Mathematical interpretation of dose-response curves

The dose-response of an individual organism can be described by a step functions if the organism survives when the dose is below a certain lethal level and dies when this level is exceeded. If, in a population of organism, the lethal dose for an individual has a unimodal distribution, the latter's properties will determine the shape of the population's response in the following manner. If the distribution is symmetric the dose-response curve has a symmetric sigmoid shape when plotted on linear coordinates. The location of the inflection point and the curve's slope around it are determined by the distribution's mode and variance. When the distribution is skewed, the dose-response curve has an asymmetric sigmoid shape which becomes reminiscent of an exponential decay when the distribution is strongly skewed to the right. The population's dose-response curve can be constructed by integration of the step changes over the distribution range. The step function representing the dose-response of an individual organism can be approximate by a Fermi function, and the distribution of an lethal doses can be represented by the Weibull distribution function. When the two functions are combined, the resulting dose-response of the populationS(X)), which is the fraction of survivors after exposure to a doseX, is given by:S(X)=∫ ₀ ¹ [1/{+exp{(X-X _c (φ))/a _i ]}]dω whereX _c (ω)={(1/b)[-ln(1-ω)]}^(1/n),n andb being the constants of the Weibull distribution anda _i an arbitrarily small number, i.e.a _i ≪[X−X _c (ϕ)], whose actual magnitude is of little significance. This model can be used to determine the underlying distributions of experimental dose-response relationship. It was applied to published survival data of microorganisms exposed to pulsed electric field, X-ray radiation and ozone to show that the different observed shapes of the dose-response curve, and shifts between them, can be expressed in terms of the correponding distribution parameters, namely the mode, variance and skewness.     

Evaluation of a proposed surface colonization equation usingThermothrix thiopara as a model organism

The colonization equation shown below was evaluated usingThermothrix thiopara as a model organism. $$N = (A/\mu )e^{\mu t} - A/\mu $$ where: N=number of cells on surface (cells field^?1); A = attachment rate (cells field^?1 h^?1); M=specific growth rate (h^?1); t=incubation period (h). Previous studies of microbial surface colonization consider attachment and growth independently. However, the proposed colonization equation integrates the effects of simultaneous attachment and growth. Using this equation, the specific growth rate ofT. thiopara was found to be 0.38±0.3 h^?1 during in situ colonization. Estimates ofμ were independent of incubation period after 4 h (2 generations). Shorter incubations were inadequate to produce sufficient microcolonies for accurate determination of specific growth rate. Empirical data for the time course of colonization fell within the 95% confidence interval of predicted values. The attachment rate, although assumed to be constant, was found to continuously increase with time. This increase may have been an artifact due to the continuous deposition of travertine on the surface, or may indicate the need for a function to replace A in the colonization equation. Using the exponential growth equation, the progeny of cells that attach during incubation are considered to be progeny of cells that attach initially. This erroneously inflated the growth rate by 55%.     

General characteristics of the sigmoidal model equation representing quasi-static pulmonary P-V curves.

A pulmonary pressure-volume (P-V) curve represented by a sigmoidal model equation with four parameters, V(P) = a + b[1 + exp[-(P - c)/d]](-1), has been demonstrated to fit inflation and deflation data obtained under a variety of conditions extremely well. In the present report, a differential equation on V(P) is identified, thus relating the fourth parameter, d, to the difference between the upper and the lower asymptotes of the volume, b, through a proportionality constant, alpha, with its order of magnitude of 10(-4) to 10(-5) (in ml(-1). cmH(2)O(-1)). When the model equation is normalized using a nondimensional volume, (-1 < < 1), and a nondimensional pressure, (=(p/c) - 1), the resulting - curve depends on a single nondimensional parameter, Lambda = alphabc. A nondimensional work of expansion/compression, (1-2), is also obtained along the quasi-static sigmoidal P-V curve between an initial volume (at 1) and a final volume (at 2). Six sets of P-V data available in the literature are used to show the changes that occur in these two parameters (Lambda defining the shape of the sigmoidal curve and (1-2) accounting for the range of clinical data) with different conditions of the total respiratory system. The clinical usefulness of these parameters requires further study.     

Analyses of dose-response curves to compare the antimicrobial activity of model cationic alpha-helical peptides highlights the necessity for a minimum of two activity parameters

To assess and compare different model Leu-Lys-containing cationic alpha-helical peptides, their antimicrobial activities were tested against Escherichia coli as target organism over a broad peptide concentration range. The natural cationic alpha-helical peptides magainin 2 and PGLa and the cyclic cationic peptide gramicidin S were also tested between comparison. The dose-response curves differed widely for these peptides, making it difficult to rank them into an activity order over the whole concentration range. We therefore compared five different inhibition parameters from dose-response curves: IC(min) (lowest concentration leading to growth inhibition), IC(50) (concentration that gives 50% growth inhibition), IC(max) (related to minimum inhibition concentration and minimum bactericidal concentration), inhibition concentration factor (IC(F); describing the increase in concentration of the peptide between minimum and maximum inhibition), and activity slope (A(S); related to the Hill coefficient). We found that these parameters were covariant: two of them sufficed to characterize the dose dependence and hence the activity of the peptides. This was corroborated by showing that the dose dependences followed the Hill equation, with a small, constant aberration. We propose that the activity of antimicrobial peptides can readily be characterized by both IC(50) and IC(F) (or A(S)) rather than by a single parameter and discuss how this may relate to investigations into their mechanisms of action.     

