A Mathematical Model of the Baroreflex

a mathematical model of the baroreflex

Sander de Vet

BMTE 09.31

Supervisors: Peter Bovendeerd Beatrijs van der Hout

1 People in exercise: a mathematical model of the baroreflex S.W.J. de Vet (id-nr.: 0571992)

1 Abstract In this paper a mathematical model of the arterial pressure control by the carotid baroreceptor in the human is presented. The model includes a time-varying elastance model of the left and right ventricle, the systemic and pulmonary circulations, the afferent carotid baroreceptor pathway, the sympathetic and vagal efferent activities, and the response of several effectors to these activities. Effector response is modeled through changes in the systemic peripheral resistance, systemic unstressed volumes, heart rate and the end-systolic elastances of the right and left ventricle. The model is used to get an implementation of the baroreflex in the circulatory system. Simulations are performed for a resting condition, an acute hemorrhage and during exercise. In the acute hemorrhage and exercise simulations, the properties of the circulatory system change qualitatively as expected. Acute hemorrhage causes an increase in peripheral resistance, heart rate and contractility of both ventricles, and a decrease in arterial pressure and unstressed volumes. During exercise peripheral resistance, arterial pressure and unstressed volumes decrease, while heart rate and contractility of both ventricles increase.

2 Introduction

The human circulatory system continuously adapts to the demands of the body. On the short term, three regulatory mechanisms are important: the baroreceptor reflex, also called the baroreflex, the chemoreceptor reflex (chemoreflex) and the pulmonary stretch receptor reflex. The baroreflex provides a negative feedback mechanism for maintaining the arterial blood pressure. This system relies on the baroreceptors in the aortic arch and in the carotid artery. When these baroreceptors monitor an increase in the arterial pressure, the contractility of the heart and the heart frequency decrease and the arteries vasodilate. This causes the arterial pressure to decrease. A decrease of the arterial pressure on the other hand, will initiate opposite processes. Mathematical modeling of this system may improve our understanding of the cardiovascular system. In the Cardiovascular Biomechanics group at TU/e, several models of the cardiovascular system have been designed. Until now, these models lack the baroreflex. Therefore, the aim of this study is to extend one of these models with a mathematical model of the baroreflex, as proposed by Ursino [1]. After implementation of the model, first a simulation of a normal person (70 kg) during exercise will be made. In future this model will be adapted to represent a pregnant woman during delivery, to obtain more insight into the cardiovascular physiology during delivery. The PhD-project of Van der Hout focuses on these aspects, where a mathematical model of the maternal and fetal circulation is used. This model will be incorporated in a full-body delivery simulator to practice difficult deliveries. In this way doctors can get more experience and a better insight into complications of mother and fetus during delivery.

3 Mathematical model

3.1 Hemodynamics model

3.1.1 Lumped parameter model The flow of blood through the circulatory system is governed by the equations of conservation of mass and momentum. By assuming blood as an incompressible Newtonian fluid and inserting this behavior into the momentum equation, we get the Navier-Stokes equations (1). Neglecting gravity effects, these read: r ∂v r r r r r r ρ( + v ⋅ ∇v) = −∇p +η∇ 2v (1) ∂t r with density ρ, viscosity η, velocity v , pressure p and time t. Assuming rotational symmetry of the vessel and assuming the radial and tangential velocity to be zero equation (2) can be transformed to:

∂v ∂v 1 ∂p 1 ∂ ∂v + v = − +ν (r ) (2) ∂t ∂z ρ ∂z r ∂r ∂r where ν=η/ρ is the kinematic viscosity. The term on the left of equation (2) represents the inertial forces. The term on the right is a summation of the pressure forces and viscous forces respectively. In a lumped parameter model we will artificially separate the contributions of inertial forces and viscous forces to the pressure forces with the ‘building blocks’ shown in figure 1 [2].

Figure 1: Components in the lumped parameter model Vessel resistance

2 A flow which is dominated by friction can be described by:

1 ∂ ∂v 1 dp ν (r ) = (3) r ∂r ∂r ρ dz

The solution of equation (3) gives the Poiseuille profile:

1 dp v = − (a 2 − r 2 ) (4) 4η dz where a represents the vessel radius. Equation (4) can be transformed to equation (5):

dp 8η = − q (5) dz πa 4 where q represents the flow through the vessel. Now the resistance R represents the ratio between the pressure drop over a segment with length l and the flow through that segment, like:

∆p 8ηl R = = (6) q πa 4

Vessel inertance If the flow is dominated by inertia the Navier-Stokes equation becomes:

∂v 1 dp = − (7) ∂t ρ dz Equation (7) can be transformed to:

dq πa 2 dp = − (8) dt ρ dz We now define the inertance L as the ratio between the pressure drop over a segment with length l and the derivative of the flow during the time, like:

∆p ρl L = = (9) (dq / dt ) πa 2

Vessel compliance The volume V of a vessel segment will change with time if the inflow q in does not equal the outflow q out :

dV = q − q (10) dt in out dV/dt can also be described as:

dV dV dp dp = = C (11) dt dp dt dt with C the compliance of the vessel. It can be seen as the storage capacity of the vessel. In practice the compliance may be determined from measuring the pressure-volume relation of a blood vessel. This relation is a nonlinear relation and will be linearized around the physiological working pressure. This yields:

V (t) −V V (t) p = u = eff (12) C C Here Vu is the intercept volume of the linearization which is also called the unstressed volume of the vessel. In other words it is the volume of the vessel at zero internal pressure. Also a new variable, the effective volume Veff is introduced, which is the difference between the volume at time t and the unstressed volume of the vessel.

