ventricular wall is strongly stretche, 2) the resting myocarial length is stretche, thus myocarial tension increases accoring to length-tension relation, 3) as a result stroke volume increases, an the cariac stroke work increases finally. Another important relationship on the cariac function is en-systolic pressure volume relation (ESPVR), which is expresse by the line connecting the point at en-systole of pressure-volume iagram an the points of other loas. ES- PVR of a human heart is linear in physiological conitions, an the slope is calle E max [3], which is an important clinical inex of cariac function. Pressure-volume area (PVA) is the area surroune by a pressure-volume loop of a cariac cycle an ESPVR. PVA inicates the total energy of external mechanical work an mechanical potential energy[4]. Although Frank-Starling law is an important empirical law to represent the relation between the loa of heart an cariac output, the unerlying mechanism is not well unerstoo yet. ESPVR an PVA show relationship between ensystolic pressure an cariac energy consumption. Experimental results show a linear relation between en-iastolic volume an en-systolic pressure. This linear relation is an important factor of Frank-Starling law. Therefore, a etaile analysis of the linear relation between en-iastolic volume an en-systolic pressure helps unerstaning the mechanism of Frank-Starling law. Computer simulation is an important technique to analyze an unerstan the mechanism of biological functions. In this stuy, the linear relation between en-iastolic volume an en-systolic pressure is analyze by a circulation ynamics simulation. The moel employe in the simulation is not mae with a system ientification approach, but compose of physiologically valiate moels. In this paper, the simulation results an the analysis are reporte. 2 Moel an Simulation Result 2.1 Hemoynamic moel In this stuy, a human infantile hemoynamics moel propose by Nobuaki et al.[5] (Nobuaki moel) was use with some moifications. Nobuaki moel, which can simulate baroreceptor reflex against moulations of the hea-up tilt angle of the boy, is compose of a vascular system moel, a left ventricular ynamic moel an a myocarial cell moel. The vascular system moel represents the ynamic change of systemic bloo pressure, bloo flow an left ventricular volume. The left ventricular ynamics moel efines the relationship between the left ventricular volume an pressure, where the volume is relate to the length of a myocarial cell an the pressure to tension by a myocarial cell. The myocarial cell moel expresses an the evelope tension of a myocarial cell. The vascular system moel in Nobuaki moel is base on an ault vascular system moel by Helt et al.[6] The vascular system is expresse as an electric circuit, where a bloo pressure is represente as a voltage an a bloo flow as a current. The circuit consists of twelve compartments such as the heart, the aorta, the peripheral circulation, the vein an the pulmonary circulation. The each compartment is compose of a resistance, a linear capacitance, an a non-linear an time-variable capacitance. The left ventricular volume is expresse as the ifference between the total bloo volume an the bloo volume of all compartments other than the left

2 Y4 T T* Y Y1 Z1 Y Z3 Y3 TCa Z2 Y2 TCa* Figure 1: Four chemical states of Negroni-Lascano moel ventricle. Moulation of the boy tilt angle varies the loa of the heart through change of the bloo istribution. In the current stuy, the arenergic effect of baroreceptor reflex is remove from original Nobuaki moel. In the left ventricular system moel, the relationship between the left ventricular pressure an the tension of the myocarial cell is expresse with Laplace formula: P h = F R, (1) where P is the left ventricular pressure, h is the left ventricular wall thickness, F is the myocarial cell tension an R is the left ventricular raius. The relation between the left ventricular volume an the half sarcomere length of a myocarial cell is moifie from the original formulation in orer to fit experimental ata an is formulate as L = c 1 V + c 2, (2) where L is the half sarcomere length, V is the left ventricular volume, an c 1 an c 2 are constants. Kyoto moel[7] which is a comprehensive physiological moel of a myocarial cell is use as myocarial cell moel. Kyoto moel precisely expresses membrane excitation an excitation-contraction coupling as orinary ifferential equations by moeling each cellular component function such as ion channels, calcium buffering an myofibril. Kyoto moel incorporates with a cariac muscle moel reporte by Negroni an Lascano (Negroni-Lascano moel) [8] with some moifications for excitation-contraction coupling. 2.2 Excitation-contraction coupling moel The excitation-contraction coupling moel of Kyoto moel escribes cellular contraction mechanism as a four chemical states of troponin C as shown in Fig. 1; free troponin C (T ), calcium ion boun troponin C (T Ca), calcium ion boun troponin C with attache cross-briges (T Ca ), an troponin C without calcium ion but with attache cross-briges (T ). The net rates of state transitions are expresse as following. T T Ca : Q 1 = Y 1 [ ] p(t ) Z 1 p(t Ca), (3) T Ca T Ca : Q 2 = Y 2 f OL (L) p(t Ca) Z 2 p(t Ca ), (4) T Ca T : Q 3 = Y 3 p(t Ca ) Z 3 [ ] p(t ), (5) T T : Q 4 = Y 4 p(t ) + Y f CB (X/t) p(t ), (6) T Ca T : Q 5 = Y f CB (X/t) p(t Ca ), (7) where Y 1 -Y 4, Y, Z 1 -Z 3 are rate parameters as efine below. f OL (L) is an overlap function to represent probability of state of T Ca which can be use to form cross-briges an efine as { f OL (L) = exp 20 (L 1.17) 2}. (8) f CB (X/t) is efine as f CB (X/t) = { (X/t) 2 /50 (X/t > 0) (X/t) 2 (X/t 0), (9) t X = B ((L X) h c), (10) where B is a proportional constant an h c the equivalent cross-briges elongation, an L X represents cross-brige elongation an X/t the velocity of motion of the mobile cross-brige en. The rate constants epen on ATP concentration factors as the following: Y 1 = 31.2 [mm/msec] (11) Z 1 = 0.06 [msec 1 ] (12) ( ) K P I i Y 2 = K P I i + [P I] Z 2 = [msec 1 ] (13) 1 ( ) 3 [msec 1 ] (14) 1 + K AT P i [AT P total ] Y 3 = 0.06 [msec 1 ] (15) Z 3 = 1248 [mm/msec] (16) 1 Y 4 = 0.12 ( ) K AT P i [AT P total ] [mm/msec] (17) 1 Y = 8000 ( ) K AT P i [AT P total ] [msec/µm 2 ], (18) where K P I i is a issociation constant for PI an K AT P i is a issociation constant for ATP. The erivatives of state

