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Toxicological Sciences
, Volume 165 (1) – Sep 1, 2018

11 pages

/lp/ou_press/time-to-event-modeling-of-left-or-right-censored-toxicity-data-in-HPi5XYU0lz

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of the Society of Toxicology. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 1096-6080
- eISSN
- 1096-0929
- D.O.I.
- 10.1093/toxsci/kfy122
- Publisher site
- See Article on Publisher Site

Abstract A time-to-event (TTE) model has been developed to characterize a histopathology toxicity that can only be detected at the time of animal sacrifice. The model of choice was a hazard model with a Weibull distribution and dose was a significant covariate. The diagnostic plots showed a satisfactory fit of the data, despite the high degree of left and right censoring. Comparison to a probabilistic logit model shows similar performance in describing the data with a slight underestimation of survival by the Logit model. However, the TTE model was found to be more predictive in extrapolating toxicity risk beyond the observation range of a truncated dataset. The diagnostic and comparison outcomes would suggest using the TTE approach as a first choice for characterizing short and long-term risk from nonclinical toxicity studies. However, further investigations are needed to explore the domain of application of this kind of approach in drug safety assessment. time-dose-toxicity relationship, time-to-event modeling, survival analysis The determination of the maximum safe starting dose in humans for pharmaceuticals relies on extrapolation of nonclinical toxicology results and the no observed adverse effect level (NOAEL) identified in the most sensitive and relevant animal species (Food and Drug Administration, 2005). Although there is no consistent standard definition of NOAEL, it is commonly identified from each vehicle-controlled toxicology dose-ranging study separately, as the next dose down from the lowest dose producing a biologically or statistically significant increase in adverse effects compared with the vehicle-control group (Dorato and Engelhardt, 2005). This descriptive approach for defining NOAEL is known to have several limitations (Crump, 1984; Filipsson et al., 2003; Glatard et al., 2015; Sand et al., 2002). For example, given that NOAEL is identified for a given study, it restricts the available data to a specific study duration and one of the doses used in the study. Furthermore, the simulation work from Filipsson et al. (2003) as well as from Glatard et al. (2015), showed that placing doses more densely and conducting larger experiments leads to the declaration of a lower NOAEL. In environmental safety, a model-based approach has been proposed as an alternative to the NOAEL, with the concept of the benchmark dose (Chen et al., 2007; Ritz et al., 2013). Very few examples of model-based risk assessment have been published in preclinical drug safety. Glatard et al. (2015) and Sahota et al. (2015) both compared the NOAEL descriptive approach with a probabilistic model of toxicity data using logistic regression. Those models integrated toxicity data across cohorts and studies to estimate the probability of toxicity and its confidence interval, as a continuous function of the study duration and dose level. Logistic regression is a technique for characterizing binary outcome data and the relative contribution and interaction of covariates. However, the common mathematical assumption made is that all covariates, including time, generally exert a linear effect on risk (as measured by the log of odds ratio), which is potentially problematic considering that, depending on the nature of the toxicity, susceptibility may change over time in nonmonotonic and nonlinear fashion (eg, tolerance or delayed risk). The time-to-event (TTE) approach, in contrast, estimates the relationship between elapsed time and likely outcome with a variety of possible statistical distributions, including nonconstant, nonlinear, and nonmonotonous functions allowing a greater flexibility than logistic regression (Holford, 2013). The TTE approach therefore has the potential to characterize the onset of a wide variety of events which, for many subjects, may lie beyond the timescales of observation. For example, the TTE approach is commonly considered in survival analysis, where the event of interest is the death of individuals. Other types of events in TTE can be considered, defined on efficacy criteria (eg, haematological response [Philippe et al., 2015], cure from severe acute malnutrition [Banbeta et al., 2015]), nonefficacy criteria (eg, tumor progression [Wilbaux et al., 2014], graft rejection [Frobel et al., 2013], lesion recurrence [Lim and Bae, 2015], use of rescue medication [Trocóniz et al., 2006], recurrent malaria infection [Tarning et al., 2012]), or toxicity criteria (eg, ribavirin-induced anemia; Tod et al., 2005). TTE approaches are also able to take into account patients’ drop-out from clinical studies (Goyal and Gomeni, 2013; Pilla Reddy et al., 2012; William-Faltaos et al., 2013). Those TTE models proposed in the literature are usually applied to clinical data, but they remain underutilized in toxicology (Sánchez-Bayo, 2009). Newman and McCloskey (1996) advocated TTE models to greatly enhance assessment