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Modelling the dynamical baroreflex-feedback control
MATHEMATICAL i%!ifPuTER MODELLING
PERGAMON
Mathematical
and Computer
Modelling
31 (2000) 167-173 www.elsevier.nl/locate/mcm
Modelling the Dynamical Baroreflex-Feedback Control J. T. OTTESEN IMFUFA, Roskilde University Postbox 260, DK-4000 Roskilde, Denmark johnnytif.ruc.dk
Abstract-A
comprehensive model of the baroreflex-feedback mechanism regulating the heart rate, the contractility of the ventricle and the peripheral vascular resistance is presented. The dynamics of the affector and the effector parts are modelled. For each of the effector organs the steady state behaviour is used explicitely. Scenarios of heart infarct, mechanical clamping of peripheral regions and increasing peripheral resistance by drugs are used for validation of the dynamics of the model. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Baroreceptor, Nonlinear feedback mechanism, Nonlinear oscillations, modelling, Neural biology, Cardiovascular system, Medical applications.
1.
Mathematical
INTRODUCTION
There are several known feedback mechanisms controlling the human blood pressure. One of these is the baroreflex-feedback which is critical because it is the fastest mechanism regulating the blood
pressure.
autoregulation,
The effect of other
control
mechanisms,
such as renal,
humoral,
and tissue
is much slower.
Much effort has been spent on modelling the baroreflex-feedback mechanism. Most of the efforts focus primarily either on the effector part of the feedback or on the affector part, where the input to the feedback critical dynamics at both this paper, the discussion
mechanism takes place, see Figure 1. However, there is evidence of places. Hence, both parts should be treated with equal caution. In will be limited to a simplified examination of the dynamics of the
baroreflex-feedback mechanism. The feedback the pulsatile cardiovascular system.
model will be evaluated
using a simple
model of
One reason, among other things, for studying the baroreflex-feedback mechanism is the development of an anaesthesia simulator. Education and maintenance of skills in diagnosing and training serious or infrequent events have long been a concern in anaesthesiology. Extensive use of simulation in training and education has become widespread in other fields. In aviation, technically advanced full scale simulators provide realistic surroundings for the training and certification of pilots. In anaesthesiology, the only training previously available was intubation and resuscitation manikin primarily used for teaching trainees in basic skills. For this purpose, the SIMA group’ has developed a simulator based on mathematical models including phamacody‘The SIMA group comprise Math-Tech the Simulator Section of Herlev University Hospital, S&W Medico Teknik and the BioMath group at Roskilde University. 0895-7177/00/s - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(00)00035-2
Typeset by &@-TEX
J. T. OTTESEN
168
namic, phamacokinatic, metabolic, temperature, respiratory, and cardiovascular models together with various feedback mechanisms controlling the system [l]. 2.
THE
BAROREFLEX-FEEDBACK
MECHANISM
In this section, a simple comprehensive model of the baroreflex-feedback mechanism will be presented. The feedback will be divided into three parts in serial: the affector part where the pressure is the input and firing rate is the output (a pressure-mechano-electrical
transduction),
the central nervous system where the input is the firing rate and the outputs are the sympathetic and parasympathetic tones (nerve activities), and the effector organs where the tones are inputs and heart rate, contractility 2.1.
The
Affector
and peripheral resistance are outputs.
Part
This section is focused exclusively on the affector part. This part of the feedback loop might be very important for the dynamics of the cardiovascular system when the latter is not in steady state [2]. More precisely, the phrase “the affector part” denotes the part of carotid sinus where the baroreceptors are located together with the baroreceptor nerves, including the axons, see Figure 1. When a change in carotid sinus arterial pressure occurs, the cross-sectional area of the sinuses changes, whereby the viscoelastic wall is deformed (pressure-mechanical deformation). The receptors at the nerve endings are located in the lateral wall of carotid sinus. The mechano-electrical transduction takes place in these receptors themselves by an unknown coupling mechanism [3]. For further discussion of this point, see [4,5]. Hence, a change in pressure causes deformation in the spatial structure, which results in a change of activity of the baroreceptor nerves. This nerve activity is denoted the firing rate. Common Carotid
Central
Nervous
System
(CNS)
5 Aortic
Ganglions
Arch
Contractility Figure 1. The entire baroreffex-feedback system. The baroreceptors are located in the lateral wall of carotid sinus. The afferent firing rate is transmitted to the central nervous system. Here, the information is processed and the outputs, the sympathetic and the parasympathetic tones, are transmitted to the various effector organs regulating the blood flow.
