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A Closed-Loop Lumped Parameter Computational Model for Human Cardiovascular System∗ Fuyou LIANG∗∗ and Hao LIU∗∗∗ For purpose of a better understanding of the behavior of the global hemodynamic interactions, a closed-loop lumped parameter computational model was developed for the human cardiovascular system with a detailed compartmental description of the heart and the main vascular circulations. Construction of the model was implemented based on a phenomenological characterization of hemodynamics using an electrical analog method and solution of the governing differential equations of the model was carried out by use of a fourth-order Runge-Kutta method. Most of the hemodynamic parameters predicted by the present model were either consonant with the clinical measurements or within reasonable physiological ranges. Furthermore, the present model was applied to predict the clinical cardiac hemodynamic characteristics observed in patients with heart abnormalities. Reasonable agreements between predictions and measurements indicate that the present computational model can serve as a useful assistant tool for computer-aided diagnosis and surgical treatment, as well as posttreatment prediction.
Key Words: Human Cardiovascular System, Regulatory Behavior, Closed-Loop, Lumped Parameter Computational Model, Hemodynamic Interaction
1.
Introduction
The human cardiovascular system is a combination of a series of subsystems that structurally connect to and functionally interact with each other. Therefore, it is of significance to take into account the effect of the global cardiovascular system in studies focused on hemodyamics in some specific cardiovascular segments. Due to the highly complicated structure of the cardiovascular system, high dimensional numerical approaches are always limited to studies on local hemodynamics, in contrast, lumped parameter models, which mathematically represents the characteristics of hemodynamics with a series of parameters, may provide a quantitative insight into the behavior of the global hemodynamics. There have been various such models varying in complexity and form depending on study purpose, ranging from very simple open-loop ones to complex multi-compartmental closed∗ ∗∗
∗∗∗
Received 26th May, 2005 (No. 05-4044) Graduate School of Science and Technology, Chiba University, 1–33 Yayoi-cho, Inage-ku, Chiba 263–8522, Japan. E-mail:
[email protected] Department of Electronics and Mechanical Engineering, Chiba University, 1–33 Yayoi-cho, Inage-ku, Chiba 263– 8522, Japan. E-mail:
[email protected]
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loop ones. Sun, et al. (1) proposed a computational model for right-left heart interaction and good agreements between predictions and clinical observations were obtained. Lu, et al. (2) developed a mathematical model for the human cardiopulmonary system. In their work, the cardiopulmonary system was divided into several subsystems arranged in series and in parallel to form a closed circulatory loop. A mathematical model for the cardiovascular system and its interaction with carotid baroregulation was proposed by Ursino(3) , who, in his study, separated the peripheral circulation into splanchnic and extrasplanchnic circulations in consideration of the difference in their responses to nervous control. Heldt, et al. (4) constructed a closed-loop seven-compartment computational model for the human cardiovascular system in their study on the cardiovascular response to orthostatic stress. All of their computational models were evinced to be effective in representing hemodynamic parameters within reasonable physiological ranges by good fits of predictions to clinical measurements. Nevertheless, most of their studies were either focused exclusively on left-right heart interactions by assuming a simplified representation of vessel system or concentrated on hemodynamics in some specific compartments with the remainder neglected or simplified, none of the models simultaneously embodied JSME International Journal
