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A theoretical analysis of the carotid body chemoreceptor response to O2 and CO2 pressure changes
Respiratory Physiology & Neurobiology 130 (2002) 99 – 110 www.elsevier.com/locate/resphysiol
A theoretical analysis of the carotid body chemoreceptor response to O2 and CO2 pressure changes Mauro Ursino *, Elisa Magosso Dipartimento di Elettronica, Informatica e Sistemistica, Uni6ersity of Bologna, Viale Risorgimento 2, I40136 Bologna, Italy Accepted 24 October 2001
Abstract A simple mathematical model of the carotid body chemoreceptor response is presented. The model assumes that the static chemoreceptor characteristic depends on oxygen saturation in the arterial blood and on CO2 arterial concentration. The values of O2 saturation and of CO2 concentration are computed, from pressure, using blood dissociation curves, which include both the Bohr and Haldane effects. Moreover, the O2 –CO2 static responses interact via a multiplicative term followed by an upper saturation. The dynamic response includes a term depending on the time derivative of CO2 concentration and a low-pass filter, which accounts for the time required to reach the steady state level. With a suitable choice of its parameters, the model reproduces the carotid chemoreceptor response under a variety of combined O2 and CO2 stimuli, both in steady state conditions and in the transient period following acute CO2 or O2 pressure changes. In particular, simulations show that if two hypercapnic stimuli are given in rapid succession, the response to the second stimulus is weaker than the first. Moreover, during transient conditions the effect of CO2 pressure changes prevail over the effect of O2 changes, due to the intrinsic derivative component of the response to CO2. In conclusion, the model allows present knowledge about chemoreceptor activity to be summarized in a single theoretical framework. In perspective, it may be used as an afferent block within large-scale models of the overall cardio-respiratory control system. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Carotid body, chemoreceptor, model; Chemoreceptor, carotid body, model; Control of breathing, carotid body; Hypoxia, carotid body; Model, carotid body
1. Introduction It is well known that the carotid body chemoreceptors exert an important role in the regulation of the cardiovascular and ventilatory systems in man and other mammals, in response to acute * Corresponding author. Tel.: + 39-051-209-3008; fax: + 39-051-209-3073. E-mail address: [email protected] (M. Ursino).
perturbations of O2 and CO2 concentration in blood (Marshall, 1994; Daly, 1997; Eyzaguirre et al., 1983). These receptors send their information to the central neural system, which, in turn, modulates the sympathetic and vagal efferent activities, thus modifying heart rate, cardiac contractility, total peripheral resistance and venous tone. Moreover, carotid chemoreceptors significantly contribute to the regulation of ventilation, especially during hypoxia.
1569-9048/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 4 - 5 6 8 7 ( 0 1 ) 0 0 3 3 5 - 8
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Knowledge of the quantitative relationship linking chemoreceptor activity to the composition of blood is extremely important, in order to build accurate mathematical models of the cardiorespiratory control system, and to summarize experimental data in the literature. However, despite the large amount of quantitative data on the carotid chemoreceptor response that have been gathered in the past decades (Marshall, 1994; Fitzgerald and Dehghani, 1982; Di Giulio et al., 1998; Daly, 1997; Lahiri and Delaney, 1975; Lahiri et al., 1982; Black et al., 1971; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Eyzaguirre et al., 1983), mathematical models are still rare. In the late seventies Yamamoto and Yamamoto (Yamamoto and Yamamoto, 1979) presented a mathematical model of the carotid chemoreflex, including a simple description of the afferent chemoreceptor response to hypercapnic stimuli. The authors were able to simulate various experimental results using a single set of parameters. In particular, the dynamic response of carotid chemoreceptors to CO2 pressure changes exhibits an adaptation phenomenon, i.e. sudden hypercapnia produces an overshoot in the carotid nerve discharge rate. However, their model does not incorporate several other important features of the chemoreceptor response, documented in the physiological literature. The main aspects, which deserve further mathematical analyses, can be summarized as follows: 1. The response to changes in CO2 tension, at constant PaO2, exhibits a lower threshold and is quite linear in the lower pressure range. However, this response becomes progressively flatter at increasing levels of PaCO2 (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Fitzgerald and Dehghani, 1982; Di Giulio et al., 1998). 2. Peripheral chemoreceptor activity depends greatly on O2 pressure in the arterial blood. At constant CO2 pressure, this dependence exhibits an exponential trend, being moderate during hyperoxia, but reaching a higher level during severe hypoxia (30– 40 mmHg) (Biscoe et al., 1970). 3. The sensitivity of carotid chemoreceptors to CO2 progressively increases with hypoxia
and vice versa. This implies the existence of a multiplicative interaction between the individual PaO2 and PaCO2 responses (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Eyzaguirre and Lewin, 1961). This interaction plays a protective role during asphyxia. 4. The positions of threshold and saturation of the chemoreceptor response to PaCO2 are modulated by hypoxia. Experimental data suggest that the threshold is significantly shifted to the left during severe hypoxia (Lahiri and Delaney, 1975). Likewise, the chemoreceptor response saturates at a much lower PaCO2 during hypoxia than during normoxia (Fitzgerald and Parks, 1971). The aim of the present work is to propose an extended mathematical model of the chemoreceptor response to changes in blood gas content, which embodies several findings not incorporated in Yamamoto’s model. The model aspires to represent an effective instrument for the interpretation of clinical and physiological data concerning the cardio-ventilatory control system. The paper is structured as follows. The main aspects of the model are first presented and justified. Then, the model is used to mimic experimental data reported in the physiological literature, both the static and dynamic patterns of discharge. Finally, the main results are critically commented. All equations and parameter values can be found in the Appendix A.
2. Qualitative model description A block diagram describing the main aspects of the model is shown in Fig. 1. In the following, the static response of the overall model is described first. Subsequently, the dynamic aspects are considered. The model assumes that the carotid body chemoreceptors are sensitive to CO2 concentration (CaCO2) and to oxygen desaturation (1− SaO2) in the arterial blood. However, since the input variables commonly manipulated in physiological experiments are arterial CO2 and O2 tensions (PaCO2 and PaO2, respectively), carbon dioxide concentration and oxygen saturation are
Fig. 1. Block diagram describing the main aspects incorporated in the model. Block 1 represents the O2 and CO2 dissociation curves in arterial blood, including the Bohr and Haldane effects. Block 2 represents the non-linear characteristic linking chemoreceptor activity to arterial oxygen saturation. Block 3 reproduces the linear dependence of chemoreceptor activity on arterial CO2 concentration, including a lower threshold, Ct. Blocks 4 and 5 describe the multiplicative O2 – CO2 interaction and the upper saturation for the static chemoreceptor activity, respectively. Block 6 is the rate-dependent component of chemoreceptor response to CO2, while block 7 denotes the upper saturation for the rate-dependent component. The static and dynamic component are summed in block 8, low-pass filtered to achieve the time constant of the response (block 9) then passed through a single-wave rectifier which eliminates negative values (block 10). All equations can be found in the Appendix A.
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computed from pressures using blood dissociation curves proposed by Spencer (Spencer et al., 1979) (Block 1 in Fig. 1 and Eqs. (1)– (5) in the Appendix A). These equations have been chosen since they include the Bohr and Haldane effects and can be easily inverted. The latter property can be of mathematical importance when the equations are used within large models of cardiorespiratory dynamics. Of course, alternative models for blood gas transport (see, for instance, Popel, 1989) can be used equally well without appreciable changes in the obtained results, provided the equations chosen include the Bohr and Haldane effects.
2.1. The static response According to the physiological literature (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Eyzaguirre and Lewin, 1961) the model assumes that the chemoreceptor static response is the result of a multiplicative interaction between the individual oxygen and CO2 responses (block 4 in Fig. 1 and Eq. (9) in Appendix A). The static response to CO2 is simply obtained through a linear relationship, i.e. arterial CO2 concentration is compared with a given threshold, Ct, and multiplied by a static gain (block 3 and Eq. (8)). In the first stage of our work we tried to simulate the static oxygen response through a linear relationship too, i.e. we assumed that chemoreceptor activity is a linear function of desaturation. The latter assumption, however, was unable to reproduce the experimental pattern of chemoreceptor activity versus PaO2 accurately, since chemoreceptor activity increased too much during deep hypoxia. Hence, we adopted a non-linear function of desaturation, which is quite linear during hyperoxia, normoxia and mild hypoxia, but progressively flattens when SaO2 is reduced below approximately 90% (block 2 and Eqs. (6) and (7)). Finally, as shown in Fig. 1, the two individual static responses described above are multiplied and passed through a further non-linear function with upper saturation (block 5 and Eq. (10)). Inclusion of this function is necessary to reproduce the attainment of a maximal level for
chemoreceptor activity observed (Fitzgerald and Parks, 1971; Fitzgerald and Dehghani, 1982; Di Giulio et al., 1998; Eyzaguirre and Lewin, 1961) during asphyxia. In the present model, according to the experimental observations by (Di Giulio et al., 1998), the upper saturation level is assumed to be constant, i.e. it does not depend on the oxygen level in blood. Contrasting results on this point are analyzed in the Section 4.
