Technical note: Excel spreadsheet calculation of the Henssge equation as an aid to estimating postmortem interval

Abstract

In forensic cases for which the time of death is unknown, several methods are used to estimate the postmortem interval. The quotient (Q) defined as the difference between the rectal and ambient temperature (Tr − Ta) divided by the initial difference (T0 − Ta) represents the progress of postmortem cooling: Q = (Tr − Ta)/(T0 − Ta), (1 ≥ Q ≥ 0). Henssge was able to show that with the body weight and its empirical corrective factor, Q can be reasonably predicted as a double exponential decay function of time (Qp(t)). On the other hand, actual Q is determined as Qd by measuring Tr and Ta under an assumption of T0 = 37.2 °C. Then, the t value at which Qp(t) is equal to Qd ($Qd=Qp\left(t\right)$) would be a good estimate of the postmortem interval (the Henssge equation). Since the equation cannot be solved analytically, it has been solved using a pair of nomograms devised by Henssge. With greater access to computers and spreadsheet software, computational methods based on the input of actual parameters of the case can be more easily utilized. In this technical note, we describe two types of Excel spreadsheets to solve the equation numerically. In one type, a fairly accurate solution was obtained by iteration using an add-in program Solver. In the other type (forward calculation), a series of Qp(t) was generated at a time interval of 0.05 h and the t value at which Qp(t) was nearest to Qd was selected as an approximate solution using a built-in function, XLOOKUP. Alternatively, a series of absolute values of the difference between Qd and Qp(t) (|Dq(t)| = |Qd − Qp(t)|) was generated with time interval 0.1 h and the t value that produces the minimum |Dq(t)| was selected. These Excel spreadsheets are available as Supplementary Files.