The first documented measurement of the baroreceptor response to a change in carotid sinus arterial pressure, in summary carotid pressure, was carried out by Bronk and Stella and reported in 1932 [6] and 1935 [7]. Since then, many experiments have been performed. The experiments indicate that the firing rate of the baroreceptor nerves shows several nonlinear characteristics. Firing rate increases with carotid pressure. The response exhibits threshold and saturation. Sufficiently fast decreases in pressure cause firing to drop even below the threshold. A step
Dynamical Baroreflex-Feedback
Control
169
change in pressure causes a step change in firing rate followed by a resetting phenomenon,
decay in firing rate, towards the threshold value. Resetting
is called adaptation
i.e., a
and should not
be confused with a distributed time delay phenomenon. Response curves are sigmoidal and show asymmetric behavior, like in hysteresis. However, this phenomenon is not physical hysteresis. Finally, observations indicate that the response curves for hypotensive sinus and hypertensive sinus simply are translations
to the left and right along the pressure axis, respectively,
of the
response curve for normotensive sinus. There has been some effort to model these nonlinear characteristics
of the baroreceptors.
gren in 1952 [8] and Robinson and Sleight in 1980 [9] suggested simple functional of the response to a step change in pressure only. form in 1965 [ll],
Land-
descriptions
Warner in 1958 [lo], and in a modified
Scher and Young in 1963 [12], Poitras et al. in 1966 [13], Spickler and Kezdi
in 1967 [14], Franz in 1969 [15], and Srinivasan and Nudelman in 1972 [16] suggested various models based on ordinary differential equations.
Taher et al. [17] proposed in 1988 a unified
model that embodies the known nonlinear characteristics
of the baroreceptors.
These models were able to explain some, and a few even all, of the nonlinear characteristics of the baroreceptors. However, most of the models described by differential equations make consistent use of set-points, which are not physiologically based [18]. Moreover, they incorporate each of the nonlinearities separately, implying they arise independently. In 1982, Cecchini et al. [19] demonstrated
that this is not the case.
In [2], the author presents and verifies a new comprehensive model. This simple model describes the highly nonlinear response in firing rate to changes in carotid sinus arterial pressure. Mathematically it is expressed in terms of nonlinear ordinary differential equations as follows:
A7i2
=
k2Ej,
n(M (M,2)2
n,_ ‘An
. n(M-n)
Aris = k3Pc
(M/2)2
72
(1)
2,
lA - 73
n37
where n is the firing rate, M the maximal firing rate, i), the time derivative of the pressure at carotid sinus, and Ani, Anz, and Ans denote the deviations from the threshold value N, such that Ani + Anz + Ana = n - N. The parameters 71, 72, and 7s are time constants and describe the characteristic adaptation phenomenon measured, for example, by Brown [3]. They are related to different kinds of receptors. The parameters kl , k2, and k3 are weighting constants describing the contribution from each receptor type. Both the rs and the ks can be estimated by the experiment of Brown mentioned above. In the literature, measured values of the rs, the ks, and M are reported for some animals [7-9,15,20,21]. Typical values are ~1 = 0.5, 72 = 5.0, 7s = 500 seconds, ICI = 1.0, k2 = 2.0, and kg = l.OHz/mmHg. Equivalently, (1) may be rewritten in integral form as
n=N+
t
J(
kle-((t-s)/~l)
+ k2e-((t-s)/Q’2) + k3e-((t-S)/T3)
--oo
2.2. The Central
Nervous
p’ 0W2
n(M - n)
ds
.
(2)
1
System
In this section, a model describing how the central nervous system processes the firing rate, n, as an input signal is proposed. The outputs are the sympathetic and the parasympathetic tones, T, and Tp, respectively. Since the knowledge on the central nervous system is very limited, a very simple ad hoc model is chosen (for a discussion on principles for mathematical modelling
J. T. OTTESEN
170
in physiology, see [ZZ]). The choice is based on the only well-established experimental facts, the reciprocal law
and the direct
where M is the maximal firing rate described in Section 2.1.