2.2. Dynamic response There is general agreement in the literature that the dynamic chemoreceptor response to a step CO2 pressure change exhibits a significant overshoot followed by adaptation to the steadystate level (Lahiri et al., 1982; Black et al., 1971; Torrance, 1977). This typical response implies the existence of a rate-dependent component of the CO2 response, i.e. a component that is sensitive to the time derivative of carbon dioxide concentration. By contrast, the response to a step change in oxygen pressure does not exhibit any appreciable adaptation (Black et al., 1971). The existence of a rate-dependent component of the CO2 response has been covered in the model by a first order high pass filter (block 6 and Eq. (11)). The response of this filter is sensitive to the rate of change of CO2 concentration up to a maximal frequency rate. The time derivative of CO2 concentration is computed, from knowledge of the input pressure waveform, by deriving the analytical equations of blood gas transport. In the model we assumed that the rate-dependent component is affected by carbonic anhydrase, whereas the static component represents a process independent of carbonic anhydrase. This is similar to the approach taken by Yamamoto and Yamamoto (1979). An important question is whether the rate-dependent component is affected by the oxygen level (as is the case of the static component) or not. Although differing opinions on this point can be found in the physiological literature, the majority of authors claim that the rate-dependent component is substantially unaffected by hypoxia or hyperoxia (see Cunningham et al., 1986, for a review). Moreover, according to data
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reported by Di Giulio et al. (1998), the rate-dependent component exhibits much higher saturation (about 45 impulses per sec) compared with the static component (about 20 impulses per sec). Accordingly, in the model the output of the highpass filter has been passed through a different saturation block, and then summed directly to the static response (blocks 7 and 8, and Eq. (12)). Finally, the overall response so obtained is passed through a first order low-pass filter with a time constant tpl and a single-wave rectifier (blocks 9 and 10, respectively, and Eqs. (13) and (14)). The presence of the low pass filter is necessary to mimic the progressive attainment of a steady state level. The rectifier simply disallows a negative response. All parameters in the model have been given to simulate the carotid chemoreceptor response in the cat (see Section 3), since most experimental data in the physiological literature have been acquired on this animal. Of course, different parameter values [for instance a different threshold, see (Mohan and Duffin, 1997)] might be required to simulate the response in humans.
Fig. 2. Static dependence of chemoreceptor activity on O2 arterial pressure during isocapnia. Two model simulation curves are shown, corresponding to two different isocapnic levels (32 mmHg, lower line and 48 mmHg upper line). Experimental points have been measured in the cat by (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Sampson and Hainsworth, 1972; Mulligan et al., 1981) in the same range of PaCO2.
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Table 1 Basal values of model parameters Dissociation cur6es in blood K1 =14.99 mmHg a1 =0.3836 a1 =0.03198 b1 =0.008275 mmHg−1 mmHg−1 a2 =1.819 K2 =194.4 mmHg a2 =0.05591 mmHg−1
b2 =0.03255 mmHg−1
Chemoreceptor response A =600 B =10.18 KCO2 =1 sec−1 Ct =0.36 L L−1 tph =3.5 sec tzh =600 sec Kdyn =45 sec−1
C1 =9 mM L−1 CaO2max =0.2 L L−1 C2 =86.11 mM L−1 Z= 0.0227 L mM−1 KO2 =200 Kstat =20 sec−1 tpl =3.5 sec
3. Results
3.1. Static response Fig. 2 shows the pattern of carotid chemoreceptor activity versus PaO2, predicted by the model in steady state conditions, using the parameter numerical values as in Table 1. Model results are then compared with data obtained by several investigators in the cat (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Sampson and Hainsworth, 1972; Mulligan et al., 1981). As it is clear from the figure, chemoreceptor activity increases moderately when passing from hyperoxia to normoxia, but it exhibits a disproportionate rise during severe hypoxia. At a PaO2 level as low as 25–30 mmHg, chemoreceptor activity is about 5-fold greater than during normoxia. The non-linear interaction between O2 and CO2 sensitivity is illustrated in Fig. 3. This figure plots the chemoreceptor response versus PaCO2 at different oxygen pressure levels, ranging from severe hypoxia (left upper panel) to hyperoxia (right lower panel). A good agreement can be found between model predictions and data by various authors in the cat (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971) at all pressure levels examined. In particular, this figure shows that the model is able to account for various experimental observations with sufficient accuracy. During normoxia and hyperoxia, the chemoreceptor response
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to CO2 is quite linear above a threshold. The sensitivity to CO2 is significantly increased by hypoxia (left upper panel) and reduced by hyperoxia (right lower panel). Furthermore, the threshold is shifted to the left (from about 21 to 16 mmHg) when passing from hyperoxia to hypoxia. Finally, at the lower O2 pressure levels, the receptor response to increasing PaCO2 tends to a plateau. The dependence of the CO2 pressure threshold on arterial oxygen pressure is shown in Fig. 4. The model ascribes this dependence to the Haldane effect (i.e. the upward shift of the CO2 dissociation curve during hypoxia (Guyton, 1986)). In fact, the real threshold in the model (i.e. parameter Ct in Eq. (8)) always remains constant despite hypoxia.
3.2. Dynamic response Fig. 5 shows the chemoreceptor response to a step increase in PaCO2 from 40 to about 90 mmHg (‘on response’) during hyperoxia. The upper panel displays the response obtained by using the basal values of model parameters (i.e. the values in Table 1). Chemoreceptors activity exhibits an overshoot, reaches a peak at about 2–3 sec after the stimulus, then declines down to the new steady-state level within 20 sec. The lower panel displays the same response after inhibiting the carbonic anhydrase of the carotid body with acetazolamide. In order to simulate this response, we assumed that the rate-dependent component of the chemoreceptor is suppressed by acetazolamide, (and so we set parameter tzh = 0 in Eq.
Fig. 3. Percent of maximum chemoreceptor response (MR%) vs. arterial CO2 pressure simulated with the model in steady state conditions at different levels of PaO2 ranging from severe hypoxia (left upper panel) to hyperoxia (right lower panel). Experimental points are taken from (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971) at comparable levels of PaO2. 100% in this figure denotes the maximal steady-state activity during asphyxia, which is the same as the static saturation Kstat in the model.
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Fig. 4. Apparent dependence of the threshold for chemoreceptor activity on arterial O2 pressure, according to the present model. This threshold represents the value of CO2 arterial pressure at which chemoreceptor activity begins to appear (see Fig. 3). At lower values of PaCO2, chemoreceptor activity is set to zero by the single-wave rectifier (block 10 in Fig. 1). It is worth-noting that the apparent dependence of the threshold on PaO2 is just the consequence of the Haldane effect (i.e. an upward shift of the CO2 dissociation curve during hypoxia) whereas the true threshold, Ct, remains unchanged.
(11). The temporal patterns in Fig. 5 agree with those obtained by (Black et al., 1971) in the cat. Fig. 6 investigates the effect of different maneuvers performed in rapid succession, comparing model results with analogous data by (Black et al., 1971). In particular, the upper panel displays the response to a step decrease in PaCO2 from 120 to 80 mmHg (‘off response’) during hyperoxia, followed by a step increase from 80 to 120 mmHg. It is evident that, during the off response, the discharge frequency falls to zero for a few seconds then progressively increases up to the new steady-state level. The lower panel shows the effect of a hypercapnic stimulus (from 40 to 120 mmHg) given first, then removed, and then applied again after a short time. The second response turns out much lower than the first. This effect, however, is weaker in the model than in the experimental data: in the model the separation between the two stimuli should be as low as 10 sec to evoke a conspicuous difference between the two responses, whereas the time separation was as high as 15 sec in Black’s data. Finally, Fig. 7 examines the interaction between O2 and CO2 in dynamic conditions. In these simulations we slightly reduced the rate dependent component (tzh = 350 instead of 600 sec) to provide better reproduction of experimental results. In fact, in this particular fiber of Black’s experiment (Black
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et al., 1971), the rate-dependent response appears a little lower than in the previous cases. The upper panel illustrates the effect of substituting a hypercapnic stimulus with a hypoxic stimulus which produces almost the same steady-state response (see figure legend for more details). The two stimuli are almost indistinguishable in steady-state conditions, whereas the rate-dependent CO2 component dominates the transient response. The lower panel shows the on and off response to CO2 stimuli, during maintained hypoxia.
