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Estimation of the time since death: Sudden increase of ambient temperature
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Forensic Science International 176 (2008) 196–199 www.elsevier.com/locate/forsciint
Estimation of the time since death: Sudden increase of ambient temperature Paolo Bisegna a,*, Claus Henßge b, Lars Althaus c, Giusto Giusti d a
Department of Civil Engineering, University of Rome ‘‘Tor Vergata’’, 00133 Rome, Italy b Institute of Legal Medicine, University of Duisburg-Essen, D-45122 Essen, Germany c Institute of Legal Medicine, Klinikum Duisburg, 47055 Duisburg, Germany d Department of Public Health and Cellular Biology, University of Rome ‘‘Tor Vergata’’, 00133 Rome, Italy Received 10 May 2007; accepted 11 September 2007 Available online 29 October 2007
Abstract This work presents a procedure for the postmortem interval estimation in the presence of a rapid increase of ambient temperature occurred during the cooling phase. The resulting disturbance produced on the cooling curve is proved to obey a two-exponential law and is removed from the actually measured body temperature. This yields a theoretical/modified body temperature, which enables the estimation of the time since death by means of the standard Nomogram method. # 2007 Elsevier Ireland Ltd. All rights reserved. Keywords: Postmortem interval estimation; Nomogram method; Body cooling
1. Introduction Different methods are available in the literature to estimate the time since death of a body found under suspicious circumstances (e.g., [1] and the references cited therein). This paper deals with the standard Nomogram method [2,3], based on body cooling modelled by the two-exponential equation of Marshall and Hoare [4]. The Nomogram method requires a constant (or, at most, slightly varying) ambient temperature during the whole cooling period. Actual murder cases in which the bodies had been transported from high ambient temperatures to low ambient temperatures motivated the research for methods enabling the postmortem interval estimation in the presence of a rapid change of ambient temperature occurred during the cooling phase. The case of a sudden decrease of ambient temperature was successfully considered in Refs. [5,6] but, in the case of a sudden increase of ambient temperature, no procedure could be found to model the cooling curves. The latter situation
may occur in caseworks due, e.g., to an automatic, timeroperated, activation of the heating system at the scene of crime. However, at the best of the authors’ knowledge, it cannot be managed by any available body cooling based procedure. The aim of this work is to propose and validate such a procedure, enabling the estimation of the time since death after a sudden increase of ambient temperature. To this end, the experimental data reported in Ref. [5] are analyzed. The main idea underlying the proposed procedure is to modify the actually measured body temperature, in order to remove the disturbance induced by the sudden increase of ambient temperature. This task is achieved since the cited disturbance is proved to obey a two-exponential law. The modified body temperature, therefore relevant to a constant ambient temperature, enables the estimation of the time since death by means of standard methods, e.g., nomographically or by computing according to the classical two-exponential curve. 2. Materials and methods
* Corresponding author. Tel.: +39 06 7259 7097; fax: +39 06 7259 7005. E-mail addresses: [email protected] (P. Bisegna), [email protected] (C. Henßge), [email protected] (L. Althaus), [email protected] (G. Giusti). 0379-0738/$ – see front matter # 2007 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.forsciint.2007.09.007
The theoretical expression of body cooling, according to the two-exponential equation of Marshall and Hoare [4], is given by: Q¼
Tr Ta AB t ¼ A expðB tÞ ðA 1Þ exp A1 To Ta
(1)
P. Bisegna et al. / Forensic Science International 176 (2008) 196–199
197
Fig. 1. Standardized temperature Q vs. cooling time [h], relevant to times before the sudden change of ambient temperature. Experimental and theoretical curves. Big-dummy experiments (data from measurements VGR11, VGR12, VGR13, VGR21, VGR22, VGR23, 2VGR11, 2VGR12, 2VGR21, 2VGR22 of Ref. [5]).
