Verification of the exponential model of body temperature decrease after death in pigs

Authors


Corresponding author R. Hauser: Department of Forensic Medicine, Medical University of Gdańsk, ul. Debowa 23, 80-204, Gdańsk, Poland. Email: rohauser@amg.gda.pl

Abstract

The authors have conducted a systematic study in pigs to verify the models of post-mortem body temperature decrease currently employed in forensic medicine. Twenty-four hour automatic temperature recordings were performed in four body sites starting 1.25 h after pig killing in an industrial slaughterhouse under typical environmental conditions (19.5–22.5°C). The animals had been randomly selected under a regular manufacturing process. The temperature decrease time plots drawn starting 75 min after death for the eyeball, the orbit soft tissues, the rectum and muscle tissue were found to fit the single-exponential thermodynamic model originally proposed by H. Rainy in 1868. In view of the actual intersubject variability, the addition of a second exponential term to the model was demonstrated to be statistically insignificant. Therefore, the two-exponential model for death time estimation frequently recommended in the forensic medicine literature, even if theoretically substantiated for individual test cases, provides no advantage as regards the reliability of estimation in an actual case. The improvement of the precision of time of death estimation by the reconstruction of an individual curve on the basis of two dead body temperature measurements taken 1 h apart or taken continuously for a longer time (about 4 h), has also been proved incorrect. It was demonstrated that the reported increase of precision of time of death estimation due to use of a multiexponential model, with individual exponential terms to account for the cooling rate of the specific body sites separately, is artifactual. The results of this study support the use of the eyeball and/or the orbit soft tissues as temperature measuring sites at times shortly after death. A single-exponential model applied to the eyeball cooling has been shown to provide a very precise estimation of the time of death up to approximately 13 h after death. For the period thereafter, a better estimation of the time of death is obtained from temperature data collected from the muscles or the rectum.

Determination of the time of death is frequently of utmost importance in criminology and forensic medicine. Various physiological factors are employed for that purpose, such as lividity, rigor, the mechanical and/or electrical excitability of muscles and the chemical excitability of the iris (Henssge et al. 2000b). Also, biochemical approaches have been reported using cerebrospinal fluid concentrations of lactic acid, urea, amino acids (Henssge & Madea, 2004) or potassium (Fraschini et al. 1963). In addition, the logarithm of sodium to potassium plasma concentration ratio, along with the logarithm of chloride concentration, were claimed to correlate with the post-mortem interval (Querido, 1990). None of the above approaches have, however, proved useful for the practical assessment of the time of death in comparison to the standard estimation from body temperature data (Green & Wright, 1985; Baccino et al. 1996; Henssge et al. 2000a).

According to Henssge & Madea (2004), the theoretical basis for using dead body temperature for estimation of the time of death had already been formulated in the mid-19th century by Rainy (1868). Rainy transferred the Newton rule of cooling to forensic medicine and calculated the time of death from the exponential formula relating the difference between the body and ambient temperatures to time.

The single-exponential model was applied by Saram & Webster (1955). That model comprised a fixed term of 45 min added arbitrarily to account for a post-mortem temperature plateau. The model was devised after an analysis of rectal temperatures of 41 judicially hanged prisoners.

The exponential model was a subject of numerous studies leading to either its simplification or its complication. The simplifications, like the so-called rule of thumb, imply a linear drop of rectal temperature with a typical value of 0.8°C h−1 (Green & Wright, 1985). Linear regressions, including multivariable regressions, combining physiological with biochemical parameters, were reported by Baccino et al. (1996). These procedures provide only a rough estimate of the time of death and are an unnecessary simplification, especially now that the measured temperature data can be processed according to even the most sophisticated algorithms.

A question arises whether the proposed extensions and complications of the single-exponential model are of real value for the reliability estimation of the time of death in individual cases. Henssge and coworkers (Henssge, 1988; Althaus & Henssge (1999); Henssge et al. 2000a) have developed a rectal temperature-based nomogram method and a relevant software package which take into account the subject's body weight as well as a number of empirical environmental correction factors. However, there are reports in the literature questioning significant correlations between the rates of cooling and the body parameters (Green & Wright, 1985).

Another question is whether the complexity of the necessary temperature data collection actually does help to improve the accuracy of estimation of the time of death. Specifically, it concerns the propositions by Green & Wright (1985) to use two rectal temperatures measured about 1 h apart and by Mall et al. (2004) to employ a continuous temperature recording during a longer time interval post mortem (4.25 h).

Another question is whether and when the rectal temperature measurements are to be preferred over measurements at the other sites of temperature recording. Technical problems of accessibility aside, there may be legal restrictions to invasive methods in some countries. Of particular importance is the dependence of the precision of estimation of time of death on the site of temperature measurement. It should also be tested whether using temperature data from several body sites, like in the triple-exponential approach of al-Alousi et al. (2002), actually improves prediction of th time of death. Perhaps such a complex model is valid in specific cases or for the ‘average’ case only.