Fig. 5. Waveform of chemoreceptor activity simulated with the model in response to a sudden increase in PaCO2 from 40 to 93 mmHg (on response) during hyperoxia (PaO2 =300 mmHg). The results are compared with data from Fig. 6 in (Black et al., 1971) (20% CO2 in air). The upper panel was obtained using all parameter values as in Table 1. The lower panel refers to the response after inhibiting the carbonic anhydrase of the carotid body with acetazolamide. The latter case was simulated by abolishing the rate-dependent component in the model (i.e. setting tzh =0 in Eq. (11)).
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may be useful in the analysis of overall cardio-respiratory regulation, where chemoreceptors represent a single afferent branch within a complex multi-input, multi-feedback, multi-output control
Fig. 6. Effect of different stimuli given in rapid succession. The upper panel shows the waveform of chemoreceptor activity following a sudden decrease in the hypercapnic stimulus (from 120 to 80 mmHg between t = 24 and 26 sec) followed by a sudden increase (from 80 to 120 mmHg between 48 and 49 sec) during hyperoxia (PaO2 =180 mmHg). The lower panel displays the model response to a hypercapnic stimulus (from 40 to 120 mmHg) given, then removed, then applied again after a few seconds during hyperoxia (PaO2 = 300 mmHg). Model results are compared with data from Fig. 3 in (Black et al., 1971). The levels of hyperoxia were chosen to have approximately the same basal activity in the model and in the experimental data. All model parameters have the same value as in Table 1. It is worth noting that, in the lower panel, a smaller delay between the two stimuli was used in the simulations (about 10 sec) compared with real data (15 sec).
4. Discussion The aim of the present work was to develop a simple empirical model of the carotid body chemoreceptor, which can help in the study of the cardio-respiratory control system. This model
Fig. 7. Interaction between CO2 and O2 in dynamic conditions. Upper panel: response of the model when alternating two stimuli (hypercapnia and hypoxia) which provides approximately the same basal level of chemoreceptor activity. PaCO2 was lowered from 120 to 40 mmHg between t =16 and 18 sec, while PaO2 was simultaneously decreased from 120 to 50 mmHg. Subsequently, PaCO2 was raised from 40 to 120 mmHg again between t = 53 and 55 sec, while PaO2 was simultaneously returned from 50 to 120 mmHg. Lower panel: effect of a rapid +10 mmHg increase in PaCO2 (from 24 to 34 mmHg) pferformed between t = 22 and 23 sec, followed by a step decrease (from 34 to 24 mmHg) between t =38 and 39 sec during hypoxia (PaO2 =50 mmHg). The initial level of PaCO2 (24 mmHg) was chosen to have the same baseline activity in the model and in the experimental points. Data points are from Fig. 5 in (Black et al., 1971). It is worth noting that, in these figures, we used a reduced value for the rate-dependent component (tzh =350 sec) to better simulate the response of this single experimental fiber.
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system. In this regard, the mathematical equations proposed here can be used in large-scale models of the entire cardio-respiratory control to drive ventilation (in synergy with medullary chemoreceptors) and to modulate vagal and sympathetic activities to heart and vessels. Of course, the model presented here is merely a functional one, i.e. its equations have been selected to mimic the observed input– output relationships, and the structure of the model may not accurately reflect the underlying physiological transduction mechanisms responsible for chemoreceptor behavior. Our model shares some important aspects with the model proposed by (Yamamoto and Yamamoto, 1979) more than 20 years ago. In fact, in both models the chemoreceptor response to CO2 includes a rate-dependent component (affected by carbonic anhydrase), a component proportional to the instant value of CO2 (assumed unaffected by carbonic anhydrase) and a single wave rectifier. However, the present model includes many new aspects too, which significantly expand its possible applications: in particular, we incorporated equations for gas transport in blood, including Bohr and Haldane effects, the chemoreceptor response to oxygen, the multiplicative O2 – CO2 interaction, and an upper saturation for both the static and the rate-dependent components. Thanks to these new aspects, the present model can simulate chemoreceptor response in several conditions not covered by the former model, such as asphyxia, or hypocapnic hypoxia. The static characteristics of the model include several different non-linearities. All of them are necessary to reproduce the well-known experimental data. A first non-linearity concerns the relationship linking chemoreceptor activity to SaO2 during isocapnia. Our analysis, based on physiological data by various authors (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Sampson and Hainsworth, 1972; Mulligan et al., 1981) confirms the previous observation by Daly (1997), p. 69), according to whom the relationship between chemoreceptor activity and percentage saturation deviates from linearity at a level of approximately 90% saturation.
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By contrast, the response to CO2 during normoxia can be described reasonably well using a simple linear relationship with a constant threshold. It is remarkable that the apparent decrease in the threshold during hypoxia (Figs. 3 and 4) is ascribed by the model to the Haldane effect (i.e. an upward shift in blood CO2 dissociation curve during hypoxia). The idea that a shift in blood dissociation curves can explain some changes in the chemoreceptor response to PaCO2 has previously been suggested by others (Di Giulio et al., 1998; Lahiri and Delaney, 1975). A further non-linearity is an upper saturation for the static response (block 5 in Fig. 1). In the present study we assumed that this upper saturation during hypercapnia is about 20 impulses per sec, independently of the oxygen level in blood. This choice agrees with the experimental results by (Di Giulio et al., 1998) who observed that the maximal response to PaCO2 stimuli is the same during hypoxia and hyperoxia. A different point of view, however, was proffered by (Fitzgerald and Dehghani, 1982): using statistical analyses, these authors reached the conclusion that the upper saturation level of the chemoreceptor response to hypercapnia is not independent of oxygen, but can be increased by lowering PaO2. However, as pointed out by Di Giulio et al. (1998), the experiments by Fitzgerald and Dehghani were limited to a maximum PaCO2 as high as 75–80 mmHg; at this level of hypercapnia, a plateau might not have actually been reached during hyperoxia and normoxia, making their conclusion doubtful. When building the model, we used particular care in the reproduction of the dynamic chemoreceptor response. This is very important not only to arrive at a correct reproduction of transient phenomena [such as the on or off-responses following acute perturbations in blood gas content (Black et al., 1971)] but also since the dynamical properties of a non-linear system may affect the average values as well (Cunningham et al., 1986). The most important feature of the chemoreceptor dynamical response is its strong dependence on the CO2 rate of change. There is a general consensus in the physiological literature that a step increase in PaCO2 causes an overshoot in
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chemoreceptor activity followed, within 10– 20 sec, by an adaptation toward the new steady-state level (Marshall, 1994; Lahiri et al., 1982; Black et al., 1971; Torrance, 1977; Cunningham et al., 1986). We made use of a first-order high-pass filter to mimic the rate-dependent component of the CO2 response. This simple and rational choice is able to reproduce the transient behavior quite well during both on and off-responses with the assignment of only two parameters. Nevertheless, two aspects of the dynamical response are critical and deserve attention. First, as shown in the block diagram of Fig. 1, the model assumes that the rate-dependent component of the chemoreceptor response to CO2 (i.e. the output of the high-pass filter) is not affected by the oxygen level in blood. Several experimental data support this assumption, although there is no complete agreement in the literature. First, the amplitude of the naturally occurring oscillations in chemoreceptor activity, caused by respiration, does not increase during hypoxia (Band and Wolff, 1978). Moreover, a strong dynamic response is still evident when a sudden increase in CO2 is performed with 100% O2 (Black et al., 1971; Torrance, 1977), that is, hyperoxia does not suppress the dynamic response whereas it significantly abates the steadystate level. Conversely, (Lahiri et al., 1982) observed that hypoxia increases both the overshoot and the steady state response to hypercapnia. However, looking at Fig. 5 in their paper, one can observe that the transient response is just a little higher in hypoxia (from 10 to 34 impulses per sec) compared with hyperoxia (from 0 to 18 impulses per sec), corresponding to a 33% increase. By contrast, the final steady state level is dramatically increased by hypoxia compared with hyperoxia (18 vs. 5 impulses per sec, that is a 260% increase). Hence, in the data by Lahiri et al. too, it is quite reasonable to assume that the rate sensitivity to CO2 is less dependent on oxygen than is the steady state CO2 response. A second important assumption concerns the use of two different saturation levels for the static and rate-dependent components: we assumed that the rate-dependent component exhibits much higher saturation (45 impulses per sec) than the static component (20 impulses per sec). There are
several justifications for this choice. First, the use of a single saturation does not allow the experimental data in Figs. 5 and 6 to be reproduced with sufficient accuracy (unpublished simulations). Second, the values used in this work agree with those measured by (Di Giulio et al., 1998) in the cat. Finally, the use of different saturation levels may reflect the existence of different processes, the first dependent on carbonic anhydrase, the second independent of it. In conclusion, the present model significantly extends the validity of the model by (Yamamoto and Yamamoto, 1979) and allows several physiological results to be summarized into a single theoretical framework. In future work the model may be further extended, including a few additional features not considered here: among them, the dependence of chemoreceptor activity on the potassium level during exercise, a distinction between the response to CO2 and H+, or the effect of pressure or flow changes at the chemoreceptor level (Cunningham et al., 1986; Marshall, 1994).