Graphical aids to interpretation of Scatchard plots and dose-response curves

General expressions are derived for the limiting slopes and intercepts of graphical representations of experimental binding data in either the Scatchard or the “dose-response” co-ordinate systems. We apply a previously formulated general model that includes heterogeneity and/or cooperativity of receptor affinity. One must establish or assume a physical chemical mechanism in order to fully interpret these limiting slopes and intercepts. However, they do provide satisfactory initial estimates for the binding parameters, for use in a non-linear least squares curve fitting approach using an exact model.     

Monotone smoothing with application to dose-response curves and the assessment of synergism

A spline-based procedure for monotone curve smoothing is proposed and is illustrated by application to dose-response curves. It is then shown how such smoothing can be applied to assess possible synergism or antagonism of two drugs.     

Interactions between prostaglandins and calcium: the importance of bell-shaped dose-response curves

Biological responses to PGs show two basic forms of dose/response relationship, plateau and bell-types. Although bell-shaped dose/response curves are well documented their possible occurrence is almost always ignored in the design and interpretation of experiments on PGs and related substances. This may lead to serious errors, several types of which are described. The ignoring of a well-documented phenomenon may take place because there is no accepted hypothesis which attempts to explain the bell-type curves. A hypothesis is proposed which accounts for both plateau and bell type responses. It is developed primarily with respect to PG-calcium interactions but may be applicable to some PG-cyclic nucleotide interactions as well. The model leads to precise predictions which can be experimentally tested in many systems.     

The use of non-linear regression analysis and the F test for model discrimination with dose-response curves and ligand binding data

Computer fitting of binding data is discussed and it is concluded that the main problem is the choice of starting estimates and internal scaling parameters, not the optimization software. Solving linear overdetermined systems of equations for starting estimates is investigated. A function, Q, is introduced to study model discrimination with binding isotherms and the behaviour of Q as a function of model parameters is calculated for the case of 2 and 3 sites. The power function of the F test is estimated for models with 2 to 5 binding sites and necessary constraints on parameters for correct model discrimination are given. The sampling distribution of F test statistics is compared to an exact F distribution using the Chi-squared and Kolmogorov-Smirnov tests. For low order modes (n less than 3) the F test statistics are approximately F distributed but for higher order models the test statistics are skewed to the left of the F distribution. The parameter covariance matrix obtained by inverting the Hessian matrix of the objective function is shown to be a good approximation to the estimate obtained by Monte Carlo sampling for low order models (n less than 3). It is concluded that analysis of up to 2 or 3 binding sites presents few problems and linear, normal statistical results are valid. To identify correctly 4 sites is much more difficult, requiring very precise data and extreme parameter values. Discrimination of 5 from 4 sites is an upper limit to the usefulness of the F test.     

The SMM model as a boundary value problem using the discrete diffusion equation

A generalized single-step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.     

Efficient design for estimation of median lethal dose and quantal dose-response curves

The results of quantal dose-response experiments are often summarized by an estimate of the "median lethal dose," denoted LD50, and many sequential designs have been proposed for efficient estimation of LD50. These designs strive to produce a sequence of trials at dose levels that get closer and closer to LD50. Consequently, they may not provide very good estimates of the overall shape of the dose-response curve. In this paper we propose guidelines for the design of experiments that estimate LD50 fairly efficiently and that also allow for efficient global estimation of the curve.     

Interaction between prostaglandins and calcium: The importance of bell-shaped dose-response curves

Biological response to PGs show two basic forms of dose/response relationship, plateau and bell-types. although bell-shaped dose/response curves are well documented their possible occurrence is almost always ignored on the design and interpretation of experiments on PGs and related substances. This may lead to serious errors, several types of which are described. The ignoring of a well-documented phenomenon may take place because there is no accepted hypothesis which attempts to explain the bell-type curves. A hypothesis is proposed which accounts for both plateau and bell type responses. It is developed primarily with respect to PG-calcium interactions but may be applicable to some PG-cyclic nucleotide interactions as well. The model leads to precise predictions which can be experimentally tested in many systems.     

Design of experiments for the precise estimation of dose-response parameters: the Hill equation

Optimal experimental designs were evaluated for the precise estimation of parameters of the Hill model. The optimally effective designs were obtained by using the criterion of D-optimization. For the Hill model, optimal designs replicate 3 sampling points. These points were shown to be quite sensitive to the behavior of the experimental error. Since an investigator is often uncertain about error conditions in biological studies, a practical approach would use the sampling scheme calculated for an intermediate error condition. Thus, if the behavior of error variances is not known, precise parameters of the Hill model are obtained by choosing concentrations which yield fractional responses (responses divided by their asymptotic, maximum value) of 0.086, 0.581 and 1.0. When experimental constraints limit the maximum attainable concentration and response, all design points are lowered. Appropriate designs can be constructed based on the design which is optimal when constraints result in a maximum attainable fractional response of 0.5. The optimal designs were found to be robust when the parameter values assumed by the investigator did not equal their true values. The estimating efficiencies obtained by using two frequently applied designs were assessed. Uniformly spaced concentrations yielded imprecise parameters. Six-point, geometrically spaced designs gave generally good results. However, their estimating efficiency was generally exceeded by the recommended sampling schemes even in the presence of uncertainty about error conditions. The method exemplified in this paper can be used for other models.