485 a detailed description of both peripheral circulation and cardiac-pulmonary circulation. Inasmuch as the fact that the human cardiovascular system is an integrated closedloop system, a better understanding of the nature of the hemodynamic interactions and the mechanisms by which the system operates and regulates in response to hemodynamic perturbations requires simultaneously synthesizing the main cardiovascular compartments in one computational model. In this sense, these models mentioned above are not likely fit for investigating the dynamic hemodynamic interactions in rigorously quantitative terms due to the lack of a comprehensive inclusion of the main cardiovascular subsystems. The purpose of the present study is to develop a comprehensive computational model that provides reasonable predictions of hemodynamics in the main cardiovascular circulations and yields a mathematical tool for quantitatively investigating the behavior of the global hemodynamic interactions. The present model also contributes a basis for further work in which the function of the autonomic nervous system is involved to form an integrated computational model capable of predicting hemodynamic regulations against various perturbations. The right-left heart interaction model proposed by Sun, et al. (1) was adopted as the basis for developing the present model. The peripheral circulation was partitioned into four parallel regional circulations branching from the aorta at four specific sites. In modeling each regional circulation, distinctions were made among arteries, arteriolar and capillary beds and veins to account for the time-dependent transfer of hemodynamic parameters through different vascular systems. In addition, several modifications were made for the venous elastance model and the time-variant elastance heart model to reflect the volume related change in venous compliance and the variation in systolic duration with a variable heart rate. Moreover, considering the influence of intrathoracic pressure on venous transmural pressure, the vena cava was separated into two portions, one inside and another outside the thoracic chamber. Where available, values of the parameters involved in the present model are mainly determined on the basis of previous reports, whereas values parameters not derivable from literature have been estimated based on either the anatomical data or the general physiological knowledge. Varying degrees of adjustments have been made for some parameters to achieve reasonable fits of predictions to clinical measurements. 2.
Methods
The whole cardiovascular system can be approximately divided into seven subsystems including the left and right hearts, the upper limb and the cerebral circulations, the renal circulation, the splanchnic circulation, the lower limb circulation and the pulmonary circulation as JSME International Journal
illustrated in Fig. 1. Each of the above circulations is a combination of arteries, arteriolar and capillary beds, and veins that connect to each other in series. The pulsating heart provides power for the whole circulatory system by rhythmically pumping blood to the peripheral circulation where through arteries, arterioles and capillaries nutrient blood is distributed to organs, tissues and skeletal muscles, then finally is returned back to the right heart by veins. Between the right and left hearts is the pulmonary circulation where blood is oxygenated. These circulations are not functionally independent but closely associated with each other via concurrent hemodynamic interactions. In order to construct an efficacious computational model for the system, a lot of simplifications have to be made because the details of the system are too complicated to be modeled quantitatively. The simplest and most effective way to mathematically represent this complex system is the lumped parameter modeling method where an electrical analog is frequently employed since hemodynamic parameters such as blood pressure, flow and resistance can be well mimicked by the corresponding electrical parameters including voltage, current and resistor. Figure 2 (b) shows a three-dimensional computer model of the arterial tree constructed based on anatomical data, which includes 266 arteries with diameters ranging from 2.4 cm to less than 1.5 mm.Such a model may favor a direct unstanding of the structure of the cardiovascular system and provide an anatomical evidence for compartmentalizing the cardiovascular system in the model devel-
Fig. 1
A sketch of the human vascular system(6) Series C, Vol. 48, No. 4, 2005
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Fig. 2 An electrical analog of the human cardiovascular system and a computer model of the arterial tree Table 1 Parameters in the electrical analog circuit
opment. Based on the arterial tree computer model and the seven-compartment concept(4) , the hemodynamic characteristics of the human cardiovascular system are represented by a closed-loop circuit as illustrated in Fig. 2 (a). Herein the hearts are represented by the time-variant elastances, and the peripheral circulations are each mimicked by an assembly of electric elements that characterize the hemodynamics in arteries, arteriolar and capillary beds, and veins, respectively. Other than some previous pure mathematical models, each element in the present model has a specific physical meaning, which may facilitate a physiological or anatomical data based parameter value identification and adjustment. Considering that the number of the parameters and variables involved in the present model is over 153, for the sake of simplification, a glossary is outlined in Table 1 instead of a detailed statement for each of the parameters. LH, RH refer to the left heart and right hearts, AA, DA refer to the ascending and deSeries C, Vol. 48, No. 4, 2005