In order to address the questions above, a controlled experiment needs to be carried out under conditions close to real life. Such an experiment would, for many reasons, be difficult to perform in humans but may be done in pigs, which are animals with similar physiology. Here, such an experiment has been designed and performed.

Methods

The experiment was carried out in a licensed industrial slaughterhouse with all the necessary certificates and in compliance with the formal requirements of the European Union.

The pigs were not killed solely for the sake of the experiment. They had been randomly selected from the animals being killed as part of the regular manufacturing process. A surplus supply of pigs is normally used for the manufacture of feeding stuff and such was the final purpose of the experimental animals used here.

The study was conducted in 19 pigs: 13 sows and 6 boars of the Great White Polish species (Wielka Biala Polska) weighing between 81 and 124 kg. Five two-channel thermometers P655 were used for the measurements, connected with pin probes Pt100, class B 1/3 DIN, 100 × 1.4 mm, ending in a a 20 mm temperature sensor, and with probes Pt100, class B, 150 × 3 mm, ending with a 40 mm temperature sensor. The measuring equipment was manufactured by Dostmann electronic GmbH (Wertheim-Reicholzheim, Germany) with the following catalogue numbers: thermometers: 5000-0655; pin probes Pt100, class B 1/3 DIN: 6000-9999; and pin probes Pt100 class B: 6000-1001. Each of the five measuring kits (1 thermometer and 2 probes) was calibrated according to the manufacturer's instructions.

Straight after killing (by electrical current), the animals were placed in a specially assigned room of the slaughterhouse with the abdominal surface facing 150 mm-high wooden gratings, in an isolated room approximately 200 m3 in volume.

After applying an eyelid retractor in order to obtain a wide lid slit, the eyeballs were stabilized with stabilization pincets. Two 100 mm pin probes were inserted into the sclera, into the nasal quadrants of the left eyeballs, 3 mm away from the corneal limbus, passing through the pars plana of the ciliary body into the vitreous chamber, and further on, posteriorly and laterally from the optic nerve head, until a depth of 22 mm was reached.

The next two 100 mm pin probes were inserted into the soft tissues of the right orbits at the medial canthus, passing along the medial rectus muscle towards the superior orbital fissure, until a depth of 25 mm was reached. After the probes had been inserted, the pincets and the eyelid retractor were removed and the eyelids were closed.

The entire lengths of two 150 mm probes were inserted into the muscles of the left rumps from the insertion site at the central portion of the rump.

Two other 150 mm probes were inserted into the rectum up to their handles.

Ambient temperature was measured using a probe located 500 mm above the ground.

The nine probes used in each set of measurements were located in parallel to the ground, and their handles were stabilized in the grips of the stands. It was also ensured that the probes, after their settlement, did not cause the eyelids to open, which were naturally closed in all the animals tested.

The thermometers were connected to a computer preset to record the transmitted temperature values at a frequency of one sample every 30 s (for construction of graphs every tenth time point was used).

The recording device was switched on 75 min following a necessary unified preparation procedure. The registration of the measurements was completed 23–25 h after the animals were killed.

The results were processed with Microsoft Excel 2000 (Microsoft Corporation, USA) using appropriate statistical procedures: Matlab® Software version 7.0 (The Math Works, Inc., Natick, MA, USA).

Results

The post-mortem body temperature measurements in pigs were taken at typical room temperatures during summer (August and September) in Poland. The time courses of ambient temperature changes during the experiments are shown in Fig. 1 for the days of experiment and the pigs numbered as in Table 1. The mean ambient temperature for a given experimental day was used in calculations.

Figure 1.

Time course of environmental temperature changes during 10 consecutive experiments
Each experiment involved two pigs, except experiment 5 which was done on one pig.

