## Abstract

The purpose of this study was mathematical modeling of the heart rate (HR) response to isoproterenol (Iso) infusion. We developed a computerized system for the controlled increase of HR by Iso, based on a modified proportional-integral controller. HR was measured in conscious, freely moving rats. We found that the steady-state HR can be described as a hyperbolic power function of the steady-state Iso flow rate. This dependence was coupled with a first-order difference equation to form a pharmacodynamic model that reliably describes the relationship between HR and Iso flow for any arbitrary form of Iso flow function. In simulation studies, we showed that the model continued to follow the HR curve from real-time experiments far beyond the initial “learning interval” from which its parameters were calculated. Our results suggest that the predictive ability and the simplicity of calculating the parameters render this pharmacodynamic model appropriate for use within future advanced, model-based, adaptive control systems and as a part of larger cardiovascular models.

- pharmacodynamics
- mathematical model
- computer models
- computer simulation

the infusion of cardioactive medications in clinical settings, in pharmacodynamic evaluations, and in physiological research is usually done by applying a series of different constant flow rates until the desired effect is reached (4,6, 13, 16, 19). Stabilization of the effect with this method is relatively slow, and it is difficult to predict the eventual steady-state physiological response. Attempts to increase or decrease the dose too rapidly may result in overshoot or undershoot of the effect. The pharmacodynamic models used (4, 6, 13, 16, 19) relate the steady-state physiological response to the steady-state drug flow or plasma concentration and neglect the transients between the levels, so information on short half-life processes is lost.

The use of computer-controlled drug administration systems has gained much interest in recent years. It has been successfully applied in lowering of blood pressure (9, 11), controlling the depth of anesthesia (14), physiological cardiac pacing (8), and insulin injection (9). The issue of heart rate (HR) acceleration was addressed only recently by a group who used arbutamine in stress echocardiography (17).

We chose to control HR increase by the use of isoproterenol (Iso) infusion. Iso, a very powerful drug, was used extensively in the past in diverse conditions such as extreme bradycardia, heart block, and asthma and has been largely replaced by other sympathomimetic amines because of the difficulty in the fine-tuning of its effect. It is still used in electrophysiological studies for the facilitation of induction of arrhythmias (1) and in orthostatic tilt tests (15).

We successfully controlled HR increase in eight rats by using a proportional-integral (PI) control algorithm (3). Stable HR values within ±9 beats/min of target HR were achieved in all eight rats, with an average settling time of 6 ± 4 min (10). For the system we used, the PI controller worked adequately, as reflected by a short settling time and good stability around the target HR. However, for better control, more advanced, adaptive controllers are needed, which are based on a pharmacodynamic model of the system (20). Such a model should be dynamic and should have predictive ability. It should be built in such a way that its parameters are able to be calculated rapidly on-line during the control process. The construction of such a pharmacodynamic model is the purpose of this study.

## METHODS

#### Experimental protocol.

We developed a computerized system for HR increase by closed-loop control of Iso infusion (10). Rats were anesthetized by intraperitoneal injection of Avertin (200 mg/kg; tribromoethanol dissolved in tertiary amyl alcohol; Ref. 7). We preferred Avertin because of its rapid induction of deep anesthesia, relatively short duration of action, and only minimal hemodynamic compromise. The catheters were implanted according to the method of Chieuh and Kopin (2), whereby the catheters and the electrocardiogram (ECG) electrodes were tunneled subcutaneously from the insertion point to the nape of the neck, where they were exteriorized and protected with a flexible 30-cm-long stainless steel spring. Blood pressure was measured from the catheter in the caudal artery; drug infusion was performed via a catheter in the jugular vein. ECG was measured from electrodes implanted subcutaneously.

After surgery, the catheters were filled with a solution of heparinized saline (500 U/ml). During the experiments, the arterial catheter was flushed with heparinized saline (50 U/ml). All tests were performed on freely moving, conscious rats operated on under anesthesia at least 24 h before the test.

The C programming language was used for most computer routines. A first routine continuously sampled the ECG and pressure signals through an analog-to-digital converter and calculated and stored the instantaneous HR and systolic and diastolic blood pressure peaks. A second, control routine was based on a modified PI controller. Its input was the HR value. This routine activated a computer-controlled infusion pump that injected Iso and stabilized HR on a predetermined target level. With our controller, the average settling time, i.e., the time until steady-state HR level was achieved, was 6.4 ± 4.3 min. Target HR was changed every 10 min through the keyboard (10). The data from these control experiments were used for the development and validation of our pharmacodynamic model of HR response to Iso infusion. A total of 10 experiments were performed on eight rats; two of the rats were tested twice. This study has been approved by the institutional animal care and use committee of Tel Aviv University (study no. 11-97-073).

