Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 250

Effect of Varying Observation Frequency

We explore the effect of varying the period between assimilation steps. To do so
our first experiment is repeated with synthetic observations generated every 1, 7,
30, 60, 90, and 180 day(s). Each simulation is still conducted over a period of
180 days beginning and ending with a data assimilation cycle. Figure 9 shows
the final analysis mean cell density. The subplot titled “Daily” corresponds to
observations everyday. “Weekly” corresponds to observations every 7 days. The
other subplots are titled analogously. In every case but the quarterly run, the final
ensemble mean gives a good approximation of the truth tumor core (region in red)
with varying degrees of satellitosis. The edema (blue region) surrounding the main
tumor mass varies significantly over the experiments and in some cases gives a
poor approximation to the truth (see the truth in Fig. 4 ). This is because when
observations are available frequently, for example in the daily run, the errors in
the model do not sufficiently propagate between assimilation cycles. The resulting
background ensemble has little variance in the edema region causing the updated
analysis to essentially ignore the observations locally. Despite varying the frequency
of observations, the simulations show that the LETKF data assimilation scheme can
still provide an improved approximation to the true tumor state. This is clinically
relevant because usually patient images are collected at irregular intervals and
typically only at initial diagnosis and before and after surgery.

6 Conclusion and Discussion

The results of this chapter demonstrate the use of ensemble forecasting and data
assimilation to make improved estimates of future growth of a simulated glioblas-
toma given synthetically generated observations of the tumor. The two experiments
presented explore different models of the relationship between the tumor state and
the contrast enhancement in an MR image. In both cases the ensemble mean had
improved tracking of the truth when data assimilation is performed despite the
substantial sources of model and observational error. This demonstrates the potential
feasibility of this framework for use in human cases of glioblastoma with real patient
MR image data.
Several related problems must be considered before the data assimilation
approach can be used as a clinical aid. First the effect of surgery, radiotherapy,
and chemotherapy on the glioma cell population needs to be incorporated into the
forecast model. Similarly any ensemble forecast must account for the uncertainty
and variance in key treatment parameters such as the size of the surgical resection
cavity, treatment dosage and timing, the application field for radiotherapy or
chemotherapy, and development of resistance by glioma cells.
A third problem concerns the use of patient imaging data in an ensemble forecast.
Here we have modeled the relationship in an ad hoc way where we assume the
pixel intensity directly relates to the tumor state under the statistical assumption of