Heart valves The heart valves are assumed to be ideal valves, which means that there is no leakage in the valve and no pressure drop when the valves are open. This can be characterized like an ideal diode with the following relations:

∆p = 0 open valve (13) q = 0 closed valve

Ventricles The ventricles are modeled with the time-varying elastance model of Suga et al. [3]. Suga et al. defined E(t) as the slope of the pressure-volume relation at a certain moment t, in the cardiac cycle. This line intersects the volume axis at a volume V0. In other words:

p(t) E(t) = (14) (V (t) −V0 ) They also defined the slope of the passive pressure-volume line as Epas and the slope of the line through all end-ejection points as Emax . Now E(t) can also be write in terms of Epas and Emax , like:

E(t) = E pas + a(t)( Emax − E pas ) (15) Here a(t) is a dimensionless normalized activation function, varying from 0 in the passive state to 1 in the active state. Now equation (15) becomes:

p(t) = (E pas + a(t)( Emax − E pas ))( V (t) −V0 ) (16)

3.1.2 Ursino’s model Parameter values in our model were partly based on Ursino’s model. The model of Ursino has also a lumped-parameter approach. The hydraulic analogue of the lumped cardiovascular system is shown in figure 3. The systemic and pulmonary arteries are

4 modeled with a compliance, an inertance and a resistance. In the peripheral circulation and the venous circulation the same compartments are used except the inertance. Ursino assumed the inertance has a negligible effect on the capillaries and the veins. The peripheral circulation and venous circulation are divided in two parts: the splanchic and extrasplanchic circulation. The atria are described as linear capacities characterized by constant values of compliance and unstressed volumes, in other words, the contractile activity of the atria are neglected. The atria and ventricles have a viscous resistance. The contractile activities of the ventricles are modeled with a time-varying elastance model. Ursino assumed that the passive pressure-volume relation during diastole is not a constant value but an exponential relation, like:

kE ⋅V (t) p(t) = p0 ⋅ (e − )1 ⋅ 1( − a(t)) + a(t) ⋅ Emax ⋅ (V (t) −V0 ) (17) Actually equation (17) is the same as equation (16) but with Epas defined as:

kE ⋅V (t) p0 ⋅ (e − )1 E pas (t) = (18) V (t) −V0 The dimensionless activation function a(t) is defined as:

 2 π ⋅t s (t) 0≤ t ≤ t sin ( ) s act a(t) =  tact (t) (19)  t ≤ t ≤ 1 0 act s where ts is a dimensionless variable, ranging between 0 and 1, that represents the fraction of the cardiac cycle. Here tact represents the duration of the systole. The duration of the systole depends on the duration of the heart cycle, according to:

1 t (t) = T − k ⋅ (20) act sys 0, sys T (t) where Tsys,0 is the duration of the systole in the first heart cycle and ksys a constant gain factor.

3.1.3 Our model The model used in this research is a generalization of the model of Ursino [1]. The main differences are as follows. 1) Because the splanchnic and extrasplanchnic circulation are not distinguished in the results of Ursino, the essence of this separation is not clear. For this reason, the splanchnic and extrasplanchnic are taken together as the whole systemic circulation. 2) The inertia in the arteries is neglected. 3) The compliances of the atria are taken together with the compliances of the veins. Also the compliances of the peripheral circulation are fused with the compliances of the arteries. 4) The resistances of the atria are neglected. In the model of Ursino the resistance is caused by the mimic viscosity of the ventricles. It’s physiological meaning is not quite clear, therefore these resistances have been neglected in our model. The lumped parameter model of this research and the hydraulic model of Ursino are shown in figure 2 and figure 3, respectively.

Figure 2: Lumped parameter model of complete Figure 3: The hydraulic analoge of Ursino. P, pressures; circulation. With qtv (flow over triscupid valve), prv (right R, resistance; C, compliances; L, inertances; F, flows; sa, ventricle pressure), E rv (right ventricle varying elastance), systemic arteries; sp, splanchic peripheral circulation; ep, qpv (flow over pulmonary valve), Z pulart (pulmonary arterial extrasplanchic peripheral circulation; sv, splanchic venous impedance), ppulart (pulmonary arterial pressure), C pulart circulation; ev, extrasplanchic venous circulation; ra, right (pulmonary arterial compliance), R pul (pulmonary atrium; rv, right ventricle; pa, pulmonary arteries; pp, resistance), qpul (pulmonary flow), C pulven (pulmonary pulmonary peripheral circulation; pv, pulmonary venous venous compliance), ppulven (pulmonary venous pressure) circulation; la, left atrium; lv, left ventricle; ol and or, and Z pulven (pulmonary venous impedance). See table 1 for output from left and right ventricle respectively; P max,rv and other parameter descriptions. Pmax,lv , ventricle pressures in isometric conditions.

3.2 Baroreflex model

This paragraph gives a description of the mechanisms of the baroreflex and is based on the paper of Ursino [1]. The input parameter is the arterial pressure. The output parameters are new values of the heart period, contractility of the ventricles, unstressed volumes of the veins and the peripheral resistance. The block diagram of the control system of the baroreflex is implemented according to Ursino. The block diagram is shown in figure 4. We will discuss each block in detail.

Figure 4. Relationships between afferent information, efferent neural activities, and effector responses according to the present mathematical model. With P b (baroreceptor pressure, f ab (afferent activity from the arterial baroreceptors), f v (activity in the vagal efferent fibers), T (heart period), E max,rv and E max,lv (end-systolic elastance of the right and left ventricle), V u,v (unstressed volume of the systemic venous circulation) and finally R p (systemic peripheral resistance).

