## Abstract

Baroreceptor reflex curves are usually analyzed using a symmetric four-parameter function. We wished to ascertain the validity of assuming symmetry in the baroreflex curve and also of constraining the curves to pass through the resting blood pressure and heart rate (HR) values. Therefore, we have investigated the suitability of a new five-parameter asymmetric logistic model for analysis of baroreflex curves from rabbits and dogs. The five-parameter model is an extension of the usual four-parameter model and reduces to that model when the fitted data are symmetrical. Using 30 data sets of blood pressure versus renal sympathetic nerve activity (RSNA) and HR from six conscious rabbits, we compared the five-parameter curves with the four-parameter model. We also tested the effect of forcing these baroreflex curves through the resting point. We found that the five-parameter model reduced the unexplained variation and gave small but important improvements to the estimates of plateaus for RSNA and HR and the HR gain. Although forcing the HR curves through the resting values had little effect, this procedure, when applied to RSNA, produced a worse curve fit by increasing the unexplained variation with alteration to most of the estimated curve parameters. The mean arterial pressure-HR baroreflex relationship from six conscious dogs was also analyzed and showed clear evidence of systematic asymmetry. We conclude that the asymmetric model is a valuable extension to the symmetric logistic model when examining baroreceptor reflexes, giving improved estimates of the parameters and a new approach to examining the mechanisms contributing to baroreflex curve asymmetry. Furthermore, forcing the curves through the resting value is a statistically questionable practice when analyzing RSNA, because it affects the parameter estimates.

- sigmoidal curve
- nonsymmetrical fitting
- blood pressure
- heart rate
- renal sympathetic nerve activity
- rabbits
- dogs

the first method that allowed quantification of baroreflex properties in humans was the ramp method, which involved rapid changes to blood pressure and was originally developed by Sleight and colleagues (30). Pressor drugs were injected in a bolus or rapid infusion to induce a relatively linear “ramp” rise in blood pressure. In the original technique, the systolic blood pressure of each pulse was plotted against the heart period of the succeeding cardiac beat, which gave a straight line relationship. This was usually parameterized by the resting value and the slope, which is also called the baroreflex sensitivity. The method has been modified for rats to allow for the very much higher heart rate (HR) (32). Apart from not assessing the contribution from the cardiac sympathetic nerves, the method assumes that the blood pressure-HR or blood pressure-heart period relationship is linear. However this is only true over a narrow range of blood pressures. If the blood pressure range examined is widened there are obviously limits to HR responses, such that the relationship more closely approximates an S-shaped or sigmoidal curve. Koch (18) established in his 1931 book that the relationship between carotid sinus pressure and systemic arterial pressure exhibits a negative sigmoidal relationship in several species.

The “steady-state” sigmoidal method was developed as an alternative to the original ramp method (21). With this technique, alterations in blood pressure from resting are maintained for ∼30 s and are related to the mean tachycardia and bradycardia responses over the last 10 s of that period. In addition, a wide range of blood pressure steps are examined. This method has advantages over linear fitting in that it gives an estimate of the plateau values at the extremes of blood pressure and assesses gain independently of resting pressure. It has become one the major analytic tools in the study of baroreceptor reflexes. The steady-state method has been used for studying cardiac baroreceptor reflex interactions in normotensive and hypertensive humans, rabbits, rats, and, most recently, in dogs (1, 13,19-22, 27, 38). The original method involved fitting two hyperbolas (21) to the S-shaped curve and thus allowed the upper and lower curvature to be calculated quite separately. An alternative to the double hyperbola was the logistic equation that had been first applied to carotid-sinus-blood pressure curves by Kent and colleagues (16). This logistic method was then applied to the blood pressure-heart period relationship (24, 37) and later to HR and renal sympathetic nerve activity (RSNA) baroreflexes in rabbits (4) as well as HR baroreflexes in conscious rats (13). Although the single logistic model is now the most widely used method, it only provides a single estimation of the baroreflex gain.

