IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 6 . JUNE 1992
629
Optimal Controller for Intraaortic Balloon Pumping Ofer Bamea, Member, IEEE, Brian T . Smith, Member, IEEE, Stephen Dubin, Thomas W . Moore, Senior Member, IEEE, and Dov Jaron, Fellow, IEEE
Abstract-An optimal control algorithm was adapted to identify and track the optimal deflation time of the intraaortic balloon pump (IABP)' Routines for physiologically imposed constraints were added to the algorithm which was implemented in a computer-controlled system. The system was designed to provide real time optimization for the clinical setting. The controller continuously maximizes a performance index while observing the constraints. The index is composed of clinically available hemodynamic variables which indicate changes in myocardial energy balance. Proper values for the algorithm parameters were determined and the system was tested in animal experiments. The results indicate that controlling deflation time relative to the R wave, which precedes the next ejection phase, reduces the time required for optimization when the heart rate varies.
INTRODUCTION WENTY-THREE years since its first clinical appliT c a t i o n , the intraaortic balloon pump remains the only cardiac-assist device in widespread clinical use [ 11, [2]. The IABP, an elongated polyurethane balloon, is inserted transcuteneously through the femoral artery into the descending thoracic aorta. 1t is inflated during diastole to augment coronary flow by increasing coronary driving pressure. It is deflated just before the beginning of systole to reduce left ventricular (LV) work by reducing the pressure against which the left ventricle pumps. While afterload reduction reduces oxygen consumption, the augmented diastolic pressure in the aorta increases the availability of oxygen to the LV. These two hemodynamic effects produce changes in the oxygen balance of the failing left ventricle. Oxygen balance is defined here as the difference between oxygen consumption and oxygen availability, where availability is the amount of oxygen which could potentially be extracted from the coronary circulation. The extent to which these two effects are achieved depends on physical parameters of the IABP, its location, the rate of gas displacement, and on its timing relative to the cardiac cycle [31-[81. However, in the clinical setting only timing can be controlled. Significant changes in indices of cardiac oxygen balance have been observed in response to changes in balManuscript received July 5 , 1990; revised November I , 1991. 0. Barnes is with the Biomedical Engineering Program, School of En. gineering. Tel Aviv University, Tel Aviv 67796, Israel. B. T . Smith. S . Dubin. T. W. Moore, and D. Jaron are with the Biomedical Engineering and Science Institute, Drexel University, Philadelphia, PA 19104. IEEE Log Number 9108128.
loon timing [9]. The change in oxygen balance has been argued to result primarily because of changes in oxygen availability. Oxygen consumption cannot be substantially reduced by IABP9 but it may greatly increase with improper timing of the balloon [lo], [ 111. Thus, a linear combination of variables reflecting oxygen availability and oxygen consumption can yield a useful performance index for closed-loop optimal control. IABP manufacturers recommend the maximal reduction of end diastolic aortic pressure (EDP) as an indicator of decreased ventricular workload for the adjustment of deflation time. This is based on a good correlation between oxygen consumption and EDP reported by Monroe et al. [12]. They also recommend reduction of peak systolic pressure (PSP). The generally accepted proper timing of inflation time is end systole. This timing produces a pressure increase starting at the dicrotic notch. Bamea et al. assessed the effects of the IABP in terms and Oxygen consumption Of cardiac Oxygen [ 111. They used both a numerical model and animal extiming affect periments to study how changes in these variables. They also examined other factors to determine which clinically available hemodynamic variables best reflect changes in oxygen consumption and oxygen availability. In theoretical studies, oxygen consumption was obtained from the pressure-volume area (PVA) and oxygen availability ,was calculated from corOnay flow. The performance index proposed by Barnea et al. is 2
Z(T,, T d ) = Kmdp MDP - Kpqp PSP
(1)
where Tj is time of inflation, Td is the deflation time, MDP is mean diastolic aortic pressure, and PSP is peak systolic pressure. The driving pressure of coronary flow during diastole is assumed to be the difference between aortic pressure and intramyocardial pressure. It is assumed that under conditions of cardiac failure the autoregulatory mechanism is exhausted and the coronary circulation is fully dilated. Under this condition, and the assumption that most coronary flow takes place during diastole, coronary flow and MDP have a straight line relationship [ 131. The relationshb between MDP and coronary flow during IABP treatmen; in the dog were measured-by He et al. 14i. Their results indicated that MDP is a good index Of coronary flow and, therefore, of oxygen availability. Peak systolic pressure is related to maximal ventricular wall stress. This parameter, when multiplied by heart rate, has been shown to be closely related to ventricular oxygen
0018-9294/92$03,00 0 1992 IEEE
630
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 6. JUNE 1992
