Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 247

tumor mass vs time

x 10 7

4.5

Analysis
Truth
Free Run

4

3.5

3

2.5

2

1.5

1

0.5

0

1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Time (days)

Fig. 6 Plot of tumor mass vs. time for first observing system simulation experiment. See text for
additional details

is applied we see a reduction in uncertainty in the estimated tumor mass as well as
improved quantitative agreement with the truth.
We perform a second experiment where we assumed the enhancement level is
linearly related to the time derivative of the growing cell population (proliferating
cells for solutions to Eikenberry model). That is, regions of highest enhancement
correspond to the areas with the greatest cell growth rate. The formulation for h k
in this case is derived from the fact that solutions to the logistic equation have
maximal derivative at half carrying capacity. Thus to find the upper bound on the
time derivative of the growing cells we simply evaluate the right-hand side of the
logistic equation, g g 1

T max ,at g

g T max

2

= α − = . The upper bound on the derivative

is α T max
4

. Hence the analogue of Eq. ( 65 )is

max 0 min 1

4 ∂ u k ( x , t )
∂ t

α k T max

h k ( x )= , , + η ( x ) , (66)