Appendix A We assumed that peripheral chemoreceptors respond to changes in arterial carbon dioxide concentration (CaCO2) and arterial oxygen saturation level (SaO2). SaO2 and CaCO2 are computed, as functions of arterial O2 and CO2 partial pressures (which are inputs for the model), using the expressions for dissociation curves in blood proposed by Spencer et al (Spencer et al., 1979). These equations offer the advantage of taking the Bohr and Haldane effects into account. We have CaO2 = ZC1 FaO2 = PaO2 SaO2 =
1 FaO1/a 2 1/a1 1+ FaO2
1+ i1PaCO2 K1(1+h1PaCO2)
CaO2 CaO2max
CaCO2 = ZC2
2 FaCO1/a 2 1/a2 1+ FaCO2
(1) (2) (3) (4)
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FaCO2 =PaCO2
1+i2PaO2 K2(1+ h2PaO2)
(5)
where PaO2 and PaCO2 represent the oxygen and carbon dioxide tensions in arterial blood, FaCO2 and FaO2 are mathematical functions, which allow for the representation of the Bohr and Haldane effects, and Z is a conversion factor which transforms O2 concentration from mmol L − 1 to L L − 1. a1, C1, a1, b1, K1 and a2, C2, a2, b2, K2 are constant parameters, which were determined by Spencer et al. (Spencer et al., 1979) to fit experimental data. Finally, CaO2max is arterial oxygen concentration when blood is completely saturated. The output of the model is obtained as the sum of two components: a steady state component and a rate dependent (dynamic) component. The former allows the relationships between chemoreceptor discharge frequency and changes in arterial PO2 and PCO2 to be described in static conditions. The main features of the steady-state response include a non-linear dependence on SaO2, a linear dependence on CaCO2 above a given threshold, below which chemoreceptor activity is set to zero, a multiplicative interaction between the O2 and CO2 responses, and the presence of saturation. Hence the static response is described by the following equations:
n
xO2 =A(1 −SaO2)+B O2 = KO2
(6)
−xO2 1−exp KO2
(7)
CO2 = KCO2(CaCO2 −Ct)
(8)
F = O2CO2
(9)
n
−F Á ÃKstat 1− exp Kstat fcstat = Í Ã 0 Ä
CaCO2 \Ct
for the static response. The ability of the model to reproduce chemoreceptor steady-state responses to independent changes in arterial PO2 and PCO2 results from Eqs. (6)–(10) combined with equations for dissociation curves (Eqs. (1)–(5)). In particular, fcstat, plotted against PaO2 at different constant levels of PaCO2, displays an exponential trend; fcstat increases quite linearly with PaCO2, at normal or elevated PaO2 values, while at lower PaO2, fcstat tends to flatten out as PaCO2 increases. The rate-dependent component of the chemoreceptor response, 8CO2dyn, is obtained by means of a high-pass filter, with a real pole and a real zero. We can write ~ph
dCO2dyn dCaCO2 + CO2dyn = ~zh dt dt
(10) fcstat is the chemoreceptor frequency discharge in static conditions, 8O2 and 8CO2 represent the contributions of oxygen and carbon dioxide to the static response, F is the result of their multiplicative interaction, Ct represents the CO2 threshold for the response and A, B, KO2, KCO2, Kstat are constant parameters tuned to fit experimental data. In particular, Kstat is the upper saturation
(11)
where tph and tzh are the time constants of the pole and the zero in the high-pass transfer function, and the time derivative of CO2 arterial concentration is computed via derivation of Eqs. (1)–(5). Finally, the static and rate-dependent components are passed through two distinct saturation blocks and summed. The signal so-obtained is low-pass filtered and sent to a single-wave rectifier, which cuts out negative values. Hence, the following equations hold:
n n
¯ c = Kstat 1− exp
−F Kstat
+ Kdyn 1− exp ~pl
dc + c = ¯c dt
fc =
CaCO2 BCt
109
!
c 0
c \ 0 c B 0
− CO2dyn Kdyn
(12) (13)
(14)
Kstat and Kdyn are the different upper saturation levels for the static and for the rate-dependent components, respectively, and 8c is the output of the low-pass filter. fc is chemoreceptor activity, while tpl is the time constant of the real pole in the low-pass filter. Values assigned to the three time constants in Eqs. (11) and (13) reproduce well the transient chemoreceptor response to a square CO2 pulse.
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Finally, it is worth noting that in static conditions (i.e. then dCaCO2/dt =0, d8CO2dyn/dt =0, d8c/dt =0) fc in Eq. (14) coincides with fcstat.
