  The availability of cancer measurements over time enables the personalised
assessment of tumour growth and therapeutic response dynamics. However, many
tumours are treated after diagnosis without collecting longitudinal data, and
cancer monitoring protocols may include infrequent measurements. To facilitate
the estimation of disease dynamics and better guide ensuing clinical decisions,
we investigate an inverse problem enabling the reconstruction of earlier tumour
states by using a single spatial tumour dataset and a biomathematical model
describing disease dynamics. We focus on prostate cancer, since aggressive
cases of this disease are usually treated after diagnosis. We describe tumour
dynamics with a phase-field model driven by a generic nutrient ruled by
reaction-diffusion dynamics. The model is completed with another
reaction-diffusion equation for the local production of prostate-specific
antigen, which is a key prostate cancer biomarker. We first improve previous
well-posedness results by further showing that the solution operator is
continuously Fr\'echet differentiable. We then analyse the backward inverse
problem concerning the reconstruction of earlier tumour states starting from
measurements of the model variables at the final time. Since this problem is
severely ill-posed, only very weak conditional stability of logarithmic type
can be recovered from the terminal data. However, by restricting the unknowns
to a compact subset of a finite-dimensional subspace, we can derive an optimal
Lipschitz stability estimate.
