Critical Assessment of the C-Optimality Design Criteria for Estimating the Median Effective Dose in Quantal Dose-Response Curves

In this paper the properties of C-optimal designs constructed for estimating the median effective dose within the framework of two-parametric linear logistic models are critically assessed. It is well known that this design criterion which is based on the first-order variance approximation of the exact variance of the maximum likelihood estimate of the ED50 leads to a one-point design where the maximum likelihood theory breaks down. The single dose used in this design is identical with the true but unknown value of the ED50. It will be shown, that at this one-point design the asymptotic variance does not exist. A two-point design in the neighbourhood of the one-point design which is symmetrical about the ED50 and associated with a small dose-distance would be nearly optimal, but extremely nonrobust if the best guess of the ED50 differs from the true value. In this situation the asymptotic variance of the two-point design converging towards the one-point design tends to infinity. Moreover, taking in consideration, that for searching an optimal design the exact variance is of primary interest and the asymptotic variance serves only as an approximation of the exact variance, we calculate the exact variance of the estimator from balanced, symmetric 2-point designs in the neighbourhood of the limiting 1-point design for various dose distances and initial best guesses of the ED50. We compare the true variance of the estimate of the ED50 with the asymptotic variance and show that the approximations generally do not represent suitable substitutes for the exact variance even in case of unrealistically large sample sizes. Kalish (1990) proposed a criterion based on the second-order asymptotic variance of the maximum likelihood estimate of the ED50 to overcome the degenerated 1-point design as the solution of the optimization procedure. In fact, we are able to show that this variance approximation does not perform substantially better than the first–order variance. From these considerations it follows, that the C-optimality criterion is not useful in this estimation problem. Other criteria like the F-optimality should be used.     

Application of the isobologram technique for the analysis of combined effects with respect to additivity as well as independence

Combined actions of two substances with similar effects are frequently expressed by pairs of doses that produce a fixed response, usually 50%, in so-called isobolograms (ED50 isobolograms). In addition to the dose scales in such graphs we propose the addition of effect scales, where possible, to indicate the effect at certain doses, e.g., the ED30. We further propose to construct isoboles for expected independent interaction, in addition to the additivity line, for which purpose a simple procedure is delineated. In practice, an independent isobole for 50% effect passes through the point formed by the ED30s of A and of B in ED50 isobolograms. Thus, the ED30s constitute the "zenith" of an independent isobole in ED50 isobolograms. It is shown that theoretical independent isoboles can either represent additive, overadditive, or underadditive interactions, depending on the steepness of the dose-response curves of the components. Hence, drugs with shallow dose-response curves exhibit overadditive independent effects, compounds with exponentially steep curves show additive independent interactions. Substances with very steep dose-response curves, producing lethal effects, exhibited marked underadditive effects which could be ascribed largely to an independent mechanism of action of the components. Hence, the inclusion of independent isoboles into conventional isobolograms provides new insights into the mechanisms of interactions and into the actions of the components. Interactions can thus be characterized better and more completely, and misinterpretations appear less likely than with conventional isoboles.     

Likelihood-based experimental design for estimation of ED50

In selecting the best dosage choice for the estimation of ED50, it is natural to try to minimize the length of the confidence intervals. In this presentation, the dose allocation that minimizes the length of the likelihood-based confidence intervals is presented and compared with alternative allocations that have been proposed based on the length of different types of confidence intervals, such as those based on the asymptotic variance or on Fieller's Theorem. Effective strategies to deal with the parameter dependence of these allocations are explored. A series of experiments to evaluate the effect of small doses per fraction on the radiation tolerance of the rat cervical spinal cord provide the motivation and an illustration for the proposed procedures.     

