IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 37, NO. 1 , JANUARY 1990
I07
Communications Algorithm for Optimal Linear Model-Based Control with Application to Pharmacokinetic Model-Driven Drug Delivery JAMES R . JACOBS
Abstract-Computerized pharmacokinetic model-driven administration of intravenous anesthetic agents has been implemented using a variety of algorithms to control the drug infusion regimen. AI1 such algorithms are similar to the extent that they use a linear pharmacokinetic model of the drug being administered to determine drug infusion rates to theoretically achieve and maintain plasma drug concentrations (setpoints) specified by the physician. Since the behavior of the pharmacokinetic model can be computed for any input, it should be possible to achieve regulation of the drug infusion rates that is flexible (i.e., the physician can interactively adjust the setpoint), practical, and analytically optimized; these objectives are realized by the algorithm described in this communication.
INTRODUCTION
model of the drug being infused [3], [9], [12], [13]. Typically, Ar is chosen to be 10-30 s. Any difference between Cp, and Cp, is acted on by a pumpcontrol algorithm to generate an infusion rate to achieve or maintain the setpoint. The infusion rate is transmitted electronically to the drug infusion pump, which delivers drug to the patient. The drug infusion rate used by the pump is fed back into the pharmacokinetic simulation, again at intervals of A t , and Cp, is updated. The user should be able to adjust cpd as frequently as necessary to titrate the infusion to meet the patient’s requirements. All other considerations (such as the flow-delivery accuracy of the infusion device, the adequacy of the numerical technique used to simulate the pharmacokinetic model, pharmacokinetic variability, etc.) considered equal, the ability of a system designed to implement pharmacokinetic model-driven intravenous drug delivery to efficiently and accurately provide particular plasma drug concentrations is determined to a large extent by the efficacy of the control algorithm. Presented below is a control algorithm for linear pharmacokinetic models that offers optimal and flexible control of the plasma drug concentration but which also explicitly accommodates some of the practical limitations of the clinical environment and infusion pump technology.
Model-based control strategies are utilized when an adequate feedback signal is not available to implement closed-loop control but where a satisfactory mathematical model of the process to be controlled has been devised. Computerized pharmacokinetic modeldriven administration of intravenous anesthetic drugs, wherein the goal is to achieve physician-specified blood concentrations of the agent being administered, is an example of linear model-based control that has attracted a large number of investigators during the past five years [I]-[lo]. It is well known that the effects of most intravenous anesthetic drugs can be related better to the plasma drug concentration than to the drug dosage; depending on the relevance and accuracy of the pharmacokinetic model used, computerized pharmacokinetic model-driven delivery should be the most efficient and reliable means of manipulating the concentration of drug in the plasma and thus of controlling the magnitude of the drug effect. Presented in this communication is an algorithm for linear pharmacokinetic model-driven delivery of intravenous drugs that provides optimal control of the plasma drug concentration using a computer-controlled drug infusion pump. This algorithm may offer advantages over approaches proposed previously.
INFUSION ALGORITHM The first stage of the proposed pump-control algorithm for linear pharmacokinetic models is illustrated in Fig. 2. A time r* [Fig. 2(a)] at the end of the current infusion period A t , the pharmacokinetic simulation predicts the plasma drug concentration to be Cp,*. T o determine the drug infusion rate k,, to be used during the next infusion period to achieve or maintain the setpoint plasma concentration cpd: I ) Use the pharmacokinetic simulation to determine the plasma that would result if an infusion rate of x, were to drug level CppC, be used during the next infusion interval [Fig. 2(a)]. 2) Use the pharmacokinetic simulation to determine the plasma that would result if an infusion rate of x2 were to drug level Cppc2 be used during the next infusion interval [Fig. 2(a)]. 3 ) Using infusion rate as the independent variable and resultant plasma concentration as the dependent variable, find the slope and y-intercept determined by the two data pairs obtained in steps 1 and 2 above (i.e., slope = ( C p p a . CpPt,)/(x2 - .xi), y-intercept = CpPc,= Cppl,- slope . xi = CppX2 - slope . x,) [Fig. 2(b)]. 