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CONSIDERATIONS ABOUT THE LUMPED PARAMETER WINDKESSEL MODEL APPLICATIVITY IN THE CARDIOVASCULAR SYSTEM STRUCTURE VASILE MANOLIU

Electrical Engineering Faculty, POLITEHNICA University of Bucharest, Splaiul Independentei 313, 060032, Romania E-MAIL: [email protected]

Key words: Windkessel model, cardiovascular interactions. The Windkessel model enlarges his applicativity on the whole circulatory dynamics studies. Then, the lumped parameter models allow global representation of blood flow circulation. In the vascular tree, peripheral branches are terminated by a resistance term representing smaller vessels. Blood flow and pressure are expressed by the intensity of current and voltage in an electrical analogue based on the Navier-Stokes equations for fluid flows in elastic tubes. Although the lumped parameters can be expressed according to the physical properties of the arterial tree and the rheology of the blood, the paper deal, also, the influence of various factors (infusion of the drugs and sympathetic stimulation) on the peripheral resistance. In a non-pulsatile cardiovascular model, the cardiac output, CO, is expressed as stroke volume times heart rate. Using a three-element Windkessel model as arterial load impedance of the heart, the mean arterial pressure variations is investigated. Also, some considerations relating the influence of an inotropic and a vasoactive drugs infusion rate variations upon the changes of systemic resistance is made. The sympathetic influence on peripheral resistance is a weighted sum of the afferent activities from baroreceptors, chemoreceptors and lung stretch receptors. The implementation in Matlab-Simulink allows the study of correlations between pressure and peripheral resistance variations.

1. INTRODUCTION In recent years many models of the cardiovascular system have been extensively studied. These models are described by a set of mathematical relations between some of the variables of the system, mainly blood pressures and flows. These variables are interrelated by a set of parameters (vessel dimensions, elasticity of the arterial wall, blood viscosity,…) which are difficult or impossible to measure directly. The different parameters are generally given an average value which reasonably fits the experimental findings.

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One of the major questions in cardiovascular physiology and physiopathology is to distinguish between a change in ventricular performance and a change in vascular system (or afterload) and to provide quantitative evaluation of both cardiac and vascular functions. Afterload is the resistance to ventricular ejection. This is caused by the resistance to flow in the systemic circulation and is the systemic vascular resistance. Resistance (&, therefore, blood flow) varies as a result of vasodilation & vasoconstriction. The resistance (dyne.sec/cm5) is determined by the diameter of the arterioles and pre-capillary sphincters. The level of systemic vascular resistance is controlled by the sympathetic system which, in turn, controls the tone of the muscle in the wall of the arteriole, and hence the diameter. The Windkessel theory explains why the pressure fluctuations in the aorta have a much smaller amplitude than that in the left ventricle. In this theory, the aorta is represented by an elastic chamber and the peripheral blood vessels are replaced by a rigid tube of constant resistance. Although the arterial compliance depends strongly on pressure, the estimation techniques allows constant values, especially for shorten arterial segments. The paper apply the Windkessel models to evaluate the mean arterial pressure; for the cardiovascular system model with lumped parameters, the influence of various factors (infusion of the drugs, respiratory activity and sympathetic stimulation) on the peripheral resistance, is analyzed. 2. EVOLUTION OF WINDKESSEL MODELS The first lumped-parameter arterial model was the two-element windkessel, introduced by Otto Frank in 1899. In this model (Fig. 1) the whole arterial tree is modeled as an elastic chamber (WK) with a constant compliance (C = dV/dp, V being systemic blood volume and p pressure) and a resistance (R) representing the total resistance of the arterial tree.

Fig. 1. Two-element Windkessel.

The governing equations in the frequency domain and time domain are given as: dp p Q + = (1) dt R ⋅ C C where Q is the flow.

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Considerations about the lumped parameter windkessel model

For a given ejection pattern (aortic flow), systolic and diastolic pressures were very accurately predicted by the 2-element Windkessel. Therefore, total arterial compliance and peripheral resistance seem to be the only 2 important arterial parameters determining aortic pulse pressure. The 2-element Windkessel matches well the true input impedance at low frequencies. Because the first few harmonics contain the major part of the information on the wave shape, these harmonics mainly determine the systolic and diastolic pressure. The systolic and diastolic pressures do not depend on the distribution of the wave reflection sites. From the measured ascending aortic pressures and flows, peripheral resistance (R) was obtained as the ratio of mean pressure and flow. Total arterial compliance (C) was estimated by the decay time method, i.e., fitting the diastolic part of the pressure wave with a single RC time.

Fig. 2. Typical aortic pressure pulse used for decay time method.

