  We present a novel formulation for the calibration of a biophysical tumor
growth model from a single-time snapshot, MRI scan of a glioblastoma patient.
Tumor growth models are typically nonlinear parabolic partial differential
equations (PDEs). Thus, we have to generate a second snapshot to be able to
extract significant information from a single patient snapshot. We create this
two-snapshot scenario as follows. We use an atlas (an average of several scans
of healthy individuals) as a substitute for an earlier, pretumor, MRI scan of
the patient. Then, using the patient scan and the atlas, we combine
image-registration algorithms and parameter estimation algorithms to achieve a
better estimate of the healthy patient scan and the tumor growth parameters
that are consistent with the data. Our scheme is based on our recent work
(Scheufele et al, "Biophysically constrained diffeomorphic image registration,
Tumor growth, Atlas registration, Adjoint-based methods, Parallel algorithms",
CMAME, 2018), but apply a different and novel scheme where the tumor growth
simulation in contrast to the previous work is executed in the patient brain
domain and not in the atlas domain yielding more meaningful patient-specific
results. As a basis, we use a PDE-constrained optimization framework. We derive
a modified Picard-iteration-type solution strategy in which we alternate
between registration and tumor parameter estimation in a new way. In addition,
we consider an $\ell_1$ sparsity constraint on the initial condition for the
tumor and integrate it with the new joint inversion scheme. We solve the
subproblems with a reduced-space, inexact Gauss-Newton-Krylov/quasi-Newton
methods. We present results using real brain data with synthetic tumor data
that show that the new scheme reconstructs the tumor parameters in a more
accurate and reliable way compared to our earlier scheme.