of ecological risk and demonstrated their utility with various ecotoxicity data (Dixon, 2001). The dichotomy of presence or absence of an event at a fixed time (ie, the end of the experiment) is a specific type of censoring in TTE analysis. For example, if the event of interest is death, live animals are right-censored animals; the only information available about their time to death is that it is longer than the study duration. A left censored event is an event which has occurred before the observation time, but its exact time is not known. Censoring reduces the information in a TTE dataset and it is of interest to explore whether a TTE approach which can handle censoring, can be relevant and provide value compared with a traditional analysis based on logistic regression analysis for nonclinical toxicological data. In this work, we propose a TTE modeling approach to characterize the toxicity incidence of a test compound, using left- and right-censored toxicity data and we compare its simulation and extrapolation properties to the traditional logistic regression analysis. MATERIALS AND METHODS Data Histopathology data on testicular toxicity generated in rats for a test compound were used for the TTE analysis. A total of 7 nonclinical safety studies were conducted for this test compound at various doses, study durations and sample sizes, to determine NOAEL values and to guide the first dose in humans. Detailed data characteristics are provided elsewhere (Glatard et al., 2015) and a summary is presented in Table 1. Table 1. Summary of Data Characteristics Study Total Number of Animals Tested Doses (mg/kg/day) Time of Sacrifice (Weeks) Study 1 103 CTL, 100, 500, 1000 13, 26 Study 2 24 CTL, 1000 13 Study 3 48 CTL, 100, 300, 1000 26 Study 4 36 CTL, 50, 500, 1000 4, 26 Study 5 36 CTL,1000 13 Study 6 30 CTL [5–20]a Study 7 40 CTL, 1000 2.6, 6, 10 Study Total Number of Animals Tested Doses (mg/kg/day) Time of Sacrifice (Weeks) Study 1 103 CTL, 100, 500, 1000 13, 26 Study 2 24 CTL, 1000 13 Study 3 48 CTL, 100, 300, 1000 26 Study 4 36 CTL, 50, 500, 1000 4, 26 Study 5 36 CTL,1000 13 Study 6 30 CTL [5–20]a Study 7 40 CTL, 1000 2.6, 6, 10 a Time: 5.3, 8.7, 9.7, 10.7, 11.7, 12.7, 14.7, 15.7, 16.7, 19.7. Abbreviation: CTL, Control group. Table 1. Summary of Data Characteristics Study Total Number of Animals Tested Doses (mg/kg/day) Time of Sacrifice (Weeks) Study 1 103 CTL, 100, 500, 1000 13, 26 Study 2 24 CTL, 1000 13 Study 3 48 CTL, 100, 300, 1000 26 Study 4 36 CTL, 50, 500, 1000 4, 26 Study 5 36 CTL,1000 13 Study 6 30 CTL [5–20]a Study 7 40 CTL, 1000 2.6, 6, 10 Study Total Number of Animals Tested Doses (mg/kg/day) Time of Sacrifice (Weeks) Study 1 103 CTL, 100, 500, 1000 13, 26 Study 2 24 CTL, 1000 13 Study 3 48 CTL, 100, 300, 1000 26 Study 4 36 CTL, 50, 500, 1000 4, 26 Study 5 36 CTL,1000 13 Study 6 30 CTL [5–20]a Study 7 40 CTL, 1000 2.6, 6, 10 a Time: 5.3, 8.7, 9.7, 10.7, 11.7, 12.7, 14.7, 15.7, 16.7, 19.7. Abbreviation: CTL, Control group. The histopathology toxicity data were collected from each animal once, at the time of sacrifice and were analyzed by a qualified pathologist. The animal experiments were conducted in accordance with the GSK Policy on the Care, Welfare, and Treatment of Laboratory Animals and were reviewed by the Institutional Animal Care and Use Committee at GSK and/or by the ethical review process at the institution where the work was performed (eg, GSK, Stevenage and Ware, United Kingdom; GSK, King of Prussia, United States; Charles River Laboratories, Horsham, United States, and Korea Institute of Toxicology, Daejeon, Korea). The analyzed toxicity was described by 4 different histopathology descriptors which characterized the overall spectrum of the testicular toxicity. In our analysis, the presence or absence of at least 1 of the 4 descriptors was treated as a binary variable (1 or 0, respectively). The exact time when the toxicity became detectable was unknown, and not all animals developed the toxicity at the end of the experiments. Therefore, all toxicity events were considered to be either right or left censored. If the toxicity was observed at the time of sacrifice, the toxicity must have developed before the animal sacrifice, and therefore the event was left censored. If the toxicity was absent at the time of sacrifice, the toxicity had not developed but, could have occurred later if the study had continued; these events were considered as right censored. TTE modeling The histopathology toxicity data was modeled using a TTE approach, considering the elapsed time of occurrence of an event T (eg, onset of toxicity) since the beginning of the study. The distribution of the event times was characterized by the hazard and the so-called “survival” probability functions (defined as the probability of nontoxicity). The hazard function h(t) is the instantaneous risk of event at time t. It is defined as the probability of having an event at time t+Δt given no event occurred until time t, and can be written as follows: ht=limΔt→0pt<T≤t+Δt | T>tΔt (1) where t denotes the time (weeks), T the time of the event occurrence, p(t<T≤t+Δt|T>t) is the probability of observing an event at time t+Δt given that the event had not occurred at time t, Δt is an infinitely small interval of time. The probability of nontoxicity function, also referred to