3 pressure[mmhg] P time[msec] V volume[ml] Figure 2: The time courses of pressure an volume at α = 0 probabilities are expresse as t p(t ) = Q 1 + Q 4 + Q 5 (19) t p(t Ca) = Q 1 Q 2 (20) t p(t Ca ) = Q 2 Q 3 Q 5 (21) t p(t ) = Q 3 Q 4. (22) The evelope tension of elongation element (F b ) is expresse as a prouct of cross-briges concentration an cross-brige elongation: F b = A [T t ] (p(t Ca ) + p(t )) (L X), (23) where A is a constant an [T t ] is total troponin concentration. The force evelope by the parallel elastic element (F p ) is expresse as F p = { K l (0.97 L) (L < 0.97) K l (0.97 L) K p (L 0.97) 5 (L 0.97), (24) where K l (= 20[mN/mm 2 /µm]) an K p (= 140, 000 [mn/mm 2 /µm 5 ]) are the intracellular passive stiffness. Finally, the tension of myocarial cell (F ext ) is expresse with F b an F p as F = F b F p. (25) 2.3 Simulation result Simulations were conucte uner multiple boy tilt angles (α) from 0 to 90 at every 15. The change of tilt angle alters the systemic bloo pressure an cariac workloa. To convergent to the steay state, 40 cariac cycles were simulate an the final cariac cycles were analyze. The simulations were performe with a simulation software simbio [9]. Fig. 2 shows the time course of pressure an volume at α = 0. Fig. 3 shows pressure-volume curve uner iffer- pressure[mmhg] P(t es )[mmhg] ESPVR volume[ml] Figure 3: The pressure-volume curve an ESPVR α= V(t e )[ml] Figure 4: The relationship between V (t e ) an P (t es ) ent values of α. ESPVR is illustrate with a soli line as the relation between pressure an volume at en-systolic time (t es ). The obtaine ESPVR is linear with the coefficient of etermination R 2 = Fig. 4 shows a relationship between en-iastolic volume V (t e ) an en-systolic pressure P (t es ). The ots in Fig. 4 represent the result of each α, an the straight line is the regression line. The relation between V (t e ) an P (t es ) is linear with R 2 = Analysis The etails of the simulation results were analyze to explain the linear relation between en-systolic pressure an en-iastolic volume. The left ventricular volume has a linear relation with the half sarcomere length of the myocarial cell as efine in Eq. 2. Although the left ventricular pressure epens not only on the tension of the myocarial cell but also on the wall thickness an the raius as efine in Eq. 1, the wall thickness an the raius have a weak epenency on the myocarial cell tension. Consequently, the relation between the half sarcomere length at en-iastole an the tension of the myocarial cell at en-systole is approximately linear (R 2 = 1.000) as shown in Fig. 5. As efine in Eq. 25, the tension of the myocarial cell is a ifference of F b an F p. Because the non-linear term of