Fig. 2. Standardized temperature Q vs. cooling time [h], relevant to times before the sudden change of ambient temperature. Experimental and theoretical curves. Small-dummy experiments (data from measurements VKL11, VKL12, VKL13, VKL21, VKL22, VKL23, 2VKL11, 2VKL12, 2VKL21, 2VKL22 of Ref. [5]).
where t is the time after death, Q is the standardized temperature, Tr is the rectal temperature at time t, To is the rectal temperature at death, Ta is the ambient temperature, assumed to be substantially constant, A is a parameter accounting for the postmortem temperature plateau, and B is the logarithmic (or Newtonian) cooling rate prevailing after the phase of plateau has elapsed. After Henßge [2,3], the constant B is estimated as follows:
This variation strongly influences the cooling of the body, so that Eq. (1) does not hold for times t greater than tc. However, Eq. (1) can be used to infer what the body temperature would have been if no sudden change of ambient temperature had taken place. This theoretical/modified body temperature is denoted by T 0r, and, at any time t > tc, it is obtained by rearranging the terms in Eq. (1), as follows:
B ¼ 1:2815 ð½corrective factor body weight ðkgÞ0:625 Þ þ 0:0284
T 0r ¼ T a1 þ Q ðT o T a1 Þ
(2)
The recommended value for the constant A is 1.25 for ambient temperature up to 23 8C.
It is emphasized that, according to the definition of temperature in Eq. (4) is held fixed to the former value Ta1.
(4) T 0r ,
the ambient
2.1. Analysis of experimental data The procedure proposed in this paper is described and validated by using the results of the cooling experiments on dummies reported in Refs. [5,6]. They are briefly described in what follows. The dummies were shaped like a human trunk, coated with a caoutchoucfoil and filled with a gel-mixture. Two different dummies were used, whose ‘‘virtual weights’’ [3] were determined as 58.7 and 24.6 kg, for the big and the small one, respectively. The dummies were heated in an incubator to a uniform body-like temperature of about 37 8C. Then they were first stored in a cold room (about 4.8 8C) for 7.5 h (big dummies) or 3.5 h (small dummies) and, thereafter, at an ambient temperature of about 21.8 8C. Eight test coolings are available from Ref. [5]. The cooling of dummies is satisfactorily described by Eq. (1) up to the time of the sudden increase of ambient temperature, just as in the further twelve test coolings [5] with a sudden decrease of ambient temperature. This issue is shown in Figs. 1 and 2, relevant to big and small dummies, respectively. In fact, one may observe from Fig. 2 that the cooling of small dummies could be better approximated by slightly modifying the value of the plateau constant A from 1.25 to 1.2. Accordingly, in what follows, the value A = 1.2 is chosen in the computations for small dummies, whereas the standard value A = 1.25 is adopted for big dummies.
2.2. Cooling after the sudden increase of ambient temperature It is assumed that, at some unknown time tc after death, the ambient temperature undergoes a sudden increase, from the value Ta1 to the value Ta2 (Fig. 3): DT a ¼ T a2 T a1
(3)
Fig. 3. Cooling curve of a big dummy (corresponding to 58.7 kg body) for the first 7.5 h at a mean ambient temperature of Ta1 = 4.2 8C and, for the next 15.5 h, at Ta2 = 21.1 8C. The ambient temperature undergoes a sudden increase DTa = 16.9 8C. At time t = 20 h, the measured temperature is Tr = 21.8 8C. If no sudden increase of ambient temperature had taken place, the dummy temperature, theoretically computed according to Eq. (1), would have been T 0r ¼ 14:1 C. Accordingly, the difference DT = 7.7 8C can be attributed to the sudden change of ambient temperature (data from measurement 2VGR21 of Ref. [5]).
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P. Bisegna et al. / Forensic Science International 176 (2008) 196–199
Fig. 4. Standardized ratio R vs. cooling time [h], relevant to times after the sudden change of ambient temperature. Experimental curves compared to theoretical curves (Eq. (7) or (8)). Big-dummy experiments (data from measurements 2VGR11, 2VGR12, 2VGR21, 2VGR22 of Ref. [5]).