as the Survival function S(t) in standard survival analysis is the probability to not have observed any event until time t. It can be expressed as a function of the integrated hazard function over time, as follows: St=pt<T=exp-∫0th(u)du (2) Subsequently, the probability of toxicity function, PTox(t) is the probability to have observed an event until time t, and can be expressed as follows: PToxt=pt≥T=1-St=1-exp-∫0th(u)du (3) Parameters are then obtained by maximum likelihood estimation, where the likelihood of left-censored events (where event occurred before time of sacrifice) and right-censored events (where event did not occur before time of sacrifice) are given by p0<T≤t=1-S(t) and pt<T=S(t), respectively. Model development Model building consisted of 2 steps: (1) choice of parametric hazard function to describe the distribution of individual event data over time and (2) testing for potential covariate effect on hazard parameters. Several hazard functions h(t) were tested with model discrimination performed using the Akaike information criteria (AIC) (Wagenmakers and Farrell, 2004), the Bayesian information criteria (BIC) (Schwarz, 1978) and the survival goodness-of-fit plot. The 3 best performing distributions were the Weibull, log-normal and log-logistic with almost undistinguishable AIC and BIC values as well as predicted survival curves (see Supplementary Appendix 1). The exponential model, assuming constant hazard model, was associated with higher AIC and BIC values (see Supplementary Appendix Table 1). The Weibull distribution was selected as the most appropriate model due to proportional hazard model assumptions, making the covariate-related effect on hazard (eg, drug effect) easier to interpret. The hazard and probability of toxicity and nontoxicity functions of the Weibull distribution can be written as follows: ht= shapescale×tscaleshape-1 (4) PToxt=1-St=1- exp-tscaleshape (5) St= exp-tscaleshape (6) The Weibull distribution is described by 2 positive parameters (shape and scale), which allow various monotonic hazard patterns and various decreasing probability of nontoxicity curves (see Weibull distribution example in Figure 1). The parameter shape (dimensionless) corresponds to a slope: when shape = 1, the model is reduced to a constant hazard model; when shape < 1, the hazard decreases over time; when shape >1, the hazard increases over time. The parameter scale (expressed in weeks) corresponds to a time scaling factor and determines the spread and location of the time values: when time = scale, the probability of nontoxicity equals to exp-1∼0.37, independently of the value of shape. Figure 1. View largeDownload slide Hazard and probability of toxicity and nontoxicity functions for a Weibull distribution with a scale of 10, and various shape values of 0.5 (- - -), 1 (—), and 1.5 (—). Figure 1. View largeDownload slide Hazard and probability of toxicity and nontoxicity functions for a Weibull distribution with a scale of 10, and various shape values of 0.5 (- - -), 1 (—), and 1.5 (—). The potential effect of covariates, including the dose level (as continuous or categorical covariate), was tested on the scale parameter of the Weibull hazard function, as follows: scale=scale0×expβcontXcont+βcat2Xcat2+⋯+βcatkXcatk (7) where scale0 is the reference value for the scale parameter, Xcont is a continuous covariate and βcont is its associated effect on scale, Xcat is a categorical covariate of k categories, Xcati is an indicator variable (0/1) for the ith category defined for each of the k categories, and βcati its associated effect on scale. The covariate effect can be interpreted as hazard ratio: exp(βcont) is the expected change in scale for a 1-unit increase of Xcont; exp(βcati) is the expected change in scale when Xcat is equal to the ith category compared with the reference category. The covariate selection procedure was based on the BIC criteria, which is derived from a statistical fitting test accounting for both the quality of fit and a penalty for the number of model parameters. A lower BIC reflects a more parsimonious model for which all relevant covariates of interest had been evaluated and retained. Data analyses, parameter estimations and graphical representations were performed in R version 3.3.1 (R DCT, 2010.), using the parametric function survreg in the package survival. The Surv function was used to handle the right- or left-censored data, with a specific type of censoring (type=“interval2”). Model evaluation The selected model was evaluated by using goodness-of-fit plots, simulation-based diagnostics (visual predictive check [VPC]) and a bootstrap analysis. Goodness-of-fit plots graphically compared the predicted survival curves to the Kaplan-Meier (KM) curves for the observations (Karlsson, 2013). To perform the simulation-based diagnostics, 1000 replicates of the original dataset were simulated according to the final model structure and parameters. For each animal, an exact time of toxicity event was simulated; and categorized into left- or right-censored event based on the study duration. VPC graphically compared the 5th, 50th, and 95th percentiles of the KM curves computed from the simulated data, to the observed KM curves. The bootstrap analysis was performed in order to evaluate parameter uncertainty, on the basis