4) Exploiting the linearity of the pharmacokinetic model, use MODEL-DRIVEN DRUGDELIVERY the point-slope formula and the results of steps 1 and 3 above to In its most general form, computerized pharmacokinetic modelcompute [Fig. 2(b)] the infusion rate k , for the next infusion period driven delivery (“CACI” [ l 11) of intravenous drugs can be sche(i.e., k, = ( c p d - Cppc,)/slope) [Fig. 2(b)]. matized by the control diagram in Fig. 1. Based on monitored and The choice of x, and x2 in steps 1) and 2 ) above, respectively, anticipated patient response, on the current estimate of the plasma is arbitrary as long as xi # x 2 . In step 4) above, if k,) < 0, then k , drug concentration, and on the pharmacological properties of the = 0 in drug delivery applications, since the infusion device cannot drug being administered, the physician specifies a desired (setremove drug from the body. point) plasma drug concentration ( Cpd). At frequent intervals ( A r ) , A complete pump control paradigm incorporating the above inwhich may or may not be evenly spaced, the setpoint is compared fusion rate algorithm is illustrated in Fig. 3. Before being transwith the current prediction of the plasma drug concentration ( CPp), mitted to the computer-controllable infusion pump, the ideal infuwhich is computed by real-time simulation of a pharmacokinetic sion rate ( k , ) must be converted to appropriate units ( e . g . , ml . h-’), truncaxd to match the resolution of the pump (e.g., integral rates), and limited to not exceed the minimum o r maximum infuManuscript received September 19, 1988; revised March 10, 1989. This sion rates of the pump. This is then the desired infusion rate. The work was supported in part by Abbott Laboratories, Abbott Park, IL. infusion device used in this type of critical application should be The author is with the Departments of Anesthesiology and Biomedical capable of providing the host computer with information about its Engineering. Duke University Medical Center. Durham, NC 27710. IEEE Log Number 8931543. current and cumulative operation and about its alarm status. After 0018-9294/90/0100-0107$01.OO 0 1990 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 37, NO. I , JANUARY 1990
108
I
I
Infusion Algorithm
1
knowledge of approximate IherapeutE Dlasma levels
At
r
infusmn
Pharmacokinetic Simulatlon
Computer
i mnitored and antcipated i paten! response
reponed miuson
1 t
i____-
U Fig. 1. Generalized control diagram for computerized pharmacokinetic model-driven delivery of intravenous drugs. Cp, is the desired plasma drug concentration, Cp, is the plasma drug concentration predicted by the pharmacokinetic simulation, and A t is the sampling interval. In some implementations, such as that described in this paper, the pump control algorithm also makes use of additional state variables from the pharmacokinetic simulation.
Pharmacokinetic simulation
1
1
cp,x2 ............. ............/ cpp
k,
X1
x2
Infusion Rate (b)
Fig. 2. Infusion rate control algorithm for linear pharmacokinetic models. (a) Given that the current predicted plasma level is Cp,., use the pharmacokinetic simulation to determine the plasma levels, Cp,,,, and Cp,,,, respectively, that would result if infusion rates of xI or x2 were used during the next sampling interval A f , which is typically 10-30 s . (b) Use the previous results and interpolation to find the infusion rate k,, necessary to achieve Cp,. it has been converted back into units appropriate for the pharmacokinetic simulation ( e . g . , pg . kg-' . min-I), the infusion rate actually used by the pump should be fed into the pharmacokinetic simulation. The infusion rate reported by the pump should differ from the desired infusion rate only when the pump has detected an error condition (e.g., occlusion of the intravenous cannula or an empty drug reservoir) causing it to suspend drug administration, but consideration of these occurrences may be critical in the clinical performance of a device implementing these algorithms. When the error condition has been corrected and the pump will again accept the desired infusion rate, the infusion algorithm described in this communication is inherently capable of compensating for the decline in the theoretical plasma drug level that will have occurred during the period while drug infusion had been discontinued. Since this infusion rate control algorithm utilizes pharmacokinetic simulations of its own, the state of all of the state variables calculated within the pharmacokinetic simulation in the feedback loop must be provided to it.