Arterial baroreceptor The block in the afferent pathway describes the relationship between baroreceptor pressure ( Pb) and the activity of the afferent fibers of the baroreceptors ( fab ). Ursino modeled this with a sigmoidal relation with an upper saturation ( fab,max ), a lower threshold (fab,min ) and a central value Pn. Since the baroreceptor frequency is also sensitive for the ~ rate of change of the baroreceptor pressure [4], the intermediate pressure P is introduced. This is the same as the mean value of the baroreceptor pressure during a period with length τz,b . This results in equations (21) and (22) ~ dP dP ~ τ ⋅ = P +τ ⋅ b − P (21) p,b dt b z,b dt ~ P − Pn f ab ,min + f ab ,max ⋅ exp( ) k ab f ab = ~ (22) P − P 1+ exp( n ) kab

7 where τp,b and τz,b are the time constants for the real pole and the real zero of the differential equation and kab is the slope of the sigmoidal function at the central point. ~ Figure 5 shows the relationship between the intermediate pressure (P ) and fab .

Figure 5: Afferent information from arterial baroreceptors (baroreceptor frequency vs. intermediate pressure) with a sigmoidal relationship according to equation (22). Here the minimim baroreceptor frequency (f ab,min ) is 2.52 Hz and the maximum baroreceptor frequency (f ab,max ) is 47.78 Hz.

Efferent vagal pathway The relation between fab and the frequency in the efferent vagal fibers ( fv) is also a sigmoidal function. Increasing fab results in an increasing value of fv. Equation (23) holds for the vagal frequency.

f − f f + f ⋅ exp( ab ab 0, ) ev 0, ev ,∞ k f = ev (23) v f − f 1+ exp( ab ab 0, ) kev Here fev,0 , fev, ∞ and kev are constant parameters. Figure 6 shows the relationship between fv and fab .

Figure 6: Efferent information from vagal fibers (baroreceptor frequency vs. vagal frequency) according to equation (23). Here the maximum and minimum vagal frequency (fev,0 and fev,∞) will never be reached because the baroreceptor frequency will get values between fab,min and fab,max.

Efferent sympathetic pathway The frequency in efferent sympathetic fibers and fab shows a monoexponential decreasing curve, described by:

f sh = f es ,∞ + ( f es 0, − f es ,∞ ) ⋅ exp( −kes ⋅ f ab ) (24)

Where fsh is the frequency in the efferent sympathetic pathway and kes , fes,0 and fes, ∞ are constants. Figure 7 shows the relationship between fsh and fab .

Figure 7: Efferent information from sympathetic fibers (baroreceptor frequency vs. efferent sympathetic frequency) according to equation (24). It shows a decreasing monoexponential relationship.

Regulation of the effectors The response of the resistances, unstressed volumes, and cardiac elastances to the sympathetic fibers is modeled in the same way as Ursino [1] and results in equations (25) through (27). Either of the controlled parameters Emax,rv , Emax,lv , Rp and Vu,v , generally represented by the parameter θ changes by θ (t) from reference value θ0 according to:

θ (t) = ∆θ (t) +θ 0 (25) The Parameter change θ (t) is governed by a first order differential equation:

d∆θ (t) 1 = ⋅[]− ∆θ (t) + σ θ (t) (26) dt τ θ where τθ is the time constant of the mechanism and σθ is the steady state change of the parameter. Now σθ depends on fsh according to:

G ⋅ ln[ f (t − D ) − f + ]1 θ sh θ es ,min if f sh ≥ f es,min σ θ (t) =  (27) (27) 0 if f sh < f es,min where Dθ is the time delay of the mechanism and fes,min is the minimum sympathetic stimulation. Finally, Gθ is a constant positive gain factor for mechanisms working on Emax,rv , Emax,lv and Rp but a negative gain factor for Vu,v . As an example, the steady state results of E max,lv are shown in figure 8.

10 Figure 8: Efferent information from the sympathetic fibers versus the maximal cardiac elastance of the left ventricle, according to equation (27) and equation (26) in steady state. This relation is identical for other output parameters but with different slopes or in some cases a negative slope with a negative output.

the response of the heart period.is determined by The balance between the vagal and sympathetic activities The new value of the heart period is calculated by a summation of the heart period changes induced by the sympathetic stimulation ( Ts), the heart period changes induced by the vagal stimulation ( Tv) and a reference value ( T0) according to:

T = ∆Ts + ∆Tv + T0 (28) Now Ts(t) and Tv(t) are governed by a first order differential equation:

d∆Ts (t) 1 = ⋅[]− ∆Ts (t) + σ T ,s (t) (29) dt τ T ,s and

d∆Tv (t) 1 = ⋅ []− ∆Tv (t) + σ T ,v (t) (30) dt τ T ,v where τT,s and τT,v are the time constant of the sympathetic pathway and the vagal pathway, respectively. Finally σT,s and σT,v are the steady state changes of the heart period due to the sympathetic stimulation and the vagal stimulation, respectively. These steady state changes depend on fsh according to:

≥ GT ,s ⋅ ln[ f sh (t − DT ,s ) − f es ,min + ]1 if f sh f es,min σ T ,s (t) =  (31) 0 if f sh < f es,min

and

σ T ,v (t) = GT ,v ⋅ f ev (t − DT ,v ) (32) where DT,s and DT,v are the time delays due to the sympathetic stimulation and the vagal stimulation, respectively. Finally, GT,s and GT,v are constant gain factors, positive for the vagal pathway and negative for the sympathetic pathway.

3.3 Parameter settings

In the model of Ursino all the parameters values are based on literature and rescaled for a subject with a 70-kg body weight.

Since the model in this study is not exactly the same as the model from Ursino, some parameters are chosen differently. Table 1 shows the parameters used in this model and the equivalent parameter from Ursino’s model. The statistical significance of the data from Ursino is very high and variable in the parameters. Therefore the same statistical significance of all values is chosen.

