Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 233

In Sect. 5 we illustrate the use of ensemble forecasting adapted to the two models
of glioblastoma discussed in Sect. 2 . While both models exhibit simple, non-
chaotic dynamics where the cancer cell population grows to carrying capacity while
diffusing outward, there are still several sources of uncertainty. Some of these
include the initial population density and measurement of model parameters. Thus
in our simulation we consider both an ensemble of models distinguished by slightly
different parameter values and an ensemble of slightly different initial conditions
assigned to each of these models.

4 Data Assimilation

In this section we derive the LETKF data assimilation procedure authored by Hunt
et al. [ 13 ]. This method is used to update an ensemble of initial conditions in light of
new observations. The formulation uses elements from [ 9 , 13 ]. The general approach
may be stated as follows: Given an imperfect forecast model which advances a
model trajectory from time t n − 1 to t n , u t n = F ( u t n − 1 , t n − 1 ) , and noise-corrupted

observational data, y 1 , y 2 ,..., y n , estimate solution trajectory, u

( t ) , that best fits the

observations. Assume now that at each time t i , i ∈ 1 ... n , the observation is related to

the system state, u ε i is a Gaussian random vector

with mean 0 and covariance matrix, R i . Then the problem can be stated precisely
by seeking the maximum-likelihood estimator of the trajectory that best fits the
observational data. In other words, we wish to maximize the likelihood function

( t i ) ,by y i = H i ( u ( t i ))+ ε i ,where

j = 1 exp

n

1
2 [

y j −

T R − 1
j

y j − L [ u ( t )] = − H j ( u ( t j ))] [ H j ( u ( t j ))] . (10)

Taking the log of Eq. ( 10 ) we see that its maximizer corresponds to minimization of
the cost function defined by

n
j = 1 [ y j − H j ( u ( t j ))]

T R − 1
j

y j − J [ u ( t )] = [ H j ( u ( t j ))] . (11)

When the forecast model and observation operator are nonlinear, cost function
J may not have a unique minimizer, and even if it does, finding it can be
computationally difficult. The LETKF combats this problem by approximating the
minimum in a manner based on the Kalman filter [ 14 , 15 ]. We proceed to derive
the Kalman filter below, which produces the minimizer analytically for the case of
a linear model and observation operator under the assumption that the minimizer
represents Gaussian distribution.