One of the major assumptions about this model is that the curvature at all points of the curve is constant; thus even when dual effectors are contributing, they are assumed to operate symmetrically. This assumption may seriously limit the investigator’s ability to examine mechanisms that may influence the baroreflex curve in a nonsymmetrical manner, namely those that affect only the upper or lower part of the curve. For example, certain cardiac baroreceptors have high pressure thresholds and only contribute to reflex bradycardia when arterial pressure is markedly or very rapidly elevated (6, 25). Dorward and colleagues (4) showed that there was a nonadditive interaction between arterial and cardiac baroreceptors influencing the renal sympathetic reflex and that the inhibitory effect of cardiac baroreceptors increased progressively with reductions in arterial pressure. Thus there is much experimental evidence to suggest that underlying mechanisms contributing to the baroreflex may not be uniform over the entire blood pressure spectrum. An attempt to examine whether the assumption of symmetry of baroreceptor-HR curves in humans and rabbits was valid was made by Kingwell and colleagues (17), who used a compound logistic method where two separate logistic curves were applied, one to the blood pressures greater than resting and another to the blood pressures less than resting. The upper half of one curve is combined with the lower half of the other, which results in a curve that passes discontinuously through resting, with two curvature parameters and two plateau values. The method uses “mirrored data” to ensure that sensible plateaus are obtained (17). This method has the advantage of giving independent assessments of the baroreflex gain for tachycardia and bradycardia responses, but is of limited value when resting is near one of the plateaus, such as in dogs (38) or hypertensive rats (11). In addition, the model produces two range parameters as well as two gain parameters and also replaces the blood pressure value at which 50% of the range is attained (BP_{50}) with mean resting pressure, making it difficult to compare parameters from one situation to another where the resting value changes but not necessarily the properties of the baroreflex. Other than this study, we have not been able to find any other attempts to determine the asymmetry or otherwise of baroreflex sigmoidal curves for HR or other variables such as sympathetic nerve responses. This limits the ability to investigate any baroreflex mechanisms that specifically affect only the gain of the upper or lower part of the baroreflex curve. Given the increasing application of nonlinear curve fitting for baroreflex studies, a simple method for determining asymmetry or otherwise needs to be developed.

The other condition that is often applied to baroreflex curves is to force the curve through the resting points on the assumption that the resting point ought to fit exactly on the curve. Resting point constraint has been applied in a number of studies (4, 13, 17, 23, 35) but not in others (9, 33). This may appear to be a logical constraint, because perturbations (drug injections, etc.) that are used to evoke the baroreflex start from this point. However, because it is necessary to add together the responses to pressor and depressor stimuli, each with possibly differing resting values, and because there may be an abrupt change in slope at the resting point, fitting a sigmoidal curve that must go through the resting point may be less accurate, resulting in suboptimal estimates of the curvature and midpoint. Head and McCarty (13) applied two normalization procedures and found that the method of deltas (normalization by change from control) was the model that gave the lowest error sum of squares. This effectively meant that absolute blood pressure and HR values were not used, but only individual changes in these parameters were used. This eliminated the variance produced by changes in basal values during the course of the experiment, which seems logical in view of the ability of baroreceptors to rapidly reset (2). However, to date there has been no systematic examination as to the effect of resting point constraint in any baroreflex studies.

The purpose of the present study was to test the assumption of constancy of curvature and resting point constraint in two commonly used baroreflexes, namely the HR baroreflex and RSNA baroreflex. Although there are a number of five-parameter asymmetrical logistic equations that could be used, such as that supplied with Sigmaplot (SPSS, Chicago IL, formerly Jandel Scientific Software; seediscussion,*equation 17*, and
), we have developed a new method that permits a more flexible transition of curvature (over a wide or narrow blood pressure range), thereby permitting its use in a wide range of nonsymmetrical circumstances. In addition, this equation represents a logical extension of the symmetrical model, because it is a modification to the curvature parameter alone and reduces to the four-parameter model when the curvature is constant, i.e., when the two curvature parameters are not different from one another, they can simply be averaged and the result is the four-parameter model. Our approach was to compare this new equation with the previously published four-parameter model and to examine whether the forcing of the curve through resting exacted any statistical penalty by fitting both constrained and nonconstrained four- and five-parameter equations. The main analysis was performed on HR and RSNA baroreceptor reflex curves from conscious rabbits produced using a slow ramp increase or decrease in blood pressure. With the use of this method, the HR responses are mainly due to changes in the activity of the cardiac vagus, which is appropriate because HR responses in conscious rabbits have only a small contribution from the cardiac sympathetic nerves (5, 36). We have also included a series of HR baroreflex curves derived using the steady-state method in conscious dogs (38). These curves are particularly difficult to analyze because the resting point is very close to the bradycardia plateau. They serve in the current context to provide a good example of the versatility and usefulness of this new method.

## METHODS

### Physiological Methods, Operations, and Protocols for Rabbits

Data for analysis were taken from experiments performed using conscious rabbits at the Baker Medical Research Institute as part of another study (29). The experiments were performed in conscious male and female rabbits crossbred from Baker Medical Research Institute stock weighing 2.6–2.9 kg in accordance with the statement on animal experimentation by the National Health and Medical Research Council of Australia. All rabbits were housed in an animal room under controlled temperature, humidity, and constant dark-light cycle. Separate preliminary surgical operations were performed on all animals before each experimental series to implant a renal nerve electrode (4) and an indwelling fourth ventricular catheter (12). The latter was used as part of another protocol involving central drug treatment. At least 4–5 days of recovery were allowed after the final operation before the experiment.