When excluding time delay of
the system (which occurs in the sympathetic part primarily), there is no need for incorporating the two tones separately, For a detailed discussion on the time delay in the baroreflex-feedback mechanism, see [23]. Hence, in this description, the tones will not be directly involved as inputs to the effector organs. Instead the firing rate will be used. 2.3. The Effector Parts In this section, the effector parts of the feedback are treated. Since the firing rate is used as the input, the central nervous system is lumped into this model. The modelling is based on well-established experimental knowledge on the input-output relation in the static situation for each of the effector organs, together with a simple unifying approach describing the dynamics. As effector organs, the heart and the arterioles are considered, see Figure 1. The effects are a change in heart rate, H, contractility of the ventricle, S, and peripheral resistance to flow, R. Here, contractility describes the strength of the individual heart contractions [24-261. In steady state heart rate, the contractility and the peripheral resistance are sigmoidal decreasing functions in firing rate [24-311. Hence, they are of the form
with x = H,S, R and
Cl=
Gnax - x0 X0
-
Xmin
’
where ze is the desired value of x at n = N, xmrn and xmaX are the minimal and maximal values of x, respectively, n is the time dependent firing rate, given by (2), and N is the threshold value described in Section 2.1. The steady state dependencies are well known from experiments, while the dynamics are only vaguely determined. Hence, the most simple but physiologically justifiable model is chosen to describe the dynamics
k(t) = $ heady(n)
- X(t))
,
(4
where 7%is the characteristic time constant describing the transient. Equation (4) states that the time-derivative of x(t) is given by the difference between the steady state value, x&&.(n), and the actual value, z(t). Notice, that xstedy(n) depends on the time-dependent firing rate, given by (2), and hence is implicitly time-dependent. From [23,32], it follows that 7~ and 7s are two seconds and that 7~ is six seconds, approximately. However, it should be emphasized that there is large intervariation as well as intravariation in most of the parameters in human beings. The model of the baroreflex-feedback mechanism is given by (1) and (4), and it requires 20 parameters, all physiologically interpretable and measurable.
Dynamical
Figure in [33]. arterial various
Baroreflex-Feedback
Control
171
2. Electrical analogue diagram of the simple Windkessel-like model proposed Here H, S, R, and P denote heart rate, contractility, peripheral resistance and pressure, respectively, and rl, r2, cl, and c2 are fixed parameters describing resistances and compliances.
3. THE
CARDIOVASCULAR
SYSTEM
As a model of the cardiovascular system, an extended but simple Windkessel model will do. This model must have heart rate, contractility and peripheral resistance as parameters, and arterial pressure as variable. The arterial pressure will be identified with the carotid sinus pressure in close loop models (as is usual in experiments). The explicit choice of model of the cardiovascular system is not critical and the model developed in [33] is chosen. A diagram of the model is shown in Figure 2. The model will not be described further in this sparse paper.
4. SIMULATION,
VALIDATION
AND
DISCUSSION
There are several ways to validate the model, for example, quantitatively, qualitatively and sensitivity analysis. Quantitative validation may be done by comparing simulation results with experiments, such as the change in the steady state of average arterial pressure and heart rate under an acute 10% hemorrhage. However, due to the limited amount of space here, only a qualitative validation will be made.
=
E
120
2
allO
i
El00 E a 90
21.5 E
90
u1 I? 0
0
10 t&1
5 t[sec]
10
1.3 2 = E 21.5
1.2 F1.1 E. =1
2 E' ul
0.9 0.9!\\:..I 0
5 t m-1
10
0.5 0 0
5
10
t WI
Figure 3. The time course in peripheral resistance, R, arterial pressure, P, heart rate, H, and contractility, S, due to a step increase in peripheral resistance.
Figure 3 shows the result of a simulation where the peripheral resistance increases in one step by a factor 2.5. As a consequence, a raise in arterial pressure over a few seconds follows, The baroreflex-feedback mechanism reacts by lowering the heart rate, the contractility of the ventricle,
