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INNOVATIVE METHODOLOGY

Calculation of threshold and saturation points of sigmoidal baroreflex function curves

School of Medical Sciences (Physiology) and Bosch Institute for Biomedical Research, University of Sydney, Australia

Submitted 1 March 2006 ; accepted in final form 15 May 2006

	ABSTRACT

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The logistic sigmoid function curve provides an accurate description of the baroreflex input-output relationship and is the most commonly used equation for this purpose. The threshold (Thr) and saturation (Sat) values for the baroreflex are commonly defined as the values of mean arterial pressure (MAP) at which the reflexly controlled variable (e.g., heart rate or sympathetic nerve activity) is within 5% of the upper or lower plateau, respectively, of the sigmoid function. These values are referred to here as Thr_5% and Sat_5%. In many studies, Thr and Sat are calculated with the equations Thr = A₃ – 2.0/A₂ and Sat = A₃ + 2.0/A₂, where A₃ is the value of MAP at the point where the reflexly controlled variable is at the midpoint of its range and A₂ is the gain coefficient. Although it is commonly stated that the values of Thr and Sat calculated with these equations represent Thr_5% and Sat_5%, we show here that instead they are significantly greater and less than Thr_5% and Sat_5%, respectively. Furthermore, the operating range (difference between Thr and Sat) calculated with these equations is 32% less than the difference between Thr_5% and Sat_5%. We further show that the equations that provide correct values of Thr_5% and Sat_5% are Thr_5% = A₃ – 2.944/A₂ and Sat_5% = A₃ + 2.944/A₂. We propose that these be used as the standard equations for calculating threshold and saturation values when a logistic sigmoid function is used to model the open-loop baroreflex function curve.

mathematical modeling; input-output relationship; baroreflex operating range; regression analysis

Kent et al. (13) first proposed that a logistic sigmoid function curve be used to describe baroreflex function, according to the following equation:

(1)

where Y is the output (dependent) variable (e.g., heart rate, sympathetic nerve activity, or systemic arterial pressure), X is the input (independent) variable (e.g., MAP or carotid sinus pressure), A₁ is the Y range (value of Y at the top plateau – value of Y at the bottom plateau), A₂ is the gain coefficient, A₃ is the value of X at the midpoint (which is also the point of maximum gain), and A₄ is the value of Y at the bottom plateau. Kent et al. (13) also showed that the maximal gain of the reflex, i.e., the maximum slope of the curve, can be derived directly from Eq. 1, as follows:

(2)

The logistic sigmoid function curve has been found to provide a very good fit of the baroreflex input-output relationship (1, 13) and has become the most commonly used equation for this purpose.

Two other very important parameters that describe baroreflex function are the threshold and saturation values for the input X (i.e., Thr and Sat). For the baroreceptor reflex, Thr is generally defined as the value of X at which no further increases in the output variable occur despite further reductions in X, whereas Sat is the value of X at which no further decreases in the output variable occurs despite further increases in X. Defined in this way, Thr and Sat cannot be determined from the logistic sigmoid function curve because Y changes continuously as X increases or decreases, approaching the lower and upper plateau values, respectively. To avoid this problem, Thr and Sat have been defined as the values of X at which Y is below or above the upper and lower plateau values of Y, respectively, by some arbitrary small value. Very commonly (see, e.g., Refs. 6–8, 12, 17–23, 28), this arbitrary value is taken as 5% of the response range for Y, and we shall follow this convention here.