consumption ( r = 0 . 9 5 8 ) [14]. O2 consumption is affected by other variables in the system as well. However, the mechanism by which the IABP attempts to unload the left ventricle is pressure reduction. Therefore, changes from beat to beat, produced by pressure unloading, may be indicated by PSP. Using a model, the proposed index has been shown to reflect changes in cardiac energy balance. It has also been shown to possess a maximum within the operational range of the intraaortic balloon pump at the timing for which O2 balance is maximized. The index is a linear combination of two variables which can be calculated from the aortic pressure waveform. This performance index is available in the clinical setting. An optimization scheme is required to continuously identify the IABP timing which maximizes the performance index. Few published optimization algorithms possess characteristics which enable their adaptation for operation in the time-varying nonlinear physiological system. The common methods for locating a maximum or a minimum of a function use either approximation methods, search methods, or a combination of the two. In the first method the performance index is approximated by a low-order polynomial and the maximum is calculated. Approximation methods are usually good only when the value of the performance index is near its extremum. Search methods locate the maximum by using fixed or time-varying step sizes or arbitrary search patterns. Some search methods require knowledge of the derivatives of the performance index. These are not suitable for our application and therefore were not considered. Also ruled out were search methods which require large nonprogress h e or random steps. These were not considered because of possible adverse effect on the physiological system. In the time-varying nonlinear system the optimization routine operates continuously and, therefore, the number of steps is not known a priori. This constraint rules out gradient methods as well as the Fibonacci search method. A combination of search and approximation methods was suggested by Davis, Swann, and Campey [15]. They used a linearly progressing search which produces four equally spaced points of which three enclose the extremum. An approximation method is then applied to estimate the extremum point. An extension of this method was used in our present study. A microprocessor-based, univariate, optimizing controller was developed. The controller was tested in animal experiments with the proposed performance index [see ( l ) ] , where the index is only a function of balloon deflation time.
METHODS Control System A microprocessor-based control system was constructed to control deflation time both manually and automatically [ 161. The controlled parameter was deflation offset time (Ad). Ad is the time when deflation is to start
relative to the time of the next R wave. Ad may be either positive, for deflation starting after the next R wave, or negative, for deflation starting before the next R wave. To control the balloon, the time from the R wave to the beginning of deflation, Td, was required. When Ad is negative, Td was calculated as the sum of the R-R interval (obtained from the previous beat) and Ad. When Ad was positive, Td was set equal to Ad. By controlling deflation offset time Ad rather than actual deflation time Td we remained close to the optimal timing when the heart rate varied. Inflation time was always set manually. In the automatic mode, deflation offset time was optimized using the algorithm shown in Fig. 1 . The algorithm proposed by Box et al. [ 151 was modified to continuously search for the time-varying optimal point while closing in on the extremum. The search was subject to four constraints: both deflation offset time and step size had upper and lower limits. The theoretical effects of different values of the limits were previously reported [ 171. Starting with a preset initial deflation offset time, normally Adm,,,and the smallest permitted step size (&,,,), the algorithm evaluates the performance index at continuously increasing (or continuously decreasing) deflation times Td(n).The step-size is doubled after each searchstep. This continues as long as all the following conditions are met: the step-size is not greater than Smax,the index is increasing, and Ad(n) is within the permitted range. If the step-size upper limit is reached, the search continues with S = S,,,. If the deflation offset time limit is reached (block A in Fig. l ) , the search direction is reversed and the step size is decreased, but the search is confined to an interval adjacent to the limit. This continues until a positive gradient of the index is detected. When a decrease in the index is detected during a normal search (increasing step-size), the index is evaluated at a point midway between the last two points resulting in four equally spaced points. The three which enclose the maximum are used for quadratic estimation of the optimal offset time which is assigned to A d ( l ) and the process starts again with step-size equal to half the last step-size. If the step-size limit was reached before the decreased index has been detected, there is no need for the last point and the evaluation midway between the last two points is omitted (line C in Fig. 1). If a decrease in the index is detected in the second step, at A d ( 2 ) (block B in Fig. I), the direction is reversed and a point is taken at the other side of the initial point. If this index value is greater than the first, a regular search is resumed. If not, we have three points enclosing the maximum and the estimation process is executed, followed by a new search. Once the algorithm closes in on a timeinvariant or slowly shifting maximum, this sequence is repeated and the step-size is reduced to minimum.
