Estimating the time of death is crucial in forensic investigations. One of the most reliable and widely accepted methods is to use the cooling of the corpse to infer the time of death. The prevailing approach is based on a purely phenomenological model of corpse cooling, however this leads to a lack of applicability to non-standard situations and difficulties in integrating multiple measurements.\\
Newer models are mechanistic and based on the computational simulation of cooling of the corpse by solving the heat equation on CT-generated personalised anatomies. This approach promises to provide higher accuracy as well as more reliable uncertainty quantification [2]. Moreover, this approach has the advantage that different situations can be taken into account by adjusting the physical parameters. These include, for example, the thermal properties of the different tissues, the environmental temperature and the exact point of measurement [1]. However, the exact values are not known. To address this, we formulate the time of death estimation as a Bayesian inverse problem [3]. Solving the inverse problem requires the repeated solution of the mechanistic model, using the finite element simulation to do so is computationally too expensive for real life applications.\\

We construct a projection-based reduced order model to replace the expensive finite element simulation. This is done by the method of snapshots and proper orthogonal decomposition. As the temperature at the point of measurement is the important quantity for estimating the time of death, we consider the point evaluation to be our output of interest. Then, an adjoint problem can be defined to derive an a posteriori error estimator for the output of interest. This error estimator can be used in a greedy algorithm to build the reduced model, such that the construction is problem specific and a certain accuracy can be guaranteed for all possible parameter values. Still, the reduced model makes an error compared to the finite element simulation. We investigate the covariance of this error and incorporate it in the formulation of the inverse problem.\\

References:

[1] J. S. Subramaniam, M. Hubig, H. Muggenthaler, S. Schenkl, J. Ullrich, G. Pourtier, M. Weiser, and G. Mall. Sensitivity of temperature-based time since death estimation on measurement location. International Journal of Legal Medicine, 137:1815 – 1837, 2023.

[2] M. Weiser, B. Erdmann, S. Schenkl, H. Muggenthaler, M. Hubig, G. Mall, and S. Zachow. Uncertainty in temperature-based determination of time of death. Heat and Mass Transfer, 54(9):2815– 2826, 2018.

[3] M. Weiser, Y. Freytag, B. Erdmann, M. Hubig, and G. Mall. Optimal design of experiments for
estimating the time of death in forensic medicine. Inverse Problems, 34(12), 2018.