Table 1.  Day of the experiment, body mass and coefficients of exponential equations describing body temperature decrease in individual pigs tested
Day of
expt and
pig no.
Mean
environmental
temp. (°C)
Body
mass
(kg)
EyeballOrbit tissuesRectumMuscles
a b r 2 r.m.s.e. a b r 2 r.m.s.e. a b r 2 r.m.s.e. a b r 2 r.m.s.e.
1_121.110316.4−0.1240.9970.20219.9−0.0520.9940.28520.9−0.0610.9980.171
15.1−0.1150.9960.218
1_221.19116.6− 0.1411.0000.07522.1−0.0670.9900.46519.3−0.0720.9990.146
15.2− 0.1420.9990.100 
2_121.18817.0−0.1191.0000.04921.6−0.0550.9960.27320.8−0.0650.9990.149
2_221.112416.3−0.0930.9970.18221.1− 0.0480.9970.22921.2−0.0581.0000.077
3_121.18814.8−0.1300.9980.13317.0−0.1180.9970.19419.3− 0.0630.9990.11318.6−0.0700.9980.171
3_221.111214.7−0.1110.9990.12317.0−0.1040.9970.20719.6−0.0510.9990.09720.0−0.0540.9990.104
4_121.58115.1−0.1070.9990.10316.0−0.1030.9980.15721.0−0.0630.9990.14319.2−0.0680.9990.135
4_221.58713.6−0.1010.9990.11013.9−0.1020.9970.15420.7−0.0561.0000.06220.6−0.0600.9990.128
5_122.010214.8−0.0980.9980.12915.3−0.0980.9990.11920.0−0.0510.9940.27519.0−0.0600.9980.183
6_121.18714.7−0.1130.9950.21815.0−0.1190.9970.19320.6−0.0640.9980.17220.6−0.0661.0000.045
6–221.18314.3−0.1240.9990.11014.1−0.1360.9990.11518.2−0.0701.0000.05519.0−0.0691.0000.067
7_120.87816.6−0.1150.9950.25520.6−0.0621.0000.06419.7−0.0691.0000.024
15.1−0.1140.9960.220
7_220.89016.0−0.1050.9940.27620.2−0.0551.0000.06121.0−0.0611.0000.099
14.8−0.1000.9970.192
8_120.910214.6−0.0990.9970.17414.7−0.1070.9950.22121.2−0.0520.9960.25520.9−0.0620.9980.168
8_220.98215.2−0.1240.9990.12215.3−0.1171.0000.06720.0−0.0610.9980.15819.9−0.0640.9990.148
9_120.510814.2−0.1100.9920.26714.7− 0.1280.9810.42422.3−0.0610.9970.23222.5−0.0640.9990.165
9_220.59915.1−0.1040.9960.18916.0−0.1290.9890.34820.3−0.0641.0000.07321.1−0.0710.9990.146
10_120.19316.8−0.0960.9980.17915.7−0.1000.9970.18121.6−0.0531.0000.05622.1−0.0580.9990.108
10_220.110916.1−0.1120.9950.24915.9−0.1130.9960.21820.4−0.0600.9990.13019.8−0.0630.9990.139

Figure 1 demonstrates that ambient temperature variations during individual experimental runs were up to about 1.5°C. The mean temperatures deviated within 2°C. Thus, our experimental conditions are typical of forensic cases, where dead bodies remain at normal housing conditions. Nonetheless, as rules of thermodynamics hold generally, our findings should also apply to other ambient temperatures.

Figure 2 represents the records at 5 min intervals of the temperature difference between the individual pig eyeball (T) and the environment (TE) for 21 eyeballs tested. It must be noted here that for both pigs tested on days 1 and 7 we measured the temperature in both eyeballs, whereas on day 2 we did not use the eyeballs for temperature measurements (see Table 1).

Figure 2.

Recording of the difference between measured pig eyeball temperature (T) and environmental temperature (TE) assesed at 5 min intervals after death for 17 animals tested
Mean equation coefficients are shown, together with their 95% confidence intervals and statistical parameters (see text for definitions) of the monoexponential model describing the temperature of the pig eyeball after death.

Figure 2 clearly shows a very regular exponential fall of temperature difference, TTE, with time over the whole measurement period, i.e. starting 75 min after the pigs' death. Coefficients a and b of the individual equations of the form

display math(1)

are presented in Table 1 for each pig separately.

In Fig. 2, a mathematical model is given along with the plots of the mean exponential relationships between the eyeball and ambient temperature difference and time for 21 eyeballs. The statistical quality of the model derived is very high, as quantified by the values of s.s.e., r2 and R.M.S.E. The meaning of these parameters is as follows. The s.s.e. denotes the sum of squares due to error. This statistics measures the total deviation of the actual value from the fit to the calculated one. The r2 measures how successful the fit is in explaining the variation of the data. The r.m.s.e. is the root mean squared error.

The physical meaning of the coefficients of the exponential model is as follows. Coefficient a is a starting temperature difference, i.e. the extrapolated difference between the temperature of the eyeball and the environment at the time of death (t= 0). Coefficient b denotes the rate constant of the temperature decrease.

Analysing the data from Table 1, it may be noted that, for example, pig 1 studied on day 4 (pig 4_1) would have an extrapolated to the time of death temperature of eyeball, TD, of 36.6°C, because a= 15.1°C when TE= 21.5°C. Hence, TDTE= 15.1°C means that TD= 36.6°C. That value appears to be reasonable. Unfortunately, we were unable to find any literature report on normal eyeball temperature in pigs. We refrained from performing an experiment in vivo, assuming that the mean values of a= 15.2°C and TE= 21.0°C provide a reliable TD or normal physiological temperature in pig eyeball of 36.2°C. That temperature might be a bit higher in pigs than in humans since the body temperature in pigs is reported (Prost, 1985; Klont & Lambooy, 1995; Hanneman et al. 2004) to be within 38–40°C, whereas in humans it is commonly recognized as 36.6–37.5°C.