scending aortas, KID, SPL, LLB, ULB indicate the kidney, the splanchnic organs, the lower limbs and the upper limbs, respectively, V, VC denote the systemic veins and vena cava, and PU means the pulmonary circulation. Subscripts attached to them have general meanings, e.g. a, al, c, v indicate artery, arteriole, capillary and vein, respectively. Value identification of these parameters is in two ways: the first way is to refer to previous literature where available, and the second to roughly estimate the values based on the anatomical data or the general physiological knowledge at the first step and subsequently repeatedly adjust them until reasonable agreements between predictions an in vivo clinical data are obtained. The present model encompasses a number of descriptions of not only arteries and veins but also arteriolar and capillary beds. These vessel systems operate at transmural blood pressures of different levels and with different blood volume baselines, exhibiting remarkable differences JSME International Journal
487 in vascular pressure - volume (P-V) relationship. Moreover, it is also not true that vascular compliance is fixed because of the nonlinear hemodynamic properties. To account for the nonlinear vascular elastic properties of different vascular systems, in this paper, vascular compliance is assumed to have an exponential P-V relationship(1) , which is written as: (1) p = E0 ev/Z · Z −1 Where E0 [mmHg·mL ] denotes the zero-volume elastance, Z [mL] refers to the volume constant, and v [mL] indicates the blood volume contained in vessel. According to Eq. (1), the different vascular elastic characteristics of different vascular systems can be reflected by means of defining different values for E0 and Z. Such a mathematical description of P-V relationship is found to well fit the vascular elastance curves reported by Dahn, et al. (7) 2. 1 Representation of the heart In the present model, the four chambers of the heart are each represented by the use of a time-variant elastance, which varies over a cardiac cycle according to an exponential charge-discharge waveform characterized by a baseline and an amplitude component. The mathematical heart model was initially constructed by Sun, et al.(1) and is adopted in this paper with several modifications made to facilitate the adjustment of Emax (maximum systolic elastance) [mmHg·mL−1 ] and systolic duration, etc. The mathematical representations of the four cardiac chambers are basically in a similar form. Two examples are given below: For the LV (left ventricle), it is given by, −t s0 /τlvc −t/τlvc 1−e · F · 1−e · E lva s 1−e−ts /τlvc +Elvb /F s 0 ≤ t ≤ t s (2) elv = −(t−t s )/τlvr −E / e · e lv t s lvb F s +Elvb /F s t s < t < tr And for the LA (left atrium), it is written as, Elaa 1 − e−(t−tac)/τlac tac ≤ t ≤ tar +E lab ela = (3) −(t−tar )/τlar ela tar − Elab · e +E tar < t < (tr + tac ) lab Where t s , tr [sec] denote the moment of the peak systolic elastance and a cardiac cycle, respectively, tac [sec] refers to the moment when the atrium begins to contract, tar [sec] indicates the moment when the atrium begins to relax, and F s is a scaling factor introduced to reflect the load dependence of myocardial contractility and nolinearity of the starling curve of the left ventricle. A simple first order linear function of F s is defined as: (4) F s = 1 − vlvde /Vmax Where vlvde [mL] indicates the end diastolic volume of the left ventricle and Vmax [mL] is a volume constant. Moreover, the interrelationship between the left and right ventricles is noteworthy inasmuch as that the left and right JSME International Journal
Table 2 Parameters in the heart model(1)
Table 3 Parameters in the assistant equations of the heart model