Fig. 5. Standardized ratio R vs. cooling time [h], relevant to times after the sudden change of ambient temperature. Experimental curves compared to theoretical curves (Eq. (7) or (8)). Small-dummy experiments (data from measurements 2VKL11, 2VKL12, 2VKL21, 2VKL22 of Ref. [5]).
Consequently, one may compute the difference DT between the measured temperature Tr and the theoretical/modified temperature T 0r (Fig. 3):
gave the best fit. It is emphasized that d is a positive quantity despite the minus sign appearing in its definition, since B is a negative quantity.
DT ¼ T r T 0r
(5)
This difference can be attributed to the sudden increase of ambient temperature from Ta1 to Ta2. At the time of change tc, no difference in the body temperature can yet have developed, so that DT = 0. On the other hand, when the cooling process has come to an end, the measured temperature Tr equals the latter ambient temperature Ta2, whereas, by definition, T 0r equals the former ambient temperature Ta1: hence, for large times, the difference DT tends to approach the increase DTa of the ambient temperature. It appears quite natural to standardize the difference DT with respect to its cause, the increase DTa, so that the standardized ratio swings between 0 and 1: R¼1
DT DT a
(6)
Here the complement to 1 of the ratio DT/DTa is taken to let R vary from 1 at time tc, to 0 after the cooling process has reached its end. Figs. 4 and 5 report the standardized ratio R versus the cooling time t tc elapsed after the sudden change of ambient temperature. It appears that all the curves for the considered cases approximately coalesce into one curve, especially for big dummies. Hence, the question arises whether a mathematical expression can be found, able to describe the coalesced curves. Reasoning from a theoretical point of view, the most natural candidate would be the Marshall–Hoare two-exponential equation, provided that the cooling time after death t has been replaced by the cooling time after the sudden change of ambient temperature t tc. This expression is: AB ðt tc Þ R ¼ A exp½B ðt tc Þ ðA 1Þ exp (7) A1 However, Figs. 4 and 5 show that a better fit can be obtained by introducing a delay d in the previous equation, which leads to the following expression: AB ðt tc dÞ (8) R ¼ A exp½B ðt tc dÞ ðA 1Þ exp A1
C B
(9)
where the value C ¼ 0:07
The ability to theoretically describe the experimental coalesced curves (6) is valuable for the purpose of estimating the time since death after a sudden increase of ambient temperature. To this end, the following four-step procedure is here proposed. 2.3.1. First step of the procedure: estimation of R The theoretical expression (8) is used to compute an estimate of R at the measurement time t: AB ðDt dÞ R ¼ A exp½B ðDt dÞ ðA 1Þ exp A1
(10)
(11)
where Dt = t tc is the known time elapsed between the sudden change of ambient temperature and the body temperature measurement. The value of the delay d given by Eq. (9) is used. It is here assumed that Dt is greater than d. In other words, Dt is assumed to be large enough to allow the sudden increase of ambient temperature to significantly influence the body cooling. 2.3.2. Second step of the procedure: estimation of DT The perturbation DT on the measured body temperature due to the variation of ambient temperature DTa is then computed by recasting Eq. (6) as follows: DT ¼ DT a ð1 RÞ
(12)
2.3.3. Third step of the procedure: estimation of T 0r In turn, the previous estimate of DT yields the theoretical/modified body temperature T 0r that would have been measured if no sudden change of ambient temperature had taken place (Fig. 3), from Eq. (5): T 0r ¼ T r DT