J. T.
172 2.5, =
1201
I
2
110
’
--,I00 I E 90 E p 90
Zl.5 z E ul
OTTESEN
0.5 0 0
I
70 5 t [set]
60
10
0
5 t [SW
IO
5
10
1.35 ,0.94 E
1.3
P 0.92
Zl.25. I
I” E u
1.2 1.15 P-m 0
5
t [see]
10
0.9
0.89
0
t [see]
Figure4. The time coursein contractility,S, arterialpressure,P, heartrate, H, and peripheralresistance,R, due to a drop in contractility.This scenariomight be used to describeheart infarcts. and the increased peripheral resistance which cause the arterial pressure to be lowered. Hence, the pressure, heart rate, and contractility show a little overshooting before they approach the new “steady state” level. The slower effect of the peripheral resistance does not exhibit overshooting but gradually decreases toward a level which is approximately two times the original level. Notice that the increase in systolic pressure is less than 10 mm Hg, but the increase in diastolic pressure is 20mmHg. This behaviour is in harmony with clinical observations, where the peripheral resistance is raised either by drugs or by mechanical clamping of vessels. Figure 4 shows the result of a simulation where the contractility of the heart decreases in one step by a factor l/1.8. As a consequence, a drop in arterial pressure from the actual heartbeat to the next follows. The baroreflex-feedback mechanism reacts by raising the heart rate and the peripheral resistance. The effect in contractility is vague but a minor increase appears. The arterial pressure stabilizes at a new “steady state” level after a little overshooting. The heart rate shows some overshooting before it approaches the new “steady state” level. As before, the slower effect of the peripheral resistance does not exhibit overshooting but gradually increases toward a level which is approximately 50 mm Hg set/ml higher. Notice that the decrease in diastolic pressure is less than 3mmHg, but the decrease in systolic pressure is 10mmHg. Also, in this case the results agree with clinical observations in the study of heart infarct in pigs. In both simulations, the overshootings are due to the slow effect in the dynamics of the peripheral resistance. Furthermore, it should be emphasized that the heart rate is not a continuous quantity; it is discrete. In the figures, the local maxima represent the actual heart rates. It is concluded that the results of the simulations agree sufficiently with the literature qualitatively as well as quantitatively. In the scenarios above, the effect of the detailed affector model, given in (l), is not pronounced. A simpler one will do as long as it embraces saturation and adaptation. However, for several other scenarios, the affector model is critical, such as in open loop experiments and in measurements of the firing rate directly.
REFERENCES 1. J. Jacobsen, P.F. Jensen,V. Andreasen,J. Larsenand J. Ottesen,The sima project: Developmentof an anaesthesiasimulator,European Journal of Anaesthesiology (submitted). 2. J.T. Ottesen,Non-linearityof the baroreceptornerves,Surveyson Mathematicsfor Industry(to appear).
Dynamical
Baroreflex-Feedback
Control
173
3. A. Brown, Receptors under pressure: An update on baroreceptors, Circulation Research 46 (1980). 4. M.W. Chapleau and F.M. Abboud, Modulation of baroreceptor activity, Brazilian Journal of Medical and Biological Research 27 (1994). 5. J.K. Kimani, Electron microscopic structure and innervation of the carotid baroreceptor region in the rock hyrax, Journal of Morphology 212 (1992). 6. D.W. Bronk and G. Stella, Afferent impulses in the carotid sinus nerve, Journal of Cellular and Compamtive Physiology 1 (1932). 7. D.W. Bronk and G. Stella, The response to steady state pressure of single end organs in the isolated carotid sinus, American Jounzal of Physiology 110 (1935). 8. S. Landgren, On the exitation mechanism of the carotid baroreceptors, Acta Physiology Scandinavic 26 (1952). 9. J.L. Robinson and P. Sleight, Single carotid sinus baroreceptor adaptation in normotensive and hypertensive dogs, In Arterial Baroreceptors and Hypertension, (Edited by P. Sleigt), Oxford Medical Publications, (1980). 10. H.R. Warner, The frequency-dependent nature of blood pressure regulation by the carotid sinus studied with an electric analog. 11. H.R. Warner, Some computer techniques of value for study circulation, In Computers in Biomedical Research, (Edited by R.W. Staycy and B.D. Waxman), Academic Press, New York, (1965). 12. A.M. Scher and A.C. Young, Servoanalysis of carotid sinus reflex effects on peripheral resistance, Circulation Research 12 (1962). 