Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 244

Tabl e 3 Ensemble
parameter value ranges

Parameter Range of values

α [ 0 . 15 , 0 . 35 ]

T max [ 8 , 000 , 12 , 000 ]

10 − 4

D w [ 6 . 5 × , 0 . 16 ]

evolved under the Logistic Swanson model, Eq. ( 2 ). Tumor heterogeneity and error
due to parameter estimation are approximated by assigning each ensemble tumor a
unique logistic growth rate (

), carrying capacity ( T max ), and diffusion rate in white
matter ( D w ) (the preferred path of migration [ 27 ]), from a uniformly distributed
random variable in the intervals reported in Table 3 . The parameter values are
assigned at initialization and remain fixed over the entire simulation (In reality
parameters are likely to vary in time.) All other parameter values for the ensemble
tumors are identical to those reported in Table 1 . The truth and ensemble tumors
are initialized by integrating each under their respective models until they reach an
approximate diameter between 15 mm and 20 mm, measured over grid points with
cell density over 80 % of carrying capacity.
The range of parameter values in Table 3 is chosen for several reasons. First
because the models used here exhibit simple dynamics, where the cell population
generally grows to carrying capacity and diffuses outward, it is necessary numer-
ically to vary parameters such as the carrying capacity over the ensemble. This
prevents the background covariance matrix from becoming ill-conditioned and its
inverse in Eq. ( 47 ) tending to infinity which can cause the filter to diverge or neglect
the observations. Second, as our results will show, we wish to demonstrate the
ability of the LETKF to better estimate the true tumor state even in the presence
of significant model parameter error. For example, we vary D w in Table 3 by several
orders of magnitude. The result is a set of tumors that vary greatly in degree of
invasion. This is clinically relevant because diffusion rates may vary greatly on a
patient by patient basis resulting in substantial uncertainty in the tumor growth.
Our OSSE then begins with generation of a synthetic observation of the truth
and the LETKF data assimilation procedure performed using a local region size of
7 mm by 7 mm. The choice of region size is motivated by the empirical assessment
that the areas of greatest forecast uncertainty are along the edges of the tumor,
meaning the boundary of the enhancing region. This yields an updated analysis after
which the truth and ensemble are integrated for 30 days, and the process is repeated
for 6 cycles totaling 180 days worth of simulation. The process is finite due to the
fatal nature of GBM tumors.
In our simulations the observation operator, H

α

, represents the MRI that would
be observed if the tumor state were x . In our initial experiment we regard the tumor
state to be the cell density at each point on the model geometry. As previously
discussed in the introduction, many details about the relationship between the
tumor state and the contrast enhancement are not well characterized, and there is
intrinsic variability arising from uptake of the contrast agent and other aspects of
MR image generation such as interrater reliability. For our purposes we assume the
enhancement varies linearly with the tumor cell density at each point on the domain,
up to a random error.

( x )