determinants of stroke volume and systolic and diastolic ... of stroke volume and systolic and...

Determinants of stroke volume and systolic and diastolic aortic pressure

NIKOS STERGIOPULOS, JEAN-JACQUES MEISTER, AND NICO WESTERHOF Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland; and Laboratory for Physiology, Institute for Cardiovascular Research, Free University, 1081 BT Amsterdam, The Netherlands

Stergiopulos, Nikos, Jean- Jacques Meister, and Nice Westerhof. Determinants of stroke volume and systolic and diastolic aortic pressure. Am. J. PhysioZ. 270 (Heart Circ. PhysioZ. 39): H2050-H2059, 1996.-We investigated how parameters describing the heart and the arterial system contribute to the systolic and diastolic pressures (Ps and Pd, respectively) and stroke volume (SV). We have described the heart by the varying-elastance model with six parameters and the systemic arterial tree by the three-element windkes- se1 model, leading to a total of nine parameters. Application of dimensional analysis led to a total of six dimensionless parameters describing dimensionless Ps and Pd, i.e., pressures with respect to venous pressure (P& and P&). SV was normalized with respect to unloaded ventricular volume (Vd). Sensitivity analysis showed that PJP”, P&, and SVNd could be accurately described by four, three, and three dimensionless parameters, respectively. With this limited number of parameters, it was then possible to obtain empirical analytical expressions for PJPv, Pd/P”, and SVN+ The analytic predictions were tested against the model values and found to be as follows: Ps predicted = (1.0007 5 0.0062) Ps, r = 0.987; Pd predicted = (1.016 ? 0.0085) Pd, r = 0.992; and SV predicted = (0.9987 2 0.0028) SV, r = 0.996. We conclude that aortic Ps, Pd, and SV can be accurately described by a limited number of parameters and that, for any condition of the heart and the arterial system, Ps, Pd, and SV can be presented in analytical form.

varying-elastance model; three-element windkessel model; dimensional analysis; coupling of heart and arterial system

THE PRESSURE AND FLOW in the ascending aorta arise from the interaction between the heart and the arterial system; i.e., both heart and arterial load determine aortic pressure and cardiac output. The determinants of aortic systolic and diastolic pressures (Ps and Pd, respectively) and stroke volume (SV) are not quantita- tively known. A number of studies have been performed to obtain insight into the effects of changes in parameters such as peripheral resistance (RJ and total arterial compliance (C) on aortic pressure and flow (3, 8). However, a general and quantitative approach to obtain all determinants of Ps, Pd, and SV has, to the best of our knowledge, not been carried out. This is in part due to the fact that a model of both the heart and the arterial system could not be formulated in a quantitative way with a limited number of parameters. Such a quantitative description is achieved by use of the varying-elastance model for the heart and the three- element windkessel model for the arterial system, both of which contain a minimal number of essential parameters. The three-element windkessel model is a generally accepted and rather accurate description of the

arterial system (9, 16). The varying-elastance model of the heart has also been shown to be an acceptable limited-parameter model (10, 14). Thus heart and arterial system can be described with a limited number of independent parameters, and heart and load can be coupled to yield aortic pressure and flow. From these, Ps, Pd, and SV are determined and used to quantify their sensitivity to cardiac and arterial parameters. We will use dimensional analysis to reduce the number of independent parameters by combining them into dimensionless groups (7). Subsequently, we will study the effects of these dimensionless groups of variables on pressures and SV. This analysis will allow us to decide which parameters are important determinants of Ps, Pd, and SV This information will be used to derive empirical formulas for Ps, Pd, and SV.

METHODS

MathematicaZ modeZ. The mathematical model for the heart and the arterial system (14) consists of the varying- elastance model for the heart (10) and the three-element windkessel model (16) representing the arterial load. An electrical analog of the entire heart-arterial system model is shown in Fig. 1A.

The varying-elastance model for the heart consists of a pressure source (venous pressure, Pv) emptying into the left ventricle with a time-varying elastance, E(t). Elastance relates ventricular pressure, P, to ventricular volume, V, through the formula

where Vd is the unloaded volume defined as the intercept of the end-systolic pressure-volume relationship with the volume axis (10).