usion mi
,
I
Infusion Device
Fig. 3. Pump control algorithm for pharmacokinetic model-driven delivery of intravenous drugs. The infusion rate algorithm is as in Fig. 2. The infusion rate ( k , ) specified by the infusion rate algorithm is descretized and converted into units (e.g., ml . hK') appropriate for the infusion pump. This desired pump rate is transmitted electronically to the pump, which infuses drug to the patient, and the rate used by the pump is transmitted back into the control loop. The desired and reported pump rates should be different only when the pump has entered a zero-infusion rate state, such as when it has detected an I.V. line occlusion. The actual pump rate is converted back into units appropriate for the pharmacokinetic model, and the simulation is updated. Note that the state of all of the state variables calculated as part of the pharmacokinetic simulation in the feedback loop must be provided to the infusion rate control algorithm. DISCUSSION Any control system implemented in software and realized using an actuator with finite response time (e.g., drug infusion pump) can only be optimized with respect to the specific sampling interval. The proposed pump control algorithm provides optimal control of the drug infusion rates for any choice of A t in that it generates infusion rates that bring the theoretical plasma level as close to the setpoint as is possible by the end of A t within the limitations imposed by the mechanical characteristics of the pump and the concentration of the infusate. Decreasing the duration of A t increases the precision of the control, but intervals much smaller than 10 s are probably not reasonable, considering the capabilities of most infusion devices. Presumably, the performance of a CACI system utilizing this algorithm could be slightly improved by using analytical equations [13], which are independent of sampling rate, rather than discrete approximations [ 121, to implement the pharmacokinetic simulation. This would allow setpoint changes or the occurrence of error conditions to be dealt with as they occur rather than at the end of the sampling period. This algorithm may offer advantages over those used previously to deliver intravenous drugs that apparently: have made assumptions about the speed with which bolus doses can be administered
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 37, NO. I , J A N U A R Y 1990
[I]-[6],[8],have not considered the limited resolution of the infusion device [ I ] , [2],[6]-[8],have not been able to compensate for the actual (self-reported) performance of the infusion device [ I ] , [ 2 ] , [ 5 ] , [6],[8],or which have contained an element of empirically derived proportional control [31, 141, [7], 191. Furthermore, with the algorithm proposed here, setpoint changes can be made as frequently as the physician desires, which is an advantage over implementations permitting only a single preset plasma level [6],[8]or predetermined series of several plasma levels [5].Where clinical results have been reported [1]-[3],[5],[6],however, they have been consistently satisfactory. The algorithm described in this communication has been incorporated into a CACI device that we are using to administer fentanyl, sufentanil, alfentanil, propofol, and midazolam [IO], [ 141, [ 151. Although applied in this communication specifically to pharmacokinetic model-driven drug delivery, the control algorithm presented here is of general applicability and could be utilized in any linear model-based control strategy.
109
A New Method for the Estimation of the Left Ventricular Pressure-Volume Area OFER BARNEA
AND
DOV JARON
Abstract-We present a new method for obtaining the pressure-volume area (PVA) as defined by Suga. The method allows calculation of the PVA from pressure and flow waveforms of ejecting beats and requires only one isovolumic ventricular contraction performed at any end diastolic volume.
INTRODUCTION The “varying elastance” model of the left ventricle relates ventricular pressure and volume. The varying elastance E ( t ) was defined by Suga and Sagawa [ l] as
