Abstract
BackgroundData assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor.ResultsWe apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise.ConclusionsThe mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling.ReviewersThis article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck).
Highlights
Data assimilation refers to methods for updating the state vector of a complex spatiotemporal model by combining new observations with one or more prior forecasts
2.3 Data assimilation we briefly describe the rationale and algorithmic implementation of the Local Ensemble Transform Kalman Filter (LETKF) for data assimilation. (See Hunt et al [17] and Ott et al [18] for a detailed mathematical justification.) The basic problem may be stated informally as follows: Given a forecast model consisting of a coupled system of ordinary differential equations, u = F(u, t), find the trajectory u(t) that best fits the observations
The immediate application is to glioblastoma, the design and implementation of the Local Ensemble Transform Kalman Filter (Sec. 2.3.3) do not depend on the particular equations of a given mathematical model
Summary
Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. Mathematical models, typically a system of ordinary or partial differential equations, can provide considerable insight into the dynamics of biological systems For initial investigations, it suffices to determine whether a model provides good qualitative agreement with the dynamical process under study. We address the question of how differences between the predicted state of a biological system can be reconciled with noisy measurements to correct the forecast in view of new information; this process is called data. The data assimilation procedure updates the background in light of the new observations to produce an analysis, which, under suitable assumptions, is the maximum likelihood estimate of the model state vector. Section 2.3.3 outlines one state-of-the-art procedure for performing the state update in complex spatiotemporal models
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