Changes in elastance are due to the contraction of the myocardium. A typical heart cycle of the elastance of an isolated cat heart is depicted in Fig. 1B. The minimum value of the elastance, Emin, together with Pv, determine filling and thus end-diastolic volume (V&. The maximum value of the elastance, the end-systolic pressure-volume relationship or E max7 is considered a measure of the contractility of the heart. For a given cardiac state (heart rate and contractility constant), the elastance-time curve is assumed to remain unchanged and independent of alterations on the load (10).

We found that the shape of the elastance curve is best approximated by the following periodic “double-Hill” function

HZ050 0363-6135/96 $5.00 Copyright 0 1996 the American Physiological Society

DETERMINANTS OF STROm VOLUME AND AORTIC PRESSURE HZ051

mitral valve

aortic valve

RP

E max

E min

‘P 0.2

Time [s]

Fig. 1. A: electrical representation of model of heart and arterial system. Pv is venous pressure and Rv a (small) resistor over which filling takes place. E(t) is time-varying elastance. Three parameters of windkessel (Rc, characteristic resistance; Rp, peripheral resistance; and C, total arterial compliance) are indicated. B: waveform of time-varying elastance with its minimum (Emin), maximum (J&&, and peak time (TJ values pertaining to cat heart.

The first term in the brackets (Hill function) describes the ascending part of the E(t) curve and the second (inverted Hill function) the descending part. The curve is shown in Fig. l& where only a single heartbeat is presented. Dimensionless shape parameters, al and CX~, define the relative appearance time of each curve within the heart period, T, and exponents, nl and n2, determine the steepness of the ascending and descending parts, respectively.

The arterial system is represented with a linear three- element windkessel (16). Thus the whole arterial system is conceived as a point load with the characteristic impedance, &, which models the inertial and compliant effects of the ascending aorta, connected in series with a parallel combina- tion of the peripheral resistance, Rp, and the mean total arterial compliance, C. With proper choice of parameters, the three-element windkessel can model the arterial input impedance well and can thus provide an accurate description of the heart load. Nonlinearities arising from the pressure dependence of C have minor effects on aortic pressure and flow waveforms and will thus be neglected in this analysis (11).

For the purposes of this study, we will use as a typical example the human circulation. For the validation of the model, we will use data from earlier experiments on isolated cat hearts (3). The control parameters for the human and cat heart-arterial system model are given in Table 1.

Circulation of the four phases of the heart cycle (isovolumic contraction, ejection, isovolumic relaxation, and filling) is done in a straightforward manner, as explained in previous studies (1). Equation 1 is coupled to the equation of the three-element windkessel only in the ejection phase. In this phase, a solution was obtained numerically by use of a fourth-order Runge-Kutta integration scheme. During ejection, ventricular and aortic pressures were assumed equal. During isovolumic contraction and relaxation, aortic flow was zero (no incisura), and Eq. 1 applies for ventricular pressure while aortic pressure decayed exponentially, as dictated by the three-element windkessel. In the filling phase, ventricular volume relates to pressure through the formula dV/dt = P - P)/..“, which is solved with Eq. 1 by use of a Runge- Kutta integration scheme.

Dimensional analysis. For a given shape of the elastance- time curve, the above heart-arterial system contains eight basic independent parameters, three of which (Rc, Rp, and C) pertain to the arterial system. Four parameters (z Em=, Emin, and Vd) pertain to the heart. The eighth parameter is venous (filling) pressure, Pv. For a complete description of the heart, however, one should include the four parameters that define the shape of the E(t) curve, namely, CQ, cxz, nl and n2. For simplicity, we will assume that the shape of the E(t) curve is approximately triangular and sufficiently determined by specifying its peak time, Tp (see Fig. 1B). The limitations of this assumption will be considered in DISCUSSION. The system is now assumed to be well defined, and any other important determinants of cardiac or arterial function (such as humoral or neural factors) are assumed to be external to the system; i.e., their influence is simply to modify one or more of the nine basic independent parameters. From this assumption, any variable resulting from the heart-arterial system interaction will be a single function of the nine basic parameters. In this study we will focus on Ps, Pd, and SV, which can be expressed in general terms as

where fl, f& and fs are the unknown functions still to be defined.

