Rate sensitivity of blood pressure to hypoxia

J. theor. Biol. (1985) 112, 839-845

Rate Sensitivity of Blood Pressure to Hypoxia VOJTECH LI~KO? AND HERSHEL RAFF~

Cardiovascular Research Institute, Liver Center and Department of Physiology, University of California, San Francisco CA 94143, U.S.A. (Received 17 April 1984, and in revised form l l September 1984) A biochemical kinetic model is used to describe changes in mean arterial blood pressure in dogs to three different rates of fall of arterial partial pressure of oxygen. The model is a linear loop with one variable rate coefficient (parametric control) which has been previously shown to characterize the rate sensitivity to presented stimuli. A three component model was identified under a least squares criterion and it showed that a unique (stimulation independent) representation can be obtained which can serve as a conceptual framework for the study of this phenomenon.

Introduction Although the steady state response of neural, endocrine and circulatory systems to hypoxia has been studied extensively, there is little information available on the transient response of these systems. In a recent investigation, one of us participated in an experimental determination of b l o o d pressure, vasopressin, corticotrophin (ACTH) and corticosteroid responses to a decrease of arterial partial pressure of oxygen (Pao2) in anesthetized dogs (Raft et al., 1983). To be able to characterize the nature of the response mechanism, three different rates o f the decrease o f Pao2 were used as a stimulus for the circulatory and humoral changes. The data suggested that blood pressure responses are sensitive not only to the level but also to the rate of fall of arterial partial pressure of oxygen. In this communication, we wish to explore in some detail the structure o f the response mechanism. In so doing, we will not employ the formal a p p r o a c h o f setting-up an i n p u t - o u t p u t system and assume that the response is some combination of "level" and " r a t e " sensitive processes. Rather, we will proceed by using a system modeling a p p r o a c h relying on a parametric excitation model enabling a molecular interpretation which has been shown to naturally incorporate "rate sensitivity" (Ekblad & Licko, 1984). t To whom correspondence should be addressed. :~Current address: Department of Internal Medicine, Medical College of Wisconsin, Milwaukee, Wisconsin 53215, U.S.A. 839

0022-5193/85/040839+07 $03.00/0

(~) 1985 Academic Press Inc. (London) Ltd

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Observations Studies from seven dogs (details of the experimental design are given by Raft et at. (1983)) are summarized in Fig. 1. Each dog took part in at least two experiments. The upper panel shows the least squares best fitted Pao2 data: the stimulation functions. The lower panel shows the mean values and standard errors of the blood pressure responses. The lines on this part of the figure are the results of the least squares fit of the model described below.

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FIG. 1. Experimental data and best fitted curves of arterial partial pressure of oxygen ( P a o 2 - - u p p e r panel) and mean arterial blood pressure (MABP---Iower panel). Three stimulation patterns (Pao2) were presented and corresponding response patterns (MABP) were recorded. The continuous curves for the latter were obtained by fitting model (ii) to the data, the best parameters are shown in equations (4).

Both panels contain three different curves and data sets which correspond to the three rates of the decline in Pao2. The overall average control Pao2 was 79.2+0-6 torr and was lowered by the experimenter to an overall average of 28-0+0.5 torr in the following fashion: a) an "instantaneous" drop within the first two minutes; b) an exponential decline with the half-life of approximately 1.5 minutes; c) an exponential decline with the half-life of approximately 6 minutes. The "instantaneous" drop of Pao2 caused the fastest and most pronounced change in the blood pressure, namely, an

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overshoot. With the slower decrease in Pao2, progressively slower and less pronounced changes in blood pressure were recorded. With increasing time, the blood pressure changes approached similar values at all three rates of Pao2 decline. Note that the data analyzed here are more complete sets (n.b. the existence of values at 15 minutes) than those presented in Fig. 1 of the experimental report (Raft et al., 1983) and several additional dogs are used in the present analysis. Model