13. J.W. Poitras, N. Pantelakis, C.W. Marble, K.R. Dwyer and G.O. Barnett, Analysis of the blood pressure-barorecepter nerve firing relationship, In Engineering in Mcdicina and Biology, Proceedings of lgth Annual Conference, Vol. 105, (1966). 14. J.W. Spickler and P. Kezdi, Dynamic response characteristics of carotid sinus baroreceptors, American Journal of Physiology 212 (2) (1967). 15. G.N. Franz, Nonlinear rate sensitivity of carotid sinus reflex es a consequence of static and dynamic nonlinearities in baroreceptor behavior, Annual New York Academy of Science 156 (1969). 16. R. Srivivasan and H. Nudelman, Modeling the carotid sinus baroreceptor, Biophysics Journal 12 (1972). 17. M.F. Taher, A.B.P. Cecchini, M.A. Allen, S.R. Gobran, R.C. Gorman, B.L. Guthrie, K.A. Lingerfelter, S.Y. Rabbany, P.M. Rolchio, J. Melbin and A. Noordergraaf, Baroreceptor responses derived from a fundamental concept, Annals in Biomedical Engineering 16 (1988). 18. A.B.P. Cecchini, J. Melbin and A. Noordergraaf, Set-point: Is it a distinct structural entity in biological control?, Jountal of Theoretical Biology 93 (1981). 19. A.B.P. Cecchini, K.L.A. Triplitz, J. Melbin and A. Noordergraaf, Baroreceptor activity related to cell properties, In Engineering in Medicine and Biology, Proceedings of 35th Annual Conference, (1966). 20. W.B. Clarke, Static and dynamic characteristics of carotid sinus baroreceptors, Ph.D. Thesis, University of Rochester, (1968). 21. G.N. Franz, A.M. Scher and C.S. Ito, Small signal characteristics of carotid sinus baroreceptors of rabbits, Journal of Applied Physiology 30 (1971). 22. J. T. Ottesen and A. Noordergraaf, The virtue of creating mathematical models, Surveys on Mathematics for Industry (1997) (to appear). 23. J.T. Ottesen, Modelling of the baroreflex-feedback mechanism with time-delay, Journal for Mathematical Biology (1997) (to appear). 24. H. Suga, Controls of ventricular contractility assessed by pressure-volume ratio, Cardiovascular Research 10 (1976). 25. H. Suga, K. Sagawa and D.P. Kostiuk, Control of ventricular contractility assessed by pressure-volume ratio, Critical Reviews in Biomedical Engineering, (1976). 26. K. Sunugawa, Models of ventricular contraction based on time-varying elastance, Critical Reviews in Biomedical Engineering, (1982). 27. D.L. Eckberg, Nonlinearities of human carotid barorecteptor-cardiac reflex, Circulation Research 47 (1980). 28. P.I. Korner, M.J. West, 3. Shaw and J.B. Uther, “Steady-state” properties of the baroreceptor-heart rate reflex in essential hypertension in man, Clin. Ezp. Pharmacol. Physiol. 1 (1974). 29. M.N. Levy and H. Zieske, Autonomic control of cardiac pacemaker, Journal of Applied Physiology 27 (1969). 30. W.E. McAdam, An analogue simulation of the mammalians cardiovascular system, M.S. thesis. 31. A. Noordergraaf, Circulatory System Dynamics, Academic Press, CA, (1978). 32. M. Ursino, M. Antonucci and E. Belardinelli, Role of active changes in venous capacity by the carotid baroreflex: Analysis with a mathematical model, American Journal of Physiology 36 (1994). 33. V.C. Rideout, Mathematical and Computer Modeling of Physiological Systems, Medical Physics Publishing, (1991).
PERGAMON
Mathematical
and Computer
Modelling
31 (2000) 167-173 www.elsevier.nl/locate/mcm
Modelling the Dynamical Baroreflex-Feedback Control J. T. OTTESEN IMFUFA, Roskilde University Postbox 260, DK-4000 Roskilde, Denmark johnnytif.ruc.dk
Abstract-A
comprehensive model of the baroreflex-feedback mechanism regulating the heart rate, the contractility of the ventricle and the peripheral vascular resistance is presented. The dynamics of the affector and the effector parts are modelled. For each of the effector organs the steady state behaviour is used explicitely. Scenarios of heart infarct, mechanical clamping of peripheral regions and increasing peripheral resistance by drugs are used for validation of the dynamics of the model. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Baroreceptor, Nonlinear feedback mechanism, Nonlinear oscillations, modelling, Neural biology, Cardiovascular system, Medical applications.
1.
Mathematical
INTRODUCTION
There are several known feedback mechanisms controlling the human blood pressure. One of these is the baroreflex-feedback which is critical because it is the fastest mechanism regulating the blood
pressure.
autoregulation,
The effect of other
control
mechanisms,
such as renal,
humoral,
and tissue
is much slower.