The delay is given by: d¼
2.3. Procedure to estimate the time since death after a sudden increase of ambient temperature
(13)
where Tr is the actually measured body temperature. 2.3.4. Fourth step of the procedure: estimation of t The theoretical/modified body temperature T 0r corresponds to a constant ambient temperature Ta1. This enables to estimate the time since death t by using
P. Bisegna et al. / Forensic Science International 176 (2008) 196–199
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Fig. 6. Histograms of errors [h] of the calculated cooling times according to the proposed procedure. Panel (a): small-dummy experiments. Panel (b): big-dummy experiments. the standard method, either nomographically or by computing as usual according to Eq. (1). 2.3.5. Example of application of the proposed procedure The cooling experiment reported in Fig. 3 is considered. The measured dummy temperature is Tr = 21.8 8C, and the measurement is performed Dt = 12.5 h after the sudden increase of ambient temperature, from Ta1 = 4.2 8C to Ta2 = 21.1 8C, so that DTa = 16.9 8C. The initial dummy temperature is To = 37.7 8C. The parameters are as follows: B = 0.0721 h1, from Eq. (2), corresponding to a virtual weight of 58.7 kg; A = 1.25. First step. From Eq. (9) one computes d = 1 h. Hence, from Eq. (11) it results: R ¼ 1:25 exp½B 11:5 0:25 exp½5 B 11:5 ¼ 0:542
(14)
evidence that the period since death can be calculated by the above-mentioned four-step procedure if the ambient temperature undergoes a rapid increase at a known time from a known level to a higher level, e.g., due to a timer-operated activation of the heating system at the scene or by transportation of the body from the scene of death to the place where it was found. The histograms of errors of the calculated cooling times (Fig. 6) demonstrate no systematic error and a distribution well within the known confidence limits of the method [7]. References
Second step. From Eq. (12) it results: DT ¼ 16:9 ð1 0:542Þ ¼ 7:7 C
(15)
Third step. From Eq. (13) it results: T 0r ¼ 21:8 7:7 ¼ 14:1 C
(16)
Fourth step. From Eq. (1), after replacing Tr with T 0r and assuming a constant ambient temperature Ta1, it results: 14:1 4:2 ¼ 1:25 expðB tÞ 0:25 expð5 B tÞ (17) 37:7 4:2 which yields t = 20 h. The same value is obtained by the Nomogram. In this example the estimated value coincides with the effective cooling time. Q¼
3. Discussion It is well known [7] that the results of dummy experiments can be transferred on body cooling. Therefore, the argument exposed in Section 2, based on the experiments on dummies reported in Ref. [5], may provide
[1] C. Henßge, B. Madea, Estimation of the time since death in the early postmortem period, Forensic Sci. Int. 144 (2004) 167–175. [2] C. Henßge, Die Pra¨zision von Todeszeitscha¨tzungen durch die mathematische Beschreibung der rektalen Leichenabku¨hlung, Z. Rechtsmed. 83 (1979) 49–67. [3] C. Henßge, Death time estimation in case work. I. The rectal temperature time of death nomogram, Forensic Sci. Int. 38 (1988) 209– 236. [4] T.K. Marshall, F.E. Hoare, Estimating the time of death. I. The rectal cooling after death and its mathematical expression, J. Forensic Sci. 7 (1962) 56–81. [5] L. Althaus, Zur Todeszeitbestimmung aus der Leichenabku¨hlung bei sprunghaft wechselnder Umgebungstemperatur, Master’s Thesis, Universita¨t Essen, 1997. [6] L. Althaus, C. Henßge, Rectal temperature time of death nomogram: sudden change of ambient temperature, Forensic Sci. Int. 99 (1999) 171–178. [7] C. Henßge, Themperature-based methods II, in: B. Knight (Ed.), The Estimation of the Time Since Death in the Early Postmortem Period, Arnold, London, 2002, pp. 43–102.