hearts do not actually work independently but interact with each other through a so-called “talk pressure” across the interventricular septum. In addressing this problem, Maughan, et al. (8) proposed a mathematical model consisting of three elastic compartments, namely, erv , elv and E s [mmHg·mL−1 ]. According to this model, pressures in the left and right ventricles can be expressed as: elv elv · vlv + · prv (5) plv = E s · E s + elv E s + elv and erv erv prv = E s · · vrv + · plv (6) E s + erv E s + erv where E s refers to the elastic compartment representing the shift of the septum. In addition, systolic duration must be redefined when the heart works at a variable heart rate. With this in mind, a sigmoidal function is proposed to relate the moment of peak systolic elastance to a cardiac cycle in the present model; the function is shown below: β4 · (T s,max − T s,min ) (7) T s = T s,min + 4 β 4 +θ T where T s [sec] is the moment of peak systolic elastance, T s,min [sec] and T s,min [sec] are the minimum and maximum values of T s , respectively, β and θ [sec] are the constants controlling the shape of the sigmoidal curve. Listed in Tables 2 and 3 are the values of the coefficients used in the heart model. 2. 2 Venous compliance Other than the arteries, the systemic veins and the vena cava often operate in low-pressure condition. They may even collapse when transmural pressure is reduced to a negative value. In fact, the compliance of vein shows an approximately constant value under low transmural pressures ranging from 0 up to 10 mmHg(11) . However, if the venous transmural pressure is forced to increase to a higher level, for example, during the Valsalva maneuver, vein will stiffen significantly to prevent excessive venous congestion caused by venous blood pooling such that necessary blood return to the heart can be maintained. ThereSeries C, Vol. 48, No. 4, 2005
488 Table 4 Parameters in the venous elastance model
fore, the role of venous compliance regulation is of great importance in determining cardiac output and that maintaining arterial blood pressure. In the present study, several empirical formulas relating venous elastance to venous volume are developed to account for the nonlinear venous P-V relationship. In the formulas, the change of venous elastance is characterized as a series of arrangement of a sigmoidal static characteristic and a linear firstorder dynamic block in time domain such that both volumetric and temporal variations in elastance can be accounted for. Furthermore, we distinguish between the systemic veins and the vena cava regarding their differences in vascular elastic properties. Two groups of equations are constructed. For the systemic veins, they are written as:
5 v(t) − Vvn ) Ev0min + Ev0max · exp( kv (8) Ev0 (v,t) =
5 v(t)− Vvn 1 + exp( ) kv dEv (v,t) −Ev (v,t) + E v0 (v,t) = dt τev
(9)
And for the vena cava, they are given by:
8 v(t)− Vvcn ) Evc0min + Evc0max · exp( kvc Evc0 (v,t) = (10)
8 v(t)− Vvcn 1 + exp( ) kvc dEvc (v,t) −Evc (v,t) + E vc0 (v,t) = dt τevc
(11)
Where subscripts v and vc denote the systemic veins and the vena cava, respectively. Vvn , Vvcn [mL] are the volume constants. Kv , and Kvc [mL] are two parameters controlling the slopes of the static functions at the central point. τev , τevc [sec] are the time constants indicating the timevariant rate. Ev0min , Evc0min [mmHg·mL−1 ] are the minimum elastances, and Ev0max , Evc0max [mmHg·mL−1 ] the maximum elastances. Ev0 (v,t), Evc0 (v,t) [mmHg·mL−1 ] are the static elastances determined by venous volume, and Ev (v,t), Evc (v,t) [mmHg·mL−1 ] are the time-variant elastances. Values of the parameters in the equations are determined by matching the computed volume-elastance curve with the reported data(11) . The parameters are summarized in Table 4. 2. 3 Governing equations and numerical method Based on the electrical analog circuit, the governing equations of the present model can be derived by means of enforcing mass conservation and force equilibrium at the Series C, Vol. 48, No. 4, 2005
(a)
(b)
Fig. 3 Schematics of the basic units in the electrical analog model
nodes of elastance (E) and inertance (L), respectively. In total, 62 differential equations are derived at the 62 corresponding nodes. Because of the limitation of the length of this paper, herein only the general forms of these equations are elucidated. Typically, the general forms can be described by two equations, one is the mass preservation equation and another is the force equilibrium equation, according to Fig. 3 (a) and (b), they are written as follows: At node E: dv = qin − qout (12) dt And at node L:
dvup dq = Eup Zup + S up · − qR dt dt dvdown /L (13) −Edown Zdown − S down · dt Where v [mL] represents the vascular volume, q [mL·sec−1 ] represents the blood flow through a vascular compartment. Definitions of the other parameters can be found in Table 1. Allowing for the nonlinear coupling of the 62 differential equations, the solution of these equations is performed by the use of a fourth-order Runge-Kutta method with adjustable step ranging from 0.5 ms to 2 ms. Equations for other nonlinear elements such as the elastances of the heart, the cardiac and venous valves, etc. are solved in intervals of 5 time steps. Such a numerical scheme is proved numerically stable. The simulation software is completely written in a C++ language and runs on a DELL workstation. 3. Results and Discussion 3. 1 Hemodynamics of the heart Figure 4 illustrates the computed atrioventricular pressure waveforms in the left and right hearts over a cardiac cycle by the use of the present computational model. The shapes of the pressure waveforms match well with the general in vivo measurements, and in addition, the predicted values in terms of both the systolic and the diastolic pressures are within reasonable physiological ranges. JSME International Journal