Many authors have used an equation derived by Chen and Chang (2) to calculate Thr and Sat, as follows:

(3)

(4)

It is commonly stated (see, e.g., Refs. 6–8, 12, 17–23, 28) that the values of Thr and Sat calculated according to Eqs. 3 and 4 are the values of X at which the corresponding Y values are 5% (of the response range) below and above the upper and lower plateau values of Y, respectively. We refer to these values here as Thr_5% and Sat_5%. In their original paper, however, Chen and Chang (2) did not claim that Thr_5% and Sat_5% could be calculated with Eqs. 3 and 4. Instead, in their analysis they approximated the sigmoid curve as three straight lines, consisting of two parallel horizontal lines that are the asymptotes of the upper and lower parts of the sigmoid curve and a third line passing through the midpoint of the curve (i.e., at X = A₃) and with a slope equal to the slope of the curve at that point (i.e., –A₁A₂/4), as shown in Fig. 1. Chen and Chang (2) defined Thr and Sat as the X values at the points where this third straight line intersects with the upper and lower horizontal lines, respectively (Fig. 1). At these intersection points, X is equal to A₃ – 2.0/A₂ and A₃ + 2.0/A₂ (Eqs. 3 and 4). The derivation of these equations is given in the APPENDIX.

View larger version (14K): [in this window] [in a new window]   Fig. 1. Baroreflex logistic sigmoid function curve showing the threshold and saturation points as defined by Chen and Chang (2). They defined the threshold value (ThrC) as the X value at the point where the straight line passing through the midpoint (where X = A3) and with a slope equal to the slope of the curve at that point (i.e., –A1A2/4) intersects with the asymptote of the upper plateau portion of the curve, i.e., the line Y = A4 + A1. Similarly, Chan and Cheng (2) defined the saturation value (SatC) as the X value at the point where this straight line intersects with the asymptote of the lower plateau portion of the curve, i.e., the line Y = A4. The operating range (ORC) is the difference between ThrC and SatC.

The aim of this study, therefore, was to determine new equations that yield correct values of Thr and Sat as defined by ourselves (16) and many others (see, e.g., Refs. 6–8, 12, 17–23, 28), i.e., that when substituted into Eq. 1 result in values for Y that are 5% below and above the upper and lower plateau values of Y, respectively.

	ANALYSIS

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As defined above, Sat_5% is the value of X when the value of Y is 5% of the Y range above the bottom plateau, i.e., when

(5)

Substituting Eq. 5 into Eq. 1 gives:

Subtracting A₄ from both sides and then dividing both sides by A₁,

Dividing both sides by 0.05 and then multiplying both sides by {1 + exp[A₂(Sat_5% – A₃)]},

Therefore,

Taking the logarithm (to the base e) of both sides:

Rearranging to make Sat_5% the subject:

i.e.,

(6)

Similarly, determination of the threshold by the same process (using Y = A₄ + 0.95 x A₁) yields the equation

(7)

With the use of a similar process in reverse, the values of Y that correspond to the values of Sat and Thr according to the equations derived by Chen and Chang (2) (Eqs. 3 and 4) can also be determined as follows. Starting with Eq. 1, i.e.,

Substituting for X = A₃ + 2.0/A₂ [according to Chen and Chang (2)]

Canceling out and rearranging:

Therefore,

(8)

Thus the value of Y corresponding to the value of X as derived from the equation of Chen and Chang (2) is at the point 11.9% (of the Y range) above the lower plateau (Fig. 2). By a very similar procedure, the value of Y corresponding to Thr as determined from the equation of Chen and Chang (2) can be shown to be at the point 11.9% (of the Y range) below the upper plateau (Fig. 2).

View larger version (23K): [in this window] [in a new window]   Fig. 2. Baroreflex sigmoid function curves showing the threshold point (Thr), saturation point (Sat), and operating range (OR) as calculated according to 3 different methods. In these examples, the independent (X) value is mean arterial pressure (MAP) and the dependent (Y) value is heart rate (HR). In A, the values of the parameters for this equation are A1 (Y range) = 200 beats/min (bpm), A2 (gain coefficient) = 0.1, A3 (midpoint) = 120 mmHg, and A4 (bottom plateau) = 300 bpm. In B, the values of the parameters for this equation are A1 = 100 bpm, A2 = 0.06, A3 = 125 mmHg, and A4 = 350 bpm. Subscripts indicate the origin of the equation used to calculate the variables: C, Chen and Chang (2); K, Kent et al. (13); 5%, present study (Eqs. 6 and 7).