Quantitative comparison of ohmefentanyl isomers induced conditioning place preference in mice

Differences of analgesia and withdrawal response among ohmefentanyl stereoisomers have been studied. In the present study, Quantitative comparison of reinforcing effects of ohmefentanyl stereoisomers and morphine was performed by using a conditioned place preference design in mice. Results showed that morphine and ohmefentanyl stereoisomers were able to increase significantly the time spent in the drug-paired side with respect to vehicle treated animals. A good linear correlation between doses of drugs and number of mice with place preference was found within a given dose range. On the basis of the dose-response curve analysis, ohmefentanyl stereoisomers displayed a significant difference in place preference ED50. The addictive index (analgesic ED50/place preference ED50) was used to assess the addictive potential of drugs. It was demonstrated that the addictive potential of ohmefentanyl stereoisomers did not exhibit a large difference as addictive index. Among these stereoisomers, the addictive potential of compound F9208 was markedly lower than that of morphine.     

Anticonvulsant profile of nimodipine and nitrendipine against pentylenetetrazole induced seizures in rats

The activity of nimodipine and nitrendipine against pentylenetetrazole (PTZ) induced seizures in Albino rats was studied alone and in combination with valproate. The median effective dose [ED50] of valproate, nimodipine and nitrendipine were initially determined. All the 3 drugs were injected i.p. 30 min before the induction of seizures. Seizures were induced by PTZ 85 mg/kg i.p., and subsequently the effect of combining ED50 doses of nimodipine and nitrendipine with ED50 dose of valproate was evaluated. ED50 of valproate and nitrendipine were 129 and 2.5 mg/kg respectively. ED50 of nimodipine could not be established since a dose-response relationship was not obtained. Hence, for the purpose of combination studies, 4 mg/kg of nimodipine was used. Both nimodipine (4 mg/kg) and nitrendipine (2.5 mg/kg) decreased the ED50 of valproate from 129 to 40 mg/kg. Both nimodipine and nitrendipine potentiate the activity of valproate against PTZ induced seizures and can be considered as potential adjuvant anticonvulsants which merit further study.     

Trimmed logit method for estimating the ED50 in quantal bioassay

Trimmed nonparametric procedures such as the trimmed Spearman-Karber method have been proposed in the literature for overcoming the deficiencies of the probit and logit models in the analysis of quantal bioassay data. However, there are situations where the median effective dose (ED50) is not calculable with the trimmed Spearman-Karber method, but is estimable with a parametric model. Also, it is helpful to have a parametric model for estimating percentiles of the dose-response curve such as the ED10 and ED25. A trimmed logit method that combines the advantages of a parametric model with that of trimming in dealing with heavy-tailed distributions is presented here. These advantages are substantiated with examples of actual bioassay data. Simulation results are presented to support the validity of the trimmed logit method, which has been found to work well in our experience with over 200 data sets. A computer program for computing the ED50 and associated 95% asymptotic confidence interval, based on the trimmed logit method, can be obtained from the authors.     

Statistical Dose-Response Models with Hormetic Effects

Monotonically increasing or decreasing functions are often used to model the relationship between the response of an experimental unit and the dose of a given substance. Of late, there has been an increased interest in dose-response relationships that exhibit hormetic effects. These effects may be characterized by an increase in response at low doses instead of the expected decrease in response that is observed at higher doses. Herein, we study the statistical implications of hormesis in several ways. First, we present a broad class of parametric mathematical-statistical models, constructed from standard dose-response models, that allow the incorporation of hormetic effects in such a way that the presence of hormesis can be tested statistically. Second, we consider the impact of model misspecification on effective dose estimation, such as the ED50 and the limiting dose for stimulation, when the hormetic effect is present but ignored in the dose-response model by the researcher (model underspecification) and when an hormetic effect is not present but incorporated into the dose-response model (model overspecification). Our simulation study reveals that it is more damaging to the estimation of effective dose to ignore the hormetic effect through model underspecification than to include the hormetic effect in the model through model overspecification. Third, we develop a nonpara-metric regression technique useful as an exploratory procedure to indicate hormetic effects when present. Finally, both parametric and nonparametric methods are illustrated with an example.     