#### Mathematical model and data analysis.

Our pharmacodynamic model was designed to relate the instantaneous filtered HR to intravenous Iso infusion. To suppress periodic and random variations, HR and Iso flow were smoothed using a moving averaging filter. Our basic requirements were as follows.*1*) The model should use the minimal possible number of parameters. On one hand, increasing the number of parameters enables the model to fit to diverse shapes of experimental curves. On the other hand, the parameters tend to lose their physiological meaning because diverse sets of parameters may give the same fitted curves. *2*) No assumption should be made concerning any parameter value; only parameters measured from our experimental data are used. We therefore chose to relate HR to Iso flow, avoiding the use of blood Iso level, which necessitates pharmacokinetic assumptions such as the volume of distribution and the structure of the pharmacokinetic model.*3*) The model should have predictive ability and should continue to function beyond the area from which its parameters are calculated (the learning interval).

The basic assumptions on which our pharmacodynamic model is based are as follows. *1*) Steady-state HR may be related to steady-state Iso flow by a hyperbolic power function.*2*) The rate of HR change at*time t* depends on the difference between the instantaneous HR and the expected steady-state HR for the specific Iso flow at *time t* − delay.

Our model has five parameters; one parameter is kept the same for all rats and does not change with time. The model takes into account the nonlinear nature of the system and a time delay between the infusion and the physiological response. First, the relationship between steady-state HR and drug flow was characterized. The response curve was linearized by a hyperbolic power transformation of the flow rate, the power of which represents the nonlinearity. The power was measured in all 10 experiments and was averaged. The averaged power served as a constant parameter of the model for all rats. The time delay between the Iso flow into the rat and the HR response was measured individually for each rat. The other three parameters are the time constant of the system (*p*
_{0}), the slope of the linear relationship between steady-state HR and the power-transformed steady-state Iso flow rate (*p*
_{1}), and basal HR (*p*
_{2}). These parameters were calculated for each rat by a least-squares routine developed by us. The model parameters were calculated using data from the first part of each experiment, the learning interval.

Characterization of the steady-state HR response and simulations for validation of the model were done by routines built by us using Matlab software (12). The equation that transforms the flow is
Equation 1where F_{t} is instantaneous Iso flow at*time t* − delay, delay is the time delay between the Iso flow into the rat and the HR response, power is the power we chose for the hyperbolic power function transforming the F_{t}, and*p*
_{f} is F_{t} transformed by the hyperbolic power function.

Our pharmacodynamic model relates the filtered raw HR to the filtered Iso flow
Equation 2
Equation 3where Δ*t* is time between successive steps, h_{r} is HR at*time t*, h_{rf} is HR at*time t* + Δ*t*, h_{rsts} is HR at the steady-state level of Iso infusion, and*p*
_{0},*p*
_{1}, and*p*
_{2} are model parameters. Rearranging yields
Equation 4
*Equations1
* and *
4
* are our model. h_{rf} as a function of h_{r} and*p*
_{f} may also be graphically represented as a spatial plane.

*p*
_{0} is normalized for the specific Δ*t* of the data set from which the parameters are calculated to compare between parameters calculated from different data sets with different lengths of Δ*t*
where*p*
_{0n} is normalized*p _{0} and Δt*

_{s}is the specific Δ

*t*of the data set from which the parameters were calculated. The general form of

*Eq. 4*is therefore Equation 5where Δ

*t*

_{a}is any arbitrary Δ

*t*.

Parameter identification was done by use of a least-squares method. The least-squares formula is
Equation 6where ndt is the number of data points used for the calculation. Substituting*u*
_{0,1,2} for*p*
_{0,1,2} and differentiating *Eq. 6
* with respect to*u*
_{0},*u*
_{1},*u*
_{2} yields a set of three linear algebraic equations
where Y is the sum of the squares, and U_{(0–2)} is the substitute of *p*
_{(0–2)}. These equations may be readily solved, and *p*
_{(0–2)}may be quickly calculated.

## RESULTS

We found high linear correlation between steady-state HR and hyperbolic power-transformed steady-state Iso flow rate (*R* ≥ 0.95 for all individual experiments). The mean power was found to be 0.65 ± 0.26 and was kept the same for all the rats.

Figure1
*A* shows the relationship between steady-state HR and steady-state Iso flow rate in one rat. Figure 1
*B* shows linearization of the curve by a hyperbolic power function transforming the flow rate. Similar curves were obtained for all rats.

Our pharmacodynamic model was successfully fitted to data from all the experiments. The model parameters were calculated using data from the first part of each experiment, the learning interval.

Figure 2 graphically presents h_{rf} as a function of h_{r} and*p*
_{f} (seemethods). The experimental points are shown fitted to the spatial plane whose coefficients are the calculated parameters of our model (*Eq.4
*,*p*
_{0,1,2}). The degree of proximity of the experimental points to the plane reflects the degree of fitting to our pharmacodynamic model. The experimental points and the plane are shown together with a three-dimensional*X*, *Y*,*Z* Cartesian system rotated around the*Z* axis. From the various projections, it may be seen that the experimental points conform to the spatial plane.