*Cardiovascular measurements*. On the day of the experiment, the rabbit was placed in a standard rabbit box. The electrode connector was retrieved from under the skin in the flank under local anesthesia with 1% prilocaine (Citanest, Astra Pharmaceuticals). The central ear artery was cannulated transcutaneously with a 22-gauge, 25 mm Teflon catheter (Jelco, Critikon, Italy) under local anesthesia. The catheter was then connected to a Statham P23Dc pressure transducer for continuous measurements of arterial pressure and HR, and the animal was allowed a 1-h recovery period before the experiment was commenced to permit stabilization of the cardiovascular parameters. Cardiovascular parameters were monitored on a Grass polygraph (model 7D; Grass Instruments). Mean arterial pressure (MAP), HR, and RSNA were digitized at 300 Hz using a Metrabyte DAS8 analog-to-digital card. HR was detected from the arterial pulse wave using an HR meter. All parameters were saved onto an IBM-compatible computer hard disk as 2-s averages for later offline analysis.

*MAP-RSNA baroreflex relationship (ramp technique)*. The renal sympathetic baroreflex was derived from slow ramp rises and falls in MAP by intravenous infusions of phenylephrine hydrochloride (Sigma, 0.5 mg/ml) and sodium nitroprusside (Fluka Chemicals, 1.0 mg/ml), respectively. Injections lasted 1–2 min, and the rate of change in MAP was controlled between 1 and 2 mmHg/s. MAP, RSNA, and HR were averaged over 2-s intervals and fitted into a sigmoid logistic function to produce MAP-RSNA and MAP-HR curves.

We applied both the four-parameter model (BARO4) and the new five-parameter model (BARO5) to 29 data sets of MAP-HR data and 30 MAP-RSNA data sets from six animals. Both models were applied with and without resting point constraint.

### Physiological Methods, Operations, and Protocols for Dogs

Baroreflex data were taken for analysis from experiments performed using six conscious male greyhound dogs at the Baker Medical Research Institute as part of another study (38). Details of the preparatory surgical procedures were published in a previous report (38). Briefly, the dogs were anesthetized with halothane and nitrous oxide and, through a flank incision, aortic catheters and thermistor and vena caval catheters were implanted, a right atrial catheter was inserted, and Doppler flow probes were placed around the mesenteric and renal arteries. For the present experiments, only the arterial and venous catheters were required.

*Cardiovascular measurements*. On the day of the experiment the conscious, unrestrained dog lay recumbent on a padded table in a quiet laboratory to which the animal was well accustomed. Phasic aortic blood pressure was measured with Statham P23Dc transducers (Oxford, CA), and phasic and mean pressures were recorded on a Neomedix Systems physiological recorder (Neotrace model 800ZF). HR was triggered from the aortic pressure signal using a Baker Institute tachograph and recorded on the Neomedix Systems recorder.

*MAP-HR baroreflex relationship (steady-state technique)*. In dogs, 12 steady-state baroreflex curves were constructed by injecting alternating infusions of phenylephrine hydrochloride (20–400 μg) or sodium nitroprusside (0.25–5 mg) intravenously to evoke changes in MAP. The increases in pressure were over the range 2–30 mmHg, whereas the reductions in pressure were over the range 5–60 mmHg. Each pressure change was maintained for ∼30–60s by an initial bolus injection followed by a slow infusion of the drug, and the mean values over the final 10–15s were taken as the steady-state MAP and HR values. The data were then fitted to the original model (BARO4) with resting constraint as previously published (38) and the new five-parameter model (BARO5) without resting point constraint.

### Mathematical Modeling

BARO4 is a four-parameter logistic model previously published (4, 13), where the fitted values are normalized by using deltas. The equation of the model is
Equation 1 where P1 is the second plateau, the value of the response at maximum blood pressure. P2 is the range of response. P1+P2 gives the first plateau. P3 is a curvature parameter. The greater the curvature, the smaller the blood pressure range over which the curve operates. A family of curves with different P3 values is shown in Fig 1. Because there is only one curvature parameter, the curvature is constant over the MAP range and therefore the curve is symmetrical. P4 corresponds to BP_{50} (Fig 1). Each ramp measurement is made after a period of rest. Because the final sigmoid curve is a combination of a ramp increase and decrease in blood pressure, an adjustment was made for any difference in resting blood pressures to avoid discontinuity at resting. This involved transforming all blood pressure values by an addition of half the difference of the two resting values according to the following equation
Equation 2 where
and
are the mean resting blood pressures before the two ramps, and*x*
_{l}
_{i}is the individual blood pressure value for ramp 1 of 2.

BARO5 is a five-parameter model extended from the four-parameter model by addition of a second curvature parameter. The second curvature parameter enables the curves to be nonsymmetrical. The same normalization was first carried out as in equation (2). The equation is
Equation 3where
Equation 4defines a logistic weighting function varying smoothly between 0 and 1, centered about the BP_{50}, and so long as P3 and P5 are of the same sign, the mean curvature of*f* is given by
Equation 5With this formula there is a smooth transition from a curvature mainly due to P3 to one mainly influenced by P5 over the linear middle part of the curve. BecauseC̅
_{f} is calculated as the reciprocal of the mean of the reciprocals of P3 and P5, this tends to bias toward the smaller of the two curvatures, which prevents the total curve from being overly influenced by an abnormally high curvature estimate. Thus there are two independent curvatures for the function, and their relative weights vary across the*x*-axis, which in this study is blood pressure.