(9)

and

(10)

When these values are substituted into Eq. 1, the Y values that correspond to these values of Sat and Thr are Y = A₄ + 0.211 x A₁ and Y = A₄ + 0.789 x A₁, respectively. Thus the Y values corresponding to Sat and Thr as defined by Kent et al. (13) are >21% (of the Y range) above and below the lower and upper plateaus, respectively (Fig. 2).

It is clear that Eqs. 6 and 7, as well as the equations derived by Chen and Chang (2) (Eqs. 3 and 4) and Kent et al. (13) (Eqs. 9 and 10) all depend only on the value of the midpoint A₃ and gain coefficient A₂ and are thus independent of the Y value at the lower plateau (A₄) or the Y range (A₁). The difference in the values of Thr and Sat calculated with the three different methods depends only, however, on the value of A₂. When the value of A₂ is decreased, the absolute magnitude of these differences increases.

For example, Fig. 2 shows two sigmoidal baroreflex function curves with different values for the parameters A₁, A₂, A₃, and A₄, and Table 1 shows, for each of these two curves, the numerical values for Thr and Sat as calculated with the equations of Chen and Chang (2) and those of Kent et al. (13), as well as the values of Thr_5% and Sat_5% as determined with Eqs. 6 and 7. When the value of A₂ is 0.1 (Fig. 2A), the difference between the value of Thr from the equation of Chen and Chang (2) and the value of Thr_5% is 9.4 mmHg (100.0 – 90.6; Table 1). When the value of A₂ is 0.06 (Fig. 2B), however, the difference is 15.8 mmHg (91.7 – 75.9; Table 1).

	DISCUSSION

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It is commonly believed (see, e.g., Refs. 6–8, 12, 17–23, 28) that the equations derived by Chen and Chang (2) produce values of Thr and Sat that are equal to Thr_5% and Sat_5%. The above analysis, however, shows that the values of Thr and Sat as calculated from the equations of Chen and Chang (2) are significantly greater and less than Thr_5% and Sat_5%, respectively, and that the correct values of Thr_5% and Sat_5% are those calculated according to Eqs. 6 and 7.

In most studies of baroreflex function in which Thr and Sat are calculated, comparisons are made between groups within the same study. In these cases, when the same equations are used to calculate Thr and Sat, such comparisons are likely to be valid, even though the absolute calculated values are different from Thr_5% and Sat_5%. The problem becomes much more significant, however, when comparisons are made between studies, and it is therefore important to have a standardized and accurate method of calculating these important parameters of the logistic sigmoid function curve.

As an illustration of this point, we have compared the Thr and Sat values of MAP for the baroreflex control of renal sympathetic nerve activity (RSNA) as determined in anesthetized rats in a recent study from our laboratory (16) with those determined in conscious rats in a previous study by Miki et al. (17). With the data from our study (16) in anesthetized rats, the mean values of Thr_5% and Sat_5% for the MAP-RSNA sigmoid curves under resting conditions were calculated (with Eqs. 6 and 7) to be 71.6 ± 3.6 and 136.9 ± 4.6 mmHg, respectively, and the operating range (difference between threshold and saturation) was 65.2 ± 3.6 mmHg. In the study by Miki et al. (17), the mean values of Thr_5% and Sat_5% for the MAP-RSNA sigmoid curves under resting conditions were reported to be 78.3 ± 4.6 and 126.5 ± 3.3 mmHg, respectively, and the operating range was 48.2 ± 4.7 mmHg. A comparison of these values suggests that in the conscious rat the threshold level for the MAP-RSNA baroreflex is higher, the saturation level is lower, and the operating range is much less, than in the anesthetized rat. In the study by Miki et al. (17), Thr and Sat were calculated according to the equations of Chen and Chang (2). When calculated according to Eqs. 6 and 7, however [with the mean values of A₂ and A₃ as determined by Miki et al. (17)], the threshold and saturation values are 69.4 and 135.4 mmHg, respectively, and the operating range is 66.0 mmHg. Thus, when calculated in the same way, it is now clear that these parameters of the MAP-RSNA baroreflex sigmoid function curve are very similar in the anesthetized and conscious states.