Kernel estimates of dose response

A nonparametric method for analyzing quantal response data from an indirect bioassay experiment is proposed. Kernel estimates of the dose-response curve are used to develop approximate confidence intervals for (i) the optimal combination dose of a drug with therapeutic effects at low doses and toxic effects at high doses, and (ii) the lethal dose levels of a toxic chemical. This nonparametric procedure was implemented on real and simulated data. The confidence interval for problem (i) has high coverage probabilities when the dose-response curve is symmetric about the optima. However, the coverage probabilities are adversely affected by asymmetry about the optima and consequently are not reliable unless the sample sizes are large. The use of kernel estimators with higher-order kernels may alleviate this sensitivity to asymmetry. The confidence interval for problem (ii) has high coverage probabilities robust with respect to the shape or symmetry of the underlying dose-response curve.     

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.     

Effects of increasing doses of glucagon-like peptide-1 on insulin-releasing phases during intravenous glucose administration in mice

The increase in insulin secretion caused by glucagon-like peptide-1 (GLP-1) and GLP-1 mimetics observed during an intravenous glucose test (IVGTT) has been reported in both normal and disease animal models, as well as in humans. In this study, a hierarchical population modeling approach is used, together with a previously reported model relating glucose to insulin appearance, to determine quantitative in vivo dose-response relationships between GLP-1 dose level and both first- and second-phase insulin release. Parameters of the insulin kinetic model were estimated from the complete set of glucose and insulin data collected in 219 anesthetized nonfasted NMR-imaged mice after intravenous injection of glucose (1 g/kg) alone or with GLP-1 (0.03-100 nmol/kg). The resulting dose-response curves indicate a difference in GLP-1 effect on the two release phases, as is also evident from the different ED(50) parameter values (0.107 vs. 6.65 nmol/kg for phase 1 vs. phase 2 insulin release parameters). The first phase of insulin release is gradually augmented with increasing GLP-1 dose, reaching saturation at a dose of ~1 nmol/kg, while the second-phase release changes more abruptly at GLP-1 doses between 3 and 10 nmol/kg and shows a more pronounced 100-fold increase between control and the high GLP-1 dose of 100 nmol/kg Moreover, separate disposition indices calculated for phase 1 and 2 insulin release, show a different pattern of increase with increasing GLP-1 dose.     

Effects of irradiation on the lip mucosa in the mice as a function of rate. Comparison with the effect of fractionation

We have investigated the effect of single doses delivered at various dose rates on the mouse lip mucosa biological system. The dose rates were: 642, 76.8, 14.1, 2.9 and 1.5 Gy/h. The incidence of desquamation in the different groups of mice was used for constructing dose effect curves. The dose leading to desquamation in 50% of the animals (ED50) was obtained by probit analysis. These ED50 were 16.5, 16.7, 19, 30.2 and 33.5 Gy for the respective dose rates. Fractionated irradiations have also been performed in the same biological system (separately published), and we have therefore been able to compare the fractionated and low dose rate irradiations. The recently published model of Dale was used for this comparison. With that mathematical approach a alpha/beta value of 7.4 Gy and a half time of repair of sublethal damage of 47 minutes have been derived. These results compare well with others from the literature on biological systems with similar characteristics (rapidly proliferating systems).     

Functional and histological changes in rat lung after boron neutron capture therapy

The motivation for this work was an unexpected occurrence of lung side effects in two human subjects undergoing cranial boron neutron capture therapy (BNCT). The objectives were to determine experimentally the biological weighting factors in rat lung for the high-LET dose components for a retrospective assessment of the dose to human lung during cranial BNCT. Lung damage after whole-thorax irradiation was assessed by serial measurement of breathing rate and evaluation of terminal lung histology. A positive response was defined as a breathing rate 20% above the control group mean and categorized as occurring either early (<110 days) or late (>110 days). The ED(50) values derived from probit analyses of the early breathing rate dose-response data for X rays and neutrons were 11.4+/-0.4 and 9.2+/-0.6 Gy, respectively, and were similar for the other end points. The ED(50) values for irradiation with neutrons plus p-boronophenylalanine were 8.7+/-1.0 and 6.7+/-0.4 for the early and late breathing rate responses, respectively, and 7.0+/-0.5 Gy for the histological response. The RBEs for thermal neutrons ranged between 2.9+/-0.7 and 3.1+/-1.2 for all end points. The weighting factors for the boron component of the dose differed significantly between the early (1.4+/-0.3) and late (2.3+/-0.3) breathing rate end points. A reassessment of doses in patients during cranial BNCT confirmed that the maximum weighted doses were well below the threshold for the onset of pneumonitis in healthy human lung.     