References Band, D.M., Wolff, C.B., 1978. Respiratory oscillations in discharge frequency of chemoreceptor afferents in sinus nerve of anaesthetized cats at normal and low arterial oxygen tensions. J. Physiol. London 282, 1 – 6. Biscoe, T.J., Purves, M.J., Sampson, S.R., 1970. The frequency of nerve impulses in single carotid body chemoreceptors afferent fibres recorded in vivo with intact circulation. J. Physiol. London 208, 121 –131. Black, A.M.S., McCloskey, D.I., Torrance, R.W., 1971. The responses of carotid body chemoreceptors in the cat to sudden changes of hypercapnic and hypoxic stimuli. Respir. Physiol. 13, 36 –49. Cunningham, D.J.C., Robbins, P.A., Wolff, C.B., 1986. Integration of respiratory responses to changes in alveolar partial pressures of CO2 and O2 and in arterial pH. In: Fishman, A.P., Cherniack, N.S., Widdicombe, J.G., Geiger, S.R. (Eds.), Handbook of Physiology, Section 3: The Respiratory System, Control of Breathing, part 2, vol. II. American Physiological Society, Bethesda, MD, pp. 475 – 528. Daly, M.B., 1997. Peripheral Arterial Chemoreceptors and Respiratory-Cardiovascular Integration. Oxford University Press, Oxford. Di Giulio, C., Huang, W., Mokashi, A., Lahiri, S., 1998. Further characterization of stimulus interaction of cat carotid chemoreceptors. J. Autonom. Nerv. Syst 71, 196 – 200. Eyzaguirre, C., Lewin, J., 1961. Chemoreceptor activity of the carotid body of the cat. J. Physiol. London 159, 222 –237. Eyzaguirre, C., Fitzgerald, R.S., Lahiri, S., Zapata, P., 1983. Arterial chemoreceptors. In: Shepherd, J.T., Abboud, F.M., Geiger, S.R. (Eds.), Handbook of Physiology, Sec-
tion 2: The Cardiovascular System, vol. III. The American Physiological Society, Bethesda, MD, pp. 557 – 621. Fitzgerald, R.S., Parks, D.C., 1971. Effect of hypoxia on carotid chemoreceptor response to carbon dioxide in cats. Respir. Physiol. 12, 218 – 229. Fitzgerald, R.S., Dehghani, G.A., 1982. Neural responses of the cat carotid and aortic bodies to hypercapnia and hypoxia. J. Appl. Physiol. 52, 596 – 601. Guyton, A.C., 1986. Textbook of Medical Physiology. Saunders, Philadelphia. Lahiri, S., Delaney, R.G., 1975. Stimulus interaction in the responses of carotid body chemoreceptor single afferent fibers. Respir. Physiol. 24, 249 – 266. Lahiri, S., Mulligan, E., Mokashi, A., 1982. Adaptive response of carotid body chemoreceptors to CO2. Brain Res. 234, 137 –147. Marshall, J.M., 1994. Peripheral chemoreceptors and cardiovascular regulation. Physiol. Rev. 74, 543 – 594. Mohan, R., Duffin, J., 1997. The effect of hypoxia on the ventilatory response to carbon dioxide in man. Respir. Physiol. 108, 101 – 115. Mulligan, E., Lahiri, S., Storey, B.T., 1981. Carotid body O2 chemoreception and mitochondrial oxidative posphorylation. J. Appl. Physiol. 51, 438 – 446. Popel, A.S., 1989. Theory of oxygen transport to tissue. CRC Crit. Rev. Biomed. Eng. 17, 257 – 321. Sampson, S.R., Hainsworth, R., 1972. Responses of aortic body chemoreceptors of the cat to physiological stimuli. Am. J. Physiol. 222 (4), 953 – 958. Spencer, J.L., Firouztale, E., Mellins, R.B., 1979. Computational expressions for blood oxygen an carbone dioxide concentrations. Ann. Biomed. Eng. 7, 59 – 66. Torrance, R.W., 1977. Manipulation of bicarbonate in the carotid body. In: Acker, H., Fidone, S., Pallot, D., Eyzaguirre, C., Lu¨ bbers, D.W., Torrance, R.W. (Eds.), Chemoreception in the Carotid Body. Springer, Berlin, pp. 286 – 293. Yamamoto, W.S., Yamamoto, P.S., 1979. A mathematical model of chemoreflex response to hypercapnic stimuli. Comp. Biomed. Res. 12, 83 – 96.
A theoretical analysis of the carotid body chemoreceptor response to O2 and CO2 pressure changes Mauro Ursino *, Elisa Magosso Dipartimento di Elettronica, Informatica e Sistemistica, Uni6ersity of Bologna, Viale Risorgimento 2, I40136 Bologna, Italy Accepted 24 October 2001
Abstract A simple mathematical model of the carotid body chemoreceptor response is presented. The model assumes that the static chemoreceptor characteristic depends on oxygen saturation in the arterial blood and on CO2 arterial concentration. The values of O2 saturation and of CO2 concentration are computed, from pressure, using blood dissociation curves, which include both the Bohr and Haldane effects. Moreover, the O2 –CO2 static responses interact via a multiplicative term followed by an upper saturation. The dynamic response includes a term depending on the time derivative of CO2 concentration and a low-pass filter, which accounts for the time required to reach the steady state level. With a suitable choice of its parameters, the model reproduces the carotid chemoreceptor response under a variety of combined O2 and CO2 stimuli, both in steady state conditions and in the transient period following acute CO2 or O2 pressure changes. In particular, simulations show that if two hypercapnic stimuli are given in rapid succession, the response to the second stimulus is weaker than the first. Moreover, during transient conditions the effect of CO2 pressure changes prevail over the effect of O2 changes, due to the intrinsic derivative component of the response to CO2. In conclusion, the model allows present knowledge about chemoreceptor activity to be summarized in a single theoretical framework. In perspective, it may be used as an afferent block within large-scale models of the overall cardio-respiratory control system. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Carotid body, chemoreceptor, model; Chemoreceptor, carotid body, model; Control of breathing, carotid body; Hypoxia, carotid body; Model, carotid body
1. Introduction It is well known that the carotid body chemoreceptors exert an important role in the regulation of the cardiovascular and ventilatory systems in man and other mammals, in response to acute * Corresponding author. Tel.: + 39-051-209-3008; fax: + 39-051-209-3073. E-mail address: [email protected] (M. Ursino).
perturbations of O2 and CO2 concentration in blood (Marshall, 1994; Daly, 1997; Eyzaguirre et al., 1983). These receptors send their information to the central neural system, which, in turn, modulates the sympathetic and vagal efferent activities, thus modifying heart rate, cardiac contractility, total peripheral resistance and venous tone. Moreover, carotid chemoreceptors significantly contribute to the regulation of ventilation, especially during hypoxia.
1569-9048/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 4 - 5 6 8 7 ( 0 1 ) 0 0 3 3 5 - 8
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Knowledge of the quantitative relationship linking chemoreceptor activity to the composition of blood is extremely important, in order to build accurate mathematical models of the cardiorespiratory control system, and to summarize experimental data in the literature. However, despite the large amount of quantitative data on the carotid chemoreceptor response that have been gathered in the past decades (Marshall, 1994; Fitzgerald and Dehghani, 1982; Di Giulio et al., 1998; Daly, 1997; Lahiri and Delaney, 1975; Lahiri et al., 1982; Black et al., 1971; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Eyzaguirre et al., 1983), mathematical models are still rare. In the late seventies Yamamoto and Yamamoto (Yamamoto and Yamamoto, 1979) presented a mathematical model of the carotid chemoreflex, including a simple description of the afferent chemoreceptor response to hypercapnic stimuli. The authors were able to simulate various experimental results using a single set of parameters. In particular, the dynamic response of carotid chemoreceptors to CO2 pressure changes exhibits an adaptation phenomenon, i.e. sudden hypercapnia produces an overshoot in the carotid nerve discharge rate. However, their model does not incorporate several other important features of the chemoreceptor response, documented in the physiological literature. The main aspects, which deserve further mathematical analyses, can be summarized as follows: 1. The response to changes in CO2 tension, at constant PaO2, exhibits a lower threshold and is quite linear in the lower pressure range. However, this response becomes progressively flatter at increasing levels of PaCO2 (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Fitzgerald and Dehghani, 1982; Di Giulio et al., 1998). 2. Peripheral chemoreceptor activity depends greatly on O2 pressure in the arterial blood. At constant CO2 pressure, this dependence exhibits an exponential trend, being moderate during hyperoxia, but reaching a higher level during severe hypoxia (30– 40 mmHg) (Biscoe et al., 1970). 3. The sensitivity of carotid chemoreceptors to CO2 progressively increases with hypoxia
and vice versa. This implies the existence of a multiplicative interaction between the individual PaO2 and PaCO2 responses (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Eyzaguirre and Lewin, 1961). This interaction plays a protective role during asphyxia. 4. The positions of threshold and saturation of the chemoreceptor response to PaCO2 are modulated by hypoxia. Experimental data suggest that the threshold is significantly shifted to the left during severe hypoxia (Lahiri and Delaney, 1975). Likewise, the chemoreceptor response saturates at a much lower PaCO2 during hypoxia than during normoxia (Fitzgerald and Parks, 1971). The aim of the present work is to propose an extended mathematical model of the chemoreceptor response to changes in blood gas content, which embodies several findings not incorporated in Yamamoto’s model. The model aspires to represent an effective instrument for the interpretation of clinical and physiological data concerning the cardio-ventilatory control system. The paper is structured as follows. The main aspects of the model are first presented and justified. Then, the model is used to mimic experimental data reported in the physiological literature, both the static and dynamic patterns of discharge. Finally, the main results are critically commented. All equations and parameter values can be found in the Appendix A.
2. Qualitative model description A block diagram describing the main aspects of the model is shown in Fig. 1. In the following, the static response of the overall model is described first. Subsequently, the dynamic aspects are considered. The model assumes that the carotid body chemoreceptors are sensitive to CO2 concentration (CaCO2) and to oxygen desaturation (1− SaO2) in the arterial blood. However, since the input variables commonly manipulated in physiological experiments are arterial CO2 and O2 tensions (PaCO2 and PaO2, respectively), carbon dioxide concentration and oxygen saturation are
Fig. 1. Block diagram describing the main aspects incorporated in the model. Block 1 represents the O2 and CO2 dissociation curves in arterial blood, including the Bohr and Haldane effects. Block 2 represents the non-linear characteristic linking chemoreceptor activity to arterial oxygen saturation. Block 3 reproduces the linear dependence of chemoreceptor activity on arterial CO2 concentration, including a lower threshold, Ct. Blocks 4 and 5 describe the multiplicative O2 – CO2 interaction and the upper saturation for the static chemoreceptor activity, respectively. Block 6 is the rate-dependent component of chemoreceptor response to CO2, while block 7 denotes the upper saturation for the rate-dependent component. The static and dynamic component are summed in block 8, low-pass filtered to achieve the time constant of the response (block 9) then passed through a single-wave rectifier which eliminates negative values (block 10). All equations can be found in the Appendix A.