Ideally, one can do parametric studies in which one of the independent parameters in the right-hand side of Eqs. 3-5 is varied while the others are kept constant. This procedure would then be repeated eight times for the remaining parameters. The result of this tedious task would be to fill a nine-dimensional parameter space, out of which it would have been very difficult to extract a global relationship.

The logical way of approaching this problem is to use the theory of dimensional analysis (7). By virtue of the Bucking- ham II theorem (7), we can rewrite Eqs. 3-5 in dimensionless form, in which the independent dimensionless parameters, often called II terms, are formed by groups of the original independent parameters. The major advantage of this approach is that the number of Il terms is equal to the number of original independent parameters (9) minus the number of basic dimensions (units) of the system (Ref. 3; namely, time, volume, and pressure). Thus the number of II terms in our equations reduces to six. Therefore, without any loss of

Table 1. Basic model parameters

Independent Cardiac and Arterial System Parameters Human Cat

Characteristic impedance (Rc), mmHg l s l ml1 0.51 1.50 Peripheral resistance (R&, mmHg l s l ml-l 1.05 21.5 Total arterial compliance (C), ml/mmHg 1.60 0.057 Heart period (79, s 1.00 0.39 Maximum elastance (&-&, mmHg/ml 2.31 41.3 Minimum elastance (Emin), mmHg/ml 0.06 2.00 Unloaded volume (Vd), ml 20.0 0.75 Venous pressure (PJ, mmHg 7.50 7.50 Time to peak elastance (Q, s 0.43 0.25 Shape factor q 0.303 0.25 Shape factor q 0.508 0.63 Exponent Q 1.32 2.00 Exponent Q 21.9 15.0

H2052 DETERMINANTS OF STROKE VOLUME AND AORTIC PRESSURE

generality, Qs. 3-5 are rewritten as A 120

sv Rc RpC 1 KIlax K -- vd -@3iP-9- - - l

Tp -

(81 P T CEmm ’ Emin ’ EminVd ’ T I

The choice of II terms in the right-hand side of Qs. 6-8 is not unique; however, it was made such that each term refers to easily identified ratios of the independent variables. The dimensionless parameter III = RJRp is a pure arterial parameter. The parameters II2 = RpC/T and lL13 = l/(C&& pertain to both heart and arterial system and are considered “coupling parameters.” The parameters IId = EmJEmin and l-I6 = Tp/T are cardiac parameters. The parameter IIs = Pv/(EminVd) gives the coupling of the heart with the venous system.

Time [s]

The dependence of P&, Pd&, and SV/Vd on the lI terms will be derived. First, a sensitivity analysis will be performed to assess which of the II terms are important in the determination of Ps, Pd, and SV By retaining only the II terms that are important, we can derive an analytic description of the functions, QI, m2, and m3.

RESULTS 0 I 2 3 4 5 6

Kilidation of’ modeZ. The model was tested by use of available data from experiments on the isolated cat heart loaded with the three-element windkessel (3). The windkessel parameters were taken directly from the original paper. The cardiac parameters were also taken from this article. By plotting the relation between mean ventricular pressure and SV and fitting this with a parabola (12), maximal SV and mean isovolumic pressure were determined. Maximal SV and Pv, together with an assumed dead volume of 0.75 ml, gave the value of Emin. Isovolumic pressure was used to estimate EmaX. The parameters of the Hill functions were chosen such that the ventricular pressure wave shape resembled the measured one (3). The resulting E(t) curve is shown in Fig. 1B. All parameter values are presented in Table 1.