of 1000 resampled datasets (with replacement, stratified on study, dose and study duration). For each dataset, model parameters were re-estimated, and the median and 95% CI were calculated for each parameter, and compared with the original parameter estimates. In order to evaluate the impact of a change in parameter values on the model outcome, the hazard ht and the probability of toxicity were calculated for each of the 1000 sets of parameters, and compared with the probability obtained from the original set of parameters. Comparison between TTE and logit approaches Recently, an analysis of the same dataset using a logistic model was reported (Glatard et al., 2015). The final model implemented using the generalized linear model function from the stats package in the R software (Nelder and Wedderburn, 1972), assumed a binomial distribution of the data (presence or absence of toxicity per animal) and used a Logit link function, including treatment duration and dose as covariates, without interaction. LogitpTox=a+b×DOSE+c×DUR (8) pTox=exp(LogitpTox)(1+exp(LogitpTox) (9) DV ∼ B n,pTox (10) where, pTox is the probability of toxicity for each animal, a is the intercept characterizing the background toxicity incidence, b and c are the regression parameters characterizing the effect of dose (DOSE) and treatment duration (DUR), DV is the dependent variable, n is the sample size per group and B is the binomial distribution. For the comparison, the variable DV is the same as the one used for the TTE analysis and corresponds to the presence or absence of at least 1 of the 4 descriptors, treated as a binary variable (1 or 0, respectively). As the TTE model and the logit model were built on the same dataset (dose from 0 to 1000 mg/kg/day and treatment duration from 0 to 26 weeks), we could compare their properties in predicting toxicity across the independent variables such as dose ( DOSE) and time ( DUR). For the comparison, the following toxicity outcomes were derived from each model: The probability of toxicity, pToxDOSE,DUR, defined by Equation (5) for the TTE model and defined by Equation (9) for the logit model The instantaneous risk of toxicity (or hazard function hToxDOSE,DUR), defined by Equation (4) for the TTE model and derived from Equation (2) for the logit model. In addition, the ability of both models to extrapolate the predicted risk of toxicity for longer studies was assessed by estimating model parameters from a reduced dataset, and truncating the data up to week 15. The KM curves predicted (up to week 15) and extrapolated (after week 15) from the truncated model, were then compared with the observed risks of toxicity from the complete dataset. RESULTS Data Characteristics The data from the 7 toxicology studies listed in Table 1 were used. Figure 2 illustrates the observed toxicity counts over time, stratified on dose level and color-coded by presence-absence of toxicity. The 4 main groups of sacrifice time points were at weeks 6, 10, 13, and 26, with low numbers of toxicity events in the control group and many more in the 1000 mg/kg/day group. Very few animals received the intermediate doses of 50, 100, 300, and 500 mg/kg/day, and their sacrifice time points were limited to 13 and 26 weeks. Figure 2. View largeDownload slide Distribution of sacrificed animal counts and toxicity outcomes (absence in gray and presence in black) over time, stratified by dose level (mg/kg/day). The control group is CTL. Figure 2. View largeDownload slide Distribution of sacrificed animal counts and toxicity outcomes (absence in gray and presence in black) over time, stratified by dose level (mg/kg/day). The control group is CTL. Model Description The final hazard model used a Weibull distribution, and the effect of dose was included on the scale parameter, which reflects a proportional increase in risk of toxicity with dose increments. The model can be written using the Equation (4) and replacing scale as follows: scale=scale0×expβDOSE×DOSE (11) where scale0 is the scale value in the control group, βDOSE is the dose effect on scale and DOSE is the continuous covariate varying from 0 to 1000 mg/kg/day. Since the proportional hazard assumptions hold for the Weibull model, the hazard ratio HR for each dose increment could thus be derived from Equations (9) and (10) as follows: HR=h(t)hCTL(t) =exp-shape×βDOSE×DOSE (12) where HR is the hazard ratio for a given dose increment and hCTL(t) is the hazard in the control group. The model parameters and their respective relative standard error (RSE) estimated from the original dataset are listed in the first 2 columns of Table 2. The value of the drug effect parameter, characterized in the Weibull model by βDOSE was estimated to be negative and led to a decrease in scale and an increase in hazard, when increasing the dose. For instance, each 100 mg/kg/day increase in dose is associated with a 34% decrease in scale, according to Equation (11), and with a 25% increase in hazard, according to Equation (12). The uncertainty characterized by RSE appeared to be large for the parameter estimate of scale0. The model values generated by bootstrap (Table 2) were consistent with those obtained for the original dataset. Table 2. Parameter Values of the Model Original Dataset Bootstrap Dataset Estimated value RSE Median 95% CI Shape (−) 