4 α=45 µ α=45 µ Figure 5: The relationship between L(t e ) an F (t es ) Figure 7: The relationship between p(t Ca ) + p(t ) at t es an L(t e ) µ Figure 6: The time course of L Table 1: The coefficients of eterminations of p(t Ca ) an p(t ) time[msec] R 2 of p(t Ca ) R 2 of p(t ) t es Eq. 24 is much smaller than the linear term in the range of the half sarcomere length (L) in the simulation results (Fig. 6), F p is approximately linear with L. Therefore, F b at ensystole (F b (t es )) has a linear relation with the half sarcomere length at en-iastole (L(t e )). With respect to Eq. 23, the value of L X at en-systole is close to h c accoring to Eq. 10, so L X can be regare as constant. Because A an [T t ] are constant, p(t Ca )+p(t ) at t es has an approximately linear relation (R 2 = 0.999) with L(t e ) as shown in Fig. 7. In the simulation results, both p(t Ca ) an p(t ) have linear relations with L(t e ) at any time from 0 to t es. The coefficients of eterminations at several points are shown in Table 1. This gives that t p(t Ca ) an tp(t ) have also linear relations with L(t e ) at any time from 0 to t es. From the moel efinition escribe in sec 2.2, t p(t Ca ) + p(t ) t = Q 2 Q 4 Q 5 = Y 2 f OL (L) p(t Ca) Z 2 p(t Ca ) Y 4 p(t ) Y f CB (X/t) {p(t Ca ) + p(t )}. (26) In this equation, the former three terms are ominant as shown in Fig. 8, 9. Y 2, Z 2 an Y 4 are almost constant as shown in Fig. 10, because [AT P total ] an [P I] are almost constant in physiological conitions in Kyoto moel. In the simulation result, p(t Ca) is nearly inepenent of α as shown in Fig. 11. As a Taylor expansion of Eq. 8 aroun a value L m, f OL (L) can be expresse as f OL (L) = f OL (L m ) 40(L m 1.17) f OL (L m ) (L L m ) + 40 { 40(L m 1.17) 2 1 } f OL (L m ) (L L m ) 2 +. (27) In the simulation results, the maximum variation of L with the change of α is as shown in Fig. 6. In this range of L, the terms of secon an more egrees in Eq. 27 can be omitte by setting L m the meian value of the range, so f OL (L) is an approximately linear function of L. Although the time courses of L epen on α, the transition shapes are almost ientical, when they are normalize as the maximum to be 1 an the minimum to be 0 as shown in Fig. 12. Thus, L can be approximately expresse as L α (t) = { a 1 L 0 (t) + b 1 } L(te ) + a 2 L 0 (t) + b 2, (28) where L α enotes L at α, an a 1, b 1, a 2 an b 2 are constants. Therefore, f OL (L) has an approximately linear relation with L(t e ). Consequently, Eq. 26 is approximately linear with L(t e ).

5 Figure 8: The time course of terms of Eq. 26 at α = 0 Figure 10: The time course of Y 2, Z 2 an Y 4 α α α Figure 9: the time course of terms of Eq. 26 at α = 90 Figure 11: The time course of p(t Ca) 4 Conclusion In orer to unerstan the unerlying mechanism of Frank- Starling law, we analyze in epth simulation results of a constructive hemoynamic moel, which successfully reprouce a linear ESPVR an a linear relation between ensystolic pressure an en-iastolic volume. The analysis inicates two important factors for the linear relation between en-systolic pressure an en-iastolic volume; 1. microscopical linearity of the force-length relationship of a myocarial cell, 2. inepenency of the transitional shape of a half sarcomere length from the cariac loa. In the present stuy, the linear relation was analyze by ecomposing the simulation results. As a future work, we plan a mathematical proof of the linear relation between the evelope tension an the half sarcomere length on the excitation-contraction coupling moel. In aition, an analysis of the linear ES- PVR is also a future work. References [1] S. W. Patterson an E. H. Starling. On the mechanical factors which etermine the output of the ventricles. J. Physiol, 48(5): , [2] Toshinori Hongo an Tsutomu Hiroshige. Hyojunseirigaku 5th eition. Igakushoin, [3] H suga, K Sagawa, an A. A. shoukas. Loa inepenence of the instantaneous pressure-volume ratio of the canine left ventricle an effects of epinephrine an heart rate on the ratio. Circulation Research, 32: , [4] Hiroyuki Suga, Miyako Takaki, Yoichi Goto, an Kenji Sunagawa. Shinzourikigaku to enajethikusu. Koronasha, [5] Yutaka Nobuaki, Akira Amano, Takao Shimayoshi, Jianyin Lu, Eun B. Shim, an Tetsuya Matsua. Infant circulation moel base on the electrophysiological cell moel. IEEE EMBS, [6] Thomas Helt, Eun B. Shim, Roger D. Kamm, an Roger G. Mark. Computational moeling of cariovascular response to orthostatic stress. J Appl Physiol, 92: , [7] Masanori Kuzumoto, Ayako Takeuchi, Hiroyuki Nakai, Chiaki Oka, Akinori Noma, an Satoshi Matsuoka. Simulation analysis of intracellular Na + an Cl homeostasis uring β1-arenergic stimulation of cariac myocyte.

6 Figure 12: The time course of normalize L Progress in Biophysics an Molecular Biology, 96: , [8] Jorge A. Negroni an Elena C. Lascano. Concentration an elongation of attache cross-briges as pressure eterminants in a ventricular moel. J Mol Cell Cariol, 31: , [9] Sarai N, Matsuoka S, an Noma A. simbio: A java package for the evelopment of etaile cell moels. Biophysics an Molecular Biology, 90: , 2006.