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(a) Right heart
(b) Left heart
Fig. 4 Simulated atrioventricular blood pressures for the right and the left hearts
(a) Right heart volumetric change (simulation)
(c) Tanstricupid and venous return flows (simulation)
(b) Left heart volumetric change (simulation)
(d) Transmitral and pulmonary venous flow (simulation)
(e) Left ventricular volumetric change (measurement) Fig. 5 Model-based simulations of cardiac hemodynamics
Moreover, the well known a wave, x descent, v wave and y descent on the atrial pressure curves are also accurately reproduced. The computed volume varying curves of the left and right hearts over a cardiac cycle are shown in Fig. 5 (a) and (b). Specifically, for ventricles, the basic property of the volumetric variation over a cardiac cycle characJSME International Journal
terized as a quick reduction in systole and a two-stage expansion in diastole has been truly reproduced by the predicted ventricular volumetric variation curve plotted in Fig. 5. For the purpose of a clinical data based examination of the predictions, measurement on the volumetric variation of the left ventricle over a cardiac cycle is presented in Fig. 5 (e)(10) . The ventricular volume is comSeries C, Vol. 48, No. 4, 2005
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(a) Aorta Fig. 7 Predicted blood flow waveforms through the arteriolar and capillary beds of the upper limb and cerebral circulations
(b) Large arteries Fig. 6 Predicted blood flow waveforms through the aorta and large arteries
puted from the three-dimensional left ventricle model constructed based on a set of ultrasonic medical images taken in a healthy man. According to Fig. 5 (b) and (e), the simulated result is satisfactorily consistent with the measurement. Figure 5 (c) and (d) also shows the model-based simulations of the transtricupid and transmitral flows and the venous flows to the left and right atriums. The typical E, A waves of the ventricular perfusion flow and the S, D waves of the atrial perfusion flow have been reasonably predicted. 3. 2 Transfer of blood flow and pressure waveforms in the peripheral circulation Shown in Fig. 6 (a) and (b) are the computed blood flow waveforms through the aorta and the big arteries branching to the upper limbs and brain, the splanchnic organs, the kidneys, as well as the lower limbs. The computed results are approximately in phase with the general in vivo measurements(12) in terms of the shapes of flow waveforms, the phase difference among blood flows to different circulations and the reverse flows occurring in early diastole. In order to gain a better understanding of the transfer and the change in waveform shape of blood flow through the vascular systems, the model generated blood flow Series C, Vol. 48, No. 4, 2005
Fig. 8 Predicted blood pressures in the aorta and the upper limb arteriolar bed
waveforms through the arteriolar and capillary beds of the upper limb and cerebral circulation are illustrated in Fig. 7. The pulsatile features of the blood flows are found to be greatly dissipated at the arteriolar level and be further damped to a nearly constant flow pattern in the capillary bed. This phenomenon can be mainly ascribed to the presence of large vascular resistances to blood flows in the arteriolar and capillary beds. Evidence can also be found in terms of the change in blood pressure waveform along the vascular systems as shown in Fig. 8, where the mean value of the arterial pressure is reduced from 100 to about 45 mmHg in the upper limb arterioles accompanied by a reduction in the amplitude of pulsatile pressure from about 40 mmHg to approximate 1.5 mmHg. Leading from the capillary beds to the right heart are the systemic veins and the vena cava. Simulated blood flow waveforms through these vascular systems are plotted in Fig. 9, where blood flows are observed to again become more and more pulsatile along the venous system JSME International Journal