The bimolecular nature of biochemical interactions which mediate stimulus-response coupling argues for a parametric control (Ekblad & Licko, 1984). Even though nonlinear effects can be observed in parametrically controlled pathways, linear catenary systems and/or loops with at least one stimulus dependent parameter can produce rate sensitive phenomena. In our approach, we have tried to identify the smallest loop (a closed system) and the simplest functional description of a stimulus sensitive parameter p(S) which would account simultaneously for all available data. Thus, we considered a model described diagramatically below in (i): p(s)

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xi stands for an ith dynamically distinguishable (biochemical) component of the regulatory system mediating transformation of stimulus S into a response proportional to x2. The relationship between the stimulus and the response in the case under consideration is a reciprocal one: a decrease of Pao2 leads to an increase in blood pressure. This implies that p(S) is a decreasing function of S, the arterial partial pressure of oxygen. A basal blood pressure is associated with a normal P a o 2. Bearing in mind again the bimolecular character of oxygen interaction with (chemo)receptors/enzymes and assuming such interaction is instantaneous in the time scale of the experiment, the steady state relationship between the value of the rate coefficient p(S) and the Pao: is a rational function with the Hill coefficient c, possibly greater than unity: K p[S(t)] = r, (1) K+SC(t)

where

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where r~ = p[0] is the maximal value of the stimulatory rate coefficient for Pao2 approaching zero. The value of K characterizes the stimulation level leading to half-suppression of the value of p ( S ) and is related to the commonly used pharmacological parameter IC50. The time function S ( t ) is the exponential function describing the decline of Pao2 during the experiment. Specifically, S ( t ) = S, exp ( - m t ) + S ~ .

(2)

In this formula~ S~ is the final value of Pao2 (28 torr, see above), S~ is the amount that the initial level of Pao2 exceeds So (51.2±0-4 torr), m has the following values: 0.458/min in experiment b and O. l l5/min in experiment c. In experiment a, p[S] becomes a step function at t = 0 from a value p = r, K / ( K + 80c) to another value p = r~K / ( K + 28c). The corresponding values of Pao2---80 and 28 torr are the directly measured average values of the arterial partial pressure of oxygen. The model is represented by a set of linear differential equations with one parameter variable ( p ( S ) ) and all other parameters constant, namely, Yq = rnx, - p[S]xl yc~_ = p [ S ] x l

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(3)

Yc. = r . - l x . - ~ - r.x..

This system (3) is now complete when considered in conjunction with equations (1) and (2) which define the time variable rate coefficient of x, transformation into x2 and initial conditions obtained from setting all time derivatives equal to zero simultaneously (p[S] having the value p[S(0)]). Identification of the model

The process is based on an iterative adjustment of the parameters r0, • • •, r, (where ro is a scaling factor) such that the sum of unweighted squared differences between the experimental data and those obtained by numerical (Euler) integration of equations (3) is smaller. Each step of iteration consists of solving equations (3) three times for the three different rates of Pao2 decline with the fitted Pao2 curves serving as the stimulation functions. Then, the sum of squares is minimized for one set of parameters simultaneously for all three stimulation functions. Both the Hill coefficient c and the size of the system n were varied in separate runs using e Hewlett-Packard HP 9825B desktop computer. The best size of the system was n =3 and the Hill coefficient value c = 1. Differences in comparison with values higher by one (n = 4 and/or c--2)

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were not significant (by F-test). Since such an increase of complexity of the system is not justified and a graphical comparison was the most favorable at the above values of n and c, the identified system is diagramatically described by (ii): x,

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with r0= 361 +38 [mm Hg] rl = 22.3 + 1.2 [min -I] r2 = 0.121 +0.011 [min -~]