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computed from pressures using blood dissociation curves proposed by Spencer (Spencer et al., 1979) (Block 1 in Fig. 1 and Eqs. (1)– (5) in the Appendix A). These equations have been chosen since they include the Bohr and Haldane effects and can be easily inverted. The latter property can be of mathematical importance when the equations are used within large models of cardiorespiratory dynamics. Of course, alternative models for blood gas transport (see, for instance, Popel, 1989) can be used equally well without appreciable changes in the obtained results, provided the equations chosen include the Bohr and Haldane effects.
2.1. The static response According to the physiological literature (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Eyzaguirre and Lewin, 1961) the model assumes that the chemoreceptor static response is the result of a multiplicative interaction between the individual oxygen and CO2 responses (block 4 in Fig. 1 and Eq. (9) in Appendix A). The static response to CO2 is simply obtained through a linear relationship, i.e. arterial CO2 concentration is compared with a given threshold, Ct, and multiplied by a static gain (block 3 and Eq. (8)). In the first stage of our work we tried to simulate the static oxygen response through a linear relationship too, i.e. we assumed that chemoreceptor activity is a linear function of desaturation. The latter assumption, however, was unable to reproduce the experimental pattern of chemoreceptor activity versus PaO2 accurately, since chemoreceptor activity increased too much during deep hypoxia. Hence, we adopted a non-linear function of desaturation, which is quite linear during hyperoxia, normoxia and mild hypoxia, but progressively flattens when SaO2 is reduced below approximately 90% (block 2 and Eqs. (6) and (7)). Finally, as shown in Fig. 1, the two individual static responses described above are multiplied and passed through a further non-linear function with upper saturation (block 5 and Eq. (10)). Inclusion of this function is necessary to reproduce the attainment of a maximal level for
chemoreceptor activity observed (Fitzgerald and Parks, 1971; Fitzgerald and Dehghani, 1982; Di Giulio et al., 1998; Eyzaguirre and Lewin, 1961) during asphyxia. In the present model, according to the experimental observations by (Di Giulio et al., 1998), the upper saturation level is assumed to be constant, i.e. it does not depend on the oxygen level in blood. Contrasting results on this point are analyzed in the Section 4.
2.2. Dynamic response There is general agreement in the literature that the dynamic chemoreceptor response to a step CO2 pressure change exhibits a significant overshoot followed by adaptation to the steadystate level (Lahiri et al., 1982; Black et al., 1971; Torrance, 1977). This typical response implies the existence of a rate-dependent component of the CO2 response, i.e. a component that is sensitive to the time derivative of carbon dioxide concentration. By contrast, the response to a step change in oxygen pressure does not exhibit any appreciable adaptation (Black et al., 1971). The existence of a rate-dependent component of the CO2 response has been covered in the model by a first order high pass filter (block 6 and Eq. (11)). The response of this filter is sensitive to the rate of change of CO2 concentration up to a maximal frequency rate. The time derivative of CO2 concentration is computed, from knowledge of the input pressure waveform, by deriving the analytical equations of blood gas transport. In the model we assumed that the rate-dependent component is affected by carbonic anhydrase, whereas the static component represents a process independent of carbonic anhydrase. This is similar to the approach taken by Yamamoto and Yamamoto (1979). An important question is whether the rate-dependent component is affected by the oxygen level (as is the case of the static component) or not. Although differing opinions on this point can be found in the physiological literature, the majority of authors claim that the rate-dependent component is substantially unaffected by hypoxia or hyperoxia (see Cunningham et al., 1986, for a review). Moreover, according to data
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reported by Di Giulio et al. (1998), the rate-dependent component exhibits much higher saturation (about 45 impulses per sec) compared with the static component (about 20 impulses per sec). Accordingly, in the model the output of the highpass filter has been passed through a different saturation block, and then summed directly to the static response (blocks 7 and 8, and Eq. (12)). Finally, the overall response so obtained is passed through a first order low-pass filter with a time constant tpl and a single-wave rectifier (blocks 9 and 10, respectively, and Eqs. (13) and (14)). The presence of the low pass filter is necessary to mimic the progressive attainment of a steady state level. The rectifier simply disallows a negative response. All parameters in the model have been given to simulate the carotid chemoreceptor response in the cat (see Section 3), since most experimental data in the physiological literature have been acquired on this animal. Of course, different parameter values [for instance a different threshold, see (Mohan and Duffin, 1997)] might be required to simulate the response in humans.
Fig. 2. Static dependence of chemoreceptor activity on O2 arterial pressure during isocapnia. Two model simulation curves are shown, corresponding to two different isocapnic levels (32 mmHg, lower line and 48 mmHg upper line). Experimental points have been measured in the cat by (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Sampson and Hainsworth, 1972; Mulligan et al., 1981) in the same range of PaCO2.
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Table 1 Basal values of model parameters Dissociation cur6es in blood K1 =14.99 mmHg a1 =0.3836 a1 =0.03198 b1 =0.008275 mmHg−1 mmHg−1 a2 =1.819 K2 =194.4 mmHg a2 =0.05591 mmHg−1
b2 =0.03255 mmHg−1
Chemoreceptor response A =600 B =10.18 KCO2 =1 sec−1 Ct =0.36 L L−1 tph =3.5 sec tzh =600 sec Kdyn =45 sec−1
C1 =9 mM L−1 CaO2max =0.2 L L−1 C2 =86.11 mM L−1 Z= 0.0227 L mM−1 KO2 =200 Kstat =20 sec−1 tpl =3.5 sec
3. Results
3.1. Static response Fig. 2 shows the pattern of carotid chemoreceptor activity versus PaO2, predicted by the model in steady state conditions, using the parameter numerical values as in Table 1. Model results are then compared with data obtained by several investigators in the cat (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Sampson and Hainsworth, 1972; Mulligan et al., 1981). As it is clear from the figure, chemoreceptor activity increases moderately when passing from hyperoxia to normoxia, but it exhibits a disproportionate rise during severe hypoxia. At a PaO2 level as low as 25–30 mmHg, chemoreceptor activity is about 5-fold greater than during normoxia. The non-linear interaction between O2 and CO2 sensitivity is illustrated in Fig. 3. This figure plots the chemoreceptor response versus PaCO2 at different oxygen pressure levels, ranging from severe hypoxia (left upper panel) to hyperoxia (right lower panel). A good agreement can be found between model predictions and data by various authors in the cat (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971) at all pressure levels examined. In particular, this figure shows that the model is able to account for various experimental observations with sufficient accuracy. During normoxia and hyperoxia, the chemoreceptor response
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to CO2 is quite linear above a threshold. The sensitivity to CO2 is significantly increased by hypoxia (left upper panel) and reduced by hyperoxia (right lower panel). Furthermore, the threshold is shifted to the left (from about 21 to 16 mmHg) when passing from hyperoxia to hypoxia. Finally, at the lower O2 pressure levels, the receptor response to increasing PaCO2 tends to a plateau. The dependence of the CO2 pressure threshold on arterial oxygen pressure is shown in Fig. 4. The model ascribes this dependence to the Haldane effect (i.e. the upward shift of the CO2 dissociation curve during hypoxia (Guyton, 1986)). In fact, the real threshold in the model (i.e. parameter Ct in Eq. (8)) always remains constant despite hypoxia.
3.2. Dynamic response Fig. 5 shows the chemoreceptor response to a step increase in PaCO2 from 40 to about 90 mmHg (‘on response’) during hyperoxia. The upper panel displays the response obtained by using the basal values of model parameters (i.e. the values in Table 1). Chemoreceptors activity exhibits an overshoot, reaches a peak at about 2–3 sec after the stimulus, then declines down to the new steady-state level within 20 sec. The lower panel displays the same response after inhibiting the carbonic anhydrase of the carotid body with acetazolamide. In order to simulate this response, we assumed that the rate-dependent component of the chemoreceptor is suppressed by acetazolamide, (and so we set parameter tzh = 0 in Eq.