Table 1. Parameters characterizing the hemodynamic system in basal condition. * the value between the brackets is the value of Ursino. ** here the initial values are not the basal values of Ursino but the initial value calculated by the linear differential equation. Name parameter in Name parameter in Value Unit this model Ursino’s model ml/kPa Combination of C sa , Cart * 60 (30) Csp and C ep 3 ml/kPa Combination of C sv , Cven 1.1•10 Cev and C ra ml/kPa Combination of C pa Cpulart 49 and C pp 2 ml/kPa Combination of C pv Cpulven 3.4•10 and C la 3 Vblood 5.0•10 ml Vt 2 ml Combination of V u,sa , Vart,0 6.1•10 Vu,sp and V u,sa 3 ml Combination V u,ev , Vu,v ** 2.2•10 Vu,sv and Vu,ra 2 ml Combination of V u,pa Vpulart,0 1.2•10 and V u,pp 2 ml Combination of V u,pv Vpulven,0 1.5•10 and V u,la -1 Zao 8.0 kPa•ms•ml Rsa 2 -1 Rp ** 1.5•10 kPa•ms•ml Rpp -1 kPa•ms•ml Combination of R sv Zven 1.5 and R ev -1 Zpulart 3.1 kPa•ms•ml Rpa -1 Rpul 12 kPa•ms•ml Rpp -1 Zpulven 0.75 kPa•ms•ml Rpv 2 tact,0 5.0•10 ms Tsys,0 4 2 ksys 7.5•10 ms ksys V0,lv 17 ml Vu,lv p0,lv 0.20 kPa p0,lv -1 kE,lv 0.014 ml kE,lv V0,rv 41 ml Vu,rv p0,rv 0.20 kPa p0,rv -1 kE,rv 0.014 ml kE,rv Emax,lv ** 0.059 kPa/ml Emax,lv Emax,rv ** 0.035 kPa/ml Emax,rv 2 tcycle ** 7.5•10 ms T

Calculation of the compliances To calculate the compliances in our model one has to be aware of the fact that all compliances are parallel. This leads to the following equations:

12 Cart = Csa + Csp + Cep C = C + C + C ven ev sv ra (33) C pulart = C pa + C pp

C pulven = C pv + Cla

Calculation of the unstressed volumes The unstressed volumes of the splanchic and extrasplanchic circulation are combined. The peripheral unstressed volumes from Ursino are added to the arterial volumes. The same applies to the unstressed volumes of the atria which are added to veins in front of the atria. In brief:

Vart 0, = Vu,sa +Vu,sp +Vu,ep

V pulart 0, = Vu, pa + Vu, pp (34)

V pulven 0, = Vu, pv +Vu,la

The initial value of the unstressed volume of the systemic veins is calculated using the differential equation of Vu,v as mentioned in the qualitative description of the mathematical model. Therefore the basal value from Ursino is not used.

Calculation of the resistances In this model one resistance is used for the systemic veins instead of the three resistances (Rev , Rsv and Rra ). These are two parallel resistances in series with Rra . Here the resistance Rra will be neglected. So the equivalent resistance ( Zven ), which will be used in this model, is:

1 Z = (35) ven 1 1 + Rev Rsv

For Rp not the basal value is taken but the initial value is calculated as is done with the unstressed volumes of the systemic veins.

Parameters describing the heart All the basal values of these parameters are equal to the basal values of Ursino, except for the values of the maximum elastance of the left and right ventricle ( Emax,lv and Emax,rv ) and the heart period ( tcycle ). The initial values of these parameters are calculated using equations (25) through (32). Actually this is done in the same way as with Rp and Vu,v . The basal values of Ursino are shown in table 3 of the appendix.

3.4 Simulations

Within this research three phenomena will be simulated to test the model behavior: the steady state situation, an acute hemorrhage and a circulation response during exercise.

13 The simulation of the steady state will be made to create a reference for the results with the other two simulations. The simulation of acute hemorrhage will be done to compare the output of the model with respect to the output of Ursino’s model. The simulation of a circulation response during exercise will be made to get a better understanding in this phenomenon. Moreover the model can be used in the PhD-project of Van der Hout for pregnant women during delivery. At first, the initial values in the simulation of the normal steady state situation are the same as in table 1. With these values a simulation is made. After that, the initial values of the output parameters of the baroreflex model are adapted to the steady state value of that simulation. In the acute hemorrhage simulation, an immediate blood volume loss of 10% will be prescribed. Therefore the value Vblood will be set to 4500 ml at the beginning. In contrast with this model Ursino simulated an acute hemorrhage of 10% performed in 5 s. In spite of this difference the output will be compared with the output of Ursino’s model. We assumed this difference has a negligible effect. The simulation will also be compared with a simulation of an acute hemorrhage without the baroreflex. In this way the influence of the baroreflex will be clear. When a person is doing exercise, the cardiac output can increase from 5 l/min to up to 20 l/min [5]. So the blood flow to the skeletal muscles has to increase with factor 4. To achieve this, the pre-capillary sphincters before the skeletal muscles will relax to increase the blood flow. In the model we simulated moderate and severe exercise by -1 reducing Rp within 30 s from the original value of 150 kPa•ms•ml to a reduced value of 100 kPa•ms•ml -and 38.75 kPa•ms•ml -1, respectively.

4 Results

Here the results of the simulations, defined above, are presented.

