Dose-response relationships curve-fitting

For non-threshold mechanisms of genotoxic carcinogenicity, the dose-response relationship is considered to be linear. The observed dose-response curve in some cases represents a single ratedetermining step however, in many cases it may be more complex and represent a superposition of a number of dose-response curves for the various steps involved in the tumor formation (EC 2003). Because of the small number of doses tested experimentally, i.e., usually only two or three, almost all data sets fit equally well various mathematical functions, and it is generally not possible to determine valid dose-response curves on the basis of mathematical modeling. This issue is addressed in further detail in Chapter 6. 

Mathematical modelling of the dose-response relationship is an alternative approach to quantify the estimated response within the experimental range. This approach can be used to determine the BMD or benchmark concentration (BMC) for inhalation exposure, which can be used in place of the LOAEL or NOAEL (Crump, 1984). The BMD (used here for either BMD or BMC) is defined as the lower confidence limit on a dose that produces a particular level of response (e.g., 1%, 5%, 10%) and has several advantages over the LOAEL or NOAEL (Kimmel Gaylor, 1988 Kimmel, 1990 USEPA, 1995 IPCS, 1999). For example, (1) the BMD approach uses all of the data in fitting a model instead of only data indicating the LOAEL or NOAEL (2) by fitting all of the data, the BMD approach takes into account the slope of the dose-response curve (3) the BMD takes into account variability in the data and (4) the BMD is not limited to one experimental dose. Calculation and use of the BMD approach are described in a US EPA... 

However, there are practical difficulties associated with fitting pharmacological data with appreciable scatter to a curved arithmetic line (Figure 6.2A). For example, a plot of arithmetic dose reaches a maximal asymptote value when the drug occupies all of the receptor sites. In addition, the range of concentrations needed to fully depict the dose-response relationship is usually too wide to be useful in the format shown. 

Another method of detecting a dose-response relationship is to fit the data to various models for dose-response curves. This method statistically determines whether or not a dose-response model (such as a Logistic function) fits the data points more accurately than simply the mean of the values this method is described fully in Chapter 12. The most simple model would be to assume no dose-response relationship and calculate the mean of the ordinate data as the response for each concentration of ligand (horizontal straight line parallel to the abscissal axis). A more complex model would be to fit the data to a sigmoidal dose-response function (Equation 11.2). A sum of squares can be calculated for the simple model (response — mean of all response) and then for a fit of the data set refit to the four parameter Logistic shown... 

A least-squares fit for a nonlinear function of the form y(x) = a(exp - x) with the Marquardt-Levenberg algorithm, where a is the consumption at 0% concentration, 5 is the rate of change, and x is the concentration of the compound, to fit curves for the dose-response relationship. The two-parameter model was used because it provided the best fit for most of the compounds tested over the broadest range of concentrations. 

Similar to cancer dose-response relationships (figure 10.1), the best-fit curve steepness and location are uncertain. Therefore, a 95UCL curve is identified based on the variability of the laboratory data. An EDjg is also identified from this 95UCL curve. This lower (e.g., more health-protective) EDjo is then selected as the BMD. This is shown in figure 10.3. 

The operational model, as presented, shows dose-response curves with slopes of unity. This pertains specifically only to stimulus-response cascades where there is no cooperativity and the relationship between stimulus ([AR] complex) and overall response is controlled by a hyperbolic function with slope = 1. In practice, it is known that there are experimental dose-response curves with slopes that are not equal to unity and there is no a priori reason for there not to be cooperativity in the stimulus-response process. To accommodate the fitting of real data (with slopes not equal to unity) and the occurrence of stimulus-response cooperativity, a form of the operational model equation can be used with a variable slope (see Section 3.13.4) ... 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

Y is the expected response and X is the corresponding concentration. A, B,C and D are the four parameters of the equation, where A gives an estimate of expected response at zero dose, B is the slope (e.g., response/concentration) in the middle of the calibration curve, C is the IC50 and D is the expected response at infinite dose. This curve satisfies all the conditions specified for the response-concentration relationship and also closely approximates the mass-action equations [20]. Weighting of the results is recommended for fitting dose-response data from immunoassay, in order to compensate the heterogeneity of response variances in the response-error relationship [17,18]. 

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