1. Introduction
Many authors have suggested that a single-exponential thermodynamic model predicts the time of death (TOD) or postmortem interval (PMI) based on evidence from pigs, as shown in an early study of Rainy [
1], and recently that of Kaliszan et al. [
2]. While the general trend is not in dispute, predicting the desired precision needs improvement to ascertain the PMI.
Noakes et al. [
3] used eight methods to estimate the postmortem interval, wherein two rule-of-thumb methods compared well with six other mathematical models published by independent investigators [
4,
5,
6,
7,
8,
9]. Marshall and Hoare [
6] and Marshall [
10,
11], proposed a double-exponential model to describe the cooling curve’s Sigmoidal shape. Al-Alousi et al. offered a triple-exponential model for 117 forensic cases in a two-part article [
12,
13]. Mall and Eisenmenger presented a complex heat flow model using a finite-element numerical solution [
14,
15]. Subsequently, a Laplace-transform approach of Rodrigo proposed a compartment-based model and addressed the convection from skin according to Newton’s law of cooling [
16]. This method provided the results within half an hour of PMI.
Using a wooden cylinder model and following the conductive heat-transfer principle, a Fourier series model of Smart provided a credible approach for estimating the PMI and suggested that the outer ear’s temperature measurements lead to reasonable estimation [
17]. Similarly, Baccino et al. preferred the exterior-ear temperature based on the rule-of-thumb correlations and field experience [
18]. Another study by Smart and Kaliszan also pointed out PMI estimation complications due to the temperature plateau effect and suggested constructing a temperature decline curve for better assessment [
19].
Although more than one exponential term in the temperature-decline trend appeared in the literature, one article by Kaliszan demonstrated that one-exponential term suffices [
20]. In subsequent studies, Kaliszan proposed measuring the eyeball temperature because of its faster decline rate [
21,
22]; this approach resulted in the PMI estimation accuracy of ±1 h for a 95% confidence interval [
21]. Perhaps the rapid temperature decline rate ensures higher accuracy in measurements due to increased fidelity requiring fewer data points. Nelson suggested average-based methods for short-term estimates of PMI [
23]. Still, it utilizes many parameters, which leads to improved fitting, but the process loses the estimation efficiency. A very recent study by Laplace et al. [
24] showed that the average PMI estimation turned out to be 4.5 ± 2.5 h on 100 inpatient bodies. The Henssage [
9] nomogram and Baccino’s [
18,
25] formulae produced these results. Overall, the use of conventional methods produces a large degree of uncertainty in PMI estimation.
More recently, new methodologies are emerging for PMI estimation based on corneal thickness and aqueous humor measurements. For instance, Napoli et al. [
26] showed that the central corneal thickness measured by optical coherence tomography correlates strongly with PMI. In contrast, Locci et al. [
27] used a
1H NMR metabolomics approach to estimate the PMI from aqueous humor (AH) in an ovine model. Based on the spectral data analysis with multivariate statistical tools, this approach involving postmortem biological modifications provided an error of about 1 + h of PMI on animals. Their subsequent study [
28] showed a general solution alignment with expectation, but the error bar appears large. These studies offer an attractive platform given the eye compartment resists postmortem modifications. Also, studies have appeared relating postmortem vitreous concentrations of sodium and chloride. For example, Zilg et al. [
29] studied 3000 cases to demonstrate that vitreous sodium and chloride levels decline at about 2.2 mmol/L per day upon death.
This study explores the feasibility of extrapolating the late-time temperature data to find the PMI with improved accuracy and identifies the body part that ensures reliable solution quality. Data from the literature helped augment our case in that the heat transfer rate, a result of that temperature decline rate, is specific to an individual body. For instance, the temperature decline rate depends on body weight, meaning lower weight exhibits a higher cooling rate. Overall, data from an internal organ (IO) or rectum formed the basis of this investigation. We showed that the heat transfer rate could provide an excellent physical perspective on the temperature decline rate through the cooling rate constant, which is the time-derivative of temperature. In this context, estimating the heat-flow rate from a given organ provides the necessary insight.