The resting condition To get the results in rest, all the parameters of table 1 are set to their basal value. The simulation is performed during 200 cycles and from the beginning, the baroreflex adapts the hemodynamic parameter values in the body. In figure 9, the input ( part ) and the output (heart rate, Rp, Vu,v , Emax,lv and Emax,rv ) of the baroreflex model are shown. In the model of Ursino, heart rate continuously adapts during the cycle. In our model, the heart rate has the value at the beginning of the cycle during the whole subsequent heart cycle,. Then, at the start of the next cycle, the heart rate is updated. In other words, the continuous signal of Ursino has been transformed to a discrete signal. All the output parameters have a damping and oscillating behavior. They do not converge to a steady state value, but keep oscillating with a time constant of typically 10 seconds in a narrow interval around a mean value. These oscillations are most likely the Mayer waves [13]. At the end of the simulation the heart rate varies between 70 bpm and 76 bpm, which are typical values of a person in rest [6]. The other parameters vary as -1 -1 follows: Rp varies between 148 kPa•ms•ml and 158 kPa•ms•ml , Vu,v between 2280 ml and 2320 ml, Emax,lv between 0.420 kPa/ml and 0.435 kPa/ml and finally Emax,rv varies between 0.245 kPa/ml and 0.255 kPa/ml.

Figure 9: The input and the output of the baroreflex model in rest as a function of time. Top left: p art ; top right: heart rate; middle left: R p; middle right: V u,v ; bottom left: E max,lv and finally bottom right E max,rv .

Also P-V loops of the systemic circulation have been made. Figure 10 shows the left ventricular pressure versus the left ventricular volume. The P-V loops from the steady state are plotted. We assumed the system is in steady state after 40 seconds. An interesting result is the oscillating behavior of the P-V loops in the blue area. The oscillating behavior is an effect of the baroreflex on the circulatory system. The black line shows the P-V loop of the last heart cycle. The stroke volume of this cycle is about 65 ml and the heart rate at that moment is about 73 bpm. This results in a cardiac output 4745 ml/min which is close to the expected 5 l/min [7].

Figure 10: P-V loops of the systemic circulation. The blue lines are the P-V loops of all the heart cycles except the first, second and last heart cycle. The last heart cycle is shown with a black line

Acute hemorrhage with baroreflex The simulation of an acute hemorrhage is performed during 200 cycles. From the beginning, the baroreflex adapts the hemodynamic parameter values in the body. To mimic hemorrhage, V blood is set to 4500 ml. So in this case a blood volume loss of 10% is applied, similar as in the simulation of Ursino. All parameter values are set as shown in table 1. Following hemorrhage, the human circulatory system adapts some hemodynamic parameters to keep arterial pressure at the normal level. The results of these adaptations are shown in figure 11. Due to the loss of blood the arterial pressure will drop initially. To compensate the decrease of arterial pressure there is an increase of the peripheral resistance ( ≈15%), heart rate ( ≈16%), the contractility of left ( ≈8%) and right ventricle (≈8%). The unstressed volume of the veins decreases (≈9%). At the left top of figure 11 it can be seen that the arterial pressure can not be entirely compensated by the baroreflex. It has a decrease of approximately 4%. Another interesting point is the decrease of oscillations at the end of the simulation. The parameters p art , R p, V u,v , E max,lv and E max,rv oscillate with a time period of about 700 ms. That’s exactly the heart rate of the circulatory system (86 bpm) at that moment.

Figure 11: The input and the output of the baroreflex model as a function of time. Top left: p art ; top right: heart rate; middle left: R p; middle right: V u,v ; bottom left: E max,lv and finally bottom right E max,rv . The red line shows the results before hemorrhage (in rest) and the blue line shows the results after hemorrhage.

The mean changes of the input and output parameters, averaged over the cardiac cycle, are visualized in figure 12. Here our model is compared with the model of Ursino and the results obtained by Kumada et al. [8] in intact dogs. One difference is that our hemorrhage is achieved instantaneously while in the model of Ursino this is modeled over 5 seconds. The agreement with the experiment is quite satisfactory. The heart rate increases less than in the experimental data.

Figure 12: Percent changes in mean value of the main hemodynamic quantities (Rp, heart rate, cardiac output and part) caused by acute hemorrhage. Simulation results where compared with results obtained by Kumada et al. in intact dog and with the simulation results of Ursino.

The P-V loops of the systemic circulation after hemorrhage are shown in figure 13. Here only a couple of heart cycles are plot, to make the “evolution track” of the P-V loops to the steady state clear. The P-V loops until the 22nd cycle are attributed to the initial transient to the new situation, with a blood volume of 4500 ml instead of 5000ml. The P- V loops of the steady state show again an oscillating behavior. The dashed black line shows the P-V loop of the last heart cycle. The stroke volume of this cycle is about 47 ml and the heart rate at that moment is about 85 bpm. This results in a cardiac output 3995 ml/min. So an acute hemorrhage results in decrease of cardiac output by approximately 16%. To compare the results of an hemorrhage with respect to the situation in resting condition, the last cycle in rest is also plotted (the continuous black line). Using this line, it can be seen that hemorrhage increases contractility (E max,lv ) and decreases stroke volume. Also a decrease in left ventricular pressure (top of the graph) and thus a decrease in arterial pressure is visible.

Figure 13: Some P-V loops of the systemic circulation to make the “track” to the steady state clear. The light blue line shows the P- V loops in steady state and the dashed black line is the P-V loop of the last heart cycle. The red line is the P-V loop of the first cycle.

Acute hemorrhage without baroreflex To make the influence of the baroreflex clear, also a simulation of an acute hemorrhage without baroreflex is performed. These results are compared to the simulation of an acute hemorrhage with baroreflex. The new simulation is performed over 200 s. All other values are the same as in the simulation of hemorrhage with baroreflex. Figure 14 shows the influence of the baroreflex. Since the baroreflex is inactive in this simulation, the variables heart rate, R p, V u,v , E max,lv and E max,rv are now constants. The arterial pressure drops due to the loss of blood, meaning that the effective volume (total blood volume minus all unstressed volumes) drops as well.