Uncertainties in the risk assessment of three mycotoxins: aflatoxin, ochratoxin, and zearalenone

The risk assessment of mycotoxins is made up of two major components: an exposure assessment and a hazard assessment. There are many uncertainties in both of these components. This paper will briefly discuss the various aspects of the risk assessment process as it applies to mycotoxins and will then focus mainly on some of the uncertainties in the hazard assessment component of several carcinogenic mycotoxins. To arrive at an estimated "safe dose" (end point of the hazard assessment), we have previously used two major approaches: the no observed effect level (NOEL) divided by a safety factor approach and a mathematical (robust linear) extrapolation to a "virtual safe dose." Both of these approaches use only points from the no observed effect region of the dose-response curve and ignore valuable data from the response region. It is proposed to use the dose at which 50% of the animals would have developed tumors (the TD50) divided by a large safety factor of 50,000 as an additional estimate of "safe dose". For many studies, the TD50 lies within the observed response region of the dose-response curve and may have more validity. It is also suggested in certain cases that some of the uncertainties regarding the NOEL can be reduced if one uses a statistically derived no effect level (NEL).     

More Accurate Dose-Response Estimation Using Monte-Carlo Uncertainty Analysis: The Data Cube Approach

The traditional q₁ ^* methodology for constructing upper confidence limits (UCLs) for the low-dose slopes of quantal dose-response functions has two limitations: (i) it is based on an asymptotic statistical result that has been shown via Monte Carlo simulation not to hold in practice for small, real bioassay experiments (Portier and Hoel, 1983); and (ii) it assumes that the multistage model (which represents cumulative hazard as a polynomial function of dose) is correct. This paper presents an uncertainty analysis approach for fitting dose-response functions to data that does not require specific parametric assumptions or depend on asymptotic results. It has the advantage that the resulting estimates of the dose-response function (and uncertainties about it) no longer depend on the validity of an assumed parametric family nor on the accuracy of the asymptotic approximation. The method derives posterior densities for the true response rates in the dose groups, rather than deriving posterior densities for model parameters, as in other Bayesian approaches (Sielken, 1991), or resampling the observed data points, as in the bootstrap and other resampling methods. It does so by conditioning constrained maximum-entropy priors on the observed data. Monte Carlo sampling of the posterior (constrained, conditioned) probability distributions generate values of response probabilities that might be observed if the experiment were repeated with very large sample sizes. A dose-response curve is fit to each such simulated dataset. If no parametric model has been specified, then a generalized representation (e.g., a power-series or orthonormal polynomial expansion) of the unknown dose-response function is fit to each simulated dataset using “model-free” methods. The simulation-based frequency distribution of all the dose-response curves fit to the simulated datasets yields a posterior distribution function for the low-dose slope of the dose-response curve. An upper confidence limit on the low-dose slope is obtained directly from this posterior distribution. This “Data Cube” procedure is illustrated with a real dataset for benzene, and is seen to produce more policy-relevant insights than does the traditional q₁ ^* methodology. For example, it shows how far apart are the 90%, 95%, and 99% limits and reveals how uncertainty about total and incremental risk vary with dose level (typically being dominated at low doses by uncertainty about the response of the control group, and being dominated at high doses by sampling variability). Strengths and limitations of the Data Cube approach are summarized, and potential decision-analytic applications to making better informed risk management decisions are briefly discussed.     