REFERENCES [I] J. Schuttler, H. Schwilden. and H. Stoekel, “Pharmacokinetics as applied to total intravenous anaesthesia: Practical implications,” AnOD. aesthesia. vol. 38. sumlement .. .. 53-56. 1983. M. E. Ausems, D. R. Stanski, and C. C. Hug, “An evaluation of the accuracy of pharmacokinetic data for the computer assisted infusion of alfentanil,” Brit. J . Anaesth., vol. 57, pp. 1217-1225, 1985. J . M. Alvis, J. G. Reves, J . A . Spain, and L. C. Sheppard, “Computer-assisted continuous infusion of the intravenous analgesic fentanyl during general anesthesia-An interactive system,” IEEE Trans. Biomed. Eng., vol. BME-32, pp. 323-329, 1985. T. C. Jannett. “Comments on ‘Computer-assisted continuous infusion of the intravenous analgesic fentanyl during general anesthesiaAn interactive system,” IEEE Trans. Biomed. Eng., vol. BME-33, pp. 722-723, 1986. R. W. Martin, H. F. Hill, H. C. Yee, L. C. Saeger, M. H. Walter, and C. R. Chapman, “An open-loop computer-based drug infusion system,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 642-649, 1987. D. P. Crankshaw, M. D. Boyd, and A. R. Bjorksten, “Plasma drug eWux-A new approach to optimization of drug infusion for constant blood concentration of thiopental and methohexital,” Anesthesiol., vol. 67, pp. 32-41, 1987. A. Tavernier, E. Coussaert, A. D’Hollander, and F. Cantraine, “Model-based pharmacokinetic regulation in computer-assisted anesthesia: An interactive system: CARIN,” Acta Anaesth. B e l g . , vol. 38, pp. 63-68, 1987. W. L. Briggs and T. Currey, “A further note on computer-assisted continuous infusion,” Anesrhesiol., vol. 68, pp. 165-167, 1988. S. L. Shafer, L. C. Siegel, J. E. Cooke, and J. C. Scott, “Testing computer-controlled infusion pumps by simulation,” Anesthesiol., vol. 68, pp. 261-266, 1988. J . R. Jacobs, E. D. Hawkins, P. Glass, and J. G. Reves, “CACI 11: Pharmacokinetic model-driven infusion of 1.V. anesthetic agents,” in Proc. AAMI 23rd Ann. M t g . , Washington, DC, 1988, p. 50. P. S . A. Glass, J. R. Jacobs, and J. G. Reves, “Intravenous anesthetic delivery,” in Anesthesia, 3rd ed. New York: Churchill Livingstone. 1990. J . R. Jacobs, L. C. Sheppard, and J . G. Reves, “Simulation of compartment models,” in Proc. IEEE Eng. Med. Biol. Soc., Forth Worth, TX, 1986, pp. 702-704. J. R. Jacobs, “Analytical solution to the three-compartment pharmacokinetic model,” IEEE Trans. Biomed. Eng., vol. 35, pp. 763765. 1988. P. Glass, J. R. Jacobs, E. D. Hawkins, B . Ginsberg, T. J. Quill, and J . G. Reves, “Accuracy and efficacy of a pharmacokinetic modeldriven device to infuse fentanyl for anesthesia during general surgery,” Anesthesiol., vol. 69, p. A290, 1988. P. Glass, B. Ginsberg, E. D. Hawkins, K . Markham, and J. R. Jacobs, “Comparison of sodium thiopentaUisoflurane to propofol (delivered by means of a pharmacokinetic model-driven device) for the induction, maintenance, and recovery from anesthesia,” Anesrhesiol., vol. 69, p. A575, 1988.
where PLv( t ) and VLv ( t ) are, respectively, left ventricular pressure and volume as functions of time. E ( r ) is the line connecting isochronal P-V points of differently loaded beats. The maximum of E ( t ) is designated by E,,,. V, is the volume axis intercept of the E,,,,, line which is tangent to the P - V curve. t,,, is the time for which E ( t ) is maximum in the E-r plane and E ( t,,,) = E,,,. It is the time which corresponds to the point on the ventricular pressure-volume trajectory where the E ( t ) line is tangent to this curve. Suga and Sagawa demonstrated that E ( t ) is not greatly affected by loading conditions [l],[2].It was shown by Shroff et al. [3]and by Campbell et al. [4]that when P L v ( t ) was predicted from ( l ) , it was necessary to add a term representing flow dependent pressure drop.
Using this model, the slope of the line tangent to the pressureintercepting the volume axis at V, is no volume curve (ETAN), Thus, E,,, cannot be obtained directly from longer equal to E,,,. the pressure-volume plane. When estimated from (2),the E ( r ) function was found to be less affected by the loading conditions [5].Moreover, when E ( t ) was normalized in magnitude and in time, the contracting portion of the resulting function (prior to t,,,) was shown to be independent of loading conditions and to have insignificant animal to animal variations [2].T o normalize the function in magnitude and in time, the value of each point on the E ( t ) curve is divided by E,,, yielding the function & ( I ) . Each time value is then divided by t,,, yielding the function E N (tN ) where tN = t/t,,,. Using the normalized function, E ( t ) may be represented as
(4) The varying elastance model gave rise to a new approach to the estimation of ventricular oxygen consumption. Suga described the relationship between the pressure-volume area and cardiac oxygen consumption [6].PVA is the area bound by the end systolic pressure volume relationship line, the diastolic pressure volume relationship line and the systolic portion of the P - V loop in the P - V plane. It was demonstrated that PVA is highly correlated with carManuscript received June 15, 1988; revised June 29. 1989. The authors are with the Biomedical Engineering and Science Institute. Drexel University, Philadelphia, PA 19104. IEEE Log Number 8932634.