The parameters of BARO5 have the same meanings as in BARO4 except for the addition of the extra curvature parameter, and BARO5 converges to BARO4 as P3 approaches P5. This can be demonstrated by making these two parameters equal in *equation 3*. The asymmetric nature of the five-parameter model is shown for a family of curves in Fig.2.

The mean curvature for the curve that is the value of curvature at BP_{50} is given by
Equation 6This model permits asymmetry to appear in either direction i.e., high or low curvature at both ends of the MAP range.

Gain, which is the derivative of the curve, is computed from
Equation 7where*g* and*h* are functions of*x*
Equation 8and
Equation 9
*g*′ and *h*′ are their derivatives
Equation 10and
Equation 11and the derivative of *f* is
Equation 12Computing the point of maximum gain (G_{max}), however, requires the second derivative, which contains increasing numbers of terms. We calculated the G_{max} by using the bisection method rather than directly. In the case of a symmetric curve, G_{max} is the same as
, but in the above curve it is displaced along the*x*-axis a small distance in the direction of the curvature of larger magnitude. G_{max} is therefore larger than
.

*Curve-fitting method*. All curves were fitted using a modified Levenberg-Marquardt method (26, 28). The requirement for the mean resting values to be included on the curve is met by the following procedure. The curves were repeatedly estimated with one parameter held constant, and then, at completion of an estimation, that parameter was adjusted so that the mean resting point was on the curve. This process was carried out until there was negligible change in the weighted sum of squares after consecutive estimations and adjustments. In alternating cycles, P1 and then P3 (and P5 if applicable) were held constant. This process serves to almost entirely prevent the premature termination of the calculation. Figure3 shows two example data sets fitted to both models with and without resting constraint. The software used was custom written for IBM-compatible computers and is available from the authors. Alternatively, information is supplied in the
to enable readers to use a general curve-fitting routine, such as that found in Sigmaplot (SPSS).

### Statistical Methods

For the main study we analyzed data for 59 (29 HR and 30 RSNA) response curves representing the control periods in two series of ramped baroreflex experiments from conscious rabbits. These curves were analyzed four ways with each of two models (4 or 5 parameters) and with or without constraint.

During curve fitting, an ANOVA table was produced, and, for*n* points fitted, includes degrees of freedom due to the model (3 for BARO4, 4 for BARO5), residual degrees of freedom (df; *n* − 4 or*n* − 5), and the sums of squares due to the model and residuals, respectively. From these, an*F* ratio can be formed (model/residual) for which a probability of <0.05 is taken as evidence that the model explained more of the variation.

To address the question of whether the fifth parameter made a difference to the fit of individual curves, we performed a χ^{2} test using information from the same ANOVA. Another *F* statistic for each data set was calculated for the increase in the regression sums of squares when an additional parameter was added (7, 34). This was calculated for each data set from the ANOVA table generated during curve fitting, using the regression sum of squares for the five-parameter curve and the four-parameter curve (SSRBARO5 and SSRBARO4 respectively) and the mean residual sum of squares for the five-parameter curve (MSEBARO5)
Equation 13with d*f* = (1,d*f*
_{Baro5}). These *F* values were then assigned to the categories significant and not significant according to the expected *F* value at the 0.05 significance level.

The categorized *F* values themselves, each with 1 d*f,* then formed a distribution that was further tested for conformation to the expected proportion. This statistic is χ^{2} distributed as
Equation 14with degrees of freedom being number of categories − 1 (34). To minimize type 1 error, we incorporated Yates correction for continuity (34). Formally, we proposed the null hypothesis (H0) “the number of significant *F* values does not differ at a 0.05 significance level from the number of apparently significant values due to chance alone,” and we rejected H0 (at*P* < 0.05) if the calculated value exceeded the appropriate critical value. An exactly analogous argument pertained to the question of whether forcing the curves through resting makes a difference to the variation explained by either model.

Altogether we computed four probabilities for each RSNA and HR. For both the four-parameter and the five-parameter models, we were able to assess the effect of omitting the forcing through resting constraint. For curves forced through resting, and with that constraint omitted, we were able to assess the effect of allowing asymmetry to be expressed.

To address the question of whether the apparent asymmetries are systematic, we calculated an index of asymmetry for the five-parameter model
Equation 15and compared this for all curves. This was tested with a simple*t*-test with H0 “the mean asymmetry index was not different from zero.” If the asymmetry index was distributed around a mean of zero then we would have to conclude that asymmetries detected were properties of the individual measurements and not a property of the group.

Significance of effects of the different models on the curve parameters was assessed by two-way ANOVA with orthogonal contrasts for each of the parameters, comparing BARO4 and BARO5 for each of the curves forced and unforced through resting (7, 31). The curvature parameter P3 for BARO4 was compared with the mean curvature for the BARO5 model given by (P3 + P5)/2. We also included in this set of comparisons,the first plateau (P2 + P1) and the gain at BP_{50}(
).