Also, we present a hyperbolic method for estimating the PMI, regardless of the body organ. We show that the hyperbolic trend can successfully describe the cooling trends from many body parts using a single expression. Although the temperature data from skin or eyeball are limited, synthetic data show the value proposition of monotonic or near-monotonic behavior. This trend thereby facilitates reliable extrapolation to the actual value of the PMI. We found that the proposed hyperbolic method yields PMI solutions primarily within 1.65 °C for the scope of this investigation. Given the limited accessibility of modern forensic data in actual cases, we consider the results of this study to be a proof of concept.
2. Materials and Methods
Given the dominance of convective heat transport, many investigators described the temperature of a human body after death using Newton’s law of cooling. The body temperature at time t after death can be quantified using the following equation:
where T is the temperature of the human body at time t, T
a is ambient temperature, T
i is the temperature at the time of death, and k
c represents cooling constant.
Considering body temperature measured at times t
1 and t
2, we can rewrite Equation (1) in the following forms:
and
Dividing Equation (3) by Equation (2) and simplifying, we have
Rearranging Equation (4) for the cooling constant, k
c, leads to the following expression:
Now, Equation (5) allows k
c estimation using the body temperature measured at times t
1 and t
2. We present two case studies involving Equation (5) for actual and synthetic data in
Section 3.2. This cooling constant paved the way for understanding the temperature data collection from various body parts and the body’s physical characteristics, such as weight.
With a 3-D whole body heat-transfer model, a recent study of Bartgis et al. [
30] has shown that we can write the heat-flow rate in the postmortem period written as
where q
c is the heat loss for a unit volume of the tissue, ρ
t is the density, and C
t is the heat capacity of the tissue. Bartgis et al. [
30] reported the density and heat capacity for the human internal organ are 1000 kg/m
3 and 3500 J/kg-°C, respectively. We show the value proposition of q
c estimation for different organs to understand the PMI estimation.
Given that Newton’s law of cooling implies exponential temperature decay with time, we explored other data-fitting options to enlarge the scope of this investigation. A recent article by Sharma et al. [
31] suggests that the hyperbolic trend effectively captures the decline behavior of fluid and heat flow in porous media. Following that approach, we can write the temperature-decline behavior as:
In Equation (6), the time-depended temperature, T (t), relates to time, t, involving three parameters, T
i, b, and D. Note that T
i reflects the starting point of data fitting and is user input. The parameter D represents the hourly temperature-decline rate (1/h), and b is the time-derivative of D, which is dimensionless. Note that for exponential temperature decay or when Newton’s law of cooling applies, b equals zero, and Equation (6) takes the following form:
So, the exponential temperature decline is a particular case of hyperbolic decay, encompassing the entire decline trend domain. Synthetic data in the modern era suggest that temperature response varies with the point of body measurement. For instance, the temperature measured on the skin declines faster when compared with the brain and rectum or IO.
Figure 1 displays the temperature responses from the model, as presented in Bartgis et al. [
30], showing the apparent differences in trend, depending on the body part. In particular, the skin curve exhibits a monotonic trend with the steepest decline, which is exponential. In contrast, the other two responses show a slow decline trend at early times.
Our curiosity stemmed from these variable trends, and we learned how a single-temperature measurement could lead to PMI estimation. In our view, potential obstacles may surface, given that the earlier response (<5 h) differs from the latter. Perhaps this reality propelled previous investigators to use double-exponential terms, as shown by Marshall [
10,
11], or triple-exponential formulation, as in Al-Alousi et al. [
12,
13] models. Still, as Henssge [
9] showed, the double-exponential model yielded the PMI estimation within ±3.2 h.