0018-9294/90/0100-0109$01 .OO
0 1990 IEEE
I07
Communications Algorithm for Optimal Linear Model-Based Control with Application to Pharmacokinetic Model-Driven Drug Delivery JAMES R . JACOBS
Abstract-Computerized pharmacokinetic model-driven administration of intravenous anesthetic agents has been implemented using a variety of algorithms to control the drug infusion regimen. AI1 such algorithms are similar to the extent that they use a linear pharmacokinetic model of the drug being administered to determine drug infusion rates to theoretically achieve and maintain plasma drug concentrations (setpoints) specified by the physician. Since the behavior of the pharmacokinetic model can be computed for any input, it should be possible to achieve regulation of the drug infusion rates that is flexible (i.e., the physician can interactively adjust the setpoint), practical, and analytically optimized; these objectives are realized by the algorithm described in this communication.
INTRODUCTION
model of the drug being infused [3], [9], [12], [13]. Typically, Ar is chosen to be 10-30 s. Any difference between Cp, and Cp, is acted on by a pumpcontrol algorithm to generate an infusion rate to achieve or maintain the setpoint. The infusion rate is transmitted electronically to the drug infusion pump, which delivers drug to the patient. The drug infusion rate used by the pump is fed back into the pharmacokinetic simulation, again at intervals of A t , and Cp, is updated. The user should be able to adjust cpd as frequently as necessary to titrate the infusion to meet the patient’s requirements. All other considerations (such as the flow-delivery accuracy of the infusion device, the adequacy of the numerical technique used to simulate the pharmacokinetic model, pharmacokinetic variability, etc.) considered equal, the ability of a system designed to implement pharmacokinetic model-driven intravenous drug delivery to efficiently and accurately provide particular plasma drug concentrations is determined to a large extent by the efficacy of the control algorithm. Presented below is a control algorithm for linear pharmacokinetic models that offers optimal and flexible control of the plasma drug concentration but which also explicitly accommodates some of the practical limitations of the clinical environment and infusion pump technology.
Model-based control strategies are utilized when an adequate feedback signal is not available to implement closed-loop control but where a satisfactory mathematical model of the process to be controlled has been devised. Computerized pharmacokinetic modeldriven administration of intravenous anesthetic drugs, wherein the goal is to achieve physician-specified blood concentrations of the agent being administered, is an example of linear model-based control that has attracted a large number of investigators during the past five years [I]-[lo]. It is well known that the effects of most intravenous anesthetic drugs can be related better to the plasma drug concentration than to the drug dosage; depending on the relevance and accuracy of the pharmacokinetic model used, computerized pharmacokinetic model-driven delivery should be the most efficient and reliable means of manipulating the concentration of drug in the plasma and thus of controlling the magnitude of the drug effect. Presented in this communication is an algorithm for linear pharmacokinetic model-driven delivery of intravenous drugs that provides optimal control of the plasma drug concentration using a computer-controlled drug infusion pump. This algorithm may offer advantages over approaches proposed previously.
INFUSION ALGORITHM The first stage of the proposed pump-control algorithm for linear pharmacokinetic models is illustrated in Fig. 2. A time r* [Fig. 2(a)] at the end of the current infusion period A t , the pharmacokinetic simulation predicts the plasma drug concentration to be Cp,*. T o determine the drug infusion rate k,, to be used during the next infusion period to achieve or maintain the setpoint plasma concentration cpd: I ) Use the pharmacokinetic simulation to determine the plasma that would result if an infusion rate of x, were to drug level CppC, be used during the next infusion interval [Fig. 2(a)]. 2) Use the pharmacokinetic simulation to determine the plasma that would result if an infusion rate of x2 were to drug level Cppc2 be used during the next infusion interval [Fig. 2(a)]. 3 ) Using infusion rate as the independent variable and resultant plasma concentration as the dependent variable, find the slope and y-intercept determined by the two data pairs obtained in steps 1 and 2 above (i.e., slope = ( C p p a . CpPt,)/(x2 - .xi), y-intercept = CpPc,= Cppl,- slope . xi = CppX2 - slope . x,) [Fig. 2(b)]. 