Volume [ml]

Fig. 2. A: left ventricular pressure, aortic pressure, and aortic flow for heart-arterial model of cat. B: pressure-volume loops obtained from heart-arterial model of cat. Three loops are shown for changes in BP (fully drawn lines). An increase in filling is given by stippled line. Two pressure-volume loops during increased contractility are given by dashed lines.

putational effort considerably without large gain in information. It may be seen that Ps mainly depends on III, l&, IIs, and l-Id. Pd depends on l&, l-Is, and IId. SV depends on l&, IIs, and l-Is. It may also be seen that the relation between Ps and Pd with II4 is linearly proportional. The same is true for SV in relation to l-Is.

Based on the results, Eqs. 6-8 are rewritten in the following simplified form

The cat model produces ventricular and aortic pressures and aortic flow as given in Fig. 2A. Pressure- volume loops for control, increased contractility (EmaX X 2), and increased filling (Pv from 7.5 to 8.5 mmHg) are given in Fig. 2B. In Fig. 3, the changes in Ps, Pd, and SV for changes in Rp and C in the model and the isolated cat heart are compared. We conclude that model predictions and experimental data compare very well in many aspects.

Human model The model was then converted to the human cardiovascular system. The parameters are given in Table 1. The dependence of the variables P&, Pd/P”, and SV& on the Il terms are given in Figs. 4 and 5. We have calculated only a limited number of points (0.5, 0.75, 1, 1.5, and 2 of control) of the dimensionless parameters. More data would have increased the com-

P S Emax -- @ ---- - P 1

V ’ T ’ CEmm’ Emin

P d -- - P

@ ’ V

sv 1 P -- @ P - - - V ’ d ’ CEma ’ EmiIVd

Equations 9-11 were used to fit the data presented in Figs. 4 and 5, following a standard procedure described in the appendix. The analysis was limited to II values in the range of half to twice the control values, fitting the data over this range only with arbitrary formulas. The empirical formulas for Ps, P& and SV resulting from the

DETERMINANTS OF STROKE VOLUME AND AORTIC PRESSURE H2053

150

100

50

0

150

100

50

0

0 50 100

Systolic pressure [mm Hg]

150

B I 1 ,- .’ . : .

0 50 100 150

Diastolic pressure [mm Hg]

C I . .’ . . . . .- : .’

0 1 2

Stroke volume [ml]

Fig. 3. A: relationship between systolic blood pressure (Ps) measured in an isolated cat heart loaded with 3-element windkessel and model prediction. Changes are obtained through variations in &,, increased to 2.14 and 4.81 of control and decreases of total arterial compliance (C) to 0.326 and 0.084 of control. B: as in A but for diastolic blood pressure (P&. C: as in A but for stroke volume (SV). Dashed lines are lines of identity.

fit are

P S E max -- - b 1 .-

P V E min (b&Em=

. b 6

sv K -- - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ vd

-dI.r.e min

(14)

where the last term in the exponent of Eq. 14 is the product of I& and IIS. The values of the empirically determined coefficients bi (i = l,..., 8), ci (i = l,..., 6), and di (i = l,..., 4) are given in Table 2.

The predictions of these analytical expressions for Ps, P& and SV were tested against their actual values (given by model) in Fig. 6. Ps predicted = (1.0007 2 0.0062) Ps, r = 0.987; Pd predicted = (1.016 2 0.0085) P& r = 0.9%; sv

predicted = (0.9987 2 0.0028) SV, r = 0.996. We conclude that the formulas are accurate over a wide range of pressures and svs.

Based on Eqs. B-14, we have derived the sensitivity of Ps, P& and SV to cardiac and arterial parameters. Sensitivities are presented in dimensionless form around the control values in Table 2.

DISCUSSION

The main objectives of this study were, first, to obtain insight into the sensitivity of Ps, Pd, and SV to cardiac and arterial parameters and, second, to derive empirical formulas that give an accurate description of Ps, Pd, and SV Using dimensional analysis, we identified the major cardiac and arterial parameters that contribute to Ps, Pd, and SV We found that all three variables (Ps, Pd, and SV) could be described by a limited number of parameters. The fact that three to four dimensionless parameters were sufficient to describe the variables made it possible to derive empirical formulas for Ps, Pd, and SV Obviously, with heart rate known, the predictions on SV can be easily extended to predictions of cardiac output.