Much effort has been spent on modelling the baroreflex-feedback mechanism. Most of the efforts focus primarily either on the effector part of the feedback or on the affector part, where the input to the feedback critical dynamics at both this paper, the discussion
mechanism takes place, see Figure 1. However, there is evidence of places. Hence, both parts should be treated with equal caution. In will be limited to a simplified examination of the dynamics of the
baroreflex-feedback mechanism. The feedback the pulsatile cardiovascular system.
model will be evaluated
using a simple
model of
One reason, among other things, for studying the baroreflex-feedback mechanism is the development of an anaesthesia simulator. Education and maintenance of skills in diagnosing and training serious or infrequent events have long been a concern in anaesthesiology. Extensive use of simulation in training and education has become widespread in other fields. In aviation, technically advanced full scale simulators provide realistic surroundings for the training and certification of pilots. In anaesthesiology, the only training previously available was intubation and resuscitation manikin primarily used for teaching trainees in basic skills. For this purpose, the SIMA group’ has developed a simulator based on mathematical models including phamacody‘The SIMA group comprise Math-Tech the Simulator Section of Herlev University Hospital, S&W Medico Teknik and the BioMath group at Roskilde University. 0895-7177/00/s - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(00)00035-2
Typeset by &@-TEX
J. T. OTTESEN
168
namic, phamacokinatic, metabolic, temperature, respiratory, and cardiovascular models together with various feedback mechanisms controlling the system [l]. 2.
THE
BAROREFLEX-FEEDBACK
MECHANISM
In this section, a simple comprehensive model of the baroreflex-feedback mechanism will be presented. The feedback will be divided into three parts in serial: the affector part where the pressure is the input and firing rate is the output (a pressure-mechano-electrical
transduction),
the central nervous system where the input is the firing rate and the outputs are the sympathetic and parasympathetic tones (nerve activities), and the effector organs where the tones are inputs and heart rate, contractility 2.1.
The
Affector
and peripheral resistance are outputs.
Part
This section is focused exclusively on the affector part. This part of the feedback loop might be very important for the dynamics of the cardiovascular system when the latter is not in steady state [2]. More precisely, the phrase “the affector part” denotes the part of carotid sinus where the baroreceptors are located together with the baroreceptor nerves, including the axons, see Figure 1. When a change in carotid sinus arterial pressure occurs, the cross-sectional area of the sinuses changes, whereby the viscoelastic wall is deformed (pressure-mechanical deformation). The receptors at the nerve endings are located in the lateral wall of carotid sinus. The mechano-electrical transduction takes place in these receptors themselves by an unknown coupling mechanism [3]. For further discussion of this point, see [4,5]. Hence, a change in pressure causes deformation in the spatial structure, which results in a change of activity of the baroreceptor nerves. This nerve activity is denoted the firing rate. Common Carotid
Central
Nervous
System
(CNS)
5 Aortic
Ganglions
Arch
Contractility Figure 1. The entire baroreffex-feedback system. The baroreceptors are located in the lateral wall of carotid sinus. The afferent firing rate is transmitted to the central nervous system. Here, the information is processed and the outputs, the sympathetic and the parasympathetic tones, are transmitted to the various effector organs regulating the blood flow.
The first documented measurement of the baroreceptor response to a change in carotid sinus arterial pressure, in summary carotid pressure, was carried out by Bronk and Stella and reported in 1932 [6] and 1935 [7]. Since then, many experiments have been performed. The experiments indicate that the firing rate of the baroreceptor nerves shows several nonlinear characteristics. Firing rate increases with carotid pressure. The response exhibits threshold and saturation. Sufficiently fast decreases in pressure cause firing to drop even below the threshold. A step
Dynamical Baroreflex-Feedback
Control
169
change in pressure causes a step change in firing rate followed by a resetting phenomenon,
decay in firing rate, towards the threshold value. Resetting
is called adaptation
i.e., a
and should not
be confused with a distributed time delay phenomenon. Response curves are sigmoidal and show asymmetric behavior, like in hysteresis. However, this phenomenon is not physical hysteresis. Finally, observations indicate that the response curves for hypotensive sinus and hypertensive sinus simply are translations
to the left and right along the pressure axis, respectively,
of the
response curve for normotensive sinus. There has been some effort to model these nonlinear characteristics
of the baroreceptors.
gren in 1952 [8] and Robinson and Sleight in 1980 [9] suggested simple functional of the response to a step change in pressure only. form in 1965 [ll],
Land-
descriptions
Warner in 1958 [lo], and in a modified
Scher and Young in 1963 [12], Poitras et al. in 1966 [13], Spickler and Kezdi
in 1967 [14], Franz in 1969 [15], and Srinivasan and Nudelman in 1972 [16] suggested various models based on ordinary differential equations.
