Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 227

mapping the patient MR scan onto the associated brain atlas (model geometry) of the
forecast model. Some sources of registration error are individual variances in patient
brain geometry and the mass effect. Radiation necrosis resulting from treatment,
which may appear similar to active tumor on the image, can further complicate MR
image interpretation [ 25 ].
The goal of this chapter is to demonstrate the use of ensemble forecasting and
data assimilation to make improved short-term (30-day) estimates of the growth
and spread of a simulated brain tumor. The remainder of the chapter is organized
as follows. Section 2 provides a brief overview of two models of glioblastoma
employed to simulate virtual brain tumors. Section 3 introduces the concept of
ensemble forecasting and the history and rationale underlying this method. Section 4
introduces the mathematical aspects of data assimilation and presents a detailed
derivation of both the basic Kalman filter for linear dynamical systems and a
state-of-the-art extension to nonlinear problems known as the local ensemble
transform Kalman filter (LETKF). Section 5 describes numerical experiments that
demonstrate implementation of the LETKF on a brain tumor models where synthetic
observations of a simulated tumor are generated. The results illustrate the potential
feasibility of this approach to better forecast the evolution of individual patient
lesions.

2 Models of Glioblastoma

GBM is the most common primary malignant brain tumor, and its prognosis is very
poor; patient survival is less than 15 months, on average, from initial diagnosis [ 23 ].
GBM tumors are highly aggressive, typically develop resistance to chemotherapy
and radiotherapy [ 1 ], and can quickly invade large and sensitive regions of the
brain, making complete surgical resection of the tumor impossible and postsurgical
recurrence inevitable [ 6 ]. Because little clinical progress has been made against
GBM, its biology remains the subject of intense study.
In this chapter we use two mathematical models to simulate tumor growth and
expansion. These models are chosen because they each represent syntheses of some
previous GBM modeling efforts and reflect increasing levels of complexity. Both
models are simulated on a static but realistic brain geometry where diffusion rates
are differentiated between white matter, gray matter, and other intracranial tissues.
The first model, initially proposed by Swanson et. al. [ 27 , 28 ], considers a single
class of glioma cells exhibiting exponential growth with cell motility governed by
Fick's law. These assumptions yield the equation

t = ∇ · D

g + α

∂ g

( x ) ∇ g . (1)

∂

The diffusion rate of GBM cells is assumed to be faster in white matter than
in gray. Each tissue type is further assumed to be homogeneous with respect to its