We explored an understanding of some of the well-known studies involving the pioneering work of de Saram et al. [
4] and Lyle and Cleveland [
32] and those in the modern era [
12,
13,
20,
22,
30,
33,
34]. Except for the recent studies of Bartgis et al. [
30] and Kanawaku et al. [
33] with models, other studies involved human bodies. Most recently, Wilk et al. [
34] provided real-time temperature measurements and validations of their numerical heat-transfer modeling approach for four bodies, wherein real-time temperature data gathering occurred in the morgue. We explored the overall results in a two-step approach. The first step attempted to gain insights into the overall results involving human bodies and synthetic data for 102 cases. In the second step, we present four examples illustrating the merit of collecting the time-dependent temperature data rather than just one data point using the hyperbolic approach.
4. Discussion
Although our exposure to modern datasets may be limiting, this investigation paved the way for learning a few valuable lessons for estimating the PMI. For instance, the cooling constant is person-specific; therefore, predicting the temperature-decline behavior demands time-variant data over several hours. Also, the early-time data (<7 h) behaves differently for the brain and IO than at late times involving more than 11 h. The change in temperature behavior can be detected by only taking its time-derivative. In contrast, the skin temperature declines monotonically, leading to fitting one exponential expression for a given body.
Bartgis et al. [
30] show that the time constant also depends on the ambient condition. For instance, it can change from 8 h to 31 h when the brain’s ambient air temperature varies from 30 °C to 10 °C. They showed that the heat loss is lower for the IO. Overall, the ambient condition dictates the PMI window.
Lack of time-lapse data appears problematic on many fronts, mainly because the k
c-parameter appears person-specific, in terms of age, gender, height, weight, clothing, among others, when the measurements occur in the internal organ. Groups of data, characterized by the k
c-parameter in the de Saram et al. [
4] datasets, support this notion. The body weight appears to be the most critical variable, meaning a body with a lower weight declines more rapidly than the one with a heavier weight.
We applied the well-known Henssge [
9] nomogram technique for Bartgis et al. [
30] internal organ and brain data. For this dataset, the body weight was 68 kg, and the ambient temperature of 20 °C. When we applied the nomogram method for the body temperature measurement of 31 °C, the PMI estimation turned out to be 10.1 ± 2.8 h, when the correct PMI corresponds to 19 h. Similarly, for the brain, the nomogram estimated PMI of 4.5 ± 1.5 h diverged from the correct PMI of 11.75 h. Overall, our limited use of the Henssge [
9] nomogram suggests that the PMI solutions generated with this tool offer considerable uncertainty.
This study also suggests that the early-time data fitting can lead to realistic solutions when the temperature measurements occur in any body organ. A monotonic signature on the temperature-derivative plot involving a single-exponential trend appears for the skin. However, data fitting and extrapolation to time zero for the other body parts after 11 h present serious challenges, given the V-shaped temperature-derivative signature.
Given that our findings on applying the hyperbola anchors on synthetic data, the time-lapse data on actual human bodies become necessary to prove this conceptual approach. We note that the body’s movement to another temperature environment, such as a morgue, becomes an obstacle for applying this method. In that situation, the body imaging data reveals the necessary temperature data for the prior environment, as shown in Wilk et al. [
34,
35]. Then, one can use the proposed simple tool to validate the numerical model results in the pre-morgue situation.
Indeed, the Wilk et al. [
34] article provided the necessary high-frequency data with new imaging technology to explore the application of other tools, such as the one proposed here. In contrast to the prior studies, Wilk et al. did not find any issue with any body part delivering non-monotonic signature with their numerical modeling approach. Perhaps a closer investigation needs doing to provide clarity on this issue. Their follow-up article [
35] found that the average PMI solutions for five measurement locations to be −0.17 ± 1.63 h, whereas the reconstructed PMIs deviate no more than ±2.8 h. In this context, the results of this study show that the hyperbolic relation confines the PMI solution to within ±1.65 h. The overall PMI error for the 25 cases studied turns out to be 0.244 h.