Data-Driven Method to Estimate Nonlinear Chemical Equivalence

There is great need to express the impacts of chemicals found in the environment in terms of effects from alternative chemicals of interest. Methods currently employed in fields such as life-cycle assessment, risk assessment, mixtures toxicology, and pharmacology rely mostly on heuristic arguments to justify the use of linear relationships in the construction of “equivalency factors,” which aim to model these concentration-concentration correlations. However, the use of linear models, even at low concentrations, oversimplifies the nonlinear nature of the concentration-response curve, therefore introducing error into calculations involving these factors. We address this problem by reporting a method to determine a concentration-concentration relationship between two chemicals based on the full extent of experimentally derived concentration-response curves. Although this method can be easily generalized, we develop and illustrate it from the perspective of toxicology, in which we provide equations relating the sigmoid and non-monotone, or “biphasic,” responses typical of the field. The resulting concentration-concentration relationships are manifestly nonlinear for nearly any chemical level, even at the very low concentrations common to environmental measurements. We demonstrate the method using real-world examples of toxicological data which may exhibit sigmoid and biphasic mortality curves. Finally, we use our models to calculate equivalency factors, and show that traditional results are recovered only when the concentration-response curves are “parallel,” which has been noted before, but we make formal here by providing mathematical conditions on the validity of this approach.


Derivation of Biphasic Model Equations Equations for the biphasic-sigmoid concentration function
We obtain the concentration-concentration correlation for a reference chemical that exhibits a sigmoid dose-response curve, and a "novel" chemical that exhibits a biphasic dose-response curve, which can be modeled using the equation: Here, C is the "novel" chemical concentration;  K and  K are concentrations that give half of the maximum concentration, max U , for each of the respective constituent sigmoids;  m and  m are parameters that help to set the slope of the constituent sigmoids at their respective inflection points; and i U and f U are the response values at the respective concentration extremes, 0  C and   C . Note that we have dropped the term "novel" from the subscript of the parameters, which is otherwise present in the main text; there is no need to distinguish between "novel" or "reference" chemicals in the calculations described here.
As explained in the main text, the dose-response function, ) (C f , must be obtained for both reference and novel chemicals from identical experiments (e.g., Mortality measured in Fathead minnow). This restriction allows for the response of the dose-response function to be used as a parameterization variable of the concentrationconcentration correlation function. Unlike the sigmoid-sigmoid relationship obtained in the main text, the biphasic function generally yields two outputs for one input, or vice versa, depending on the choice of reference and novel chemical.
To begin, we first assume that the biphasic and sigmoid functions have been fitted to dose-response data so that parameter values are given. The goal will be to use parameter values of Eq. (A1), and nothing else, to find analytic equations that estimate the concentration-concentration correlation function.

Partitioning the biphasic function using a threshold novel chemical concentration
We first partition the biphasic function into parts that individually yield one-to-one correlation functions. Depending on the fitted parameter values of the biphasic function to the dataset, it may exhibit either a local maxima or minima. Here we consider exemplary parameters that provide a local maximum, as shown in Fig. A; although, the equations we derive may be applied to the case of a local minimum. We partition the biphasic curve into halves, depending on whether the novel chemical concentration falls above or below at a threshold value,   / C , which is presented in the main text: Equations (A2) and (A3) may also be approximated with the geometric mean of the parameters  K and  K :

Component models for the decomposed biphasic curve
As given by Eq. (A4), we partition the biphasic curve into two pieces, referred to herein as its left-hand (LHS) and right-hand (RHS) sides. We model each such portion with a sigmoid equation, which are, in principle, different than the constituent sigmoid-like equations used in construction of the biphasic curve equation. Parameters associated with the LHS sigmoid are herein denoted by a minus sign, -, while parameters of the sigmoid modeling the RHS of the biphasic function are herein denoted using an addition sign, +. Thus, sigmoid equations that model the LHS and RHS are respectively given by: denoted with tildes can be written exclusively in terms of the fitted parameters of the biphasic curve. In the next section we explain how to achieve this for the LHS sigmoid, Eq. (A5). The procedure to obtain effective parameters for the RHS, Eq. (A6), is similar, but yields a different result.