The dispersion of the eyeball temperature time courses among the pigs tested is illustrated in Fig. 2. The data for only one pig cross the 95% confidence limit provided by the model given in Fig. 2. The single-exponential model appears to be reliable and can be used to estimate the time elapsing from death based on the eyeball temperature measured. For the purpose of such estimation, a simple algebraic transformation of the model into the following form seems convenient:

display math(2)

Table 2 clearly illustrates the predictive properties of our single-exponential model with regard to the time elapsing from the pig's death. It may be noted that prediction precision strongly decreases with decreasing temperature difference, TTE.

Table 2.  Lower and upper 95% confidence limits of time elapsing since death of pigs calculated from the exponential model of temperature decrease of the eyeball
   TTE (°C)LowerTime (h)Upper
  1. T, measured temperature; TE, environmental temperature.

   1.415.821.236.7
   1.814.519.028.1
   2.213.417.223.8
   2.612.415.720.9
   3.011.514.418.7
   3.410.713.317.0
   3.810.012.315.5
   4.2 9.311.414.3
   4.6 8.610.613.2
   5.0 8.0 9.912.2
   5.4 7.5 9.211.3
   5.8 6.9 8.610.5
   6.2 6.4 8.0 9.8
   6.6 6.0 7.4 9.1
   7.0 5.5 6.9 8.5
   7.4 5.1 6.4 7.9
   7.8 4.7 5.9 7.3
   8.2 4.3 5.5 6.8
   8.6 3.9 5.1 6.3
   9.0 3.6 4.6 5.9
   9.4 3.2 4.3 5.4
   9.8 2.9 3.9 5.0
   10.4 2.4 3.4 4.4
   10.8 2.1 3.0 4.0
   11.2 1.8 2.7 3.7
   11.6 1.5 2.4 3.3
   12.0 1.3 2.1 3.0
   12.4 1.0 1.8 2.7
   12.8 0.8 1.5 2.4
   13.2 0.5 1.2 2.1
   13.6 0.3 1.0 1.8
   14.0 0.0 0.7 1.5

In Table 2, the lower and upper 95% confidence limits are provided regarding the time elapsed from death estimated from the eyeball temperature. It is evident that the estimations may be especially reliable if time from death is relatively short. If more than 20 h since death have elapsed, predictions using this model become impossible.

Figure 3, illustrating data for the orbit soft tissues, shows a close similarity to the corresponding figure for the eyeball (Fig. 2). Coefficients a and b of the exponential equation for individual pigs (Table 1), as well as those of the general model (Fig. 2), are closely similar to those derived for the eyeballs. This would mean that both the extrapolated temperature at death, a, and the rate constant of temperature decrease, b, are similar, as expected because of the close vicinity of temperature measurement sites. However, the dispersion of temperature decrease time courses is evidently larger for the orbit soft tissues (Fig. 3) than for the eyeballs (Fig. 2). This is quantitatively evidenced by a larger r.m.s.e. value for the orbit soft tissues than for the eyeballs (0.705 versus 0.589). Also, the intervals between the lower and the upper death time 95% confidence limits, corresponding to the same temperature differences, TTE, are a bit larger in the case of the orbit soft tissues compared to the eyeballs (Table 3).

Figure 3.

Recording of the difference between measured pig orbit tissue temperature (T) and environmental temperature (TE) assessed at 5 min intervals after death for 15 animals tested
Mean equation coefficients are shown, together with their 95% confidence intervals and statistical parameters (see text for definitions) of the monoexponential model describing the temperature of the pig orbit tissues after death.

Table 3.  Lower and upper 95% confidence limits of time elapsing since death of pigs calculated from the exponential model of temperature decrease of the orbit tissues
   TTE (°C)LowerTime (h)Upper
  1. T, measured temperature; TE, environmental temperature.