Figure 14: The input and the output of the baroreflex model as a function of time. Top left: p art ; top right: heart rate; middle left: R p; middle right: V u,v ; bottom left: E max,lv and finally bottom right E max,rv . The red line shows the results of a hemorrhage with barorelfex and the blue line shows the results of a hemorrhage without baroreflex.

The decrease in arterial pressure leads also to a decrease in the left ventricular pressure. In figure 15 the decrease of the left ventricular pressure is also visible in the tops of the P- V loops, which go down from 18 kPa to 14 kPa. The decrease in stroke volume, as compared to the situation with baroreflex, is 4% (from 50 ml to 48 ml). With a heart rate of 75 bpm this results in a cardiac output of 3600 ml/min, which means a decrease of 24% with respect to the cardiac output at rest (4745 ml/min). So due to the baroreflex the cardiac output decreases with 16 % instead of 24%.

Figure 15: P-V loops of the systemic circulation of a hemorrhage with and without an barorelfex. The blue lines are the P-V loops of all the heart cycles except the first, second, third and last heart cycle. The last heart cycle with baroreflex is shown with a dashed black line towards the last heart cycle without baroreflex which is shown with a solid black line. The first, second and third P-V loop are left out to eliminate the initial transient.

In exercise The simulation of a person in exercise is accomplished by a decrease of the peripheral resistance from 155 kPa to 100 kPa during 30 seconds and keeping it at 100 kPa thereafter. This can only be realized when the peripheral resistance is taken out of the baroreflex model, otherwise the baroreflex will reset the peripheral resistance. Therefore during exercise it is assumed that the baroreflex is overruled by autoregulation in this part of the circulation. The decrease of R p starts at a steady state situation (t start = 100 s), so the simulation is prolonged to 400 cycles to study the influence of the baroreflex to the change in R p. Due to the decrease of the peripheral resistance the arterial pressure drops. As a consequence the output parameters of the baroreflex model (heart rate, V u,v , E max,lv and E max,rv ) will also change. After the decrease of arterial pressure, the system evolves a new steady state, which results in an increase of arterial pressure. As a consequence the output parameters also change slightly. The small increase in the unstressed volume is not perceptible in figure 16. The changes of steady state values are as follows: R p ( ≈ -35%), heart rate ( ≈ +18%), V u,v ( ≈ -11%), E max,lv ( ≈ +10% ), E maxr,rv ( ≈ +10% ) and finally p art (≈ -4%). Because the R p decreases with 35% you also expect, without the baroreflex, a decrease of 35 % in the arterial pressure. However this is just 4% as a result of the baroreflex. The parameters p art , R p, V u,v , E max,lv and E max,rv oscillate with a time period of about 700 ms. That’s exactly the heart rate of the circulatory system (86 bpm) at that moment.

Figure 16: The input and the output of the baroreflex model as a function of time. Top left: p art ; top right: heart rate; middle left: R p; middle right: V u,v ; bottom left: E max,lv and finally bottom right E max,rv . The red line shows the results of a person in rest and the blue line shows the results of a person in exercise. Comment: in the simulation of exercise the parameter R p is not a output parameter of the baroreflex model anymore.

The P-V loops of the systemic circulation of a person in exercise are shown in figure 17. Here these P-V loops are compared with the last heart cycle of a person in rest to see the effect of being in exercise. Before the reduction of R p has started, the P-V loops are in dynamic state, similar to a person in rest. After the decrease of R p takes place, the P-V loops go to the last heart cycle in exercise and oscillate at that place. In figure 17 it is immediately clear that stroke volume increases (from 65 ml to 80 ml), contractility increases and that left ventricular and thus the arterial pressure increases. With a heart rate of 86 bpm and a stroke volume of 65 ml the cardiac output is 5590 ml/min. This is a change of 18 % with respect to a person in rest.

Figure 17: P-V loops of the systemic circulation of a person in exercise. The blue lines are the P-V loops of all the heart cycles except the first, second, third and last heart cycle. The last heart cycle of a person in exercise is shown with a red line towards the last heart cycle with a person in rest which is shown with a black line. The first, second and third P-V loop are leaved out to eliminate the initial transient.

Table 2 gives a short summary of the hemodynamic quantities at new steady state in the different simulations.

Table 2: The hemodynamic values of Vstroke (stroke volume), CO (cardiac output), HR (heart rate), Vu,v (unstressed volumes of the veins), Rp (peripheral resistance), Emax,lv (maximal cardiac elastance left ventricle), Emax,rv (maximal cardiac elastance right ventricle), part (systemic arterial pressure), pven (systemic venous pressure), ppul,art (pulmonary arterial pressure), ppulven (pulmonary venous pressure), prv (right ventricular pressure), plv (left ventricular pressure) and qav (flow through aortic valve). Parameter Resting condition Acute hemorrhage Exercise Vstroke [ml] 65 47 80 CO [ml/min] 4745 3995 5590 HR [bpm] 73 85 86 Vu,v [ml] 2300 2085 2045 -1 Rp [kPa•ms•ml ] 153 176 100 Emax,lv [kPa/ml] 0.43 0.46 0.47 Emax,rv [kPa/ml] 0.25 0.27 0.28 part [kPa] 12.7 12.2 12.2 pven [kPa] 0.57 0.46 0.76 ppul,art [kPa] 1.8 1.4 2.4 ppul,ven [kPa] 0.84 0.57 1.01 prv [kPa] 2.5 1.8 3.0 plv [kPa] 10 9.0 11 qav [ml/s] 840 650 1050