Experimental Setup and Protocol Animal experiments were performed to assess the sensitivity of the performance index to balloon deflation tim-
BARNEA et al.: OPTIMAL CONTROLLER FOR IABP
63 1
A
__.
B
L
2s
C
A&) = Ad(n-l)+S
+
,
1”
Reset SmaxRag
YES S = Smin
*
*
(b) Fig. 1. (a) Optimization algorithm. The bold blocks constitute mostly the original algorithm. Block A handles timing constraints. Block B handles cases of search in wrong direction and search at the optimum. Line C handles cases of limited step size. (b) Subroutines for increasing and decreasing step size used by the main algorithm.
ing and to evaluate the usefulness of the optimization scheme and the controller. Three 18 f 3 kg mongrel dogs were used in the experiments. Each was preanesthetized with 20 mL xylazine
injected itramuscularly, and then anesthetized with pentobarbital injected intravenously (30 mg/kg) or to effect. A thoracotomy was then performed through the fifth intercostal space. The dog was instrumented with two
632
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO 6, JUNE 1992
electromagnetic flow probes, one on the aortic root and the other on the main left coronary artery. A Millar catheter tip pressure transducer was inserted into the left ventricle through the apex and secured by a purse string suture. An intraaortic balloon was inserted through a femoral artery into the descending thoracic aorta. The inflation time of the balloon was adjusted so that the rise of the balloon pressure was initiated at the dicrotic notch. This inflation time was kept constant throughout the experiment. Only deflation time of the IABP was varied in the experiment. First we obtained MDP and PSP as functions of increasing and then decreasing deflation times. For each deflation time five beats were recorded. Each measurement was performed at least 20 beats after the last change was made to allow the cardiovascular system to return to a steady state operation. Recordings were made on a Gould strip chart recorder and on an 8-channel FM tape recorder. The data were also digitized and stored on an IBM PC/AT compatible computer. In the second stage, the automatic mode of the controiler was tested. An initial Ad was set and the controller was started with Kmdp and Kpspequal one. In the third experiment the heart was paced and the effect of a sudden change in heart rate was examined. RESULTS Figs. 2-4 depict results obtained during the manual runs. Figs. 5-7 depict results obtained with the use of the automatic optimizing controller. Fig. 2 shows MDP as a function of Ad for increasing values (forward) and decreasing values (backward) obtained from dog #2. The curve was more sensitive to deflation time change in the forward direction. However, in both cases, a monotonic curve has been observed in the operational timing range of the balloon. Fig. 3 shows PSP as a function of Ad in the forward and backward directions. In both cases the minimum is at the same timing despite the change in the signal level. The sensitivity to timing in both cases is similar. Fig. 4 plots the performance index as a function of Ad for Kmdpand Kpspequal one in both directions. Changing Kmdpin the range of 0.52.0 did not affect the location of the maximum significantly. Fig. 5 shows the search pattern for dog # 1 where the limits of the step size S were set to Smi, = 10 ms and S,,, = 80 ms. A polynomial fit (seventh order) results in a curve similar to that of a step response of an underdamped second order system with a damping factor of approximately 0.6 and natural undamped frequency U, of 0.09 l/beat. Maximum over-shoot was 80 ms. The effect of using different limits in dog # 3 (S,,, = 8 ms and S,, = 32 ms) are shown in Fig. 6. { in the analog continuous second-order system would be about 0.7 and U, would be 0.08. The maximum over-shoot was 40 ms. Fig. 7 shows the effect of a sudden change in heart rate. The mean deflation time did not change significantly as a result of the sudden chance " in heart rate.