Fig. 3. Percent of maximum chemoreceptor response (MR%) vs. arterial CO2 pressure simulated with the model in steady state conditions at different levels of PaO2 ranging from severe hypoxia (left upper panel) to hyperoxia (right lower panel). Experimental points are taken from (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971) at comparable levels of PaO2. 100% in this figure denotes the maximal steady-state activity during asphyxia, which is the same as the static saturation Kstat in the model.
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Fig. 4. Apparent dependence of the threshold for chemoreceptor activity on arterial O2 pressure, according to the present model. This threshold represents the value of CO2 arterial pressure at which chemoreceptor activity begins to appear (see Fig. 3). At lower values of PaCO2, chemoreceptor activity is set to zero by the single-wave rectifier (block 10 in Fig. 1). It is worth-noting that the apparent dependence of the threshold on PaO2 is just the consequence of the Haldane effect (i.e. an upward shift of the CO2 dissociation curve during hypoxia) whereas the true threshold, Ct, remains unchanged.
(11). The temporal patterns in Fig. 5 agree with those obtained by (Black et al., 1971) in the cat. Fig. 6 investigates the effect of different maneuvers performed in rapid succession, comparing model results with analogous data by (Black et al., 1971). In particular, the upper panel displays the response to a step decrease in PaCO2 from 120 to 80 mmHg (‘off response’) during hyperoxia, followed by a step increase from 80 to 120 mmHg. It is evident that, during the off response, the discharge frequency falls to zero for a few seconds then progressively increases up to the new steady-state level. The lower panel shows the effect of a hypercapnic stimulus (from 40 to 120 mmHg) given first, then removed, and then applied again after a short time. The second response turns out much lower than the first. This effect, however, is weaker in the model than in the experimental data: in the model the separation between the two stimuli should be as low as 10 sec to evoke a conspicuous difference between the two responses, whereas the time separation was as high as 15 sec in Black’s data. Finally, Fig. 7 examines the interaction between O2 and CO2 in dynamic conditions. In these simulations we slightly reduced the rate dependent component (tzh = 350 instead of 600 sec) to provide better reproduction of experimental results. In fact, in this particular fiber of Black’s experiment (Black
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et al., 1971), the rate-dependent response appears a little lower than in the previous cases. The upper panel illustrates the effect of substituting a hypercapnic stimulus with a hypoxic stimulus which produces almost the same steady-state response (see figure legend for more details). The two stimuli are almost indistinguishable in steady-state conditions, whereas the rate-dependent CO2 component dominates the transient response. The lower panel shows the on and off response to CO2 stimuli, during maintained hypoxia.
Fig. 5. Waveform of chemoreceptor activity simulated with the model in response to a sudden increase in PaCO2 from 40 to 93 mmHg (on response) during hyperoxia (PaO2 =300 mmHg). The results are compared with data from Fig. 6 in (Black et al., 1971) (20% CO2 in air). The upper panel was obtained using all parameter values as in Table 1. The lower panel refers to the response after inhibiting the carbonic anhydrase of the carotid body with acetazolamide. The latter case was simulated by abolishing the rate-dependent component in the model (i.e. setting tzh =0 in Eq. (11)).
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may be useful in the analysis of overall cardio-respiratory regulation, where chemoreceptors represent a single afferent branch within a complex multi-input, multi-feedback, multi-output control
Fig. 6. Effect of different stimuli given in rapid succession. The upper panel shows the waveform of chemoreceptor activity following a sudden decrease in the hypercapnic stimulus (from 120 to 80 mmHg between t = 24 and 26 sec) followed by a sudden increase (from 80 to 120 mmHg between 48 and 49 sec) during hyperoxia (PaO2 =180 mmHg). The lower panel displays the model response to a hypercapnic stimulus (from 40 to 120 mmHg) given, then removed, then applied again after a few seconds during hyperoxia (PaO2 = 300 mmHg). Model results are compared with data from Fig. 3 in (Black et al., 1971). The levels of hyperoxia were chosen to have approximately the same basal activity in the model and in the experimental data. All model parameters have the same value as in Table 1. It is worth noting that, in the lower panel, a smaller delay between the two stimuli was used in the simulations (about 10 sec) compared with real data (15 sec).
4. Discussion The aim of the present work was to develop a simple empirical model of the carotid body chemoreceptor, which can help in the study of the cardio-respiratory control system. This model
Fig. 7. Interaction between CO2 and O2 in dynamic conditions. Upper panel: response of the model when alternating two stimuli (hypercapnia and hypoxia) which provides approximately the same basal level of chemoreceptor activity. PaCO2 was lowered from 120 to 40 mmHg between t =16 and 18 sec, while PaO2 was simultaneously decreased from 120 to 50 mmHg. Subsequently, PaCO2 was raised from 40 to 120 mmHg again between t = 53 and 55 sec, while PaO2 was simultaneously returned from 50 to 120 mmHg. Lower panel: effect of a rapid +10 mmHg increase in PaCO2 (from 24 to 34 mmHg) pferformed between t = 22 and 23 sec, followed by a step decrease (from 34 to 24 mmHg) between t =38 and 39 sec during hypoxia (PaO2 =50 mmHg). The initial level of PaCO2 (24 mmHg) was chosen to have the same baseline activity in the model and in the experimental points. Data points are from Fig. 5 in (Black et al., 1971). It is worth noting that, in these figures, we used a reduced value for the rate-dependent component (tzh =350 sec) to better simulate the response of this single experimental fiber.
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system. In this regard, the mathematical equations proposed here can be used in large-scale models of the entire cardio-respiratory control to drive ventilation (in synergy with medullary chemoreceptors) and to modulate vagal and sympathetic activities to heart and vessels. Of course, the model presented here is merely a functional one, i.e. its equations have been selected to mimic the observed input– output relationships, and the structure of the model may not accurately reflect the underlying physiological transduction mechanisms responsible for chemoreceptor behavior. Our model shares some important aspects with the model proposed by (Yamamoto and Yamamoto, 1979) more than 20 years ago. In fact, in both models the chemoreceptor response to CO2 includes a rate-dependent component (affected by carbonic anhydrase), a component proportional to the instant value of CO2 (assumed unaffected by carbonic anhydrase) and a single wave rectifier. However, the present model includes many new aspects too, which significantly expand its possible applications: in particular, we incorporated equations for gas transport in blood, including Bohr and Haldane effects, the chemoreceptor response to oxygen, the multiplicative O2 – CO2 interaction, and an upper saturation for both the static and the rate-dependent components. Thanks to these new aspects, the present model can simulate chemoreceptor response in several conditions not covered by the former model, such as asphyxia, or hypocapnic hypoxia. The static characteristics of the model include several different non-linearities. All of them are necessary to reproduce the well-known experimental data. A first non-linearity concerns the relationship linking chemoreceptor activity to SaO2 during isocapnia. Our analysis, based on physiological data by various authors (Lahiri and Delaney, 1975; Fitzgerald and Parks, 1971; Biscoe et al., 1970; Sampson and Hainsworth, 1972; Mulligan et al., 1981) confirms the previous observation by Daly (1997), p. 69), according to whom the relationship between chemoreceptor activity and percentage saturation deviates from linearity at a level of approximately 90% saturation.