5 Discussion

The aim of this work was to extend an existing lumped parameter model of the human circulation with a baroreflex model. In the PhD-project of Van der Hout this model will be used to simulate hemodynamics during delivery. Here the baroreflex is one part of the whole mathematical model. Also the lung stretch reflex and the chemoreflex are implemented in this model. A second reason for doing this research is the lack of a mathematical cardiovascular baroreflex model in the Cardiovascular Biomechanics group at TU/e. In all results the influence of the baroreflex is clear in one glance. In the simulations of an acute hemorrhage and a person in exercise the properties of the circulatory system change, at least qualitatively as expected. An acute hemorrhage causes an increase in peripheral resistance, heart rate and the contractility of both ventricles, and a decrease in arterial pressure and unstressed volumes. A person in exercise has a decrease in peripheral resistance, arterial pressure and unstressed volumes, and an increase in the heart rate and contractility of both ventricles. Also in figure 14 the influence of the baroreflex is made clear by looking to a simulation without baroreflex towards a simulation with the baroreflex. Concerning the development of the mathematical model two steps are made. In the first step we make a lumped parameter of the circulatory system. In a lumped parameter model the influence of viscous and inertial effects is artificially separated with ‘building blocks’. The building blocks are resistance, inertance, compliance, ideal valve and a ventricle. Our lumped parameter model deviates slightly from that of Ursino [1]. The first assumption we have made is that the inertance of the blood can be neglected. A second assumption with respect to Ursino is the neglection of the resistance of the ventricles due to the viscosity. This neglect is made since its physiological meaning is not quite clear. In the second step the same baroreflex model as in the model of Ursino is taken, with the same values of the parameters. In table 4 (Appendix) the values of these parameters are shown. The input of this system is the arterial pressure from the lumped parameter model. The output consists of new values of the peripheral resistance, heart rate, unstressed volume and contractility of both ventricles. So now a closed loop is made between the baroreflex model and the lumped parameter model. Almost all the initial values of the lumped parameter model are the same as the basal values of the model from Ursino except the output parameters of the baroreflex model, the arterial compliance and the total blood volume. In Ursino’s model there is a difference between the basal values of the output parameters of the baroreflex model and the initial values derived by the equations (25) through (32). Since this difference is not clear we take as initial value for these output parameters the value derived by the equations. The arterial compliance is doubled to get a larger RC time constant and thus less variation in the arterial pressure. This results also in less variation of the output parameters of the baroreflex model. In figure 18 the results are shown of the simulation with the arterial compliance of Ursino. This is a simulation in rest. However, this is not a realistic situation of a person in rest because the heart rate varies too much. Normally this value lies around the 70 bpm [9]

24 whereas in the original Ursino model it varies from 60 bpm to 110 bpm. The last parameter which we adapt is the total blood volume. Here we assumed the total blood volume is 5000 ml [9] instead of 5300 ml.

Figure 18: The input and the output of the baroreflex model as a function of time. Here C art has the same value as in the model of Ursino. Top left: p art ; top right: heart rate; middle left: R p; middle right: V u,v ; bottom left: E max,lv and finally bottom right E max,rv .

Although almost all the values of our model are the same as in the model of Ursino, there are some critical remarks about the parameter settings in Ursino’s model. First the high statistical significance level of these parameters is unusual. It is difficult to measure these parameters so accurately. Second, the unstressed volumes of the pulmonary arteries and the systemic arteries are zero. It seems to be unusual that when there is no blood inside the arteries, the volumes of the arteries are zero. After all it is not the distribution of the unstressed volumes over the cardiovascular system but only the sum of the volumes that determines the pressure in the circulatory system. Another limitation is the absence of local autoregulation mechanisms in the control of the peripheral systemic resistance. The vasodilation and vasoconstriction is not only regulated by the central nervous system but also by its own autoregulation [10]. During exercise the blood flow increases as a consequence of the vasodilation of the pre- capillary sphincters. A further limitation is the absence of other baroreceptors. There are also baroreceptors in the aortic arch, in the veins and in the wall of the right atria which may also have a great influence in the hemodynamical limitation. They give the output parameters of our model more or less feedback, which results in a higher or lower output parameter.

25 An interesting point is the absence of steady state values in the simulations. All the simulations oscillate around a “steady state” value. In the simulation of the resting condition this oscillation takes place with a time period of about 10 seconds, the Mayer waves. These waves are caused by pure time-delay [13]. In the other simulations the oscillation takes place over a time period of about 700 milliseconds which correspond with the heart rate. It’s unclear why the Mayer waves are only be realized in the steady state simulations and not in the other simulations. To compare our model with that of Ursino figure 12 is used. Here the results are compared after an acute hemorrhage. The decrease in cardiac output is exactly the same as in the model of Ursino. In our model the arterial pressure decreases less than in Ursino’s model. The decrease in arterial pressure due to hemorrhage will be more compensated because of a higher heart rate and a higher peripheral resistance. Maybe the difference in change of arterial pressure could also be the difference in arterial compliance. Finally, in the present model, the values of the arterial pressure are not realistic: the systolic pressure and diastolic pressure are 13.5 kPa and 12.0 kPa, respectively. Normally the systolic pressure and diastolic pressure are 16.0 kPa and 11.0 kPa, respectively. One reason for that could be the “location of measurement” of the arterial pressure. The arterial pressure in our model is measured after the arterial impedance. If one measures the arterial pressure after the aortic valve directly, then the arterial pressure has a higher systolic and lower diastolic value. Another reason could be the parameters τz,b and τp,b in equation (21). When these parameters decrease the arterial pressure is less flattened. To get a better understanding of the baroreflex further research is necessary. First of all, the parameters should be tuned such that the simulation has normal systolic and diastolic pressure of the arterial pressure. Secondly, further research should be necessary to explain the high time period of oscillation (in order to 10 seconds) of the output parameters. Finally in our model a time-varying elastance model of Suga et al. [3] is used. More recently, the group of Arts and Bovendeerd [11, 12] introduced a one-fiber model for left ventricular pressure calculation which is based on myofiber mechanics. So in the next model a one-fiber model can be used instead of the time-varying elastance model.