0.54 45% 0.55 [0.23–0.96] Scale0 (weeks) 1366 198% 1297 [194–525072] βDOSE (mg/kg/day)−1 −0.0042 45% −0.0041 [−0.01 to −0.0023] Original Dataset Bootstrap Dataset Estimated value RSE Median 95% CI Shape (−) 0.54 45% 0.55 [0.23–0.96] Scale0 (weeks) 1366 198% 1297 [194–525072] βDOSE (mg/kg/day)−1 −0.0042 45% −0.0041 [−0.01 to −0.0023] Table 2. Parameter Values of the Model Original Dataset Bootstrap Dataset Estimated value RSE Median 95% CI Shape (−) 0.54 45% 0.55 [0.23–0.96] Scale0 (weeks) 1366 198% 1297 [194–525072] βDOSE (mg/kg/day)−1 −0.0042 45% −0.0041 [−0.01 to −0.0023] Original Dataset Bootstrap Dataset Estimated value RSE Median 95% CI Shape (−) 0.54 45% 0.55 [0.23–0.96] Scale0 (weeks) 1366 198% 1297 [194–525072] βDOSE (mg/kg/day)−1 −0.0042 45% −0.0041 [−0.01 to −0.0023] The response surface of the dose—time—probability of toxicity relationship described by the selected model is depicted in Figure 3 and shows (1) the toxicity incidence was null and the dose-response was flat at baseline, (2) the incidence in the control group increased slowly over time, and (3) for a given time point (except baseline), the incidence increased with dose. Figure 3. View largeDownload slide Response surface of the probability of toxicity as a function of dose (mg/kg/day) and time (weeks) from the TTE model. Figure 3. View largeDownload slide Response surface of the probability of toxicity as a function of dose (mg/kg/day) and time (weeks) from the TTE model. Model Diagnostics The VPC is shown in Figure 4 for the control group and 1000 mg/kg/day. The VPC showed a good agreement between the observed and the simulated data. Indeed, the observed and the predicted probabilities of nontoxicity are close to each other and the 95% prediction interval of simulated nontoxicity (gray area) is mostly within the confidence interval of the observed KM curves (dotted black line) for both doses. A slight overestimation of the probability of nontoxicity and consequently, a slight underestimation of the probability of toxicity can be noted for the control group between 15 and 26 weeks. Given the limited number of time points of sacrifice for the intermediate dose levels (eg, a single time for dose 50 and 500 mg/kg/day), the data at these dose levels were part of the modeling analysis but not part of the diagnostic plots. The bootstrap results are illustrated on the left panel from Figure 5 and Figure 6 in terms of hazard and probability of toxicity respectively. The plots include the time-toxicity relationship in the control arm and after 1000 mg/kg/day and the dose-toxicity relationship at 3 and 26 weeks (observational range). The 95% CI from the bootstrap characterizing the uncertainty around the model outcome (shaded area) does not appear to be especially high, despite the high uncertainty shown for the model parameter values (cf. Table 2). In addition, the median computed from the bootstrap (dotted line) was superposed to the predicted toxicity obtained from the original set of parameters (not shown). Figure 4. View largeDownload slide VPC (1000 simulated datasets) for the control group (upper panel) and 1000 mg/kg/day dose (lower panel). The 95% predictive interval from the simulated data is the shaded area, the median of the simulations is the gray solid line. The KM plot of the observations are characterized by the solid line and the 95% CI by the dotted lines. Figure 4. View largeDownload slide VPC (1000 simulated datasets) for the control group (upper panel) and 1000 mg/kg/day dose (lower panel). The 95% predictive interval from the simulated data is the shaded area, the median of the simulations is the gray solid line. The KM plot of the observations are characterized by the solid line and the 95% CI by the dotted lines. Comparison Between TTE and Logit Approaches The toxicity outcomes in an animal at a given dose (DOSE) and for a given study duration (DUR) was expressed in terms of instantaneous risk of toxicity hToxDOSE,DUR and probability of toxicity pToxDOSE,DUR, both according to the Weibull TTE and the Logit models (see Table 3). Table 3. Equations of Instantaneous Risk and Probability of Toxicity for a Given Dose (DOSE) and Study Duration (DUR), From the Weibull and Logit Models Weibull Model Logit Model Hazard hToxDOSE,DUR shapescale×DURscaleshape-1 c*exp(L)(1+exp(L) Probability pToxDOSE,DUR 1-exp-DURscaleshape expL1+expL Covariates scale=scale0*expβDOSE*DOSE L=a+b*DOSE+c*DUR Weibull Model Logit Model Hazard hToxDOSE,DUR shapescale×DURscaleshape-1 c*exp(L)(1+exp(L) Probability pToxDOSE,DUR 1-exp-DURscaleshape expL1+expL Covariates scale=scale0*expβDOSE*DOSE L=a+b*DOSE+c*DUR Weibull parameter values ( scale0, βDOSE, and shape) listed in Table 2. Logit model parameter values: a = −3.01, b = 0.0026 (mg/kg/day)−1, and c = 0.037 (week)−1. Abbrevations: hToxDOSE,DUR, instantaneous risk of toxicity; pToxDOSE,DUR, probability of toxicity; L, logit of the probability of toxicity; DOSE, dose given; DUR, study duration; a, toxicity incidence; b and c, regression parameters characterizing the effect of DUR and DOSE; scale0, time scaling factor; shape, slope. Table 3. Equations of Instantaneous Risk and Probability of Toxicity for a Given Dose (DOSE) and Study Duration (DUR), From the Weibull and Logit Models Weibull Model Logit Model Hazard hToxDOSE,DUR shapescale×DURscaleshape-1 