Forensic Science International 176 (2008) 196–199 www.elsevier.com/locate/forsciint
Estimation of the time since death: Sudden increase of ambient temperature Paolo Bisegna a,*, Claus Henßge b, Lars Althaus c, Giusto Giusti d a
Department of Civil Engineering, University of Rome ‘‘Tor Vergata’’, 00133 Rome, Italy b Institute of Legal Medicine, University of Duisburg-Essen, D-45122 Essen, Germany c Institute of Legal Medicine, Klinikum Duisburg, 47055 Duisburg, Germany d Department of Public Health and Cellular Biology, University of Rome ‘‘Tor Vergata’’, 00133 Rome, Italy Received 10 May 2007; accepted 11 September 2007 Available online 29 October 2007
Abstract This work presents a procedure for the postmortem interval estimation in the presence of a rapid increase of ambient temperature occurred during the cooling phase. The resulting disturbance produced on the cooling curve is proved to obey a two-exponential law and is removed from the actually measured body temperature. This yields a theoretical/modified body temperature, which enables the estimation of the time since death by means of the standard Nomogram method. # 2007 Elsevier Ireland Ltd. All rights reserved. Keywords: Postmortem interval estimation; Nomogram method; Body cooling
1. Introduction Different methods are available in the literature to estimate the time since death of a body found under suspicious circumstances (e.g., [1] and the references cited therein). This paper deals with the standard Nomogram method [2,3], based on body cooling modelled by the two-exponential equation of Marshall and Hoare [4]. The Nomogram method requires a constant (or, at most, slightly varying) ambient temperature during the whole cooling period. Actual murder cases in which the bodies had been transported from high ambient temperatures to low ambient temperatures motivated the research for methods enabling the postmortem interval estimation in the presence of a rapid change of ambient temperature occurred during the cooling phase. The case of a sudden decrease of ambient temperature was successfully considered in Refs. [5,6] but, in the case of a sudden increase of ambient temperature, no procedure could be found to model the cooling curves. The latter situation
may occur in caseworks due, e.g., to an automatic, timeroperated, activation of the heating system at the scene of crime. However, at the best of the authors’ knowledge, it cannot be managed by any available body cooling based procedure. The aim of this work is to propose and validate such a procedure, enabling the estimation of the time since death after a sudden increase of ambient temperature. To this end, the experimental data reported in Ref. [5] are analyzed. The main idea underlying the proposed procedure is to modify the actually measured body temperature, in order to remove the disturbance induced by the sudden increase of ambient temperature. This task is achieved since the cited disturbance is proved to obey a two-exponential law. The modified body temperature, therefore relevant to a constant ambient temperature, enables the estimation of the time since death by means of standard methods, e.g., nomographically or by computing according to the classical two-exponential curve. 2. Materials and methods
* Corresponding author. Tel.: +39 06 7259 7097; fax: +39 06 7259 7005. E-mail addresses: [email protected] (P. Bisegna), [email protected] (C. Henßge), [email protected] (L. Althaus), [email protected] (G. Giusti). 0379-0738/$ – see front matter # 2007 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.forsciint.2007.09.007
The theoretical expression of body cooling, according to the two-exponential equation of Marshall and Hoare [4], is given by: Q¼
Tr Ta AB t ¼ A expðB tÞ ðA 1Þ exp A1 To Ta
(1)
P. Bisegna et al. / Forensic Science International 176 (2008) 196–199
197
Fig. 1. Standardized temperature Q vs. cooling time [h], relevant to times before the sudden change of ambient temperature. Experimental and theoretical curves. Big-dummy experiments (data from measurements VGR11, VGR12, VGR13, VGR21, VGR22, VGR23, 2VGR11, 2VGR12, 2VGR21, 2VGR22 of Ref. [5]).
Fig. 2. Standardized temperature Q vs. cooling time [h], relevant to times before the sudden change of ambient temperature. Experimental and theoretical curves. Small-dummy experiments (data from measurements VKL11, VKL12, VKL13, VKL21, VKL22, VKL23, 2VKL11, 2VKL12, 2VKL21, 2VKL22 of Ref. [5]).
where t is the time after death, Q is the standardized temperature, Tr is the rectal temperature at time t, To is the rectal temperature at death, Ta is the ambient temperature, assumed to be substantially constant, A is a parameter accounting for the postmortem temperature plateau, and B is the logarithmic (or Newtonian) cooling rate prevailing after the phase of plateau has elapsed. After Henßge [2,3], the constant B is estimated as follows:
This variation strongly influences the cooling of the body, so that Eq. (1) does not hold for times t greater than tc. However, Eq. (1) can be used to infer what the body temperature would have been if no sudden change of ambient temperature had taken place. This theoretical/modified body temperature is denoted by T 0r, and, at any time t > tc, it is obtained by rearranging the terms in Eq. (1), as follows:
B ¼ 1:2815 ð½corrective factor body weight ðkgÞ0:625 Þ þ 0:0284
T 0r ¼ T a1 þ Q ðT o T a1 Þ
(2)
The recommended value for the constant A is 1.25 for ambient temperature up to 23 8C.