MANUAL R U N
Fig. 2. Mean diastolic pressure (MDP) as a function of the manually varied A
629
Optimal Controller for Intraaortic Balloon Pumping Ofer Bamea, Member, IEEE, Brian T . Smith, Member, IEEE, Stephen Dubin, Thomas W . Moore, Senior Member, IEEE, and Dov Jaron, Fellow, IEEE
Abstract-An optimal control algorithm was adapted to identify and track the optimal deflation time of the intraaortic balloon pump (IABP)' Routines for physiologically imposed constraints were added to the algorithm which was implemented in a computer-controlled system. The system was designed to provide real time optimization for the clinical setting. The controller continuously maximizes a performance index while observing the constraints. The index is composed of clinically available hemodynamic variables which indicate changes in myocardial energy balance. Proper values for the algorithm parameters were determined and the system was tested in animal experiments. The results indicate that controlling deflation time relative to the R wave, which precedes the next ejection phase, reduces the time required for optimization when the heart rate varies.
INTRODUCTION WENTY-THREE years since its first clinical appliT c a t i o n , the intraaortic balloon pump remains the only cardiac-assist device in widespread clinical use [ 11, [2]. The IABP, an elongated polyurethane balloon, is inserted transcuteneously through the femoral artery into the descending thoracic aorta. 1t is inflated during diastole to augment coronary flow by increasing coronary driving pressure. It is deflated just before the beginning of systole to reduce left ventricular (LV) work by reducing the pressure against which the left ventricle pumps. While afterload reduction reduces oxygen consumption, the augmented diastolic pressure in the aorta increases the availability of oxygen to the LV. These two hemodynamic effects produce changes in the oxygen balance of the failing left ventricle. Oxygen balance is defined here as the difference between oxygen consumption and oxygen availability, where availability is the amount of oxygen which could potentially be extracted from the coronary circulation. The extent to which these two effects are achieved depends on physical parameters of the IABP, its location, the rate of gas displacement, and on its timing relative to the cardiac cycle [31-[81. However, in the clinical setting only timing can be controlled. Significant changes in indices of cardiac oxygen balance have been observed in response to changes in balManuscript received July 5 , 1990; revised November I , 1991. 0. Barnes is with the Biomedical Engineering Program, School of En. gineering. Tel Aviv University, Tel Aviv 67796, Israel. B. T . Smith. S . Dubin. T. W. Moore, and D. Jaron are with the Biomedical Engineering and Science Institute, Drexel University, Philadelphia, PA 19104. IEEE Log Number 9108128.
loon timing [9]. The change in oxygen balance has been argued to result primarily because of changes in oxygen availability. Oxygen consumption cannot be substantially reduced by IABP9 but it may greatly increase with improper timing of the balloon [lo], [ 111. Thus, a linear combination of variables reflecting oxygen availability and oxygen consumption can yield a useful performance index for closed-loop optimal control. IABP manufacturers recommend the maximal reduction of end diastolic aortic pressure (EDP) as an indicator of decreased ventricular workload for the adjustment of deflation time. This is based on a good correlation between oxygen consumption and EDP reported by Monroe et al. [12]. They also recommend reduction of peak systolic pressure (PSP). The generally accepted proper timing of inflation time is end systole. This timing produces a pressure increase starting at the dicrotic notch. Bamea et al. assessed the effects of the IABP in terms and Oxygen consumption Of cardiac Oxygen [ 111. They used both a numerical model and animal extiming affect periments to study how changes in these variables. They also examined other factors to determine which clinically available hemodynamic variables best reflect changes in oxygen consumption and oxygen availability. In theoretical studies, oxygen consumption was obtained from the pressure-volume area (PVA) and oxygen availability ,was calculated from corOnay flow. The performance index proposed by Barnea et al. is 2
Z(T,, T d ) = Kmdp MDP - Kpqp PSP
(1)
where Tj is time of inflation, Td is the deflation time, MDP is mean diastolic aortic pressure, and PSP is peak systolic pressure. The driving pressure of coronary flow during diastole is assumed to be the difference between aortic pressure and intramyocardial pressure. It is assumed that under conditions of cardiac failure the autoregulatory mechanism is exhausted and the coronary circulation is fully dilated. Under this condition, and the assumption that most coronary flow takes place during diastole, coronary flow and MDP have a straight line relationship [ 131. The relationshb between MDP and coronary flow during IABP treatmen; in the dog were measured-by He et al. 14i. Their results indicated that MDP is a good index Of coronary flow and, therefore, of oxygen availability. Peak systolic pressure is related to maximal ventricular wall stress. This parameter, when multiplied by heart rate, has been shown to be closely related to ventricular oxygen
0018-9294/92$03,00 0 1992 IEEE
630
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 6. JUNE 1992