Universality of approach and applicability of empiri- caZ formulas. The use of dimensional analysis for the derivation of the empirical formulas (Eqs. 12-14), apart from being an elegant method of reducing the number of independent variables, offers another significant advantage: the dimensionless II terms, which are now the independent parameters, are for control conditions quite similar for all mammals. For example, parameters RJRp and RpCIT were shown earlier to be the same in all mammals (6, 15). Also, the cardiac parameter (Em= /Emin) is a ratio that is independent of heart size because it gives the ratio of isovolumic pressure and Pd, a ratio which is similar in mammals. The ratio of V,d (PJEmin + Vd) and Vd is found to be the same in different mammals. Pressures in the aorta and Pv are also similar and SV and Vd make a volume ratio likely to be similar in mammals.

The empirical formulas for the Ps, Pd, and SV (Eqs. 12-14) were derived for the human and do not necessar- ily apply to all mammals. We have checked this by applying the empirical formulas to the isolated cat experiments and significant differences were found. This is mainly due to the fact that the elastance function for the human is quite different from that of


30

20

10

0

0 0.5 I 1.5 2 2.50 0.5 I 1.5 2 2.5

Rc’Rp RpC/T

0 0.5 1 1.5 l/(C E )

max

2 2.50 0.5 1 1.5 E /E max min

2 2.5

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5

'v ' tvcj 'mi"l T/T

Fig. 4. Relations between dimensionless Ps and Pd and 6 II terms. Horizontal axis is normalized with respect to control. Pv, venous (filling) pressure; V& unloaded volume; 7’, heart period.

the cat (see Table 1). Also, in Figs. 4 and 5, some dependence on Z’P that is seen was not included in the analysis. We may expect that if appropriate analysis was done to include the additional II terms resulting from the E(t) shape factors (cx~, a2, nl, and n2), one could derive global relationships for pressures and SV Fur- ther research, however, is needed to conclude whether this is feasible.

GeneraZity. It should be stressed that the model applies to all cardiovascular conditions. Nervous and humoral control will affect the cardiac and arterial parameters, but when changes in the independent parameters are known and accounted for, the correct pressures and SV will result.

Limitations. The analysis was performed with a model of the heart and arterial system. In principle, this approach can be applied to the intact animal but in such a preparation it is extremely difficult to induce variations in certain parameters while leaving other parameters unaffected. For instance, a change in C in the intact animal will result in changes in pressure, leading to baroreflex control changes in heart rate and contractility (8). It will also lead to changes in cardiac filling. Moreover, the analysis requires a large number of data points, even if the sensitivity to the parameters that are of major importance is investigated. In general, experiments would require very long measure- ment times in a stable preparation. The present study


Fig. 5. Relations between dimensionless SV and 6 II terms. Horizontal axis is normalized with respect to control.

5

4

u 3 2 > CD 2

1

0

I I I I

I I I I

0 0.5 I 1.5 2 2.5 Cl 0.5 1 1.5 2 2.5

5

2.5 0 0.5 1 1.5 2 2.5 E

max ’ ‘min

8

6

vp F$C/T

0 0.5 I 1.5 2 2.5 0 0.5 1 1.5 2 2.5

p 1 o$ Ii V min J Tp/T

gives the functional dependence of Ps, Pd, and SV on the variables Ps and Pd to the independent parameters, important parameters and forms the basis for further although the quantitative relations varied. studies in animals and the human. By assuming a triangular wave shape of E(t), we

A triangular-like wave shape in the E(t) curve was have limited the shape factors of the E(t) to a single chosen for the human model. By changing the wave parameter, Tp. If the Hill equations were taken into shape of E(t), we found similar sensitivity of the account, this would have implied four parameters, i.e.,

Table 2. Parameter values of Eqs. 12-14 estimated by least-squares fit

Parameters

0.337 1.87 0.271 0.651 0.93 0.595 0.60 0.369 0.595 1.09 0.346 0.271 0.866 0.111 1.04 0.0589 0.568 0.389

DETERMINANTS OF STROKE VOLUME AND AORTIC PRESSURE

Systolic pressure [mm Hg]

B I s . l

. .