6 Appendix

Table 3: Basal values of the right and left heart. Basal heart period (T) is 0.833 s; k sys and T sys,0 , which describe duration of systole as 2 function of heart rate are 7.5•104 ms and 500 ms, respectively; kE and P0 are parameters which describes the end-diastolic pressure- volume function of ventricle; E max , the slope of end-systolic relationship. Subscripts: la, left atrium; ra, right atrium; lv, left ventricle; rv, right ventricle. Left Heart Right Heart Cla = 144.2 ml/kPa Cra = 234.4 ml/kPa Vu,la = 25 ml Vu,ra = 25 ml P0,lv = 0.20 kPa P0,rv = 0.20 kPa -1 -1 kE,lv = 0.014 ml kE,rv = 0.011 ml Vu,lv = 16.77 ml Vu,rv = 40.8 ml Emax,lv = 0.393 kPa/ml Emax,rv = 0.233 kPa/ml

Table 4: Basal values of parameters for regulatory mechanisms. G, mechanism strength; f, frequency; τ, time constant; D, time delay. Subscript 0 indicates parameter absence of innervation (i.e., when vagal and symphatetic activities are zero). Heart period control includes both symphatetic (subscript s) and vagal (subscript v) dependence Parameter values Afferent baroreflex pathway Pn =12.27 kPa fab,min = 2.52 Hz fab,max = 47.78 Hz kab = 1.5676 kPa τz,b = 6370 ms τz,b = 2076 ms Sympathetic efferent pathway fes, ∞ = 2.10 Hz fes,0 = 16.11 Hz kes = 67.5 ms fes,min = 2.66 Hz Vagal efferent pathway fev,0 = 3.2 Hz fev, ∞ = 6.3 Hz kev = 7.06 Hz fab,0 = 25 Hz Effectors GEmax,lv = 0.0633 τEmax,lv = 8000 ms DEmax,lv = 2000 ms Emax,lv0 = 0.3189 kPa•ml -1•Hz -1 kPa/ml GEmax,rv = 0.0376 τEmax,rv = 8000 ms DEmax,rv = 2000 ms Emax,rv0 = 0.1883 kPa•ml -1•Hz-1 kPa/ml GR,sp = 92.66 τR,sp = 6000 ms DR,sp = 2000 ms Rsp,0 = 332 kPa•ms•Hz -1 kPa•ms•ml -1 GR,ep = 70.66 τR,ep = 6000 ms DR,ep = 2000 ms Rep,0 = 104 kPa•ms•Hz -1 kPa•ms•ml -1 GVu,sv = -265.4 τVu,sv = 20000 ms DVu,sv = 5000 ms Vu,sv0 = 1435.4 ml ml/Hz GVu,ev = -132.5 τVu,ev = 20000 ms DVu,ev = 5000 ms Vu,ev0 = 1537 ml ml/Hz GT,s = -130 ms/Hz τT,s = 2000 ms DT,s = 2000 ms T0 = 580 ms GT,v = 90 ms/Hz τT,v = 1500 ms DT,v = 200 ms

7 References [1] Ursino M. Interaction between carotid baroregulation and the pulsating heart: a mathematical model. Am J Physiol Heart Circ Physiol 275: H1733-H1747, 1998. [2] Bovendeerd, P.H.M., Modeling Cardiac Function 8W160 – lecture notes 2008, Eindhoven University of Technology. [3] Suga, H., Sagawa, K., and Shoukas, A.A. Load independence of the instantaneous pressure-volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio. Circ. Res. 32: 314-322, 1973. [4] Chapleau, M.W., amd F.M. Abboud. Contrasting effects of static and pulsatile pressure on carotid baroreceptor activity in dogs. Circ. Res. 61:648-658,1987. [5] Guyton, A.C., Hall, J.E., Textbook of medical physiology. Saunders company, Philadelphia, 10 th edition 2000; 21:223. [6] Guyton, A.C., Hall, J.E., Textbook of medical physiology. Saunders company, Philadelphia, 10 th edition 2000; 11:116.

27 [7] Guyton, A.C., Hall, J.E., Textbook of medical physiology. Saunders company, Philadelphia, 10 th edition 2000; 20:210. [8] Kumada, M. R.M. Schmidt, K. Sagawa, and K.S. Tan. Carotid sinus reflex in response to hemorrhage. Am. J. Physiol. 219: 1373-1379, 1970. [9] Guyton, A.C., Hall, J.E., Textbook of medical physiology. Saunders company, Philadelphia, 10 th edition 2000; 25:266. [10] Guyton, A.C., Hall, J.E., Textbook of medical physiology. Saunders company, Philadelphia, 10 th edition 2000; 17:177-179. [11] Arts, T., Bovendeerd, P., Delhaas, T., Prinzen, F., Modeling the relation between cardiac pump function and myofiber mechanics. J Biomechanics 2003; 36:731-736. [12] Bovendeerd, P.H.M., Borsje, P., Arts, T., van de Vosse, F.N., Dependence of intramyocardial pressure and coronary flow on ventricular loading and contractility: a model study. Ann Biomed Eng 2006; 34:1833-45. [13] Hatakeyama, I. Analysis of baroreceptor control of the circulation. In Physical Bases of Circulatory Transport: Regulation and Exchange, edited by E. B. Reeve and A. C. Guyton. Philadelphia, PA: Saunders, 1967, p. 91–112.