   1.614.920.538.5
   2.013.718.529.1
   2.412.716.824.6
   2.811.815.521.6
   3.211.014.319.4
   3.610.313.217.6
   4.09.612.216.1
   4.48.911.414.8
   4.88.310.613.7
   5.27.79.912.7
   5.67.29.211.8
   6.06.78.610.9
   6.46.28.010.2
   6.85.87.59.5
   7.25.46.98.9
   7.64.96.58.3
   8.04.56.07.7
   8.44.25.57.2
   8.83.85.16.7
   9.23.54.76.2
   9.63.14.35.7
   10.02.84.05.3
   10.42.53.64.9
   10.82.23.34.5
   11.21.93.04.1
   11.61.62.63.8
   12.01.32.33.4
   12.41.12.03.1
   12.80.81.72.8
   13.20.61.52.5
   13.60.31.22.2
   14.00.10.91.9

In Fig. 4, respective plots are presented referring to the temperature measurements in the pig's rectum. Marked differences may be noted between the corresponding figures illustrating post-mortem temperature decrease in the eyeballs and in the orbit soft tissues. Firstly, in Fig. 4, a lag phase is observed at the initial stages of temperature recording. This phase may continue for up to 5–6 h after death. This would be consistent with reports on the duration of that phase in humans (Saram & de Webster, 1955; Marshall & Hoare, 1962; Shapiro, 1965). However, the lag phase is evident only in the minority of the 19 pigs tested starting at 75 min after death. It is not clear whether the plateau would be present within the first 75 min after death for which the measurements were technically not feasible. That phase actually does exist because the rectal temperature at death calculated from extrapolated coefficients a (Table 2) varies for individual pigs between 39.3°C and 43.2°C, which appears a bit too high, although not dramatically so.

Figure 4.

Recording of the difference between measured pig rectum temperature (T) and environmental temperature (TE) assessed at 5 min intervals after death for 19 animals tested
Mean equation coefficients are shown, together with their 95% confidence intervals and statistical parameters (see text for definitions) of the monoexponential model describing the temperature of the pig rectum after death.

The mean rate constant of the rectal temperature change, b, is −0.058 h−1 (Fig. 3), which means a significantly slower temperature decrease rate compared to the eyeball (b=−0.113 h−1) and the orbit soft tissues (b=−0.111 h−1).

Table 4 shows that errors in estimation of the time of death from the single-exponential model employing rectal temperature are larger than in the case of the eyeballs and the orbit soft tissues at earlier periods after death. However, rectal temperature data allow estimations of the time of death after longer periods of time (up to 30 h), whereas no time estimations beyond 20 h since death are possible from temperature measurements in the eyeballs and in the orbit soft tissues.

Table 4.  Lower and upper 95% confidence limits of time elapsing since death of pigs calculated from the exponential model of temperature decrease of the rectum
   TTE (°C)LowerTime (h)Upper
  1. T, measured temperature; TE, environmental temperature.

   3.025.133.449.9
   3.423.731.344.7
   3.822.529.340.8
   4.221.327.637.6
   4.620.226.034.8
   4.819.625.333.6
   5.218.623.931.4
   5.617.622.629.5
   6.016.721.427.7
   6.415.920.326.1
   6.815.019.224.7
   7.214.318.223.3
   7.613.517.322.1
   8.012.816.420.9
   8.412.115.519.8
   8.811.414.718.8
   9.210.813.917.8
   9.610.213.216.9
   10.0 9.612.516.0
   10.4 9.011.815.2
   10.8 8.411.214.4
   11.0 8.210.814.0
   11.4 7.610.213.3
   11.8 7.1 9.612.6
   12.2 6.6 9.011.9
   12.6 6.1 8.511.2
   12.8 5.9 8.210.9
   13.2 5.4 7.710.3
   13.6 4.9 7.1 9.7
   14.0 4.5 6.6 9.1
   14.4 4.1 6.2 8.5
   14.8 3.6 5.7 8.0
   15.2 3.2 5.2 7.5
   15.6 2.8 4.8 6.9
   16.0 2.4 4.3 6.4
   16.4 2.0 3.9 6.0
   16.8 1.7 3.5 5.5
   17.2 1.3 3.1 5.0
   17.6 0.9 2.7 4.6
   18.0 0.6 2.3 4.1
   18.4 0.2 1.9 3.7

The results of temperature measurements in the muscles (Fig. 5 and Table 5) are quite similar to those concerning the rectum. In Fig. 5, the lag phase is observed to last for 75 min after death in some pigs. While coefficients a and b differ only marginally from those derived for the rectum as the temperature measurement site, the difference is, however, significant. The temperature in the muscles at death, extrapolated from the single-exponential model (Table 5), ranges from 39.7 to 43.0°C. The rate constant of temperature decrease of −0.063 h−1 suggests a slightly faster cooling of muscles than rectum (−0.058 h−1). Somewhat surprisingly, the dispersion of temperature time course curves determined in the muscles as compared to the rectum is smaller (Fig. 5versus Fig. 4), This is further confirmed by r.m.s.e. values, which are 0.862 for the muscles and 0.936 for the rectum. Also, Table 5 suggests that estimation of the time of death is a bit more precise from the muscle data, although the duration of reliable estimation of the time of death is a little shorter than that provided by rectal temperature.

Figure 5.