. .

0 100 200

Diastolic pressure [mm Hg]

C

. ci .ri .

. 0;

. ,

0 40 80 120

Stroke volume [ml] Fig. 6. Comparison between model prediction and empirical formulas (Eqs. 12-14) for Ps (A), Pd (B), and SV (C).

increasing the number of parameters by three. Better fits, i.e., fits with more parameters, will improve the final result and make it possible to accurately account for changes in the E(t) curve.

The pressure-volume relations in diastole and systole were approximated by straight lines. There is evidence that these relations are curved with convexity toward the volume axis in diastole and concavity in systole (2, 13). This second-order effect was not included to keep relations simple and the number of parameters small.

The way we derived the empirical formulas was as follows. We have started from a chosen dimensionless parameter and studied its relation to the variable of interest. Subsequently, we studied the contributions of the other parameters on this relation (see APPENDIX). By doing so, we have chosen a certain path to arrive at our results. If we had started the analysis from a different dimensionless parameter, the formulas would have had a different form. However, the same parameters would have appeared in the formulas. We have also fitted the relations with linear, exponential or power curves. The choices were made on the basis of the quality of the fit during the search for a minimal number of coefficients in the fitted curve, since the dependence of each coeffi- cient on the other parameters needed to be quantified.

We have carried out the analysis for a limited range of the dimensionless parameters (from 0.5 to 2 of their control value). Again this was done to limit the amount of work. Most variations in the parameters cover the physiological working range. For instance, during exer- cise, heart rate increases considerably (i.e., factor of 3-4) so that T decreases by the same amount, but $, decreases considerably as well. Thus the dimensionless parameter R&/T does not undergo large changes. However, for changes in a single parameter, the formulas apply only for the range 0.5-2 of control.

ImpZications. Figures 7 and 8 show the variations of Ps, Pd, and SV for changes in Rp and C (arterial) and for changes in the end-systolic pressure-volume relation and heart rate (cardiac). The values derived from pressure and flow in time are given by the symbols. The fully drawn lines are from the derived equations. It may be seen that our empirical formulas, when applied within the allowed range, predict these variables well. The sensitivity of Ps, Pd, and SV to the independent cardiac and arterial parameters is given in Table 3. It may be seen from Table 3 that characteristic impedance has a small effect on Ps only.

Increased Rp decreases SV and increases mean pressure (4). If the pressure wave shape remains the same, the increase in mean pressure would result in proportional increases in Ps and Pd. However, with an increase in Rp the arterial pressure in diastole will decrease more slowly due to increased RpC time, so that Pd remains higher. This then results in a relatively larger increase in Pd than Ps. In Table 3 of Elzinga and Westerhof (3), we indeed see that with an increase in Rp (by a factor of -2) P d increased by 50% compared with Ps. This is in good agreement with the theoretical predictions given in Table 3.

For a decrease in C, SV increases slightly and mean pressure increases somewhat. This is again in agreement with experimental findings in (3). When C decreases, the decay of pressure in diastole will be faster so that Pd will be lower. Elzinga and Westerhof (3) showed that a decrease in C (by a factor of -3) resulted in a decrease in Pd that was about twice as much as the increase in Ps. Randall et al. (8) studied Ps and Pd in the intact dog by replacing the aorta with a stiff tube while keeping Rp and heart rate unchanged. However, in


250

0 5 IO 15 20 25

Compliance, C [ml/kPa]

250 I I I I I

0 1 I I I I I I

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Maximum elastance, E,.,,aX [kPa/ml] Heart rate [bpm]

0.2 0.4 0.6

Peripheral resistance, F$, [kPa s/ml]

0.8

250 I I I I D 1

I

200

150

100

50

0 I I I I

0 50 100 150 200 250

Fig. 7. Effects of changes in C (A), $, (B), Emax (C), and heart rate (D) on Ps and Pd. Points are obtained from model and lines are representations of empirical relationships given by Qs. 12 and 13.

