# Computationally approximated solution for the equation for Henssge’s time of death estimation

Schweitzer, Wolf; Thali, Michael J (2019). Computationally approximated solution for the equation for Henssge’s time of death estimation. BMC Medical Informatics and Decision Making, 19(1):201.

## Abstract

Background: Time of death estimation in humans for the benefit of forensic medicine has been successfully approached by Henssge, who modelled body cooling based on measurements of Marshall and Hoare. Thereby, body and ambient temperatures are measured at the death scene to estimate a time of death based on a number of assumptions, such as initial body temperature and stable ambient temperature. While so far, practical use of the method resorted to paper print outs or copies of a nomogram using a ruler, increasingly, users are interested in computer or mobile device applications. We developed a computational solution that has been available online as a web accessible PHP program since 2005. From that, we have received numerous requests not so much to detail our code but to explain how to efficiently approximate the solution to the Henssge equation.
Methods: To solve Henssge’s double exponential equation that models physical cooling of a body, it is sufficient to determine a difference term of the equation that will be close to zero for the correct time of death using a discrete set of all sensible possible solutions given that the modelled time frame has practical upper limits. Best post-mortem interval approximation yields minimal difference between equation terms
Results: The solution is approximated by solving the equation term difference for a discrete set of all possible time of death intervals that are sensibly found, and by then determining the particular time of death where equation term difference is minimal.
Conclusions: The advantage of a computational model over the nomogram is that the user is also able to model hypothermia and hyperthermia. While mathematically impossible to solve in a straightforward way, solutions to the Henssge equation can be approximated computationally.

## Abstract

Background: Time of death estimation in humans for the benefit of forensic medicine has been successfully approached by Henssge, who modelled body cooling based on measurements of Marshall and Hoare. Thereby, body and ambient temperatures are measured at the death scene to estimate a time of death based on a number of assumptions, such as initial body temperature and stable ambient temperature. While so far, practical use of the method resorted to paper print outs or copies of a nomogram using a ruler, increasingly, users are interested in computer or mobile device applications. We developed a computational solution that has been available online as a web accessible PHP program since 2005. From that, we have received numerous requests not so much to detail our code but to explain how to efficiently approximate the solution to the Henssge equation.
Methods: To solve Henssge’s double exponential equation that models physical cooling of a body, it is sufficient to determine a difference term of the equation that will be close to zero for the correct time of death using a discrete set of all sensible possible solutions given that the modelled time frame has practical upper limits. Best post-mortem interval approximation yields minimal difference between equation terms
Results: The solution is approximated by solving the equation term difference for a discrete set of all possible time of death intervals that are sensibly found, and by then determining the particular time of death where equation term difference is minimal.
Conclusions: The advantage of a computational model over the nomogram is that the user is also able to model hypothermia and hyperthermia. While mathematically impossible to solve in a straightforward way, solutions to the Henssge equation can be approximated computationally.

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