consumption ( r = 0 . 9 5 8 ) [14]. O2 consumption is affected by other variables in the system as well. However, the mechanism by which the IABP attempts to unload the left ventricle is pressure reduction. Therefore, changes from beat to beat, produced by pressure unloading, may be indicated by PSP. Using a model, the proposed index has been shown to reflect changes in cardiac energy balance. It has also been shown to possess a maximum within the operational range of the intraaortic balloon pump at the timing for which O2 balance is maximized. The index is a linear combination of two variables which can be calculated from the aortic pressure waveform. This performance index is available in the clinical setting. An optimization scheme is required to continuously identify the IABP timing which maximizes the performance index. Few published optimization algorithms possess characteristics which enable their adaptation for operation in the time-varying nonlinear physiological system. The common methods for locating a maximum or a minimum of a function use either approximation methods, search methods, or a combination of the two. In the first method the performance index is approximated by a low-order polynomial and the maximum is calculated. Approximation methods are usually good only when the value of the performance index is near its extremum. Search methods locate the maximum by using fixed or time-varying step sizes or arbitrary search patterns. Some search methods require knowledge of the derivatives of the performance index. These are not suitable for our application and therefore were not considered. Also ruled out were search methods which require large nonprogress h e or random steps. These were not considered because of possible adverse effect on the physiological system. In the time-varying nonlinear system the optimization routine operates continuously and, therefore, the number of steps is not known a priori. This constraint rules out gradient methods as well as the Fibonacci search method. A combination of search and approximation methods was suggested by Davis, Swann, and Campey [15]. They used a linearly progressing search which produces four equally spaced points of which three enclose the extremum. An approximation method is then applied to estimate the extremum point. An extension of this method was used in our present study. A microprocessor-based, univariate, optimizing controller was developed. The controller was tested in animal experiments with the proposed performance index [see ( l ) ] , where the index is only a function of balloon deflation time.
METHODS Control System A microprocessor-based control system was constructed to control deflation time both manually and automatically [ 161. The controlled parameter was deflation offset time (Ad). Ad is the time when deflation is to start
relative to the time of the next R wave. Ad may be either positive, for deflation starting after the next R wave, or negative, for deflation starting before the next R wave. To control the balloon, the time from the R wave to the beginning of deflation, Td, was required. When Ad is negative, Td was calculated as the sum of the R-R interval (obtained from the previous beat) and Ad. When Ad was positive, Td was set equal to Ad. By controlling deflation offset time Ad rather than actual deflation time Td we remained close to the optimal timing when the heart rate varied. Inflation time was always set manually. In the automatic mode, deflation offset time was optimized using the algorithm shown in Fig. 1 . The algorithm proposed by Box et al. [ 151 was modified to continuously search for the time-varying optimal point while closing in on the extremum. The search was subject to four constraints: both deflation offset time and step size had upper and lower limits. The theoretical effects of different values of the limits were previously reported [ 171. Starting with a preset initial deflation offset time, normally Adm,,,and the smallest permitted step size (&,,,), the algorithm evaluates the performance index at continuously increasing (or continuously decreasing) deflation times Td(n).The step-size is doubled after each searchstep. This continues as long as all the following conditions are met: the step-size is not greater than Smax,the index is increasing, and Ad(n) is within the permitted range. If the step-size upper limit is reached, the search continues with S = S,,,. If the deflation offset time limit is reached (block A in Fig. l ) , the search direction is reversed and the step size is decreased, but the search is confined to an interval adjacent to the limit. This continues until a positive gradient of the index is detected. When a decrease in the index is detected during a normal search (increasing step-size), the index is evaluated at a point midway between the last two points resulting in four equally spaced points. The three which enclose the maximum are used for quadratic estimation of the optimal offset time which is assigned to A d ( l ) and the process starts again with step-size equal to half the last step-size. If the step-size limit was reached before the decreased index has been detected, there is no need for the last point and the evaluation midway between the last two points is omitted (line C in Fig. 1). If a decrease in the index is detected in the second step, at A d ( 2 ) (block B in Fig. I), the direction is reversed and a point is taken at the other side of the initial point. If this index value is greater than the first, a regular search is resumed. If not, we have three points enclosing the maximum and the estimation process is executed, followed by a new search. Once the algorithm closes in on a timeinvariant or slowly shifting maximum, this sequence is repeated and the step-size is reduced to minimum.
