# A Simplified Approach to Understanding Body Cooling Behavior and Estimating the Postmortem Interval

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## Abstract

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## 1. Introduction

^{1}H 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.

## 2. Materials and Methods

_{a}is ambient temperature, T

_{i}is the temperature at the time of death, and k

_{c}represents cooling constant.

_{1}and t

_{2}, we can rewrite Equation (1) in the following forms:

_{c}, leads to the following expression:

_{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.

_{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.

_{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:

## 3. Results

#### 3.1. Understanding the Significance of Cooling Constant, k_{c}

_{c}estimation in a meaningful statistical distribution. Figure 2 shows the distribution of the range of the cooling constant, k

_{c}. This distribution range of k

_{c}reflects the underlying reasons, such as body weight, height, gender, clothed, unclothed, and room temperature. Given this reality, we attempted to understand some of these variables on PMI, as shown in Appendix A.

#### 3.2. Diagnosing the Fitting Window with the Heat-Flow Rate

_{c}. The non-monotonic signatures, as in Figure 3a, for both the brain and IO provide clues about two-time domains that need honoring while estimating the cooling constant. However, that is not so for the skin, where the monotonic signature appears. This non-monotonic trend helps ascertain the time window for estimating the cooling constant. Let us point out that estimation of the heat-flow rate is not needed; just the time-derivative of temperature suffices as Figure 3b exhibits, given that only the two parameters, ρ

_{t}, and C

_{t}, differ.

#### 3.3. Application of the Hyperbolic Approach for PMI Estimation

_{c}and D, 1/h, the D parameter in the Arps expression can be a surrogate of k

_{c}. Regardless, the Arps hyperbola can fit a range of decline trends with credible PMI solutions, which appears reassuring.

## 4. Discussion

_{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.

## 5. Conclusions

- A rapid temperature decline rate occurs initially, followed by a slower pace, regardless of the body part. Therefore, the hyperbolic trend can describe the overall signature. As part of the general hyperbolic trend, the exponential decay may represent some aspects of the overall signature, but it does not appear systemic. Stated differently, the application of Newton’s law of cooling does not appear holistic in the body-temperature decline.
- Ascertaining the PMI from a single-temperature data point appears challenging. Knowing both the initial body temperature and that of the room does not suffice, given that the cooling constant, k
_{c}, is dependent on an individual’s body characteristics and surrounding conditions. In other words, gathering time-lapse data appears a requirement for a reliable solution for estimating PMI. - The hyperbolic relation fits all monotonic trends, established either by temperature derivative or the heat-flow rate, regardless of the body part. This fitting leads to a high degree of PMI accuracy, that is, 0.24 h on average for the cases reported in this study.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

b | time-derivative of D in Arps, dimensionless |

C_{t} | heat capacity of tissue, J/kg-°C |

D | log-time derivative of temperature in Arps, 1/h |

k_{c} | cooling constant, 1/h |

q_{c} | heat-flow rate, J/m3.sec |

t | time, h |

T | temperature, °C |

T_{a} | ambient temperature, °C |

T_{i} | initial temperature at time of death, °C |

dT/dt | temperature gradient, °C/h |

ρ_{t} | density of tissue, kg/m^{3} |

## Appendix A

#### Estimating the Cooling Constant with Limited Data

_{c}, the general trend suggests that a lighter body declines faster than those with heavier counterparts, as Figure A1 testifies. This outcome makes intuitive sense from the standpoint of heat transfer in that a person with a higher body mass shields the interior, thereby decreasing the rate of cooling. Of course, other variables such as body fat may play roles that may be hard to discern, given that our dataset may be limiting. Let us present a few examples showing how limited time-lapse data can reveal the k

_{c}parameter.

**Figure A1.**The overall distribution of k

_{c}(

**a**), k

_{c}appears to increase with decreasing body weight (

**b**).

**Table A1.**Reported data for Case 20 (after de Saram et al. [4]).

Time of Death | 8:30:00 a.m. |
---|---|

Temperature at time of death (T_{i}) | 38.22 °C |

Ambient Temperature (T_{a}) | 30.61 °C |

Temperature at 12 a.m. | 36.61 °C |

Temperature at 1 PM | 36.17 °C |

Temperature at 2 PM | 35.83 °C |

**Table A2.**Internal organ temperature results (after de Saram et al. [4]).

Time Interval | k_{c} (1/h) | Estimated PMI (h) | Actual PMI (h) | Difference (min) |
---|---|---|---|---|

12 a.m.–1 p.m. | 0.07696 | 3.09 | 3.5 | 25 |

1 p.m.–2 p.m. | 0.061875 | 3.84 | 3.5 | −21 |

12 a.m.–2 p.m. | 0.069418 | 3.43 | 3.5 | 4 |

Avg (12 a.m.–1 p.m.) and (1 p.m.–2 p.m.) | 0.069418 | 3.40 | 3.5 | 4 |

**Table A3.**Internal organ temperature results (after Al Alousi et al. [12]).

Time Interval | k_{c} (1/h) | Estimated PMI (h) | Actual PMI (h) | Difference (min) |
---|---|---|---|---|

2 to 4 h | 0.05 | 2.19 | 2.13 | −4 |

2 to 6 h | 0.05 | 2.21 | 2.13 | −5 |

**Table A4.**Brain temperature results (after Al Alousi et al. [12]).

Time Interval | k_{c} (1/h) | Estimated PMI (h) | Actual PMI (h) | Difference (min) |
---|---|---|---|---|

1 to 2 h | 0.17 | 0.80 | 1.20 | 24 |

1 to 4 h | 0.15 | 0.94 | 1.20 | 15 |

1 to 5 h | 0.14 | 1.02 | 1.20 | 11 |

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**Figure 1.**Skin temperature recording exhibits the steepest decline among the three measurements points (after Bartgis et al. [30]).

