Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 231

The spatial derivatives are approximated by finite differences, and the resulting
set of ordinary differential equations is integrated using the second-order (in time)
Heun's method with a fixed time step 0 5day − 1 in the case of Logistic Swanson

and a fourth-order Runge-Kutta method for the Eikenberry with fixed timestep of
0

.

1day − 1 . The numerical methods chosen here are for computational efficiency
and haven't exhibited instabilities with the chosen time steps. Given the discrete
nature of the brain geometry, location-dependent parameters (such as the diffusion
constants) are taken to be piecewise constant.
These models are chosen for this initial investigation because they are inexpen-
sive to integrate, particularly in two dimensions, and are adequate for illustrating
the basic ideas behind data assimilation, which is our principal focus here. Other
efforts have modeled important aspects of GBM growth such as various forms of
treatment [ 32 ], the mass effect of the tumor, and directed diffusion of GBM cells
along white matter tracts [ 3 ]. Such refinements would undoubtedly be part of any
data assimilation system intended for clinical application.

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3 Ensemble Forecasting

Ensemble forecasting, a technique used to assess and quantify the effect of
uncertainty in a mathematical model of a dynamical system, was developed from
early studies of chaotic behavior. A classic example, formulated by Edward Lorenz
in 1963 [ 19 ], consisted of a system of three coupled ordinary differential equations
modeling fluid flow. The system exhibited sensitive dependence on initial condi-
tions. That is, small errors in non-fixed-point initial conditions quickly propagated in
time, leading to large differences in solutions. Even though trajectories had similar
limit sets, they became uncorrelated over time even when the initial conditions were
very similar. In the case of weather forecasting, like many other systems, one cannot
sample the atmosphere at every point, observations are corrupted by noise, and any
given weather model is imperfect. This, coupled with chaotic behavior, leads to
models that offer no predictive advantage over climatological averages. Even on
timescales of a few days or less, uncertainties in the initial state of the atmosphere
may lead to substantial forecast errors.
To account for uncertainties in a forecast model, Lorenz suggested in 1965 [ 20 ]
that, instead of simulating a single initial condition under the model from a best
estimate of the state of the atmosphere, one should evolve a set or ensemble of
initial conditions, each from a statistically equivalent estimate of the true initial
state. Then the ensemble gives a Monte Carlo estimate of the uncertainty in a given
weather model. Under assumptions discussed in the next section, the ensemble
mean constitutes an empirical maximum-likelihood estimate of the true state of the
atmosphere. Ensemble forecasting became part of the routine operations at the U.S.
and European weather centers in 1992 [ 16 ].
Figure 3 shows representative ensemble forecasts of geopotential height contours
at 500 hPa. Each curve shows the result, from one initial condition on Oct. 12, 2010,