c*exp(L)(1+exp(L) Probability pToxDOSE,DUR 1-exp-DURscaleshape expL1+expL Covariates scale=scale0*expβDOSE*DOSE L=a+b*DOSE+c*DUR Weibull Model Logit Model Hazard hToxDOSE,DUR shapescale×DURscaleshape-1 c*exp(L)(1+exp(L) Probability pToxDOSE,DUR 1-exp-DURscaleshape expL1+expL Covariates scale=scale0*expβDOSE*DOSE L=a+b*DOSE+c*DUR Weibull parameter values ( scale0, βDOSE, and shape) listed in Table 2. Logit model parameter values: a = −3.01, b = 0.0026 (mg/kg/day)−1, and c = 0.037 (week)−1. Abbrevations: hToxDOSE,DUR, instantaneous risk of toxicity; pToxDOSE,DUR, probability of toxicity; L, logit of the probability of toxicity; DOSE, dose given; DUR, study duration; a, toxicity incidence; b and c, regression parameters characterizing the effect of DUR and DOSE; scale0, time scaling factor; shape, slope. The hazard predictions were graphically compared between the TTE and the logit model in the right panel of Figure 5. At the high dose of 1000 mg/kg/day, the hazard-time profiles were very different across models: a biphasic decrease with the Weibull model versus a small increase with the logit model, resulting in a large discrepancy up to weeks 15 and an overlap in predictions at week 26. In the control group, the very sharp and early hazard decrease from the TTE model led to quasi constant hazard values, very close to the hazard derived from the logit model. The hazard predictions versus dose were all characterized by an increase with various rates and amplitudes depending on the study duration and the type of models used. The hazard-dose profiles were similar at 26 weeks but largely differed at 3 weeks with a greater amplitude with the TTE model versus the logit model. Figure 5. View largeDownload slide Sections for the response surface of hazard: (1) time-hazard relation in the control group (gray) and at 1000 mg/kg/day (black), (2) dose-hazard relation at 3 weeks (gray) and 26 weeks (black). The dotted and continuous lines represent the median predictions by the Weibull and Logit model. respectively. The area corresponds to the associated 95% CI, computed from parameter sets obtained from the bootstrap analysis. Figure 5. View largeDownload slide Sections for the response surface of hazard: (1) time-hazard relation in the control group (gray) and at 1000 mg/kg/day (black), (2) dose-hazard relation at 3 weeks (gray) and 26 weeks (black). The dotted and continuous lines represent the median predictions by the Weibull and Logit model. respectively. The area corresponds to the associated 95% CI, computed from parameter sets obtained from the bootstrap analysis. Overall, the predicted toxicity profiles illustrated in Figure 6 were consistent between the 2 models except at short study duration (between 0 and 5 weeks) where predictions are mostly extrapolated based on observed outcomes (first observation at 2.6 weeks) and model properties. For example, at time 0, the TTE model assumes a null probability of toxicity, while the logit model assumes a non-null probability, expressed from 2 regression parameters: a (baseline toxicity incidence) and b (effect of dose). Figure 6. View largeDownload slide Sections for the response surface of probability of toxicity : i) time-toxicity relation in the control group (grey) and at 1000 mg/kg/day (black), ii) dose-toxicity relation at 3 weeks (grey) and 26 weeks (black). The dotted and continuous lines represent the median predictions the Weibull and Logit model respectively. The area corresponds to the associated 95% confidence interval, computed from parameter sets obtained from the bootstrap analysis. Figure 6. View largeDownload slide Sections for the response surface of probability of toxicity : i) time-toxicity relation in the control group (grey) and at 1000 mg/kg/day (black), ii) dose-toxicity relation at 3 weeks (grey) and 26 weeks (black). The dotted and continuous lines represent the median predictions the Weibull and Logit model respectively. The area corresponds to the associated 95% confidence interval, computed from parameter sets obtained from the bootstrap analysis. The VPC plots based on the truncated data up to week 15 is illustrated in Figure 7, showing the TTE approach results in the left panels and the logit approach results in the right panels. The VPC for the TTE approach is similar to the one generated with the full dataset (cf. Figure 4) and the concordance between simulations and observations remained acceptable, even after week 15. In addition, the parameter estimates generated from the truncated dataset were similar to the original dataset (see Supplementary Appendix 3). The VPC from the logit approach differed from the complete dataset, with a faster decline of the probability of nontoxicity, mainly at 1000 mg/kg/day (probability of nontoxicity is null at week 16). Figure 7. View largeDownload slide VPC (1000 replicates) based on the model results from the truncated dataset (data with sacrifice later than week 15 removed) for the control group (upper panels) and 1000 mg/kg/day dose (lower panels). The TTE approach is shown on the left and the Logit approach is shown on the right. The 95% predictive interval from the simulated data is the shaded gray area (the dark gray color indicates the extrapolation beyond week 15), the median of the simulations is the gray solid line and KM