Therapeutic evaluation of free and liposome-encapsulated atovaquone in the treatment of murine leishmaniasis

The use of drug delivery systems may reduce the toxicity and improve the activity of anti-leishmanial compounds. The activity of atovaquone (ATV)-loaded liposomes was compared by determination of median effective doses (ED(25) and ED(50)), with that of free ATV in a murine model of visceral leishmaniasis induced by Leishmania infantum. On day 0, mice were infected intravenously with 4.10(7) promastigotes and treated via the tail vein on days 15, 17 and 19 by free drug in a DMSO/cremophor/water solution (0.2 to 1.6 mg/kg body weight) or by liposomal drug (0.04 to 0.32 mg/kg body weight). Mice were killed and livers and spleens were removed and weighed on day 21 p.i. and liver parasite burdens evaluated using the Stauber method. Effective doses were determined using the Hill representation relating the percentage of parasite suppression to the dose. Liposomal ATV was significantly more effective than the free drug in reducing liver parasites (61.6% of parasite suppression at a dose of 0.32 mg/kg vs 34.9% at a dose of 1.6mg/kg). Liposomal ATV was 23 times more active than the free drug (ED(25) value=0. 02+/-0.01 mg/kg vs 0.46+/-0.15 mg/kg for free drug). It was not possible to obtain the ED(50) for free ATV because the dose-response curve reached a plateau around 33% of parasite suppression. Conversely, the ED(50) for liposomal ATV was 0.17+/-0.05 mg/kg. 100% efficacy of bound ATV could be obtained with a concentration of 1. 77+/-0.35 mg/kg. A significant decrease in spleen weights was also observed reflecting a leishmanicidal activity of ATV. These results suggest that liposome loaded ATV is more efficacious than the free drug against Leishmania infantum in this murine model.     

Dose-response curves and tolerance doses for late functional changes in the normal rat brain after stereotactic radiosurgery evaluated by magnetic resonance imaging: influence of end points and follow-up time

Late reaction of normal tissue is still a limiting factor in radiotherapy and radiosurgery of patients with brain tumors. Few quantitative data in terms of dose-response curves are available. In the present study, 99 animals were irradiated stereotactically at the right frontal lobe using a linear accelerator and single doses between 26 and 50 Gy. The diameter of the spherical dose distribution was 4.7 mm (80% isodose). Dose-response curves for late changes in the normal brain at 20 months were measured using T1- and T2-weighted magnetic resonance imaging (MRI). The dependence of the dose-response curves on the follow-up time and the definition of the biological end point were determined. Tolerance doses were calculated at several effect probability levels and times after irradiation. The MRI changes were found to be dependent on dose and progressive in time. At 20 months, the tolerance doses at a 50% effect probability level were 39.6 +/- 1.0 Gy and 42.4 +/- 1.4 Gy for changes in T1- and T2-weighted images, respectively. These dose-response curves can be used for further quantitative investigations on the influence of various treatment parameters, such as the application of charged particles, radiopharmaceuticals or the variation of tissue oxygenation.     

Enhanced airway responses to substance P after repeated challenge in guinea pigs

We performed three consecutive dose-response curves to rapid intravenous infusions of substance P (SP) in anesthetized, mechanically ventilated guinea pigs. The dose of SP required to decrease pulmonary conductance to 50% of its base-line value (ED50GL) decreased 2.8-fold (P less than 0.002) and 3.3-fold (P less than 0.001) on the second and third dose-response curves, respectively, compared with the first. SP did not alter airway responses to intravenous histamine but did cause a significant (3.7-fold) decrease in ED50GL for dose-response curves to intravenous capsaicin, an agent that causes bronchoconstriction by release of endogenous tachykinins. The neutral metalloendopeptidase inhibitor thiorphan (0.5 mg) and the angiotensin-converting enzyme inhibitor captopril (1.7 mg) both caused a marked enhancement of airway responses to SP observed on the first dose-response curve but did not alter the enhancement of SP-induced airway responses produced by repeated SP challenge. The anticholinergic atropine (5 mg/kg iv), the antihistamine mepyramine (8 mg/kg iv), and the cyclooxygenase inhibitor indomethacin (30 mg/kg ip) had no effect on the first SP dose-response curve. Atropine and mepyramine did not prevent the enhancement of SP responses observed with repeated challenge, but after pretreatment with either indomethacin or acetylsalicylic acid, dose-response curves to SP were reproducible. Our results indicate that airway responses to intravenous SP are enhanced with repeated SP challenge and suggest that cyclooxygenase products of arachidonic acid metabolism are involved in the mediation of this phenomenon.     