During diastole there is no inflow from the heart (Q = 0). Therefore the right-hand side of Eq. (2) vanishes and a direct integration yields equation in the frequency domain is given by:

p = p1 ⋅ e −(t − t1 ) / R ⋅C (2) where p1 is the pressure at time t1, a reference point in the diastolic phase. Equation (2) expresses a monoexponential decay and can be fitted to any portion of the diastole to yield the characteristic time or time constant, which is RC. When the value of peripheral resistance is known, the time constant divided by peripheral resistance gave total arterial compliance. The classic three-element Windkessel model is a suitable representation of the arterial load impedance of the heart [1]. Although these parameters are time varying because of the regulatory mechanisms, they were kept at constant values during simulation.

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Fig. 3. Schematic representation of cardiovascular interactions.

The dynamic relationship between the mean arterial pressure, p(t), and mean cardiac output, Q(t), is: dp s (t ) 1 = ⋅ ( R ⋅ Q(t ) − p s (t )) (3) dt τ p (t ) = p s (t ) + Rc ⋅ Q(t ) (4) where τ = R⋅C. The mean aortic flow, Q(t), was regarded as the input, the mean arterial pressure, p(t), as output, and the pressure, ps, loading the arterial compliance, C, as a state variable. Since the model is non-pulsatile, the flow Q(t), was simply expressed as stroke volumes times heart rate. This third term, the characteristic impedance of the aorta (Zc), accounts for the local inertia and local compliance of the very proximal ascending aorta and is based on wave transmission theory. It therefore connects lumped-parameter models with transmission-line models. Zc was connected in series with the two-element windkessel. Introduction of Zc improves considerably the medium to highfrequency behavior of the model. The three-element windkessel is thus based on hemodynamic principles and has become the most widely used and accepted lumped-parameter model of the systemic circulation. The four-element windkessel (Fig. 4), including an inertia term, describes the relations between aortic pressure and flow better than the three-element windkessel. The inertance is inversely proportional to arterial cross-sectional area. It is logical to have the inertial term in parallel with Zc. This parallel arrangement means that at very low frequencies, where the local properties of the aorta (Zc) play a negligible role, it is the total arterial inertia that dominates, whereas at high frequencies it is Zc that determines the arterial impedance. In modeling activity, a three-element Windkessel model can be used for evaluate the initial ventricular ejection dynamics.


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Fig. 4.

Using Fig. 5, with τe = La /Ra, τc = Ra⋅Ca, result:

τe ⋅ τc ⋅ where:

d 2 p ao dp + τ c ⋅ ao + p ao = pvs dt dt 2

(5)

pao = aortic pressure; pvs = left ventricular pressure.

Fig. 5. Equivalent circuit for an aortic segment.

3. CONTROL OF CARDIOVASCULAR QUANTITIES BY DRUG INFUSION In some clinical situations, for instance congestive heart failure, the simultaneous regulation of blood pressure and cardiac output (CO) is needed. In order to maintain or increase CO, and, at the same time, to decrease the blood pressure, can be used two frequently used drugs in clinical practice. An inotropic agent, Dopamine (DOP), increase the heart's contractility and cardiac output, and the vasoactive drug Sodium NitroPrusside (SNP), which dilates the vasoculature and lowers the arterial pressure. Using the model described in [2], the dynamics of changes in systemic resistance due to increments in infusion rate for DOP (I1) and for SNP (I2), in the s-domain are expressed by:

∆RA =

e − τ1 ⋅ s ⋅ ( K 21 ⋅ I1 + K 22 ⋅ I 2 ) s ⋅ T2 + 1

(6)

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where: ∆RA is the change in systemic resistance due to I1 and I2; K21 and K22 are steady-state gains with typical values of –0.09 and –0.15 respectively; τ2 represent a time delay with typical value τ2 = 30 s; T2 are time constant equating 58.75 s. The great values time constants meanings low values of propagation speed. The negative steady-state gains agree with decreased values of peripheral resistances. Because the cardiovascular dynamics is much faster than the drug dynamics, in a cardiovascular model which take account for drug dynamics, one can retain only the steady-state gains. 4. THE SYMPATHETIC INFLUENCE ON PERIPHERAL RESISTANCE The level of systemic vascular resistance is controlled by the sympathetic system which, in turn, controls the tone of the muscle in the wall of the arteriole, and hence the diameter. The mean arterial pressure (MAP) is the product of cardiac output (CO) and systemic vascular resistance (SVR): MAP = CO x SVR If cardiac output falls, for example when venous return decreases in hypovolaemia, MAP will also fall unless there is a compensatory rise in SVR by vasoconstriction of the arterioles. This response is mediated by baroreceptors, which are specialized sensors of pressure located in the carotid sinus and aortic arch, and connected to the vasomotor centre. Compared with the previous work [3], the present study follows the sympathetic activity on the peripheral resistances . The function that relates the activity in the afferent pathways to the efferent sympathetic activities has been given an exponential trend [4]. The input to the exponential is the weighted sum of the afferent activities from baroreceptors, chemoreceptors, and lung stretch receptors. However, we assumed that sympathetic activity couldn't increase above a saturation level. Hence: f sp = f es, ∞ + ( f es,0 − f es, ∞ ) exp[k es (−Wb, sp ⋅ f ab + Wc, sp ⋅ f ac − W p, sp ⋅ f ap )] if f sp < f es, max (7) and f sp = f es, max if f sp ≥ f es, max . In the above relations, fsp and fsh represent the frequency of spikes in the sympathetic efferent fibers to the vessels and to the heart, respectively, kes , fes,max , fes,∞ , and fes,0 are constants (with fes,max > fes,0 > fes,∞ ), and Wb,sp, Wb,sh, Wc,sp, Wc,sh, and Wp,sp are synaptic weights, tuned to reproduce physiological results. Also, the offset term for sympathetic neural activation, θsp is neglected. As in previous study [4], the arterial blood pressure and the alveolar ventilation was used as main determinant factors.

