gms | German Medical Science

Introduction: In low grade glioma (LGG), a primary tool for response evaluation is relative tumor size as referred to baseline volume at start of treatment. Tumor size is monitored regularly by magnetic resonance imaging. According to clinical experience the following growth behavior is expected: Following an initial phase of tumor shrinkage, the relative tumor size under treatment stabilizes around some long-term mean representing the non-responding part of the tumor. According to common response criteria in LGG, treatment is considered as successful if the long-term mean falls below some a priori fixed critical threshold. Our objective is statistical inference about the true (unknown) long-term mean using stochastic differential equation models.

Methods: In a longitudinal data setting with metric outcome variable, the linear mixed model is commonly used. It assumes a linear relationship between the known vector of observations and the unknown vector of fixed and random effects. In the underlying example from LGG, however, the relationship between dependent variable and predictor is generically non-linear, with relative tumor size stabilizing around some point of equilibrium. A possible model approach is the non-linear mixed-effects (NLME) model. Here, we instead use an approach via stochastic differential equations in order to model the time evolution of relative tumor size. The true (unknown) long-term mean is one parameter of the underlying stochastic differential equation and is estimated using maximum likelihood techniques.

Results: The likelihood function and a maximum likelihood estimate for the long-term mean are obtained in analytically closed form. This shows that a model approach via stochastic differential equations is feasible with reasonable effort in situations of clinical relevance. The variance of the maximum likelihood estimate decreases as the number of patients and the number of measurements per patient increases. The power of hypotheses tests associated with the long-term mean may thus be improved by taking into account repeated measurements. This is of interest in a personalized medicine setting when sample size may no longer be increased readily.

Discussion: Personalized medicine is a medical model that proposes the customization of healthcare using molecular analysis - with medical decisions, practices, and products being tailored to the individual patient. This implies treatment groups of small size as compared to traditional approaches. Then sample size may no longer be increased readily and repeated measurements per patient provide valuable information. In such settings stochastic differential equations (SDE) might be a viable option to model the time evolution of outcome. These models are feasible practically and analytically. Being a parametric approach, however, SDE models bear the risk of potential misspecification, since they require profound knowledge on the underlying dynamics of outcome. SDE models might thus not appear suitable in phase III trials, but might be a viable option in early clinical phases in the context of personalized medicine.

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