their preparation in contrast to the isolated heart increase in contractility resulted in roughly the same studies by Elzinga and Westerhof (3), end-diastolic changes in SV and Ps, in good agreement with our pressure increased. They found that Ps increased prediction of similar sensitivity of these parameters slightly, whereas Pd decreased. This may be explained (Table 3). from the fact that end-diastolic ventricular pressure The Pd-volume relation is a very important param- (cardiac filling) increased somewhat and contractility eter in the determination in both pressure and SV. The was also possibly increased. sensitivities to this parameter are similar for pressures

Comparison of the sensitivities to I$., and C shows and flow. A decrease in Emin will result in a larger Ved, that Ps and Pd are about four times less sensitive to resulting in changes similar to those due to increased changes in C than to changes in Rp. For SV, the Pv. The large sensitivity shows that, with poor relax- sensitivity to C is even smaller compared with the ation of the cardiac muscle as in ischemia, the pumping sensitivity to Rp. function of the heart is strongly reduced.

When heart rate decreases (and T increases), we Pv has a large effect on pressures and flow. This is the predict an increase in SV, and a decrease in Ps and Pd. Frank-Starling mechanism. The predictions (Table 3) This is in agreement with the findings on the isolated are in agreement with animal data (4). In these animal feline heart (5). With an increase in T, the pressure in experiments, similar changes in SV and pressure were diastole decreases over a longer period in time so that found for changes in Pv. Pd is decreased more than Ps. Thus the sensitivity of Pd We have presented a method to derive the effects of to T is greater than the sensitivity of Ps. the major parameters that characterize the heart and

With an increase in the end-systolic pressure-volume arterial tree on Ps, Pd, and SV. Taking into account all relation, interpreted as an increase in contractility, major determinants, one can derive to empirical analyti- both cardiac output (and thus SV) and mean pressure cal formula for these three variables under all possible are predicted to increase, in agreement with (4). Be- conditions. We have performed a limited analysis that cause the duration of diastole and the speed of decay of takes into account the major effects and arrived at Pd are not affected, Ps and Pd changes will be about the empirical formulas. The effects of changes in single same. Suga et al. (see Fig. 3 in (10)) showed that an parameters such as C and Rp or heart rate and contrac-


100

0

A I I I 1 120

cl cl 100 m cl

80

20

I I I I 0 1

B ’ I I I I I I CJ

I I I I I I I I I

0 5 IO 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Compliance, C [ml/kPa] Peripheral resistance, R,, [kPa s/ml]

100 l-l . C

80 -

60 -

40 -

20 -

0 I I I I I

100

80

60

40

20

D I I I I

0 1 I I I I

1 I !

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 50 100 150 200 250

Maximum elastance, Emax [kPa/ml] Heart rate [bpm]

Fig. 8. As in Fig. 7 but for SV and lines based on Eq. 14.

Table 3. Sensitivity to independent cardiac and arterial system parameters

Sensitivity at Control

ps pd sv

The methodology used in deriving an appropriate form of the function @ was as follows. We first determined the functional relationship between (PJPJ& and I& for a range of values of l-I2 from 0.5 to 2. The data were best fitted by a power law suggesting that Eq. Al can be written as

& 9 % 41 C -10

0 0 90 -28 22 5

= FI( II2 ) l ( l-13)F2(U2J

T -50 -90 28 E max 40 32 33 E min -100 -100 -100 P” 100 100 100

Functions @r and @z were plotted against II2 and could be well approximated by an exponential and a linear function, respectively, so that Eq. A2 is written as

Sensitivity is expressed as @Y/aX)/(Y/X) X 100, where Y is systolic pressure (P& diastolic pressure (Pd), or stroke volume (SV) and X is an independent parameter.

tility (Emax) can easily be assessed. The results can be generally applied and hold in conditions of health and disease.