The four-parameter model with resting constraint is the most “constrained” model and was used as a basis for comparison in assessment of both resting constraint and asymmetry.

## RESULTS

### Renal Sympathetic Nerve Baroreflexes in Rabbits

*Asymmetry*. The RSNA-MAP curves were sigmoidal with clear upper and lower plateaus. The four-parameter model that was forced to pass through resting on average accounted for all but 5% of the variation (*r*
^{2} = 0.950, range 0.868–0.986). Refitting the data with a five-parameter model reduced the residual variation in only three curves. This was not different from that expected merely by chance (*P* > 0.05). With four-parameter curves not forced through resting, refitting with the five-parameter model explained more variation in five data sets (*P* = 0.012).

The test for systematic asymmetry was again performed, using the asymmetry index and the Student’s*t*-test (Table1). The computed values were not different from zero (*P* = 0.754 and*P* = 0.223, respectively). Thus in the sympathetic nerve activity data there was no systematic asymmetry evident.

*Resting constraint*. Starting with the four-parameter curve, the effect of not constraining the curve through resting was to reduce the unexplained variation in 15 of 30 curves (*P* < 0.005). For the five-parameter model, 14 of 30 curves showed a reduction in the unexplained variation (*P* < 0.005). The use of both procedures (5 parameter and no resting constraint) improved the fit in 23 of 30 curves (*P* < 0.001). Of these 23 curves, 12 were improved by not forcing the fit through resting, 2 were improved due to the extra curvature parameter, 4 by either procedure, and 5 by the combination.

*Effects of model on estimates of parameters*. The major effect of the different models on the estimates of the curve parameters was the resting point constraint. This procedure affected RSNA range due to differences in both the upper and lower plateaus as well as the mean curvature and gain (*P* < 0.05, Table2, see Fig. 5). By contrast, using the model with the extra curvature parameter (BARO5) altered the estimate of the first plateau (*P* < 0.05) but made little difference to the other curve parameters.

### Heart Rate Baroreflexes in Rabbits

*Asymmetry*. In all 29 cases, four-parameter HR curves forced through resting were sigmoidal with clear upper and lower plateaus and were similar to those described previously (4; Fig. 4). This model on average accounted for 97% of the variation (*r*
^{2} = 0.973, range 0.938–0.991). Refitting the curves with five parameters reduced the residual variation (i.e., better fit) in 10 curves but not in 19. The number of improved curves (10 of 29) was, however, greater than what would have been expected by chance alone (1.45 of 29). Thus the five-parameter model can be considered to be an improvement (*P* < 0.005; Table3). In addition, given that the five-parameter model allows for asymmetry, these data indicate that just over one-third of the data sets showed evidence of asymmetry. For curves not forced through resting, 8 data sets of 29 showed improved fit with BARO5 (*P* < 0.005).

To test whether there was a systematic trend in all of the data sets of asymmetry in a particular direction, i.e., whether curvature tended to increase or decrease as pressure increased, we determined whether the mean asymmetry index was different from zero using the Student’s*t*-test (Table 1). For both constrained and unconstrained curves, the computed values were not strictly different from zero, indicating that there was no systematic asymmetry (*P* = 0.053 and*P* = 0.116, respectively). However, it is worth noting that for the constrained curves, the probability was borderline, suggesting that there was a tendency for the constrained curves to be asymmetric in one particular direction. We also noted that there was no systematic trend in the 8–10 curves that were asymmetric.

*Resting constraint*. The effect of not constraining the four-parameter curves was to explain more of the total variation in 15 of 29 curves (*P* < 0.005; 1.45 expected by chance, Table 3, Fig. 3). Thus just over one-half of the data sets showed evidence of the resting value not being on the curve. Similarly, for the five-parameter curve, 11 data sets showed improved fit without the constraint of forcing the curves through the resting point (*P* < 0.005).

The effect of both procedures (a 5-parameter model not constrained through resting) improved the fit in 20 of 29 curves compared with the four-parameter constrained model, which has been used previously or either of the procedures alone (4). In addition, the 20 curves were made up of 10 improved by not forcing the curve through the resting, 6 improved due to the five-parameter model, 4 by either procedure, and 0 by the combination.

*Effects of model on estimates of parameters*. We determined whether the means of the estimated parameters of the curves were affected by the choice of model. Despite the degree of fit being improved in the majority of curves, most of the parameters were similar for all models. However, the upper HR plateau and the gain at BP_{50} were both underestimated using the four-parameter model compared with the five-parameter model (*P* < 0.01, 0.05, respectively; Table 2, Fig. 5). Removing the constraint of the curve through the resting point did not affect the mean value of any parameter.