4) Exploiting the linearity of the pharmacokinetic model, use MODEL-DRIVEN DRUGDELIVERY the point-slope formula and the results of steps 1 and 3 above to In its most general form, computerized pharmacokinetic modelcompute [Fig. 2(b)] the infusion rate k , for the next infusion period driven delivery (“CACI” [ l 11) of intravenous drugs can be sche(i.e., k, = ( c p d - Cppc,)/slope) [Fig. 2(b)]. matized by the control diagram in Fig. 1. Based on monitored and The choice of x, and x2 in steps 1) and 2 ) above, respectively, anticipated patient response, on the current estimate of the plasma is arbitrary as long as xi # x 2 . In step 4) above, if k,) < 0, then k , drug concentration, and on the pharmacological properties of the = 0 in drug delivery applications, since the infusion device cannot drug being administered, the physician specifies a desired (setremove drug from the body. point) plasma drug concentration ( Cpd). At frequent intervals ( A r ) , A complete pump control paradigm incorporating the above inwhich may or may not be evenly spaced, the setpoint is compared fusion rate algorithm is illustrated in Fig. 3. Before being transwith the current prediction of the plasma drug concentration ( CPp), mitted to the computer-controllable infusion pump, the ideal infuwhich is computed by real-time simulation of a pharmacokinetic sion rate ( k , ) must be converted to appropriate units ( e . g . , ml . h-’), truncaxd to match the resolution of the pump (e.g., integral rates), and limited to not exceed the minimum o r maximum infuManuscript received September 19, 1988; revised March 10, 1989. This sion rates of the pump. This is then the desired infusion rate. The work was supported in part by Abbott Laboratories, Abbott Park, IL. infusion device used in this type of critical application should be The author is with the Departments of Anesthesiology and Biomedical capable of providing the host computer with information about its Engineering. Duke University Medical Center. Durham, NC 27710. IEEE Log Number 8931543. current and cumulative operation and about its alarm status. After 0018-9294/90/0100-0107$01.OO 0 1990 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 37, NO. I , JANUARY 1990
108
I
I
Infusion Algorithm
1
knowledge of approximate IherapeutE Dlasma levels
At
r
infusmn
Pharmacokinetic Simulatlon
Computer
i mnitored and antcipated i paten! response
reponed miuson
1 t
i____-
U Fig. 1. Generalized control diagram for computerized pharmacokinetic model-driven delivery of intravenous drugs. Cp, is the desired plasma drug concentration, Cp, is the plasma drug concentration predicted by the pharmacokinetic simulation, and A t is the sampling interval. In some implementations, such as that described in this paper, the pump control algorithm also makes use of additional state variables from the pharmacokinetic simulation.
Pharmacokinetic simulation
1
1
cp,x2 ............. ............/ cpp
k,
X1
x2
Infusion Rate (b)
Fig. 2. Infusion rate control algorithm for linear pharmacokinetic models. (a) Given that the current predicted plasma level is Cp,., use the pharmacokinetic simulation to determine the plasma levels, Cp,,,, and Cp,,,, respectively, that would result if infusion rates of xI or x2 were used during the next sampling interval A f , which is typically 10-30 s . (b) Use the previous results and interpolation to find the infusion rate k,, necessary to achieve Cp,. it has been converted back into units appropriate for the pharmacokinetic simulation ( e . g . , pg . kg-' . min-I), the infusion rate actually used by the pump should be fed into the pharmacokinetic simulation. The infusion rate reported by the pump should differ from the desired infusion rate only when the pump has detected an error condition (e.g., occlusion of the intravenous cannula or an empty drug reservoir) causing it to suspend drug administration, but consideration of these occurrences may be critical in the clinical performance of a device implementing these algorithms. When the error condition has been corrected and the pump will again accept the desired infusion rate, the infusion algorithm described in this communication is inherently capable of compensating for the decline in the theoretical plasma drug level that will have occurred during the period while drug infusion had been discontinued. Since this infusion rate control algorithm utilizes pharmacokinetic simulations of its own, the state of all of the state variables calculated within the pharmacokinetic simulation in the feedback loop must be provided to it.