Recording of the difference between measured pig muscle temperature (T) and environmental temperature (TE) assessed at 5 min intervals after death for 19 animals tested
Mean equation coefficients are shown, together with their 95% confidence intervals and statistical parameters (see text for definitions) of the monoexponential model describing the temperature of the pig muscles after death.

Table 5.  Lower and upper 95% confidence limits of time elapsing since death of pigs calculated from the exponential model of temperature decrease of muscle
   TTE (°C)LowerTime (h)Upper
  1. T, measured temperature; TE, environmental temperature.

   2.624.432.449.1
   3.023.030.143.3
   3.421.828.239.1
   3.820.626.435.8
   4.219.524.833.0
   4.618.423.430.7
   5.017.522.128.6
   5.416.520.926.8
   5.815.719.725.2
   6.214.918.723.7
   6.614.117.722.4
   7.013.316.821.2
   7.412.615.920.0
   7.812.015.119.0
   8.211.314.318.0
   8.610.713.517.0
   9.010.112.816.1
   9.4 9.512.115.3
   9.8 9.011.514.5
   10.6 7.910.213.0
   11.0 7.4 9.712.3
   11.4 6.9 9.111.6
   11.8 6.4 8.611.0
   12.2 6.0 8.010.4
   12.6 5.5 7.5 9.8
   13.0 5.1 7.0 9.2
   13.4 4.7 6.5 8.7
   13.8 4.3 6.1 8.2
   14.2 3.8 5.6 7.6
   14.6 3.5 5.2 7.1
   10.2 8.410.813.7
   15.0 3.1 4.8 6.7
   15.4 2.7 4.4 6.2
   15.8 2.3 4.0 5.7
   16.2 2.0 3.6 5.3
   16.6 1.6 3.2 4.9
   17.0 1.3 2.8 4.5
   17.4 1.0 2.4 4.1
   17.8 0.6 2.1 3.7
   18.2 0.3 1.7 3.3

Table 6 provides the squares of linear correlation coefficients, r2, between coefficients a and b of the single-exponential temperature decrease model at various measurement sites and the pigs' body mass. Evidently, neither a nor b correlates with the body mass, which is to be expected in the case of a. However, the lack of correlation between the rate constant of temperature decrease and the mass of the pig appears to contradict Henssge's nomogram, where the mass of the body is considered.

Table 6.  Correlations between the coefficients a and b of the single-exponential model of temperature decrease after death and pig body mass
Coefficients of
exponential
model
Square of correlation coefficient, r2
EyeballOrbit tissuesRectumMuscles
a 0.0020.0680.0270.145
b 0.0370.1280.3650.320

Discussion

The analysis of cooling curves for each of the tested body sites in the 19 individual animals separately (Figs 2–5 and Table 1) leads to the conclusion that, within the temperature measurement period, i.e. starting 1.25 h after death, the body temperature decreases with time according to a single-exponential function. In the case of only six pigs tested, the decreases of the time curves of rectal temperature show, at their initial phases, some curve flattening, suggesting a possible temperature plateau (lag time) during first 75 min after the animals' death (Fig. 4). An even less marked deviation of the initial part of the cooling curves from the monoexponential function can be observed in Fig. 5 for three pigs with respect to temperature measurements in the muscles.

The deviations from the single-exponential model, as illustrated in Figs 4 and 5, are small. However, the intersubject diversity is large enough to exclude a more complex two-exponential model claimed by Marshall & Hoare (1962) to improve the reliability of predictions of the time of death. Of course, mathematically a two-exponential model cannot produce worse predictions than a simpler single-exponential model. However, at least in the case of the data considered, the second exponential term provides no statistically significant improvement of description of the mean temperature decrease time course. In such a situation, there is no practical advantage to employing a more complex model.

It must be noted here that considering a single curve for a given animal in Figs 2–5, one would expect a better fit of the measured data points to a two-exponential model than to a single-exponential one. From our analysis (Table 1) it turns out that in fact that is not the case. Values of r2 close to 1 and low values of r.m.s.e. are evidence that the monoexponential model is good enough for individual cases. The scatter of the individual cooling curves plotted 75 min after death for 19 pigs makes the increase of correlation obtained with a second exponential term insignificant due to the lack of statistical significance of the second exponential term.

A question arises whether the missing temperature data for the first 1.25 h after death would form a plateau on the mean cooling curves derived. This question is, in fact, academic in nature, because either answer, ‘yes’ or ‘no’, would not support the practical use of the two-exponential model, leaving aside technical difficulties regarding the collection of the necessary reproducible experimental data directly after death. If, indeed, the initial plateau did exist, no estimation of the time of death could be done within that period other than the plateau length. Here, this would be the case between 0 and 1.25 h. If no plateau existed, the differences between the lower and the upper 95% confidence limits for time since death estimated from the single-exponential model would be 1.5 h at best (Tables 2–5).