### Heart Rate Baroreflexes in Dogs

The HR-MAP curves in conscious dogs were sigmoidal, with clear upper and lower plateaus similar to that found for rabbits, although the levels of HR reached at each plateau were considerably lower. One other difference was that the resting value was consistently close to the bradycardia plateau rather than being close to the middle of the curve, as is the case with rabbits (Fig. 6). The four-parameter model that was forced to pass through resting, on average, accounted for all but 6% of the variation (*r*
^{2} = 0.945, range 0.879–0.988). Refitting the data with a five-parameter model not constrained through resting reduced the residual variation in 5 curves of a total of 12 examined. This was more than the number expected merely by chance (*P* < 0.05). The unexplained variation was reduced to 4% (*r*
^{2} = 0.962, range 0.897–0.992). Furthermore, the dog baroreflex data showed evidence of systematic asymmetry with the asymmetry index of 0.69 ± 0.2 being >0 (*P* < 0.001). This was also shown in the two estimates of curvature, with the asymmetric model indicating that the upper curvature was greater and the lower curvature less than that estimated from the resting constrained symmetric model (*P* < 0.05 for each curvature compared with symmetric estimate; Fig. 6). Other parameters were not different, as estimated by both models.

## DISCUSSION

The present study examines two assumptions commonly made when assessing baroreflexes using a sigmoidal fitting procedure. First, the curvature is constant and therefore the curve is symmetrical, and second, the data should be forced to include the resting values. Using data obtained from both RSNA and HR recordings, we found that both assumptions individually imposed a statistical penalty for the goodness of fit and influenced the estimation of baroreflex parameters, particularly the estimate of curvature (normalized slope) and hence gain. In addition, it appears that these two effects are relatively independent in that an appreciable number of curves were improved by one procedure and not by the other. Relatively few were such that either procedure would give the benefit. Thus it appears that the combination of both (5-parameter model without resting constraint) gives the most benefit and the most flexibility for fitting sigmoidal curves to baroreflex HR and sympathetic curves.

On consideration that the initial starting point (4-parameter constrained model) explained >95% of the variance, there might appear to be little room for improvement. However, it must be considered that the typical variance explained by a linear model is 85%. Therefore the improvement by using the five-parameter unconstrained model is in effect ∼13% over the original model. The impact of the better fitting gave different estimates of parameters that were in some cases nearly 20% greater.

With respect to HR-MAP curves in rabbits, more than one-third of curves were improved by an asymmetrical fit but were not improved by removing resting point constraint. Therefore it appears that these curves were indeed asymmetric and that the extra parameter accommodated this feature rather than perhaps simply allowing for more flexibility to deal with the particular data set. However, the group data did not show a consistent tendency for asymmetry in any one direction, suggesting that the asymmetry did not appear to be systematic, i.e., related to the methodology or to a specific aspect of the physiological process. This contrasts with the reflex curves from dogs, which did show a clear asymmetry, with the upper curvature being greater and the lower curvature being less than the average estimated by the four-parameter model. The reason for the asymmetry in this particular case is not known, but possibilities include *1*) differing contributions from the vagal and sympathetic nerve at each end of the curve, as suggested is the case by Glick and Braunwald (8),*2*) a contribution of nonarterial afferents, possibly high-threshold cardiac afferents, which influence the high pressure side of the baroreflex curve, and*3*) a “nonspecific” action of the pharmacological agents used to elicit the baroreflex. By using the current methodology it is possible that this question can be examined further. In the wider context it can be used to further understand the underlying mechanisms contributing to asymmetry and the relevance of the shape characteristics of the baroreflex relationship for cardiovascular homeostasis.

By contrast, very few of the RSNA-MAP curves from the conscious rabbits could be considered asymmetric by the above criterion, and one might suggest that it is unnecessary to use a model that allows for asymmetry. However, if it is required for one variable such as HR, then it is convenient to use the same model for all variables in a study. These curves were more affected by the removal of the resting constraint. Thus we observed that for a large number of curves (particularly RSNA), the resting value was not located on the curve. One possibility is that we poorly estimated the resting point in the process of performing the ramp. This is partly due to the way in which the resting is chosen as a number of values before the beginning of the ramp. With conscious animals, the blood pressure and HR can readily change from moment to moment. Thus, because the baroreceptors reset to a new level of blood pressure very rapidly (3), it results in the estimated basal values sometimes being away from the curves. Our conclusion from the current RSNA and HR data is that, if one is to use a consistent baroreflex curve-fitting analysis, the same formula needs to be applied to RSNA and HR data. In this case, our results suggest that the five-parameter equation without the resting constraint is preferable, as this seems to give reliable estimates of curve parameters and the greatest flexibility to maximize the fit.