Taher et al. [17] proposed in 1988 a unified
model that embodies the known nonlinear characteristics
of the baroreceptors.
These models were able to explain some, and a few even all, of the nonlinear characteristics of the baroreceptors. However, most of the models described by differential equations make consistent use of set-points, which are not physiologically based [18]. Moreover, they incorporate each of the nonlinearities separately, implying they arise independently. In 1982, Cecchini et al. [19] demonstrated
that this is not the case.
In [2], the author presents and verifies a new comprehensive model. This simple model describes the highly nonlinear response in firing rate to changes in carotid sinus arterial pressure. Mathematically it is expressed in terms of nonlinear ordinary differential equations as follows:
A7i2
=
k2Ej,
n(M (M,2)2
n,_ ‘An
. n(M-n)
Aris = k3Pc
(M/2)2
72
(1)
2,
lA - 73
n37
where n is the firing rate, M the maximal firing rate, i), the time derivative of the pressure at carotid sinus, and Ani, Anz, and Ans denote the deviations from the threshold value N, such that Ani + Anz + Ana = n - N. The parameters 71, 72, and 7s are time constants and describe the characteristic adaptation phenomenon measured, for example, by Brown [3]. They are related to different kinds of receptors. The parameters kl , k2, and k3 are weighting constants describing the contribution from each receptor type. Both the rs and the ks can be estimated by the experiment of Brown mentioned above. In the literature, measured values of the rs, the ks, and M are reported for some animals [7-9,15,20,21]. Typical values are ~1 = 0.5, 72 = 5.0, 7s = 500 seconds, ICI = 1.0, k2 = 2.0, and kg = l.OHz/mmHg. Equivalently, (1) may be rewritten in integral form as
n=N+
t
J(
kle-((t-s)/~l)
+ k2e-((t-s)/Q’2) + k3e-((t-S)/T3)
--oo
2.2. The Central
Nervous
p’ 0W2
n(M - n)
ds
.
(2)
1
System
In this section, a model describing how the central nervous system processes the firing rate, n, as an input signal is proposed. The outputs are the sympathetic and the parasympathetic tones, T, and Tp, respectively. Since the knowledge on the central nervous system is very limited, a very simple ad hoc model is chosen (for a discussion on principles for mathematical modelling
J. T. OTTESEN
170
in physiology, see [ZZ]). The choice is based on the only well-established experimental facts, the reciprocal law
and the direct
where M is the maximal firing rate described in Section 2.1.
When excluding time delay of
the system (which occurs in the sympathetic part primarily), there is no need for incorporating the two tones separately, For a detailed discussion on the time delay in the baroreflex-feedback mechanism, see [23]. Hence, in this description, the tones will not be directly involved as inputs to the effector organs. Instead the firing rate will be used. 2.3. The Effector Parts In this section, the effector parts of the feedback are treated. Since the firing rate is used as the input, the central nervous system is lumped into this model. The modelling is based on well-established experimental knowledge on the input-output relation in the static situation for each of the effector organs, together with a simple unifying approach describing the dynamics. As effector organs, the heart and the arterioles are considered, see Figure 1. The effects are a change in heart rate, H, contractility of the ventricle, S, and peripheral resistance to flow, R. Here, contractility describes the strength of the individual heart contractions [24-261. In steady state heart rate, the contractility and the peripheral resistance are sigmoidal decreasing functions in firing rate [24-311. Hence, they are of the form
with x = H,S, R and
Cl=
Gnax - x0 X0
-
Xmin
’
where ze is the desired value of x at n = N, xmrn and xmaX are the minimal and maximal values of x, respectively, n is the time dependent firing rate, given by (2), and N is the threshold value described in Section 2.1. The steady state dependencies are well known from experiments, while the dynamics are only vaguely determined. Hence, the most simple but physiologically justifiable model is chosen to describe the dynamics
k(t) = $ heady(n)
- X(t))
,
(4
where 7%is the characteristic time constant describing the transient. Equation (4) states that the time-derivative of x(t) is given by the difference between the steady state value, x&&.(n), and the actual value, z(t). Notice, that xstedy(n) depends on the time-dependent firing rate, given by (2), and hence is implicitly time-dependent. From [23,32], it follows that 7~ and 7s are two seconds and that 7~ is six seconds, approximately. However, it should be emphasized that there is large intervariation as well as intravariation in most of the parameters in human beings. The model of the baroreflex-feedback mechanism is given by (1) and (4), and it requires 20 parameters, all physiologically interpretable and measurable.