usion mi
,
I
Infusion Device
Fig. 3. Pump control algorithm for pharmacokinetic model-driven delivery of intravenous drugs. The infusion rate algorithm is as in Fig. 2. The infusion rate ( k , ) specified by the infusion rate algorithm is descretized and converted into units (e.g., ml . hK') appropriate for the infusion pump. This desired pump rate is transmitted electronically to the pump, which infuses drug to the patient, and the rate used by the pump is transmitted back into the control loop. The desired and reported pump rates should be different only when the pump has entered a zero-infusion rate state, such as when it has detected an I.V. line occlusion. The actual pump rate is converted back into units appropriate for the pharmacokinetic model, and the simulation is updated. Note that the state of all of the state variables calculated as part of the pharmacokinetic simulation in the feedback loop must be provided to the infusion rate control algorithm. DISCUSSION Any control system implemented in software and realized using an actuator with finite response time (e.g., drug infusion pump) can only be optimized with respect to the specific sampling interval. The proposed pump control algorithm provides optimal control of the drug infusion rates for any choice of A t in that it generates infusion rates that bring the theoretical plasma level as close to the setpoint as is possible by the end of A t within the limitations imposed by the mechanical characteristics of the pump and the concentration of the infusate. Decreasing the duration of A t increases the precision of the control, but intervals much smaller than 10 s are probably not reasonable, considering the capabilities of most infusion devices. Presumably, the performance of a CACI system utilizing this algorithm could be slightly improved by using analytical equations [13], which are independent of sampling rate, rather than discrete approximations [ 121, to implement the pharmacokinetic simulation. This would allow setpoint changes or the occurrence of error conditions to be dealt with as they occur rather than at the end of the sampling period. This algorithm may offer advantages over those used previously to deliver intravenous drugs that apparently: have made assumptions about the speed with which bolus doses can be administered
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 37, NO. I , J A N U A R Y 1990
[I]-[6],[8],have not considered the limited resolution of the infusion device [ I ] , [2],[6]-[8],have not been able to compensate for the actual (self-reported) performance of the infusion device [ I ] , [ 2 ] , [ 5 ] , [6],[8],or which have contained an element of empirically derived proportional control [31, 141, [7], 191. Furthermore, with the algorithm proposed here, setpoint changes can be made as frequently as the physician desires, which is an advantage over implementations permitting only a single preset plasma level [6],[8]or predetermined series of several plasma levels [5].Where clinical results have been reported [1]-[3],[5],[6],however, they have been consistently satisfactory. The algorithm described in this communication has been incorporated into a CACI device that we are using to administer fentanyl, sufentanil, alfentanil, propofol, and midazolam [IO], [ 141, [ 151. Although applied in this communication specifically to pharmacokinetic model-driven drug delivery, the control algorithm presented here is of general applicability and could be utilized in any linear model-based control strategy.
109
A New Method for the Estimation of the Left Ventricular Pressure-Volume Area OFER BARNEA
AND
DOV JARON
Abstract-We present a new method for obtaining the pressure-volume area (PVA) as defined by Suga. The method allows calculation of the PVA from pressure and flow waveforms of ejecting beats and requires only one isovolumic ventricular contraction performed at any end diastolic volume.
INTRODUCTION The “varying elastance” model of the left ventricle relates ventricular pressure and volume. The varying elastance E ( t ) was defined by Suga and Sagawa [ l] as