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By contrast, the response to CO2 during normoxia can be described reasonably well using a simple linear relationship with a constant threshold. It is remarkable that the apparent decrease in the threshold during hypoxia (Figs. 3 and 4) is ascribed by the model to the Haldane effect (i.e. an upward shift in blood CO2 dissociation curve during hypoxia). The idea that a shift in blood dissociation curves can explain some changes in the chemoreceptor response to PaCO2 has previously been suggested by others (Di Giulio et al., 1998; Lahiri and Delaney, 1975). A further non-linearity is an upper saturation for the static response (block 5 in Fig. 1). In the present study we assumed that this upper saturation during hypercapnia is about 20 impulses per sec, independently of the oxygen level in blood. This choice agrees with the experimental results by (Di Giulio et al., 1998) who observed that the maximal response to PaCO2 stimuli is the same during hypoxia and hyperoxia. A different point of view, however, was proffered by (Fitzgerald and Dehghani, 1982): using statistical analyses, these authors reached the conclusion that the upper saturation level of the chemoreceptor response to hypercapnia is not independent of oxygen, but can be increased by lowering PaO2. However, as pointed out by Di Giulio et al. (1998), the experiments by Fitzgerald and Dehghani were limited to a maximum PaCO2 as high as 75–80 mmHg; at this level of hypercapnia, a plateau might not have actually been reached during hyperoxia and normoxia, making their conclusion doubtful. When building the model, we used particular care in the reproduction of the dynamic chemoreceptor response. This is very important not only to arrive at a correct reproduction of transient phenomena [such as the on or off-responses following acute perturbations in blood gas content (Black et al., 1971)] but also since the dynamical properties of a non-linear system may affect the average values as well (Cunningham et al., 1986). The most important feature of the chemoreceptor dynamical response is its strong dependence on the CO2 rate of change. There is a general consensus in the physiological literature that a step increase in PaCO2 causes an overshoot in
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chemoreceptor activity followed, within 10– 20 sec, by an adaptation toward the new steady-state level (Marshall, 1994; Lahiri et al., 1982; Black et al., 1971; Torrance, 1977; Cunningham et al., 1986). We made use of a first-order high-pass filter to mimic the rate-dependent component of the CO2 response. This simple and rational choice is able to reproduce the transient behavior quite well during both on and off-responses with the assignment of only two parameters. Nevertheless, two aspects of the dynamical response are critical and deserve attention. First, as shown in the block diagram of Fig. 1, the model assumes that the rate-dependent component of the chemoreceptor response to CO2 (i.e. the output of the high-pass filter) is not affected by the oxygen level in blood. Several experimental data support this assumption, although there is no complete agreement in the literature. First, the amplitude of the naturally occurring oscillations in chemoreceptor activity, caused by respiration, does not increase during hypoxia (Band and Wolff, 1978). Moreover, a strong dynamic response is still evident when a sudden increase in CO2 is performed with 100% O2 (Black et al., 1971; Torrance, 1977), that is, hyperoxia does not suppress the dynamic response whereas it significantly abates the steadystate level. Conversely, (Lahiri et al., 1982) observed that hypoxia increases both the overshoot and the steady state response to hypercapnia. However, looking at Fig. 5 in their paper, one can observe that the transient response is just a little higher in hypoxia (from 10 to 34 impulses per sec) compared with hyperoxia (from 0 to 18 impulses per sec), corresponding to a 33% increase. By contrast, the final steady state level is dramatically increased by hypoxia compared with hyperoxia (18 vs. 5 impulses per sec, that is a 260% increase). Hence, in the data by Lahiri et al. too, it is quite reasonable to assume that the rate sensitivity to CO2 is less dependent on oxygen than is the steady state CO2 response. A second important assumption concerns the use of two different saturation levels for the static and rate-dependent components: we assumed that the rate-dependent component exhibits much higher saturation (45 impulses per sec) than the static component (20 impulses per sec). There are
several justifications for this choice. First, the use of a single saturation does not allow the experimental data in Figs. 5 and 6 to be reproduced with sufficient accuracy (unpublished simulations). Second, the values used in this work agree with those measured by (Di Giulio et al., 1998) in the cat. Finally, the use of different saturation levels may reflect the existence of different processes, the first dependent on carbonic anhydrase, the second independent of it. In conclusion, the present model significantly extends the validity of the model by (Yamamoto and Yamamoto, 1979) and allows several physiological results to be summarized into a single theoretical framework. In future work the model may be further extended, including a few additional features not considered here: among them, the dependence of chemoreceptor activity on the potassium level during exercise, a distinction between the response to CO2 and H+, or the effect of pressure or flow changes at the chemoreceptor level (Cunningham et al., 1986; Marshall, 1994).
Appendix A We assumed that peripheral chemoreceptors respond to changes in arterial carbon dioxide concentration (CaCO2) and arterial oxygen saturation level (SaO2). SaO2 and CaCO2 are computed, as functions of arterial O2 and CO2 partial pressures (which are inputs for the model), using the expressions for dissociation curves in blood proposed by Spencer et al (Spencer et al., 1979). These equations offer the advantage of taking the Bohr and Haldane effects into account. We have CaO2 = ZC1 FaO2 = PaO2 SaO2 =
1 FaO1/a 2 1/a1 1+ FaO2
1+ i1PaCO2 K1(1+h1PaCO2)
CaO2 CaO2max
CaCO2 = ZC2
2 FaCO1/a 2 1/a2 1+ FaCO2
(1) (2) (3) (4)
M. Ursino, E. Magosso / Respiratory Physiology & Neurobiology 130 (2002) 99–110
FaCO2 =PaCO2
1+i2PaO2 K2(1+ h2PaO2)
(5)
where PaO2 and PaCO2 represent the oxygen and carbon dioxide tensions in arterial blood, FaCO2 and FaO2 are mathematical functions, which allow for the representation of the Bohr and Haldane effects, and Z is a conversion factor which transforms O2 concentration from mmol L − 1 to L L − 1. a1, C1, a1, b1, K1 and a2, C2, a2, b2, K2 are constant parameters, which were determined by Spencer et al. (Spencer et al., 1979) to fit experimental data. Finally, CaO2max is arterial oxygen concentration when blood is completely saturated. The output of the model is obtained as the sum of two components: a steady state component and a rate dependent (dynamic) component. The former allows the relationships between chemoreceptor discharge frequency and changes in arterial PO2 and PCO2 to be described in static conditions. The main features of the steady-state response include a non-linear dependence on SaO2, a linear dependence on CaCO2 above a given threshold, below which chemoreceptor activity is set to zero, a multiplicative interaction between the O2 and CO2 responses, and the presence of saturation. Hence the static response is described by the following equations:
n
xO2 =A(1 −SaO2)+B O2 = KO2
(6)
−xO2 1−exp KO2
(7)
CO2 = KCO2(CaCO2 −Ct)
(8)
F = O2CO2
(9)
n
−F Á ÃKstat 1− exp Kstat fcstat = Í Ã 0 Ä
CaCO2 \Ct
for the static response. The ability of the model to reproduce chemoreceptor steady-state responses to independent changes in arterial PO2 and PCO2 results from Eqs. (6)–(10) combined with equations for dissociation curves (Eqs. (1)–(5)). In particular, fcstat, plotted against PaO2 at different constant levels of PaCO2, displays an exponential trend; fcstat increases quite linearly with PaCO2, at normal or elevated PaO2 values, while at lower PaO2, fcstat tends to flatten out as PaCO2 increases. The rate-dependent component of the chemoreceptor response, 8CO2dyn, is obtained by means of a high-pass filter, with a real pole and a real zero. We can write ~ph
dCO2dyn dCaCO2 + CO2dyn = ~zh dt dt
(10) fcstat is the chemoreceptor frequency discharge in static conditions, 8O2 and 8CO2 represent the contributions of oxygen and carbon dioxide to the static response, F is the result of their multiplicative interaction, Ct represents the CO2 threshold for the response and A, B, KO2, KCO2, Kstat are constant parameters tuned to fit experimental data. In particular, Kstat is the upper saturation
(11)
where tph and tzh are the time constants of the pole and the zero in the high-pass transfer function, and the time derivative of CO2 arterial concentration is computed via derivation of Eqs. (1)–(5). Finally, the static and rate-dependent components are passed through two distinct saturation blocks and summed. The signal so-obtained is low-pass filtered and sent to a single-wave rectifier, which cuts out negative values. Hence, the following equations hold:
n n
¯ c = Kstat 1− exp
−F Kstat
+ Kdyn 1− exp ~pl
dc + c = ¯c dt
fc =
CaCO2 BCt
109
!
c 0
c \ 0 c B 0
− CO2dyn Kdyn
(12) (13)
(14)
Kstat and Kdyn are the different upper saturation levels for the static and for the rate-dependent components, respectively, and 8c is the output of the low-pass filter. fc is chemoreceptor activity, while tpl is the time constant of the real pole in the low-pass filter. Values assigned to the three time constants in Eqs. (11) and (13) reproduce well the transient chemoreceptor response to a square CO2 pulse.
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Finally, it is worth noting that in static conditions (i.e. then dCaCO2/dt =0, d8CO2dyn/dt =0, d8c/dt =0) fc in Eq. (14) coincides with fcstat.
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