APPENDIX

Dimensional analysis (Figs. 5, A-3’) showed that Pd/Pv is a function only of I&J, IIs, and II,+ Furthermore, Pd/Pv is proportional to IId which means that the relation simplifies to

with al = 172, a2 = 186, a3 = 0.581, a4 = 0.866, and as = 0.187 obtained by nonlinear and linear regression. Substituting l-I2 = R&IT, l-I3 = l/(C&&, and I& = Ema/Emin into ZQ. A3 and normalizing with respect to the II values of control (see Table 1), we obtain Eq. 13 in the text. A similar approach was followed for deriving the other two empirical formulas.

Address for reprint requests: N. Stergiopulos, Biomedical Engineer- ing Lab., Swiss Federal Institute of Technology, PSE-Ecublens, 1015 Lausanne, Switzerland (Email: [email protected]).

Received 14 August 1995; accepted in final form 9 November 1995.


REFERENCES

1. Berger, D. S., J. K.-J. Li, and A. Noordergraaf. Differential effects of wave reflections and peripheral resistance on aortic blood pressure: a model-based study. Am. J. PhysioZ. 266 (Heurt Circ. PhysioZ. 35): Hl626-H1642, 1994.

2. Burkhoff, D., E. T. Van der Velde, D.A. Kaas, J. Baan, W. L. Maughan and K. Sagawa. Contractility-dependent curvilinear- ity of end-systolic pressure-volume relations. Am. J. PhysioZ. 252 (Heart Circ. Physiol. 21): Hl218-H1227, 1987.

3. Elzinga, G., and N. Westerhof. Pressure and flow generated by the left ventricle against different impedances. Circ. Res. 32: 178-186,1973.

4. Elzinga, G., and N. Westerhof. How to quantify pump function of the heart. Circ. Res. 44: 303-308, 1979.

5. Elzinga, G., and N. Westerhof. Pump function of the feline left heart: changes with heart rate and its bearing on the energy balance. Cardiouusc. Res. 14: 81-92,198O.

6. Elzinga, G., and N. Westerhof. Matching between ventricle and arterial load. Circ. Res. 68: 1495-1500,199l.

7. Langhaar, H. L. Dimensional Analysis and Theory of Models. New York: Wiley, 195 1.

8. Randall, 0. S., G. C. Van Den Bos, and N. Westerhof. Systemic compliance: does it play a role in the genesis of essential hypertension? Cardiouasc. Res. 18: 455-462,1984.

9. Stergiopulos, N., J.-J. Meister, and N. Westerhof. Evalua- tion of methods for the estimation of total arterial compliance.

Am. J. Physiol. 268 (Heart Circ. Physiol. 37): Hl540-Hl548, 1995.

10. Suga, H., K. Sagawa, and A. A. Shoukas. 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, 1971.

11. Toorop, G. X?, N. Westerhof, and G. Elzinga. Beat-to-beat estimation of peripheral resistance and arterial compliance during pressure transients. Am. J. PhysioZ. 252 (Heart Circ. PhysioZ. 31): Hl275-Hl283,1987.

12. Van den Horn, G. J., N. Westerhof, and G. Elzinga. Optimal power generation by the left ventricle. Circ. Res. 56: 252-261, 1985.

13. Van der Velde, E. T., D. Burkhoff, R Steendijk, J. Karsdon, K. Sagawa, and J. Baan. Nonlinearity and afterload sensitivity of the end-systolic pressure-volume relation of the canine left ventricle in vivo. Circukztion 83: 315-327,199l.

14. Westerhof, N., and G. Elzinga. The apparent source resistance of heart and muscle. Ann. Biomed. Eng. 6: 16-32,1978.

15. Westerhof, N., and G. Elzinga. Normalized input impedance and arterial decay time over heart period are independent of animal size. Am. J. PhysioZ. 261 (Regulatory Integrative Camp. PhysioZ. 30): Rl26-Rl33,1991.

16. Westerhof, N., G. Elzinga, and I?. Sipkema. An artificial arterial system for pumping hearts. J. AppZ. PhysioZ. 31: 776- 781,197l.

determinants of stroke volume and systolic and diastolic ... of stroke volume and systolic and...

Documents