REFERENCES [I] J. Schuttler, H. Schwilden. and H. Stoekel, “Pharmacokinetics as applied to total intravenous anaesthesia: Practical implications,” AnOD. aesthesia. vol. 38. sumlement .. .. 53-56. 1983. M. E. Ausems, D. R. Stanski, and C. C. Hug, “An evaluation of the accuracy of pharmacokinetic data for the computer assisted infusion of alfentanil,” Brit. J . Anaesth., vol. 57, pp. 1217-1225, 1985. J . M. Alvis, J. G. Reves, J . A . Spain, and L. C. Sheppard, “Computer-assisted continuous infusion of the intravenous analgesic fentanyl during general anesthesia-An interactive system,” IEEE Trans. Biomed. Eng., vol. BME-32, pp. 323-329, 1985. T. C. Jannett. “Comments on ‘Computer-assisted continuous infusion of the intravenous analgesic fentanyl during general anesthesiaAn interactive system,” IEEE Trans. Biomed. Eng., vol. BME-33, pp. 722-723, 1986. R. W. Martin, H. F. Hill, H. C. Yee, L. C. Saeger, M. H. Walter, and C. R. Chapman, “An open-loop computer-based drug infusion system,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 642-649, 1987. D. P. Crankshaw, M. D. Boyd, and A. R. Bjorksten, “Plasma drug eWux-A new approach to optimization of drug infusion for constant blood concentration of thiopental and methohexital,” Anesthesiol., vol. 67, pp. 32-41, 1987. A. Tavernier, E. Coussaert, A. D’Hollander, and F. Cantraine, “Model-based pharmacokinetic regulation in computer-assisted anesthesia: An interactive system: CARIN,” Acta Anaesth. B e l g . , vol. 38, pp. 63-68, 1987. W. L. Briggs and T. Currey, “A further note on computer-assisted continuous infusion,” Anesrhesiol., vol. 68, pp. 165-167, 1988. S. L. Shafer, L. C. Siegel, J. E. Cooke, and J. C. Scott, “Testing computer-controlled infusion pumps by simulation,” Anesthesiol., vol. 68, pp. 261-266, 1988. J . R. Jacobs, E. D. Hawkins, P. Glass, and J. G. Reves, “CACI 11: Pharmacokinetic model-driven infusion of 1.V. anesthetic agents,” in Proc. AAMI 23rd Ann. M t g . , Washington, DC, 1988, p. 50. P. S . A. Glass, J. R. Jacobs, and J. G. Reves, “Intravenous anesthetic delivery,” in Anesthesia, 3rd ed. New York: Churchill Livingstone. 1990. J . R. Jacobs, L. C. Sheppard, and J . G. Reves, “Simulation of compartment models,” in Proc. IEEE Eng. Med. Biol. Soc., Forth Worth, TX, 1986, pp. 702-704. J. R. Jacobs, “Analytical solution to the three-compartment pharmacokinetic model,” IEEE Trans. Biomed. Eng., vol. 35, pp. 763765. 1988. P. Glass, J. R. Jacobs, E. D. Hawkins, B . Ginsberg, T. J. Quill, and J . G. Reves, “Accuracy and efficacy of a pharmacokinetic modeldriven device to infuse fentanyl for anesthesia during general surgery,” Anesthesiol., vol. 69, p. A290, 1988. P. Glass, B. Ginsberg, E. D. Hawkins, K . Markham, and J. R. Jacobs, “Comparison of sodium thiopentaUisoflurane to propofol (delivered by means of a pharmacokinetic model-driven device) for the induction, maintenance, and recovery from anesthesia,” Anesrhesiol., vol. 69, p. A575, 1988.
where PLv( t ) and VLv ( t ) are, respectively, left ventricular pressure and volume as functions of time. E ( r ) is the line connecting isochronal P-V points of differently loaded beats. The maximum of E ( t ) is designated by E,,,. V, is the volume axis intercept of the E,,,,, line which is tangent to the P - V curve. t,,, is the time for which E ( t ) is maximum in the E-r plane and E ( t,,,) = E,,,. It is the time which corresponds to the point on the ventricular pressure-volume trajectory where the E ( t ) line is tangent to this curve. Suga and Sagawa demonstrated that E ( t ) is not greatly affected by loading conditions [l],[2].It was shown by Shroff et al. [3]and by Campbell et al. [4]that when P L v ( t ) was predicted from ( l ) , it was necessary to add a term representing flow dependent pressure drop.
Using this model, the slope of the line tangent to the pressureintercepting the volume axis at V, is no volume curve (ETAN), Thus, E,,, cannot be obtained directly from longer equal to E,,,. the pressure-volume plane. When estimated from (2),the E ( r ) function was found to be less affected by the loading conditions [5].Moreover, when E ( t ) was normalized in magnitude and in time, the contracting portion of the resulting function (prior to t,,,) was shown to be independent of loading conditions and to have insignificant animal to animal variations [2].T o normalize the function in magnitude and in time, the value of each point on the E ( t ) curve is divided by E,,, yielding the function & ( I ) . Each time value is then divided by t,,, yielding the function E N (tN ) where tN = t/t,,,. Using the normalized function, E ( t ) may be represented as
(4) The varying elastance model gave rise to a new approach to the estimation of ventricular oxygen consumption. Suga described the relationship between the pressure-volume area and cardiac oxygen consumption [6].PVA is the area bound by the end systolic pressure volume relationship line, the diastolic pressure volume relationship line and the systolic portion of the P - V loop in the P - V plane. It was demonstrated that PVA is highly correlated with carManuscript received June 15, 1988; revised June 29. 1989. The authors are with the Biomedical Engineering and Science Institute. Drexel University, Philadelphia, PA 19104. IEEE Log Number 8932634.
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