(4)

r3 = 0.067 ± 0.009 [min-'] K = 0.553 ± 0.029 [torr]. This model suggests that a three-state molecular species undergoing transformation changes under the influence of hypoxia allows for a quantitative interpretation of the data in a wide range of stimulatory changes. As an example of such biochemical factor, some enzyme and/or enzymereceptor system may be invoked. However, any candidate biochemical mechanism ought to involve a parametrically controlled multi-state scheme. In this context, it is interesting to compare the results of this identification with those in other systems. A transient activation of cyclic AMP in frog gastric mucosa by histamine (Ekblad & Licko, 1984) resulted in a set of parameter values of a similar three component system which led to a near-basal level return during a sustained histamine stimulation. The distribution of the three states (equation (5)) is obtained from the steady state of the system described by equations (3)

(x'°°:x2°°:x3~)=(I'p:p "

(5)

Here p stands for the value of p[S] either before stimulation (p[S(0)]= 0.153 min -~) or at the steady state (p[S(oo)] = 0.432 min-l). Then, the prestimulatory distribution of the three states is (1:1.27:2.28) as compared with the same in the example with histamine step stimulated cyclic AMP transient changes (1:0.41:10). The pre-stimulatory state in the latter is characterized by the major portion of the enzyme state being inactive and

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unavailable for stimulation while in the present model, there is a more equal distribution among the three states. This difference in the states distribution is the reason for different forms of the response pattern: one nearly returns to the basal level while other approaches a new steady state.

Discussion

The notion of "rate sensitivity" in biological processes is old but until very recently was dealt with in a strictly formal way. Earlier, it has been merely described verbally as a phenomenon. Later, owing mostly to influence of the control theory, it has been represented in mathematical models by inclusion of a time-derivative element. Finally, analytical tools 6f linear systems have been suggested to characterize the phenomenon (Clynes, 1961 ). While such a description of the phenomenon might be considered adequate in studies with a single stimulation pattern, it fails to be instructive for the purposes of delineation of system characteristics when several stimulation patterns are used. Not only does the formal character lack any interpretative power in terms of molecular interaction, it also requires modification of the system structure itself to accomodate for different stimulation patterns. This conceptual failure is primarily due to the linear nature of the analysis. A kinetic model, as used in the present analysis, avoids both problems. This is shown by the ability of the model to describe the response of the real system to different stimulation functions while retaining both structural and quantitative descriptors (cf. the unique set of parameters shown in equations (4) for all three stimulatory patterns). The model demonstrates that a rather common though rarely considered kinetic scheme may be the underlying mechanism for the stimulus-response coupling. The importance of the latter is in providing a conceptual framework for designing experimental procedures leading to identification of biological entities participating in creating this phenomenon. In the current view of the control of hypertension associated with hypoxia, chemoreceptors such as those of the carotid body, are responsible for the chemical signal transduction (Eyzaguirre & Fidone, 1980). Recognizing the complexity of the biochemical/neural pathways, there are a number of possible nodes of transduction. The data and the model presented here are consistent with the notion of an involvement of systems like adenylate cyclase (Fitzgerald, Rogus & Dehghani, 1977) which in other tissues (Ekblad & Licko, 1984) has been shown to possess the characteristics of a three component parametrically driven system.

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To the memory of Julius H. Comroe Jr. This study was supported in part by NIH Grants AM 28172, AM 06419 and AM 06704 and in part by the University of California at San Francisco M. S. C. Clough Fund to V. Licko. REFERENCES CLYNES, M. (1961). Ann. N Y A c a d . ScL 92, 946. EKBLAD, E. B. M. & LI~:KO,V. (1984). Am. J. Physiol. 246, RlI4. EYZAGUIRRE, C. & FIDONE, S. J. (1980). Am. J. Physiol. 239, C135. FITZGERALD,R. S., ROGUS,E. M. & DEHGI-IANI,A. (1977). In: Tissue hypoxia and ischemia. (Reivich, M., Coburn, R., Lahiri, S. & Chance, B. eds). p. 245. New York: Plenum Press. RAFF, H., SHINSAKO,J., KEIL, L. C. & DALLMAN~M. F. (1983). Am~ Z Physiol. 245, E489.