plot of the observed survival data are characterized by the solid line and the 95% CI by the dotted lines. Figure 7. View largeDownload slide VPC (1000 replicates) based on the model results from the truncated dataset (data with sacrifice later than week 15 removed) for the control group (upper panels) and 1000 mg/kg/day dose (lower panels). The TTE approach is shown on the left and the Logit approach is shown on the right. The 95% predictive interval from the simulated data is the shaded gray area (the dark gray color indicates the extrapolation beyond week 15), the median of the simulations is the gray solid line and KM plot of the observed survival data are characterized by the solid line and the 95% CI by the dotted lines. DISCUSSION We propose a TTE approach to analyze toxicology data, in the context of a toxicity that cannot be constantly monitored and that can only be detected at time of animal sacrifice, which results in either right- or left-censored data. This type of censored data is not common in survival analysis, largely focusing on right-censored survival data, and may be expected to provide relatively little predictive information. However, in this example, the TTE model adequately described the toxicity and was able to extrapolate the risk of toxicity beyond the time of observation range. The TTE model selected in our example was a Weibull model, with the effect of dose on the scale parameter, accelerating the time of occurrence of toxicity. This is in line with the pharmacology-toxicology concepts which dictate that an increase in dose leads to an increase in drug effect as well as drug toxicity, no matter how small that increase may be. It is worth noting that, given the limited number of time points of sacrifice and the small sample size, the toxicity data at the intermediate dose groups (between 50 and 500 mg/kg/day) could not be properly summarized by the Kaplan Meier estimator and were not displayed in the plots. However, all data across the doses were used in the TTE analysis. The shape parameter in this work was found to be statistically different from 1 as supported by the AIC and BIC criteria (Supplementary Appendix Table 1) and estimated to be less than 1, leading to a decrease of the hazard over time (see Table 2). This hazard decrease was also confirmed by other TTE distributions able to describe a large range of hazard patterns, such as the log-normal and the log-logistic distributions (Supplementary Appendix 4). For this specific histopathology toxicity, the instantaneous risk is therefore expected to decrease over time and it is more likely to observe toxicity at early time points (ie, time points before the estimated value of scale0) than later time points. One potential reason explaining this decrease is that only a subpopulation of the animals is susceptible to the drug effects. Although the toxicity occurs early in those more susceptible animals, the remaining animals may be less prone to having toxicity effects regardless of treatment duration. Another hypothesis is the development of tolerance over time to this type of drug induced toxicity. Both the RSE and the bootstrap analysis showed a high uncertainly around parameter estimates of the TTE model, especially for the parameter scale0. This is probably due to the small amount of data informing this parameter. In the Weibull model, the scale parameter is the time associated with a probability of nontoxicity equal to exp(1) ∼0.37 (or a probability of toxicity of 1–0.37 = 0.63). At this specific time, the nontoxicity function is independent of the shape parameter as shown in Equation (6) and in Figure 1. In the control group, the observed incidence of toxicity is low (approximately 10%) and thus, the toxicity data from this group cannot robustly inform the time associated with a probability of toxicity as high as (1–0.37). The other model parameters shape and βDOSE could be informed by the toxicity data from the 1000 mg/kg/day, with a higher toxicity incidence and they could be better estimated (see Table 2). One potential option to decrease the uncertainty around the model parameters (and mainly scale0) would be to aggregate control group data within a large toxicology database. Another way would be to enrich the toxicity data from the active groups (eg, by increasing the sample size or the number of dose levels from the intermediate dose groups). Those data would support precise estimation for the parameters shape and βDOSE, which are linked to scale0 by the covariate relationship (Equation 11). With this model-based approach, it would become possible to simulate various toxicity studies, or to apply optimal design strategy (Mentreg et al., 2013) to select the most informative experimental design allowing a robust parameter estimation. Despite the high parameter uncertainty, the probability of nontoxicity resulting from a combination of scale0, βDOSE, and shape, was accurately described for the range of doses and treatment durations from the original dataset (see Figure 4). In this specific example, the TTE model could also predict well the nontoxicity probability for the entire dataset based on the truncated version of the data (see Figure 7). It is important to acknowledge that this extrapolation is successful only if the underlying mechanism of the hazard (eg, monotonic decrease with time assumed in the Weibull model) does not change over time. Mechanisms like either gradual selfrepair or accelerated tissue damage can potentially occur beyond the time points tested by the model and significantly deviate the predictions from the observed probability of toxicity. In the example presented here, the TTE and logit models characterize the same outcome (eg, the observed toxicity in animals for a given time and dose), and both captured the toxicity data collected across the different studies well. In both models, 3 parameters are to be estimated but their respective role is different. The TTE approach does not require any implementation of the time covariate as it integrates the effect of time intrinsically, and the Weibull model chosen in our example allows the modulation of this time effect with the shape parameter. In the logit model (Glatard et al., 2015), the effect of the dose and of the treatment duration were found to exert linear and independent effects on the probability of toxicity and any modulation of time effect would require supplementary parameter(s). The graphical comparison of the toxicity predictions between the 2 models showed somewhat different patterns of the hazard time profiles, mainly for the high dose group (see right panel on Figure 5). This difference across models was less noticeable in terms of probability of toxicity, with a reasonable overlap between model predictions from 6 to 26 weeks (see right panel on Figure 6). At the early time (<5 weeks), only few measures of toxicity were available and the predictions are highly dependent on their respective model assumptions. In addition, the predictions of hazard-time course derived from the Weibull model showed decreasing risk over time with the majority of hazard decline before week 15 (see first biphasic decline Figure 5). Additional data beyond this time point is thus expected to marginally contribute to the toxicity-time relationship. On the contrary, the linear time effect implemented in the logit model models an event with increasing risk over time making it more sensitive to the extreme values, including those beyond 15 weeks. Those characteristics may explain the differences between the 2 models in terms of parameter estimation and toxicity prediction based on the truncated data (see Supplementary Appendix 3 and Figure 7) The amount of the available data in our case example is quite unusual for a toxicology program. Regardless the amount of the data, a model-based approach, integrating data from multiple studies and quantifying uncertainty is more relevant than the empirical approach to determine NOAEL levels (6). In this current example we show that the TTE approach has better extrapolation properties than the logit model, due to its ability to modulate the time effect via various distributions. In our toxicity example, the instantaneous risk decreases over time but other patterns of hazard may be relevant for different types of toxicities. For example, a hyper sensibility reaction which develops over time may be captured by a Weibull distribution with a shape greater than 1 or a nonmonotonic risk of toxicity may be captured by a log-normal distribution. Depending on the nature of the toxicity and the potential indication, different TTE distributions would need to be explored and further assessed in terms of quantifiable insight to toxic dose and to toxicity time course and ultimately, in terms of relevant human dose extrapolation. Although flexible in terms of time modulation, the TTE approach typically assumes a common hazard distribution across the different covariate groups. For example, the Weibull model in our example applied a hazard decrease even for the control group, more likely to be associated with a constant hazard. Another assumption of the TTE approach is that it handles time events occurring only once: any toxicity which can be resolved by themselves before the end of the study would require another type of analysis, including for example Markov models (Abner et al., 2014). More informative data that can inform estimation of toxicity dissipation rate (eg, longitudinal data) would also be required. In conclusion, we previously advocated the model-based approach as part of the estimation of the maximum safe starting dose in humans and illustrated its utility by applying a logit model to real experimental data (12). The currently presented work is an extension of this, and compares a TTE model to the previous logit model using the same dataset. Despite the right and left censoring of the data, the results showed the better simulation and extrapolation properties of the TTE model compared with the logit model. This example indicates that the TTE model is a promising approach to characterize the toxicity incidence of a drug and should be further explored within the toxicology area. SUPPLEMENTARY DATA Supplementary data are available at Toxicological Sciences online. REFERENCES Abner E. L., Charnigo R. J., Kryscio R. J. (2014) Markov chains and semi–Markov models in time–to–event analysis. J. Biom. Biostat. Suppl 1(e001), 19522. 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Toxicological Sciences – Oxford University Press

**Published: ** Sep 1, 2018

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