491 due to the influence of the pulsating right heart downstream. 3. 3 Model representation of blood volume distribution in the cardiovascular system The model representation of blood volume distribution in the cardiovascular system is summarized in Table 5. Good agreement between the model-based predictions and the data reported in Refs. (13) and (14) implies that the values assigned for the vascular compliances are within reasonable physiological ranges. 3. 4 Heart abnormality Diastolic dysfunction is a common cardiac disease often found in the elderly, which accounts for approximate 30% of heart failure. In patients with isolated diastolic dysfunction, there is often a preserved systolic function, namely, a maintained ejection fraction comparable to the one in the normal case; however, uncontrolled progression of diastolic dysfunction may still be an important causation of the onset and progression of systolic dysfunction. Generally, diastolic dysfunction is associated with the prolonged relaxation and the stiffening of myocardium(9) of left ventricle. From the point of view of model simulation, the duration of relaxation and the stiffness of myocardium(9) of left ventricle can be represented by the left ventricular baseline elastance Elvb , and the time constant τlvr in the heart model, respectively. Via changing the
Fig. 9
Predicted blood flow waveforms through the systemic veins and the vena cava
Table 5 Model representation of blood volume distribution
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values of the two parameters and correspondingly making slight adjustments for other parameters in the heart model, cardiac hemodynamic in various grades of diastolic dysfunction can be simulated with the present model. The cardiac hemodynamics in three grades of diastolic dysfunction was simulated. In Fig. 10, the predicted results are compared with the in vivo Doppler velocity recordings of the transmitral and pulmonary venous flows(5) . Grade I is characterized by the prolonged relaxation, a decreased E wave and an increased A wave of the transmitral flow and the strengthening of the S wave accompanied by a impaired D wave of the pulmonary venous flow are observed. Grade II is characterized by the stiffening of the left ventricle, namely, pseudo-normalization. In this case, increased flow reversal into the pulmonary vein is observed due to the increased diastolic filling pressure. Grade III dysfunction is specified by a severe deterioration of the left ventricle compliance. Under this condition, the A wave of the transmitral flow is greatly shortened in duration and reduced in magnitude, correspondingly, the S wave is decreased, reflecting the presence of a high pressure in the left atrium. Overall, the predictions show reasonable agreements with the measurements. 3. 5 Limitations Limitations of the present model mainly arise from the value assignment for a host of parameters involved in the model. For purpose of reproducing the closed-loop hemodynamic properties of the cardiovascular system, the present model contains a comprehensive representation of the main vascular circulations. Data on the compliances and resistances of large veins and the aorta, and the total resistances of the main vascular circulations can be found in Refs. (2) and (4). Values of parameters in the heart model have been derived from Ref. (1). However, insufficient information on the arteriolar and capillary compartments is available in literature. In the present study, we estimated the arteriolar and capillary resistances based on the known total resistance of each circulation to predict reasonable pressure fall in these vascular levels and adjusted the arteriolar and capillary bed compliances to yield physiologically realistic predictions of blood volume and blood flow and pressure waveforms in these vascular beds. Although these estimations are not likely to pose serious problems to simulation of the global hemodynamics, they may largely limit the achievement of high fidelity prediction of local hemodynamics in arteriolar and capillary beds. In addition, there is also lack of data on the arterial resistances and compliances of the regional circulations. In the present study, arterial resistances were firstly roughly calculated according to the law of fluid mechanics based on the anatomical-data-based 3D computer model of the arterial tree. Subsequently the arterial resistances were adjusted carefully so as to obtain arterial blood flow waveform predictions comparable to the in vivo measureSeries C, Vol. 48, No. 4, 2005
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(a) Normal
(b) Grade I dysfunction
(c) Grade II dysfunction
(d) Grade III dysfunction Fig. 10 Predictions and measurements for transmitral and pulmonary venous flows under diastolic dysfunction conditions