From the practical point of view, for a forensic medicine specialist, the two-exponential model describing the body temperature fall after death is of no use. This model may of course provide a better description of a given individual body site temperature time course, especially with regard to such measurement sites as the rectum or the muscles. However, its predictive value with respect to a biological object not used to derive the specific two-exponential model is not higher than that of a simpler, thermodynamically based monoexponential model. In our opinion, the often cited advantages of the two-exponential model might be statistical artefacts.

In view of the marked scatter of the cooling curves for the body sites illustrated in Figs 2–5, the statistical significance of various correction factors to the exponential models recommended for use in forensic medicine also seems disputable (Henssge, 1988). While these correction factors could work for a given artificial phantom subjected to various controlled environmental conditions, it is highly unlikely that those corrections could significantly increase the reliability of estimation of the time of death of individually diversified mammals.

Certain authors believe that taking into account two temperature measurements about 1 h apart (Green & Wright, 1985) or a continuous recording of temperature for a certain period (4.25 h) after death (Mall et al. 2004) will improve estimation of the time of death. This might indeed work, but complicates the procedure for estimation of the time of death. The single-exponential equation describing post-mortem temperature decrease at individual body sites is readily interpretable in physical terms. Coefficient a denotes the temperature difference between a given measurement site and the environment, TTE, extrapolated to t= 0, i.e. the moment of death. Hence, the temperature at death, TD, can be calculated from a=TDTE. Coefficient a, once derived, may be adjusted to any ambient temperature if TD is known. Assuming the mean temperature of the environment during the experiments to be TE= 21°C (Fig. 1), one may calculate the mean pig eyeball temperature at death from the equation characterized in Fig. 2. Hence, a=TDTE becomes 15.2 =TD− 21 or TD= 36.2°C. Similarly, the extrapolated temperature at death would be 36.6°C in the orbit soft tissues (Fig. 3), 41.5°C in the rectum (Fig. 4) and 41.3°C in the muscles (Fig. 5).

We did not manage to find any data in the literature literature on the physiological eyeball temperature of pigs. In humans, the mean eyeball temperature is reported to be about 34–35°C (Charman & Jennings, 2000; Jurowski et al. 2004). The value of the pig eyeball temperature of 36.2°C, as estimated from our models, appears to be reliable. Also, the value 36.6°C for the orbit soft tissues is reasonable. These values seem to agree with the value of 36.6°C for tympanic temperature, which may be estimated from the graph included in the report on temperature measurement in swine by Hanneman et al. (2004). Hence, the single-exponential model might fully account for the changes in temperature of the eyeball and the orbit soft tissues from the moment of death. That is because of the lack of any plateau in the cooling plots for these body sites.

The extrapolated temperature at death in the rectum (41.5°C) and in the muscles (41.3°C) appears to be higher than physiological temperature. Veterinary reports (Prost, 1985; Klont & Lambooy, 1995; Hanneman et al. 2004) quote pig temperature as ranging from 38.0 to 40.0°C in the rectum and from 38.5 to 40.2°C in the muscles. These values are slightly lower than those obtained by extrapolation from our single-exponential model. Thus, a plateau phase on the temperature decrease time course for the rectum and muscles may exist during the early phase after death. This deviation from the single-exponential model is minor and of no practical consequence for the reliability of estimation of the time of death. Introducing a second exponential term to the cooling equation to account for the probable initial plateau would neither increase the range of the actual estimation period nor the precision of estimations.

Coefficient b of the relationships presented in Figs 2–5 denotes the rate constant of the cooling process in individual body sites. Its mean value is −0.113 h−1 for the eyeballs, −0.111 h−1 for the orbit soft tissues, −0.058 h−1 for the rectum and −0.064 h−1 for the muscles. Thus, the rate constant of temperature decrease in the pig's eyeball is nearly twice that in the rectum. This is due to the anatomical and localization differences between the temperature measuring sites and, consequently, their different thermal conductivities.

While coefficient a depends on the actual ambient temperature only, coefficient b may be affected by various body and environmental parameters such as the body mass, body surface area, obesity, environmental humidity, airflow, clothing, etc. The question is how strongly b depends on individual factors and whether the corrections due to individual factors significantly affect the estimation of the time of death where a large intersubject diversity is present.

In our opinion, the practical usefulness of the otherwise rational corrections is highly disputable. The most commonly accepted factor to affect the cooling speed is the object's body mass (Henssge, 1988). Our correlation analysis does not support the assumed practical value of that factor for improving predictions of the time of death. Table 6 shows that correlations between the pig's body mass (ranging from 81 to 124 kg) and the cooling rate constant b are generally low. In the case of the eyeball cooling equation, the correlation between b and the body mass is at the level of r2= 0.037, which is practically non-existent. There is a higher, but still very weak, corresponding correlation in the case of the rectum, r2= 0.365. Still, taking an individual case into consideration, one has little chance to improve the estimation of the time of death by using an exponential equation corrected for the individual's body mass.