Dynamical
Figure in [33]. arterial various
Baroreflex-Feedback
Control
171
2. Electrical analogue diagram of the simple Windkessel-like model proposed Here H, S, R, and P denote heart rate, contractility, peripheral resistance and pressure, respectively, and rl, r2, cl, and c2 are fixed parameters describing resistances and compliances.
3. THE
CARDIOVASCULAR
SYSTEM
As a model of the cardiovascular system, an extended but simple Windkessel model will do. This model must have heart rate, contractility and peripheral resistance as parameters, and arterial pressure as variable. The arterial pressure will be identified with the carotid sinus pressure in close loop models (as is usual in experiments). The explicit choice of model of the cardiovascular system is not critical and the model developed in [33] is chosen. A diagram of the model is shown in Figure 2. The model will not be described further in this sparse paper.
4. SIMULATION,
VALIDATION
AND
DISCUSSION
There are several ways to validate the model, for example, quantitatively, qualitatively and sensitivity analysis. Quantitative validation may be done by comparing simulation results with experiments, such as the change in the steady state of average arterial pressure and heart rate under an acute 10% hemorrhage. However, due to the limited amount of space here, only a qualitative validation will be made.
=
E
120
2
allO
i
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21.5 E
90
u1 I? 0
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10 t&1
5 t[sec]
10
1.3 2 = E 21.5
1.2 F1.1 E. =1
2 E' ul
0.9 0.9!\\:..I 0
5 t m-1
10
0.5 0 0
5
10
t WI
Figure 3. The time course in peripheral resistance, R, arterial pressure, P, heart rate, H, and contractility, S, due to a step increase in peripheral resistance.
Figure 3 shows the result of a simulation where the peripheral resistance increases in one step by a factor 2.5. As a consequence, a raise in arterial pressure over a few seconds follows, The baroreflex-feedback mechanism reacts by lowering the heart rate, the contractility of the ventricle,
J. T.
172 2.5, =
1201
I
2
110
’
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OTTESEN
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I
70 5 t [set]
60
10
0
5 t [SW
IO
5
10
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1.3
P 0.92
Zl.25. I
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t [see]
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Figure4. The time coursein contractility,S, arterialpressure,P, heartrate, H, and peripheralresistance,R, due to a drop in contractility.This scenariomight be used to describeheart infarcts. and the increased peripheral resistance which cause the arterial pressure to be lowered. Hence, the pressure, heart rate, and contractility show a little overshooting before they approach the new “steady state” level. The slower effect of the peripheral resistance does not exhibit overshooting but gradually decreases toward a level which is approximately two times the original level. Notice that the increase in systolic pressure is less than 10 mm Hg, but the increase in diastolic pressure is 20mmHg. This behaviour is in harmony with clinical observations, where the peripheral resistance is raised either by drugs or by mechanical clamping of vessels. Figure 4 shows the result of a simulation where the contractility of the heart decreases in one step by a factor l/1.8. As a consequence, a drop in arterial pressure from the actual heartbeat to the next follows. The baroreflex-feedback mechanism reacts by raising the heart rate and the peripheral resistance. The effect in contractility is vague but a minor increase appears. The arterial pressure stabilizes at a new “steady state” level after a little overshooting. The heart rate shows some overshooting before it approaches the new “steady state” level. As before, the slower effect of the peripheral resistance does not exhibit overshooting but gradually increases toward a level which is approximately 50 mm Hg set/ml higher. Notice that the decrease in diastolic pressure is less than 3mmHg, but the decrease in systolic pressure is 10mmHg. Also, in this case the results agree with clinical observations in the study of heart infarct in pigs. In both simulations, the overshootings are due to the slow effect in the dynamics of the peripheral resistance. Furthermore, it should be emphasized that the heart rate is not a continuous quantity; it is discrete. In the figures, the local maxima represent the actual heart rates. It is concluded that the results of the simulations agree sufficiently with the literature qualitatively as well as quantitatively. In the scenarios above, the effect of the detailed affector model, given in (l), is not pronounced. A simpler one will do as long as it embraces saturation and adaptation. However, for several other scenarios, the affector model is critical, such as in open loop experiments and in measurements of the firing rate directly.
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Control
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