Effective parameters for the LHS of the biphasic curve, Eq. (A5)
Equation (A5)  . Our general strategy will be to identify conditions that restrict the model sigmoid (e.g., Eq. (A5)) to "match" the response of the biphasic curve in the LHS domain ( there are three effective parameters, we must obtain three independent conditions, as we explain in the following subsections. We note that these conceptual restrictions are identical for the RHS model sigmoid; the actual form of the equations differ, which leads to slightly different results to those obtained for the LHS sigmoid. Finally, it will be useful to express the approximate value of the local maximum (or minimum) of the biphasic curve, max U : which can be found by evaluating Eq. (A1) at the point

Effective parameter: the condition for  m
The parameter  m sets the slope of the sigmoid model in the log-log scale, evaluated at its inflection point. If the sigmoid model (red line, Fig. A(b)) is to converge with the lefthand side of the biphasic response, then both slopes should be evaluated at their respective inflection points, and should necessarily match in value. Carrying out this calculation gives: (A8) Effective parameter: the condition for  K The previous section applied the restriction that the slopes for the model sigmoid (Eq. (A5)) and that of the left-hand side of the biphasic response should match, which gave an equation for  m (Eq. (A8)). We now make another restriction: that response values for the model sigmoid and the biphasic response should match, at least once between the initial and final levels of the response function; namely at the point , which is the geometric mean between the initial and maximum levels of the left-hand side biphasic response. We can first identify a concentration value that provides this particular value of the biphasic response, denoted by  K C , when put into the constituent sigmoid of the left-hand side response: which can be solved, to give: (A10) We propose to solve Eq. (A10) approximately, by noting that the local maximum (or minimum) value of the biphasic function, max U , should be reasonably close to its solution. This suggests that we expand the left-hand side using a Taylor series to first order in a logarithm scale, due to the underlying sigmoid structure. Solving the resulting approximate equation gives: wherein the variables   and   are, respectively, a prefactor and the exponent of the power-law approximation: (A13)

Effective parameters for the RHS of the biphasic curve, Eq. (A6)
Parameters for the right-hand side sigmoid model, Eq. (A6), of the biphasic response function, Eq. (A1), can be obtained using similar restrictions to each of the three variables described above in the previous section. However, we should note that these restrictions result in slightly different expressions for the effective parameters  m ,  K , and  , max eff U , than are given, respectively, by Eqns. (A8), (A9), and (A11)-(A13). Figure B compares the sigmoid models of Eqns. (A5)-(A6) with the "exact" biphasic equation, Eq. (A1), with parameter values given in Table A. As shown in Fig. B(a), the sigmoid models of Eqns. (A5)-(A6) are qualitatively very close to the response of the biphasic function, with a maximum relative error of approximately 5.6% (Fig. B(b)) for the illustrated parameter values. The absolute value of this relative error rises slightly as 1 /    K K (from above); however, the sigmoid models for both left-and right-hand sides match the biphasic response exactly in the limit ).

Validity of the sigmoid-based approximate model to the biphasic response function
Finally, we note that, in the above analyses, we have assumed that 0  can be manipulated to yield: Similarly, the negative affector can be given by: (A15) Applying the function composition operation, , yields: (A16) After some manipulation of this equation, we find that Eq. (A16) is identical to Eq. (A1), if the parameters of Eq. (A1) are subjected to the following set of transformations: The result of Eqns. (A17a-c) is intuitive, given that we have merely shifted the y-axis of the biphasic curve, which does not otherwise affect its shape. Moreover, this shifting

Normalization of the sigmoid and biphasic response functions
Some response functions, such as a population's mortality, may be experimentally measured such that the control population suffers losses from expected effects, such as the normal aging process, or through additional adverse effects, such as disease. For such cases, a fitted response curve, such as mortality, . (For an example of this effect, refer to the experimental data of Fig. 4 in the main text.) Additionally, a chemical compound experienced in large dosages or exposures may not be strictly fatal: . These effects may also depend on the time-scale of the measurements. Even so, it may be advantageous to normalize the fitted concentration-response curves to a fixed interval, from which the concentration-concentration correlation function can be reliably identified. Thus, we seek a set of equations that modify the values of the empirically fitted parameter values of both sigmoid and biphasic curves, such that the relative shape of the concentration-response function is preserved under a dilation or contraction of the response-axis.

Scaling relationships for parameters of the sigmoid curve
An example of a sigmoid curve is given by Fig. C(a) (B1) gives the transformation for any point along the y-axis: Evaluating this equation at the endpoints confirms the desired endpoints:

Scaling relationships for parameters of the biphasic curve
A similar methodology which led to Eq. B2 can be used to find scaled values for the initial and final levels of the sigmoid and biphasic concentration-response functions. As above, normalizing both curves in this way restricts descriptions to an effective response, rather than to absolute levels, which is a common practice in when the response levels are arbitrary or not standardized. In such situations a common concentration measure is the EC 50 , which is the concentration that corresponds to a median (for a population), or half-maximal state level for the (potentially) nonlethal effect.
While it is straight-forward to normalize the sigmoid equation (