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ments reported in Ref. (12). The computable arterial blood volumes based on the 3D computer model of the arterial tree also provided valuable evidences for the estimation of arterial compliances in consideration of the fact that arterial compliance determines the blood volume contained in arterial compartment. Although most of the parameters estimated partially based on the anatomical data or the general physiological knowledge have been proved able to provide reasonable predictions of hemodynamics, direct experimental data is still imperative if a more deliberate model were to be developed. As to the application of the present model, we have to proclaim that the present lumped parameter model is able to predict hemodynamic parameters within reasonable physiological ranges but not with high accuracy. In fact, we have sought to construct a computational model of the cardiovascular system capable of providing rational investigations of the global hemodynamic behaviors as well as the reciprocal hemodynamic interactions among the regional circulations. High dimensional simulation of local hemodynamics seems to possess high accuracy. Nevertheless, applications of high dimensional simulation are always limited to some specific cardiovascular segments. Moreover, allowing for the high dependence of prediction on boundary conditions and the active change in hemodynamics with physiological state, the high accuracy of high dimensional simulation may be greatly compromised if simulation is performed with fixed boundary conditions. At this point, low dimensional computational approaches like the present model, though only able to provide low dimensional, or to say the least, low accuracy predictions of hemodynamics, may favor an integrative comprehension of the global hemodynamic behaviors under various physiological conditions. Moreover, in consideration of the fact that the present model was constructed based on the baseline hemodynamic parameters, the present model is limited to predictions of hemodynamics in normal or seminormal physiological states because marked changes in physiological state will activate regulatory mechanisms, in the short term case mainly the autonomic nervous reflex, that work to maintain hemodynamics at a specific level through regulations of vascular tone, cardiac function and total blood volume, etc. A mathematical model fit for predictions of hemodynamics in various physiological states must include both a model of the cardiovascular system and other models describing various regulatory mechanisms. Such a model is currently under development in our ongoing study. 4.
Conclusions
Inasmuch as the human cardiovascular system is a closed-loop system, hemodynamics in any separate segment is determined by hemodynamic interactions throughJSME International Journal
493 out the whole system. Although there have been many numerical and experimental studies on hemodynamics of different dimensions with various degrees of accuracies, until now, most of them have been focused exclusively on some isolated cardiovascular segments, little attention has been given to the global hemodynamic interactions that naturally exist in the system. The present computational model provides a quantitative insight into the global hemodynamics, helps deepen the understanding of the dynamic hemodynamic interactions among different regional cardiovascular circulations. Despite the fact that the present model is limited to low dimensional prediction, it underscores the significance of the closed-loop conception in studies concerning hemodynamics. Most of the hemodynamic parameters predicted by the present model were within reasonable physiological ranges and some of them satisfactorily agreed with the available in vivo measurements. In addition, the successful application of the present model to simulate the cardiac hemodynamic characteristics in patients with heart abnormalities suggests that the present model may serve as a useful assistant tool for computer-aided diagnosis and treatment of cardiovascular diseases. Moreover, considering that hemodynamics may change significantly depending on the physiological state, the present computational model may provide a basis for further incorporation of models of the autonomic nervous system and other models describing regulatory mechanisms involved in hemodynamic regulations to form an integrated computational model capable of predicting hemodynamics in various physiological states.
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Acknowledgments
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This work is, in part, supported by grant-in-Aid for Scientific Research No.(B)17300141, JSPS.
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References (1)
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