Figure 3,*left*, presents simulations using the filtered flow rate as input to the model. The model parameters were calculated using data from the learning interval. It may be clearly seen that the model accurately predicts the anticipated trajectory of the HR curve using only the data from the learning interval. Figure 3,*right*, shows a clear association between HR as predicted by the model and filtered raw HR. Similar simulations and association curves were obtained for all rats. In all rats, the model continued to predict HR level far beyond the area from which the parameters were calculated.

Figure 4 shows the response of filtered HR, systolic blood pressure, and diastolic blood pressure to the infusion of Iso in one rat. Iso causes a marked increase in HR and a small decrease in blood pressure. Table 1 shows the calculated parameters from all experiments.

## DISCUSSION

The importance of mathematical modeling of the cardiovascular parameter response to the infusion of cardio- and vasoactive medications has been recognized for many years. It was used as a data compression method that characterizes drug properties using few parameters. By applying standard pharmacodynamic models, interactions with other medications, stimulants, and blockers could be evaluated and compared (16, 19). With the advent of computer-controlled infusions, modeling became even more important. Software controllers can be trained and assessed through simulations with the pharmacodynamic model. This can minimize the use of complex experimental systems with laboratory animals. Furthermore, pharmacodynamic models can serve as a part of more advanced, adaptive controllers of drug infusion. They can also serve as a part of larger theoretical models for the simulation of cardiovascular control and help us to gain more insight into the system.

The response of cardiovascular parameters such as blood pressure, cardiac output, and HR to the infusion of cardioactive medications depends on the kinetics of drug distribution, on the response at the receptor level, and on counterregulatory mechanisms of the body. Even the simplest attempt to characterize these dependencies results in a number of differential equations and a number of coefficients. Calculation of the parameters of such complex models is time consuming, and there is always the possibility that different sets of parameter values will give the same results, thereby losing their physiological meaning.

Iso accelerates HR by two mechanisms, a direct action on the heart and a secondary compensating influence caused by its blood pressure-lowering effect. The HR response to the infusion of Iso is nonlinear. There is a plateau effect at high infusion rates, the so-called “HR saturation” phenomenon (17). There is a time delay between infusion and effect, and there is a time constant of rise and fall of HR according to the infusion rate.

The most widely used method of pharmacodynamic modeling is based on the Hill equation, which is a sigmoidal function that relates the steady-state physiological response to drug plasma concentration (5,13, 16, 19). This model is static and not dynamic. It represents the steady-state physiological response to constant levels of drug infusion. Therefore, its four coefficients can be calculated only after at least three steady-state levels of concentration and effect are reached in addition to the baseline level. In the attempt to relate the physiological effect to drug infusion rate, a pharmacokinetic drug distribution model should also be coupled. The lack of dynamic information, the need for a series of steady-state infusions, and the long time it takes to characterize the coefficients make this model unsuitable for use within controllers.

In an attempt to dynamically relate cardiac output to step function (square wave)-type dopamine infusion, a first-order exponential system was coupled with a sigmoidal function resembling the Hill equation (4). A recent educational simulation simulated the dynamic response to intravenous bolus injections. This model coupled an exponential pharmacokinetic model with a transfer function for effector site drug level and a Hill equation to calculate the effect (18). Another approach was used in a study of closed-loop infusion of arbutamine (17). The pharmacodynamic model was based on a discrete time autoregressive model with time delay and gain correction for the nonlinearity.

We have shown that the relationship between HR and Iso flow may be adequately described by a hyperbolic power function with no need for a sigmoidal function. This makes the pharmacodynamic model simpler for numerical analysis. A direct solution of a set of linear algebraic equations is much quicker than the iterative process that is needed for the solution of a Hill-type system. Our pharmacodynamic model takes into account the nonlinearity of the system and a time delay between Iso infusion and HR response. It is based on a first-order difference equation whose parameters may be readily calculated rapidly on-line, without any iterations. The input into the model can be any arbitrary Iso flow function. All the parameters have intuitive physiological meaning, and we have shown that the model has predictive ability. It has the potential to be used within on-line adaptive controllers of drug infusion and within theoretical cardiovascular models.

Although the basal HR in rats is much higher than that in humans, the absolute change in HR during Iso infusion is within a similar range. We can therefore anticipate our model’s validity in humans, with the appropriate change in parameter values.

In conclusion, we have shown that the steady-state response of HR to the infusion of Iso can be adequately described by a hyperbolic power function. The model we have built provides a good approximation of the HR response to Iso infusion with predictive ability. Its structure is built in such a way that the calculation of its parameters is very fast. The possibility of simulating the system will minimize the use of laboratory animals. Further studies are needed to access its applicability in other systems of physiological response to drug infusion and its ability to serve as a part of larger cardiovascular models. This pharmacodynamic model will undoubtedly have future practical clinical and research applications.

## Footnotes

Address for reprint requests and other correspondence: R. J. Leor-Librach, The Heart Institute, Sheba Medical Center, Tel Hashomer, PO Box 744, Netanya 42107, Israel.

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- Copyright © 1999 the American Physiological Society