Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 243

The data assimilation process is completed by forming the global analysis
ensemble, u a ( i ) , which consists of the collection of local analysis ensembles, x a ( i ) ,
at the center of each local region. In principle, the only free parameters in the
LETKF scheme are the ensemble size, k , and the size of each local region. In
practice, however, the model is always an imperfect representation of the underlying
dynamics. As a result, ensemble methods tend to underestimate the actual back-
ground uncertainty, which causes them to underweight the observations when the
new analysis is computed. In severe cases, the filter can diverge. One ad hoc remedy
is to “inflate” the background ensemble covariance by a tunable parameter. The
procedure described above has the effect of multiplying the background ensemble
perturbations by √ ρ , thereby helping to ensure the analysis gives appropriate weight

to the observations.

5R su s

In meteorology, tests of proposed data assimilation systems are called observing
system simulation experiments (OSSEs). Because the weather is a complex multi-
scale process, one hopes to separate the effects of observation density and error from
model error. In a perfect model simulation, one creates a “truth run” from a fixed
initial condition with the same model that is used to make the ensemble forecasts.
At intervals, synthetic noisy observations are generated from the “truth.” The goal
of the OSSE is to determine how well a forecast ensemble tracks the truth when the
synthetic observations are assimilated using a forecast model that is identical to the
model used for the truth run. Such experiments can quantify the effect of noise and
observation density and frequency on the accuracy of the analyses, since there is no
model error.
Contrary to weather prediction, where the models are well developed, efforts to
forecast a true patient GBM are likely to have significant sources of model error
because current models, such as the ones used here, represent crude idealizations of
the true tumor dynamics. Thus our numerical experiments explore a range of sources
of model error which are likely to be found in the clinical setting. While a model
with a given set of parameter values may reasonably predict the growth of a tumor,
the underlying heterogenous and genetically unstable cell population may cause
unaccounted for perturbations in key growth and migration rates. Additionally,
GBM patients typically experience a combination of treatments including surgery,
chemotherapy, and radiotherapy whose effects on the tumor are not well understood
or considered here.
Given the above anticipated sources of error, we establish proof of concept for
the data assimilation method in the GBM setting. This is accomplished by adapting
the LETKF procedure to the two models of glioblastoma discussed in this chapter
while accounting for several sources of uncertainty. We start with a “truth” tumor
whose dynamics are assumed to be governed exactly by the Eikenberry model,
Eqs. ( 3 )-( 6 ), with parameter values given by Table 2 . An ensemble of 25 tumors is