It is emphasized that, according to the definition of temperature in Eq. (4) is held fixed to the former value Ta1.
(4) T 0r ,
the ambient
2.1. Analysis of experimental data The procedure proposed in this paper is described and validated by using the results of the cooling experiments on dummies reported in Refs. [5,6]. They are briefly described in what follows. The dummies were shaped like a human trunk, coated with a caoutchoucfoil and filled with a gel-mixture. Two different dummies were used, whose ‘‘virtual weights’’ [3] were determined as 58.7 and 24.6 kg, for the big and the small one, respectively. The dummies were heated in an incubator to a uniform body-like temperature of about 37 8C. Then they were first stored in a cold room (about 4.8 8C) for 7.5 h (big dummies) or 3.5 h (small dummies) and, thereafter, at an ambient temperature of about 21.8 8C. Eight test coolings are available from Ref. [5]. The cooling of dummies is satisfactorily described by Eq. (1) up to the time of the sudden increase of ambient temperature, just as in the further twelve test coolings [5] with a sudden decrease of ambient temperature. This issue is shown in Figs. 1 and 2, relevant to big and small dummies, respectively. In fact, one may observe from Fig. 2 that the cooling of small dummies could be better approximated by slightly modifying the value of the plateau constant A from 1.25 to 1.2. Accordingly, in what follows, the value A = 1.2 is chosen in the computations for small dummies, whereas the standard value A = 1.25 is adopted for big dummies.
2.2. Cooling after the sudden increase of ambient temperature It is assumed that, at some unknown time tc after death, the ambient temperature undergoes a sudden increase, from the value Ta1 to the value Ta2 (Fig. 3): DT a ¼ T a2 T a1
(3)
Fig. 3. Cooling curve of a big dummy (corresponding to 58.7 kg body) for the first 7.5 h at a mean ambient temperature of Ta1 = 4.2 8C and, for the next 15.5 h, at Ta2 = 21.1 8C. The ambient temperature undergoes a sudden increase DTa = 16.9 8C. At time t = 20 h, the measured temperature is Tr = 21.8 8C. If no sudden increase of ambient temperature had taken place, the dummy temperature, theoretically computed according to Eq. (1), would have been T 0r ¼ 14:1 C. Accordingly, the difference DT = 7.7 8C can be attributed to the sudden change of ambient temperature (data from measurement 2VGR21 of Ref. [5]).
198
P. Bisegna et al. / Forensic Science International 176 (2008) 196–199
Fig. 4. Standardized ratio R vs. cooling time [h], relevant to times after the sudden change of ambient temperature. Experimental curves compared to theoretical curves (Eq. (7) or (8)). Big-dummy experiments (data from measurements 2VGR11, 2VGR12, 2VGR21, 2VGR22 of Ref. [5]).
Fig. 5. Standardized ratio R vs. cooling time [h], relevant to times after the sudden change of ambient temperature. Experimental curves compared to theoretical curves (Eq. (7) or (8)). Small-dummy experiments (data from measurements 2VKL11, 2VKL12, 2VKL21, 2VKL22 of Ref. [5]).
Consequently, one may compute the difference DT between the measured temperature Tr and the theoretical/modified temperature T 0r (Fig. 3):
gave the best fit. It is emphasized that d is a positive quantity despite the minus sign appearing in its definition, since B is a negative quantity.