**Figure 3.**Heat flow rate estimation of the three body parts with the synthetic data (

**a**), and temperature derivative identifies the cooling trend (

**b**), after Bartgis et al. [30].

**Figure 4.**Temperature derivate identifies the cooling trend in the liver, after Al Alousi et al. [12].

**Figure 5.**The heat-flow rate response of the Kaliszan [21] dataset; the faster cooling rate for the eye (

**b**) than internal organ (

**a**).

**Figure 6.**Good overall fit quality appears for the brain (

**a**), and a small PMI-estimation error beyond three hours (

**b**).

**Figure 7.**Gauging overall fit quality for Case B Thigh (

**a**) and the PMI estimation error beyond five hours (

**b**).

Source | Body Part | Start of Fitting Window, h | Estimated PMI, h | Actual PMI, h | ΔPMI, h | b-Factor, Dimensionless | D, 1/h |
---|---|---|---|---|---|---|---|

Bertgis et al. | Internal Organ | 3 | 2.8 | 3.3 | 0.5 | 0 | 0.012 |

Bertgis et al. | Internal Organ | 5 | 3.7 | 5.0 | 1.3 | 0 | 0.012 |

Bertgis et al. | Internal Organ | 9 | 8.2 | 9.3 | 1.2 | 0 | 0.012 |

Bertgis et al. | Brain | 3 | 3.8 | 3.3 | −0.5 | 0 | 0.017 |

Bertgis et al. | Brain | 5 | 5.8 | 5.2 | −0.6 | 0 | 0.017 |

Bertgis et al. | Brain | 9 | 10.1 | 9.3 | −0.8 | 0 | 0.017 |

Wilk et al. | Case B-Abdomen | 3 | 3.2 | 3.5 | 0.3 | 8.5 | 0.032 |

Wilk et al. | Case B-Abdomen | 5 | 5.3 | 5.6 | 0.3 | 8.5 | 0.020 |

Wilk et al. | Case B-Abdomen | 9 | 10.0 | 9.7 | −0.3 | 8.1 | 0.012 |

Wilk et al. | Case B-Forehead | 3 | 3.6 | 3.1 | −0.5 | 5.3 | 0.039 |

Wilk et al. | Case B-Forehead | 5 | 5.4 | 5.6 | 0.2 | 5.8 | 0.027 |

Wilk et al. | Case B-Forehead | 9 | 7.8 | 9.4 | 1.6 | 6.7 | 0.018 |

Wilk et al. | Case B-Thighs | 3 | 5.1 | 3.0 | −2.1 | 4.5 | 0.030 |

Wilk et al. | Case B-Thighs | 5 | 6.7 | 5.1 | −1.6 | 4.7 | 0.024 |

Wilk et al. | Case B-Thighs | 9 | 8.8 | 9.9 | 1.1 | 6.1 | 0.017 |

Wilk et al. | Case B-Chest | 3 | 3.2 | 3.1 | −0.1 | 10 | 0.028 |

Wilk et al. | Case B-Chest | 5 | 5.1 | 5.2 | 0.1 | 10 | 0.018 |

Wilk et al. | Case B-Chest | 9 | 9.5 | 9.4 | 0.0 | 10 | 0.010 |

Wilk et al. | Case C-Forehead | 3 | 2.1 | 3.3 | 1.2 | 10 | 0.046 |

Wilk et al. | Case C-Forehead | 5 | 5.4 | 5.2 | −0.2 | 10 | 0.018 |

Wilk et al. | Case C-Abdomen | 9 | 10.8 | 10.0 | −0.8 | 5 | 0.014 |

Wilk et al. | Case D-Forehead | 3 | 3.1 | 4.2 | 1.1 | 10 | 0.032 |

Wilk et al. | Case D-Forehead | 5 | 5.1 | 5.6 | 0.5 | 10 | 0.019 |

Wilk et al. | Case D-Abdomen | 3 | 2.9 | 3.5 | 0.6 | 9.11 | 0.034 |

Wilk et al. | Case D-Abdomen | 5 | 5.8 | 6.5 | 0.8 | 9.17 | 0.018 |

Wilk et al. | Case D-Abdomen | 9 | 8.5 | 9.4 | 0.9 | 9.1 | 0.012 |

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**MDPI and ACS Style**

Sharma, P.; Kabir, C.S.
A Simplified Approach to Understanding Body Cooling Behavior and Estimating the Postmortem Interval. *Forensic Sci.* **2022**, *2*, 403-416.
https://doi.org/10.3390/forensicsci2020030

**AMA Style**

Sharma P, Kabir CS.
A Simplified Approach to Understanding Body Cooling Behavior and Estimating the Postmortem Interval. *Forensic Sciences*. 2022; 2(2):403-416.
https://doi.org/10.3390/forensicsci2020030

**Chicago/Turabian Style**

Sharma, Pushpesh, and C. S. Kabir.
2022. "A Simplified Approach to Understanding Body Cooling Behavior and Estimating the Postmortem Interval" *Forensic Sciences* 2, no. 2: 403-416.
https://doi.org/10.3390/forensicsci2020030