Modeling tumor size and survival in patients with metastatic gastric cancer

Aims: The aim of this study is to 1) model the temporal course of tumor size of stage IV gastric cancer patients enrolled into 1st line chemotherapy trials, and 2) predict progression free survival (PFS) and post-progression survival (PPS) using tumor size model parameter estimates and patient covariates.

Methods: We analyzed 69 advanced gastric cancer patients enrolled into the ToGA study – a parallel, randomized, open-label, multi-center study – conducted in Severance Hospital, Seoul, Korea, who received one of the following 3 regimens – TXP, TFP and XP (T: Trastuzumab (Herceptin), X: xeloda (capecitabine), P: cisplatin, F: 5-FU) – beginning from December 2005. The time course of tumor size, recorded as a sum of longest diameter of all target lesions, was modeled using the following assumptions. First, kg, tumor growth rate constant, was assumed to increase with tumor size with its initial rate depending on the baseline tumor size ("density dependence"). To this end, logistic growth model was used. Second, from clonal selection theory, kg was assumed to increase with time ("time dependence"). Third, tumor cell populations were partitioned based on sensitivity profiles to different chemotherapeutic regimens, where population fractions sensitive to each drug were estimated as model parameters. Given no concentration data available, drug effect of each drug was modeled as a log-linear function of a hypothetical drug amount obtained from a KPD model linked with a delay compartment. In KPD modeling, a common rate constant was assumed for all drugs assuming X and F have the same drug effect. Total drug effect, which was assumed be a simple sum of drug effect to each drug, was then assumed to inhibit kg. In modeling tumor size, the baseline tumor size was treated as a covariate considering the nature of highly asymmetric distribution thereby being prone to introducing a bias in estimating other model parameters, and the carrying capacity of tumor growth was assumed to increase with the baseline tumor size. For survival modeling, PFS was predicted using the predicted depth of response (DoR) and patient covariates. The predicted DoR was obtained from the initial tumor shrinkage rate, which was either estimated from the baseline or observed from the tumor size measured at the first visit. Finally, a survival model of PPS was fitted using the predicted PFS and patient covariates. Internal validation of the model was performed using visual predictive checks.

Results: With kg increasing linearly with time due to selection pressure, the final tumor size model was parameterized using probability of sensitivity to each drug, rate of tumor progression (which is theoretically equal to tumor heterogeneity), rate constant of KPD model, scale factor of chemotherapeutic effect, and carrying capacity of tumor growth. The fraction of cells resistant to all three drugs was estimated to be about 63.99% for HER2 < 3+ and 49.41% for HER2 3+. Roughly, half of the total tumor cells in a typical patient are sensitive to at least one of the three drugs. The inter-individual variability associated with this probability, however, was large (CV% > 160%). The estimated rate of increase of kg is 11%/month/mm. There was a strong tendency of patients with lower posthoc probability of trastuzumab resistance to be associated with HER2 3+ status. The model well described the general time course of tumor size changes of most patients, and achieved a good fit with observed data. PFS was positively correlated with higher DoR and lower histologic grade. When survival analysis of PPS was done using the predicted PFS and patient covariates, longer predicted PFS and lower baseline tumor size seemed to confer a survival benefit, but only when histologic grade was either well- or moderately-differentiated. Patients with either poorly-differentiated or signet ring cell histology showed poor prognosis regardless of PFS or baseline tumor size. Visual predictive checks of the survival models suggested that our models well predicted the observations.

Conclusion: A new concept of tumor size progression, which is dependent on both tumor density and progression time, was presented. The main findings are: (1) tumor shrinkage rate is inversely proportional to tumor size (favoring Norton-Simon hypothesis over log-kill hypothesis) and thus higher baseline size is naturally associated with poorer prognosis. (2) Probability of trastuzumab sensitivity seems to positively correlate with the positivity of HER2 receptor status(3) With regards to PPS, histologic grade is the strongest determinant. Only when histologic grade is low does PFS and baseline tumor size useful for predicting PPS. This work demonstrates the feasibility of applying a model-based approach to predict tumor growth and patient survival.