The main analysis was performed on data collected by the ramp method for evoking baroreflex curves in rabbits, as described by Dorward et al. (4), where blood pressure is raised or lowered at ∼1–2 mmHg/s. Other methods are commonly used, such as the steady-state technique that was used in the dog baroreflex curves that we have currently analyzed, where step changes in blood pressure are held constant until the responses have stabilized (13, 21). The latter method has the limitation that baroreceptors reset during this period but has the advantage that estimates of the resting values are numerous and therefore likely to be better. Thus the limitation of forcing through resting values may have more impact when using ramps than when using steady-state techniques. However, the response of the renal sympathetic nerve and the cardiac vagus HR responses are rapid and do not require stable blood pressure >1–2 s. By performing ramps, better estimates of the baroreflex gain are achieved, because there is much less baroreceptor resetting. To estimate the contribution of cardiac sympathetic nerves to the baroreflexes, the steady-state method must be used.

### Five-Parameter Asymmetric Model

To determine whether one has chosen the best model to fit baroreflex data, a number of criteria need to be considered for the model, such as*1*) it should be valid physiologically with parameters that have physiological meaning;*2*) it ought to be useful for giving good predictions about the data; *3*) it ought to have the minimum number of parameters necessary (in general, more parameters make a model more flexible without adding predictive power); *4*) it ought to be resistant to noise in the data; and*5*) it needs to be mathematically tractable.

The BARO5 model was devised to allow a direct comparison of the behavior of the curvature parameters in the four- and five-parameter models. The meanings of the other parameters are essentially identical in both models, and generally they are very similar numerically, and all have physiological meaning. The advantage of using the five-parameter model is that it readily gives an index of asymmetry in every situation. If there is no evidence of asymmetry in the curves of a particular study, the five-parameter model can very readily be reduced mathematically to the four-parameter model. The other major advantage is that the model gives an estimate of the curvature for the upper and lower parts of the curve independent of the position of the resting point. The curvature is a range-independent estimation of gain. These estimates of gain may be quite different from the tachycardia and bradycardia gains that have often been quoted in the literature, because these values can differ even if the curve is symmetrical and because this difference largely depends on the position of the resting point on the curve.

The five-parameter model, similar to most other sigmoidal curve-fitting methods, has the statistically desirable property that all points contribute to all of the estimated parameters, although the contribution is not necessarily equal. The curvatures are more sensitive to variation in the transition regions of the curve than in the linear portions, for example. Thus it remains important experimentally to have sufficient data points in all of the critical portions of the curve. The fact that there are two curvature parameters increases the importance of collecting adequate data in these sensitive regions. The plateaus are most influenced by the more extreme data. The five-parameter sigmoidal model has the mathematically convenient property that it is continuously differentiable, although it does not have simple derivatives. This means that gain can be estimated at all points including resting.

It appears that if the resting values are inappropriate for the rest of the data, the BARO5 model is able to compensate to a certain extent. This does mean, however, that the estimates of curvature in particular and BP_{50} to a lesser extent are acutely sensitive to the resting values if the curve is to be passed through resting. If these values are of particular interest, then it would seem advisable to consider omitting the constraint on resting. Using BARO5 will provide slightly improved estimates of the parameters in this sort of experiment, but we caution that any method that attempts to partition curvature will necessarily be highly sensitive to the quality of data in the regions of maximum inflection. This is often the region where data are scarcest. Although BARO5 is no exception, its virtue in these cases is that it provides good estimates of the plateaus.

### Other Asymmetric Models

The question of asymmetry in baroreflex data has been addressed previously by a compound logistic model of Kingwell and colleagues (17) that has two curvature parameters and also produces two range parameters. The curve is essentially made up of two separate BARO4 equations, one for the points that are above the resting value and one for the points below. Thus the curve also is discontinuous at the BP_{50}, and so the gain function of the whole curve is a composite of the gain of the partial curves. The authors devised a measure that they called the adjusted gain to compare with the G_{max} of the four-parameter model. The method also effectively replaces BP_{50} with resting point values. Conceptually, this model does not treat the reflex as a unified whole to the same extent as BARO5. One would expect that the estimates of the plateaus obtained by the compound logistic method would be no worse and that the estimates of curvature would be no better than BARO5. The major disadvantage is its inability to fit an asymmetrical function when the resting value is close to the plateau that can often occur. This is the case in the conscious dog HR baroreflex curves we have analyzed where the resting value lies close to the bradycardia plateau (38).

There are a number of other formulas that could have been used to show variable asymmetry, but none of them partition curvature according to blood pressure or attempt to explain asymmetry as a function of variable gain in the reflex.

In particular Equation 16has seven parameters and hence requires extremely high-quality data and a large number of data points. It is completely general, in that it makes no assumption about which data points are to contribute most to which curvature parameter (P3 or P6), but by the same token it is not guaranteed to cleanly separate them.

Another equation that was supplied with Sigmaplot (SPSS) Equation 17is sometimes used to fit reflex curves, but this equation does not explain asymmetry as a function of activation or curvature, rather as an amplification of the response nonlinearly dependent on pressure. The fifth parameter is difficult to explain in terms of the present reflex model. As such it gives no guide as to the likely curvature near either of the plateaus.