Statistical models for low dose exposure

Extrapolation of health risks from high to low doses has received a considerable amount of attention in carcinogenic risk assessment over decades. Fitting statistical dose-response models to experimental data collected at high doses and use of the fitted model for estimating effects at low doses lead to quite different risk predictions. Dissatisfaction with this procedure was formulated both by toxicologists who saw a deficit of biological knowledge in the models as well as by risk modelers who saw the need of mechanistically-based stochastic modeling. This contribution summarizes the present status of low dose modeling and the determination of the shape of dose-response curves. We will address the controversial issues of the appropriateness of threshold models, the estimation of no observed adverse effect levels (NOAEL), and their relevance for low dose modeling. We will distinguish between quantal dose-response models for tumor incidence and models of the more informative age/time dependent tumor incidence. The multistage model and the two-stage model of clonal expansion are considered as dose-response models accounting for biological mechanisms. Problems of the identifiability of mechanisms are addressed, the relation between administered dose and effective target dose is illustrated by examples, and the recently proposed Benchmark Dose concept for risk assessment is presented with its consequences for mechanistic modeling and statistical estimation.     

Estimating LD50s without a computer

Quantitative analysis of dose-related effects, such as mosquitoes killed by insecticide or parasites killed by a drug, usually involves estimating the dose which kills, on average, 50% of the subjects. This quantity is often termed the LD(50) (LD for lethal dose), or the ED(50) (ED for effective dose). Other specified response levels, such as the LD(90) - the dose that kills 90% of subjects - may also be derived. Dose-related effects of this type follow an S-shaped curve because, clearly, doses lower than those giving zero response will also give zero response, while at the other end of the curve, doses above those giving a maximum response can also only give a maximum response. In other words the curve flattens out at both ends. The mathematics of fitting a suitable S-shaped curve to such data - for example by probit analysis - is quite simple in principle but can be arduous and time-consuming without a suitably programmed computer. In this article, Michael Healy explains an alternative approach which is particularly applicable to field observations where computers are unavailable.     

Imipramine-fentanyl antinociception in a rabbit tooth pulp model

Antinociception of imipramine (I) and its effect in combination with fentanyl (F) was evaluated in rabbits using electrically-induced lick chew responses via tooth pulp stimulation as the model of nociception. Acute i.v. injections of I elicited a graded dose response comparable to i.v. morphine (M) with I ED 50 = 4.35 mg/kg (2.31-8.14, 95% CL) and M ED 50 = 1.81 mg/kg (1.11-3.90), with no differences in the slopes between the two curves. The lethal dose of I was 10 mg/kg. An i.v. dose of I twice the ED 50 elicited an antinociceptive effect of more than 50% maximum possible effect (MPE) for 90 minutes with peak effect of 82% MPE occurring at 15 minutes. These effects of I were not reversed by a morphine-reversal dose of naloxone (0.1 mg/kg i.v.) but were reversed with a ten fold dose of naloxone. F ED 50 values (mcg/kg) were lowered from 11.35 to 2.70, 0.74 and 0.33 with increasing pretreatment doses of I (1.0, 2.1 and 3.2 mg/kg). These magnitudes of potency increases of F were 4.2, 15.3 and 34.4 fold respectively. A single i.v. ED 50 dose of I extended the time to 50% MPE of an ED 90 dose of F from 26 minutes to 77 minutes; of a 2 X ED 50 dose of F from 17 minutes to 28 minutes. Data points for three different combinations of I and F fell significantly within the synergistic field of an ED 50 isobologram and a polynomial equation described the curve best fitting the data points. F alone (i.v. ED 50 dose) increased the PaCO2 values to 74% above controls and three different combinations with I showed no increases in PaCO2 values above controls. I alone did not significantly cause any change in PaCO2 values from controls.