Modeling of Longitudinal Tumor Size Data in Clinical Oncology Studies of Drugs in Combination N. Frances 1, L. Claret 2, F. Schaedeli Stark 3, R. Bruno 2, A. Iliadis 1 1 School of Pharmacy, University Méditerranée, Marseille, France; 2 Pharsight Corp., Mountain View, USA; 3 Hoffman-La Roche, Basel, Switzerland #1355 Flow chart for model development 1. Choose a model Proliferation Gompertz vs. exponential Resistance term Dynamical dose model Estimation of the initial condition Residual variability Define bounds in the parameters 2. Define the KB values in the dose model Studies on single agent data Phase II-C ( ) Phase III-D (14999) Validation on the combination (C+D) data 3. Modeling tests : Phase III-(C+D) Covariance matrix of estimates Parameter variability (Omega) Residual error (Sigma) Non usable models Validate d models 4. Comparison between retained models Objective function Residual error Final model 5. Ultimate validation Dispersion plots and histograms (Figure 1) Individual fits (Figure 2) “Posterior Predictive Check” (Figure 3a, b) Minimum number of patients (Figure 4) ABSTRACT Introduction : The analysis of tumor size measurements, obtained in clinical studies involving combination chemotherapy, remains an open modeling problem. We used retrospective clinical data in metastatic breast cancer in order to investigate whether the contribution to the anti-tumor effect of each compound in a combination setting can be estimated 1) from single agent data and combination data with or without single agent data, and 2) from datasets with a limited number of patients. Methods : Data concerning tumor size measurements and treatments characteristics were available for docetaxel (D, n=223), capecitabine (C, n=168) [1, 2] given as single agents and their combination (D+C, n=222) [3]. The developed model is an extension of already presented disturbed growth models [4, 5] and it is based on the following hypotheses: 1) Tumor growth is exponential or Gompertz; 2) K-PD model describes administration protocols; 3) Resistance is materialized by exponential decline of cell-kill rate; 4) Drugs are combined either in a linear, or Emax, or Weibull model involving a drug interaction term. Population analyses were performed using NONMEM Version 6 within a MATLAB environment. The models were validated using posterior predictive checks. Results : In the developed models, over-parameterization was the most frequent problem. K-PD models involve only one parameter expressing the dynamics of drug amounts in the cell-kill rate formulation. This parameter was obtained for D and C from the single agent studies and was fixed in the analysis using the combination data only. When using the combination data only, the contribution of each drug to the anti-tumor effect was accurately estimated and the estimates were consistent with those obtained using single- agent data. The effect of the 2 drugs was found to be additive with no drug interaction term. Situation #2 is still under investigation. Conclusion : Using combination data, the tumor size dynamic model parameters were successfully estimated. Further investigations are in progress for assessing the minimum required extent and type of clinical data for evaluating drug combinations in oncology. This model will be part of a modeling framework to simulate expected clinical response of new compounds and to support end-of-phase II decisions and design of phase III studies [6]. References: [1] Blum JL, Jones SE, Buzdar AU, et al: Multicenter Phase II Study of Capecitabine in Paclitaxel-Refractory Metastatic Breast Cancer. J. Clin. Oncol. 17: , [2] Blum JL, Dieras V, Mucci Lo Russo P, et al: Multicenter, phase II study of capecitabine in taxane pretreated metastatic breast carcinoma patients, Cancer 92: , [3] O’Shaughnessy J, Miles D, Vukelja S et al. Superior survival with capecitabine plus docetaxel combination therapy in anthracycline-pretreated patients with advanced breast cancer: Phase III trial results. J. Clin. Oncol. 12: , [4] Iliadis A, Barbolosi D: Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model. Comput. Biomed. Res. 33: , [5] Claret L, Girard P, Zuideveld KP, et al: A longitudinal model for tumor size measurements in clinical oncology studies. PAGE 15 (abstract 1004), 2006a [ meeting.org/?abstract=1004]. [6] Claret L, Girard P, O'Shaughnessy J et al: Model-based predictions of expected anti-tumor response and survival in phase III studies based on phase II data of an investigational agent. Proc. Am. Soc. Clin. Oncol, 24, 307s (abstract 6025), 2006b. Conclusion The model is built from Phase III data : two drugs (D+C) in combination, resistance parameter common to both drugs and acting by increasing proliferation term (selection of resistant cells by the treatment), interaction term not estimated (assumes additive effects). This model can be used to predict therapy efficacy in a future clinical trial [6] : using Bayesian approach, a minimum number of patient seems to be necessary, but small sample sizes typical to those in early clinical studies (e.g. 50 patients) may be enough, instead of K-PD, a PK-PD model would supply consistent information. Objectives  Elaborate the best model (parsimonious principle) fitting the longitudinal tumor size data on :  Single agent data  Combination data  Is this model able to describe the contribution of each drug in the combination data ?  Can a drug interaction term be estimated ?  What is the minimum number of patients in a study to obtain a good enough estimation of the model parameters ?  e.g. in a prospective Phase Ib or Phase II study Remove the variability added by individual designs Figure 3a. Flow chart for Posterior Predictive Check Actual data (drug combination) pdf(ratio) Predicted ratio « Observed » ratio (from actual data) Reference design Random drawn parameters n=1000 Posthoc estimated parameters n=222 NONMEM Intra- and inter-patient variability Reference design « Posterior Predictive Check » on ( at the first visit ) Probability density functions of the ratio : For 4 typical designs, predicted ratio from : posthoc estimated parameters (, ) and randomly drawn parameters (, ) Figure 3b. Posterior Predictive Check pdf Design ID n° Design ID n° Tumor size ratio pdf Design ID n° Tumor size ratio Design ID n° 333 Figure 2. Individual fits For 9 patients from the phase III combination study (C+D), observed tumor size data (o) and model predictions vs. time : population ( ), individual ( ) Patient ID n° Patient ID n° Patient ID n° Patient ID n° Patient ID n° Patient ID n° Patient ID n° Patient ID n° Patient ID n° 153 Time (weeks) Tumor size (mm) Data presentation  Retrospective analysis :  2 drugs in metastatic breast cancer Docetaxel (D) Capecitabine (C)  PD-data : observed tumor burden sum of the longest diameter of metastatic sites measured (dependent variable in NONMEM)  3 studies [1, 2, 3] :  # : phase II data on C ( )  # : phase II data on C ( )  # : phase III data on C+D ( ) vs. D ( )  Data already treated by a different growth model [4, 5] Numerical results Fixed effects : Random effects are log-normal distributed : Residual error is proportional : Objective function :(>100 models tested) Final model Model explanations  : “Effective dose” for C and D respectively  : Administration protocols for C and D respectively  : Tumor size  : K-PD elimination constants (already evaluated on single agent data)  Estimated parameters : : Proliferation parameter (max tumor size : mm, fixed) : Resistance parameter, common to both drugs : Constant cell kill rate (efficacy parameter), distinct for each drug : Initial tumor size Figure 4. Minimum number of patients (Probability density functions of model parameters) Samples were obtained from 100 random permutations of the 222 patients data in the phase III combination study. 50-patients tailed samples ( ) and 70-patients tailed samples ( ). Covariance matrix obtained : 26/100 with 50 patients and 38/100 with 70 patients KL KEC KED R n0 Figure 1. Dispersion plots and histograms KL KEC KED R n KEC KED R n