In conclusion, in the present study we have examined a new method for detecting and coping with asymmetrical curvature in sigmoidal response curves of the general form commonly used in baroreflex experiments. We have shown that the new model relates well to the commonly used four-parameter formula, being an extension that includes an additional curvature parameter and uses a novel way of transitioning between the upper and lower curvatures. In the data from rabbits, small but important differences in the nonpartitioned parameters existed in HR measurements, which may be attributed to variable asymmetry in HR baroreflex curves. These differences were not due to asymmetry in RSNA measurements. We have shown that forcing the curve through the resting point did cause an appreciable reduction in the variation explained by the curves. For HR, similar mean curve parameters were obtained, whereas the RSNA parameter estimates were affected. The HR curves from dogs showed clear evidence of systematic asymmetry, which was readily detected by the five-parameter model and which provided different estimates of the upper and lower curvature. We conclude that the asymmetric model is a valuable extension to the symmetric logistic model when examining the HR baroreflex, and it gives improved estimates of plateau values. Furthermore, forcing the curves through the resting values is a statistically questionable practice when analyzing RSNA, because it does affect the parameter estimates.

### Perspectives

The new five-parameter logistic sigmoid equation with two curvature parameters provides a much more flexible equation, giving improved fitting of nonsymmetrical data and a modest improvement in the parameter estimates. Furthermore, when there is no evidence for asymmetry, the five-parameter equation can be reduced to the four-parameter equation by simply averaging the two curvature parameters. However, this is only a limited view of the potential of this finding. Perhaps a more important aspect of this approach is that we have provided a new tool to readily determine when data are symmetrical or not symmetrical. Although it has been our experience that most baroreflex data are actually quite symmetrical, the data from conscious dogs indicate that there are situations when the data are clearly asymmetric. This approach provides very specific information about the shape of the curve, which may be important physiologically. For example, it may be advantageous for the baroreflex to respond to particular situations with a higher gain at one end of the curve than the other. However, when investigating the shape of the curve one should note that a larger number of data points needs to be collected in the shape-forming regions of the curve rather than at the plateaus. As such, it may be necessary to refine experimental protocols accordingly. We also suggest that subtle pharmacological effects may be detectable by changes in asymmetry of the baroreflex.

A second important aspect of this work relates to whether information about the shape of the curve has any physiological meaning. It well known that the baroreflexes in intact animals are compound reflexes, with input not only from both arterial baroreceptors but also from nonarterial baroreceptors such as those from the cardiac and pulmonary circulation (19). Often these receptors have quite different profiles of responsiveness to pressure with very high or low thresholds (25). Thus there is the potential to affect mainly one end of the curve and, therefore, influence the shape of the baroreflex function curve in a nonsymmetrical manner. For example, it is clear that the main effect of hypertension on the cardiac baroreflex is to attenuate the cardiac vagal nerve component rather than the sympathetic nerve component (11). We showed that this was directly correlated with the degree of cardiac hypertrophy (14, 15). We hypothesized that this was related to a reduced input from cardiac afferents, possibly high-threshold ventricular receptors, that respond to the large increase in afterload produced by phenylephrine infusion (10). Furthermore, we found that atrial natriuretic peptides facilitate cardiac vagal nerve baroreflex responses, possibly also through cardiac afferents. It remains to be determined whether such examples of nonarterial baroreceptor influences on only one of the normal cardiac effectors, namely the vagus nerve, result in measurable asymmetry of the baroreflex curve. If this turns out to be the case, then one can envision a situation in which a particular kind of treatment might induce a characteristic asymmetry that indicates the underlying mechanism.

## Acknowledgments

We are particularly indebted to Professor Frederick Sannajust and Dr. Robyn Woods, who performed the rabbit and dog experiments while at the Baker Medical Research Institute and permitted us to use the data for analysis. We thank Shirley Godwin and Alison Learmonth for valuable technical assistance.

## Appendix

So that researchers may become more familiar with the BARO5 equation, we have provided the raw data and an annotated copy of the Sigmaplot curve-fitting (baro5.fit), and transform files (baro5.xfm) for the data shown in Fig. 2. In the language below, comments start with a semicolon and continue to the end of the line. First, one would type the data into columns 1 and 2 of the Sigmaplot spreadsheet and then run the baro.fit nonlinear regression. The resulting parameters after an estimated 23 iterations would be placed in column 3 of the Sigmaplot spreadsheet.

The transform function (baro5.xfm) described below can be used to generate a smooth 41-point fitted curve and place them in column 4 (*x*) and column 5 (*y*). The code also computes the gain function as a separate later step in column 6.

## Footnotes

Address for reprint requests and other correspondence: G. A. Head, Baker Medical Research Institute, PO Box 6492, St. Kilda Rd. Central, Melbourne, Victoria 8008, Australia (E-mail:geoff.head{at}baker.edu.au).

This study was supported by a block institute grant from the National Health and Medical Research Council of Australia.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 1999 the American Physiological Society