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5. RESULTS To solve eq. (3) and (4) for various values of heart rate and stroke volume, a Simulink scheme is used. The mean arterial pressure variations are slighting affected with the modification of parameter values. For R = 1.05 mmHg⋅s/ml, Rc = 0.042 mmHg⋅s/ml and C = 1.1 ml/mmHg, and for SV = 80 ml; HR = 1.166 s-1, the arterial pressure is shown in Fig. 6.

Fig. 6. Variations of mean arterial pressure.

The solution of equation (5), considering the following values: Ca = 0,178 ml/mmHg; Ra = 0,033 mmHg/ml/s; La = 0,2278 mmHg⋅s2/ml, and assuming that at suddenly open of aortic valve, the pressure pvs = 100 mmHg, is:

p ao (t ) = 100 − 100e − 3,417t cos(12,4495t ) − 27,4412e − 3,417t sin(12,4495t )

(8) with temporal variation represented in Fig. 7.

Fig. 7. Temporal variation of aortic pressure.

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Since τc « τe, the aortic pressure waveform will be always oscillate damped. For the relationships among afferent information, efferent neural activities and effector response, a block diagram (Fig. 8) was implemented in Simulink.

pao2 20s+0.334 Ventilator

s2 +0.059s

Transfer Fcn (with initial states)

fac

VT

fap

In1 Out1

In1 Out1

In1 Out1

Chem_art

Pulm_vent.

Pulm_str_rec.

psa In1

Ql x' = Ax+Bu y = Cx+Du Qr

Vo

State-Space

In2

Out1

In3

In1 Out1

fsp

Subsystem1 Subsystem

fab

Fig. 8. Block diagram describing relationships for various vagal and sympathetic activities.

In a state-space expression of a simplified model of the cardiovascular system equations, the venous systemic, the venous pulmonary and the arterial pulmonary pressures is used as state variables. The model, implemented in MatlabSimulink allows a useful link between hemodynamics and baroreflex response. Furthermore, for arterial baroreflex response, the arterial systemic pressure variation is used as input. Because the various control mechanisms exhibit a variety of temporal lags, one can observe both tendencies of variations. As an example, the variations of tidal volume, by lung stretch receptor activity, causes a progressive decrease in total systemic resistance with increasing VT. The sympathetic and parasympathetic pathways included low pass filter and a pure time delay to describe the dynamics of the baroreflex changes.

CONCLUSIONS

For a segment, thin-walled cylindrical tube of arterial part, a transfer function relating the pressures up and down the segment, was obtained, allowing

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investigation of influences of compliance and resistive terms variations upon the pulsatility and variation rate of response. The study methods based of linear phenomena contribute greatly to the better understanding of physiological mechanisms. However, it is necessary to take into account the nonlinear nature of the systems. Moreover, it is necessary to disseminate between several mechanisms characterised by varied delays. Received on October, 22,2004

REFERENCES 1. S. Cavalcanti, Arterial baroreflex influence on heart rate variability: a mathematical model-based analysis, Med. Biol. Eng. Comput., 38, 2000, pp. 189-197. 2. G. W. Irwin, et al., Neural network – application in control, 1995. 3. Vasile Manoliu – Modeling principles of cardiopulmonar interdependences - 4-th European Symposium on Biomedical Engineering , 25 - 27 june 2004, Patras, Greece, Session 2. 4. M.Ursino, Elisa Magosso – Acute cardiovascular response to isocapnic hypoxia. I. A mathematical model, A. J. Physiol. Heart Circ. Physiol., 279, H149-H165, 2000.