Some authors propose a multiexponential equation with individual exponential terms accounting for the temperature in specific body sites (al-Alousi et al. 2002). Such equations are claimed to improve the estimation of the time of death. The approach appears problematic from the thermodynamic point of view, however. If single-exponential equations best describe the cooling of individual sites, as some authors have correctly noticed (al-Alousi et al. 2002), then perhaps the mean of times since death calculated from these simple equations could reduce the error of the estimation. However, the multiexponential equations make no clear physical sense. We managed to formally derive and test multivariable equations for estimation of the time of death taking into account the ratio of temperature differences in individual body sites and the environment. Hence, for example, denoting the actual temperature at time t in the eyeball by T1 and the corresponding temperature in the rectum by T2, we have:

display math(3)

This may be rewritten as:

display math(4)

where a′=a1/a2 and b′=b1b2. From eqn (4) one can obtain the time elapsed from death as:

display math(5)

where k1= 1/b′, k2= (ln a′)/b. We derived an appropriate equation by means of regression analysis:

display math(6)

The fit of our experimental data to the above model was very poor, r.m.s.e.= 1.600.

For the eyeball temperature data sets we also derived a two-exponential equation of the form proposed by al-Alousi et al. (2002):

display math(7)

The r.m.s.e. for that equation was 1.412. The corresponding values were 0.589 in the case of the eyeball, 0.705 for the orbit soft tissues, 0.936 for the rectum and 0.868 for the muscles. Again, the most simple and thermodynamically valid model appeared to be the best one.

The results of the present study may have direct practical consequences for forensic medicine. The analysis of the data contained in Figs 2–5 and Tables 1–5 indicates that the precision of the estimation of time of death decreases functionally with time. The single-exponential model correctly predicts the increasingly larger differences between the actual and the calculated time of death, especially in case of the upper 95% confidence limit. Therefore, with time, a larger overestimation than underestimation of the time of death is likely. Our study demonstrates that over the first 13 h after death, the highest accuracy estimations of time of death are provided by measurements of eyeball temperature. This period is characterized by the difference between its temperature (T) and the ambient temperature (TE) = 3.5 (± 1.2)°C. From 13 h post mortem onwards, this accuracy starts to be superseded by the accuracy of muscle temperature measurements, for which this period is defined by the values TTE= 8.9 (± 1.7)°C. From 14 h post mortem, the accuracy of eyeball temperature measurements is superseded by the accuracy of rectal temperature measurements, for which this period is defined by the values TTE= 9.2 (± 1.8)°C. Between 13 and 23.5 h post mortem, the accuracy of the estimation of time of death is highest for muscle temperature measurements, defined by the values TTE= 8.9 (± 1.7)°C and TTE= 4.6 (± 1.7)°C, while from approximately 24 h post mortem onwards, until the cooling of the body is complete, it becomes highest for rectal temperature measurements, for which this period is defined by the values TTE= 5.2 (± 1.8)°C. The accuracy of the measurements of orbit soft tissue temperatures throughout the entire recording period was worse than that for the eyeball measurements. Starting approximately 10.5 h post mortem, defined by TTE= 4.8 (± 1.4)°C, this started to be superseded by the accuracy of muscle temperature measurements.

In summary, the tests have shown that eyeball and orbital tissue temperature measurements may play a key role for estimation of the time of death over the several-hour-long dynamic period occurring directly after death during which the dead body cools off, and that thereafter these measurements should be replaced by muscle or rectal temperature measurements. The results prove the usefulness of the eyeball and the orbit soft tissues as sites for temperature measurement in dead bodies in order to assess the time of death.

The anatomy and physiology of the swine and the human eyeball is similar. This similarity must also apply to the time course of the post-mortem temperature decrease. Arguments to support selection of the eyeballs include: absence of any influence of clothing on their temperature; the practical lack of intersubject variability with respect to thermal capacity of the eyeballs resulting from the homogenicity of the structure and the anatomical location, which leads to identical thermal properties; and the influence of ambient temperature in different individuals depends on eyelid thickness, which is related to the degree of obesity only to a minor extent. This should enable comparison of results obtained from different corpses. The above reasoning applies for the most part to the orbit soft tissues. With longer times from death, however, temperature measurements in the rectum and in the muscles prove superior to measurements in the eyeball and the orbit soft tissues.

The proposition to use eyeball temperature in forensic medicine can be supported by the minor effect of intersubject variables (body size, clothing, etc.) on the cooling rate. The invasive nature of the test may limit its use in some countries for regulatory reasons. However, this obstacle may be omitted using non-invasive infrared laser thermometers providing that the reading will not be too much affected by external conditions, such as wind.

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