Experimental Setup and Protocol Animal experiments were performed to assess the sensitivity of the performance index to balloon deflation tim-
BARNEA et al.: OPTIMAL CONTROLLER FOR IABP
63 1
A
__.
B
L
2s
C
A&) = Ad(n-l)+S
+
,
1”
Reset SmaxRag
YES S = Smin
*
*
(b) Fig. 1. (a) Optimization algorithm. The bold blocks constitute mostly the original algorithm. Block A handles timing constraints. Block B handles cases of search in wrong direction and search at the optimum. Line C handles cases of limited step size. (b) Subroutines for increasing and decreasing step size used by the main algorithm.
ing and to evaluate the usefulness of the optimization scheme and the controller. Three 18 f 3 kg mongrel dogs were used in the experiments. Each was preanesthetized with 20 mL xylazine
injected itramuscularly, and then anesthetized with pentobarbital injected intravenously (30 mg/kg) or to effect. A thoracotomy was then performed through the fifth intercostal space. The dog was instrumented with two
632
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO 6, JUNE 1992
electromagnetic flow probes, one on the aortic root and the other on the main left coronary artery. A Millar catheter tip pressure transducer was inserted into the left ventricle through the apex and secured by a purse string suture. An intraaortic balloon was inserted through a femoral artery into the descending thoracic aorta. The inflation time of the balloon was adjusted so that the rise of the balloon pressure was initiated at the dicrotic notch. This inflation time was kept constant throughout the experiment. Only deflation time of the IABP was varied in the experiment. First we obtained MDP and PSP as functions of increasing and then decreasing deflation times. For each deflation time five beats were recorded. Each measurement was performed at least 20 beats after the last change was made to allow the cardiovascular system to return to a steady state operation. Recordings were made on a Gould strip chart recorder and on an 8-channel FM tape recorder. The data were also digitized and stored on an IBM PC/AT compatible computer. In the second stage, the automatic mode of the controiler was tested. An initial Ad was set and the controller was started with Kmdp and Kpspequal one. In the third experiment the heart was paced and the effect of a sudden change in heart rate was examined. RESULTS Figs. 2-4 depict results obtained during the manual runs. Figs. 5-7 depict results obtained with the use of the automatic optimizing controller. Fig. 2 shows MDP as a function of Ad for increasing values (forward) and decreasing values (backward) obtained from dog #2. The curve was more sensitive to deflation time change in the forward direction. However, in both cases, a monotonic curve has been observed in the operational timing range of the balloon. Fig. 3 shows PSP as a function of Ad in the forward and backward directions. In both cases the minimum is at the same timing despite the change in the signal level. The sensitivity to timing in both cases is similar. Fig. 4 plots the performance index as a function of Ad for Kmdpand Kpspequal one in both directions. Changing Kmdpin the range of 0.52.0 did not affect the location of the maximum significantly. Fig. 5 shows the search pattern for dog # 1 where the limits of the step size S were set to Smi, = 10 ms and S,,, = 80 ms. A polynomial fit (seventh order) results in a curve similar to that of a step response of an underdamped second order system with a damping factor of approximately 0.6 and natural undamped frequency U, of 0.09 l/beat. Maximum over-shoot was 80 ms. The effect of using different limits in dog # 3 (S,,, = 8 ms and S,, = 32 ms) are shown in Fig. 6. { in the analog continuous second-order system would be about 0.7 and U, would be 0.08. The maximum over-shoot was 40 ms. Fig. 7 shows the effect of a sudden change in heart rate. The mean deflation time did not change significantly as a result of the sudden chance " in heart rate.
MANUAL R U N
Fig. 2. Mean diastolic pressure (MDP) as a function of the manually varied A