Although the four-parameter logistic sigmoid function is much more commonly used, Ricketts and Head (25) derived a five-parameter logistic sigmoid function, which includes an additional parameter that takes into account asymmetry of the curve above and below the midpoint. This function reduces to the four-parameter logistic sigmoid function when the curves are symmetrical. If the five-parameter sigmoid function is used, the values of Thr_5% and Sat_5% cannot be derived with simple equations such as Eqs. 6 and 7. Similarly, the equations for Thr and Sat derived by Chen and Chang (2) and Kent et al. (13) also cannot be used, because they also depend on the assumption that the curve is symmetrical above and below the midpoint. In cases where the five-parameter logistic sigmoid function derived by Ricketts and Head (25) is used, however, the values of Thr_5% and Sat_5% could be determined instead from the plotted graph of the derived function curve.

In summary, Eqs. 6 and 7 produce accurate values for Thr and Sat, where these are defined as the values of X at which Y is below or above the upper and lower plateau values of Y, respectively, by 5% of the Y range. It is, of course, possible to define Thr and Sat as the values of X at which Y is below or above the upper and lower plateau values of Y by a different percentage (e.g., 10%). Because most authors accept 5% as the appropriate difference, however, we propose that Eqs. 6 and 7 be used as the standard equations for calculating values of Thr and Sat when a four-parameter logistic sigmoid function is used to model the open-loop baroreflex function curve.

	APPENDIX

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As defined by Chen and Chang (2), the threshold and saturation points are the X values at the points where the straight line passing through the midpoint of the logistic sigmoid curve, and with a slope equal to the sigmoid curve at this point, intersects with the upper and lower horizontal straight lines that are the asymptotes of the upper and lower parts, respectively, of the sigmoid curve (Fig. 1). We refer to these points here as Thr_C and Sat_C.

Let the equation of this straight line passing through the midpoint be:

At this point, the slope b = –A₁A₂/4 (see Eq. 2).

As shown in Fig. 1, when X = A₃, Y = A₄ + A₁/2. Therefore,

Therefore,

Thus the equation of this straight line is

At the point where this line intersects with the lower asymptote, Y = A₄ and (by definition) X = Sat_C (Fig. 1). Therefore, at this point,

Subtracting A₄ from both sides and rearranging,

Dividing both sides by A₁A₂/4,

Similarly, the threshold value (Thr_C) is defined as the point where the straight line intersects with the upper asymptote, i.e., where Y = A₄ + A₁ and (by definition) X = Thr_C (Fig. 1).

By a derivation similar to that shown above for the Sat_C value,

	GRANTS

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This study was supported by the National Health and Medical Research Council of Australia.

	ACKNOWLEDGMENTS

 
We thank Professor John Ludbrook and Associate Professor Bill Gibson for very helpful discussions.

	FOOTNOTES

 

Address for reprint requests and other correspondence: R. A. L. Dampney, Dept. of Physiology, F13, Univ. of Sydney, NSW 2006, Australia (e-mail: rogerd{at}physiol.usyd.edu.au)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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This Article Abstract Full Text (PDF) All Versions of this Article: 291/4/H2003    most recent 00219.2006v1 Alert me when this article is cited Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (7) Citing Articles via Google Scholar Google Scholar Articles by McDowall, L. M. Articles by Dampney, R. A. L. Search for Related Content PubMed PubMed Citation Articles by McDowall, L. M. Articles by Dampney, R. A. L.