DT ¼ T r T 0r
(5)
This difference can be attributed to the sudden increase of ambient temperature from Ta1 to Ta2. At the time of change tc, no difference in the body temperature can yet have developed, so that DT = 0. On the other hand, when the cooling process has come to an end, the measured temperature Tr equals the latter ambient temperature Ta2, whereas, by definition, T 0r equals the former ambient temperature Ta1: hence, for large times, the difference DT tends to approach the increase DTa of the ambient temperature. It appears quite natural to standardize the difference DT with respect to its cause, the increase DTa, so that the standardized ratio swings between 0 and 1: R¼1
DT DT a
(6)
Here the complement to 1 of the ratio DT/DTa is taken to let R vary from 1 at time tc, to 0 after the cooling process has reached its end. Figs. 4 and 5 report the standardized ratio R versus the cooling time t tc elapsed after the sudden change of ambient temperature. It appears that all the curves for the considered cases approximately coalesce into one curve, especially for big dummies. Hence, the question arises whether a mathematical expression can be found, able to describe the coalesced curves. Reasoning from a theoretical point of view, the most natural candidate would be the Marshall–Hoare two-exponential equation, provided that the cooling time after death t has been replaced by the cooling time after the sudden change of ambient temperature t tc. This expression is: AB ðt tc Þ R ¼ A exp½B ðt tc Þ ðA 1Þ exp (7) A1 However, Figs. 4 and 5 show that a better fit can be obtained by introducing a delay d in the previous equation, which leads to the following expression: AB ðt tc dÞ (8) R ¼ A exp½B ðt tc dÞ ðA 1Þ exp A1
C B
(9)
where the value C ¼ 0:07
The ability to theoretically describe the experimental coalesced curves (6) is valuable for the purpose of estimating the time since death after a sudden increase of ambient temperature. To this end, the following four-step procedure is here proposed. 2.3.1. First step of the procedure: estimation of R The theoretical expression (8) is used to compute an estimate of R at the measurement time t: AB ðDt dÞ R ¼ A exp½B ðDt dÞ ðA 1Þ exp A1
(10)
(11)
where Dt = t tc is the known time elapsed between the sudden change of ambient temperature and the body temperature measurement. The value of the delay d given by Eq. (9) is used. It is here assumed that Dt is greater than d. In other words, Dt is assumed to be large enough to allow the sudden increase of ambient temperature to significantly influence the body cooling. 2.3.2. Second step of the procedure: estimation of DT The perturbation DT on the measured body temperature due to the variation of ambient temperature DTa is then computed by recasting Eq. (6) as follows: DT ¼ DT a ð1 RÞ
(12)
2.3.3. Third step of the procedure: estimation of T 0r In turn, the previous estimate of DT yields the theoretical/modified body temperature T 0r that would have been measured if no sudden change of ambient temperature had taken place (Fig. 3), from Eq. (5): T 0r ¼ T r DT
The delay is given by: d¼
2.3. Procedure to estimate the time since death after a sudden increase of ambient temperature
(13)
where Tr is the actually measured body temperature. 2.3.4. Fourth step of the procedure: estimation of t The theoretical/modified body temperature T 0r corresponds to a constant ambient temperature Ta1. This enables to estimate the time since death t by using
P. Bisegna et al. / Forensic Science International 176 (2008) 196–199
199
Fig. 6. Histograms of errors [h] of the calculated cooling times according to the proposed procedure. Panel (a): small-dummy experiments. Panel (b): big-dummy experiments. the standard method, either nomographically or by computing as usual according to Eq. (1). 2.3.5. Example of application of the proposed procedure The cooling experiment reported in Fig. 3 is considered. The measured dummy temperature is Tr = 21.8 8C, and the measurement is performed Dt = 12.5 h after the sudden increase of ambient temperature, from Ta1 = 4.2 8C to Ta2 = 21.1 8C, so that DTa = 16.9 8C. The initial dummy temperature is To = 37.7 8C. The parameters are as follows: B = 0.0721 h1, from Eq. (2), corresponding to a virtual weight of 58.7 kg; A = 1.25. First step. From Eq. (9) one computes d = 1 h. Hence, from Eq. (11) it results: R ¼ 1:25 exp½B 11:5 0:25 exp½5 B 11:5 ¼ 0:542
(14)
evidence that the period since death can be calculated by the above-mentioned four-step procedure if the ambient temperature undergoes a rapid increase at a known time from a known level to a higher level, e.g., due to a timer-operated activation of the heating system at the scene or by transportation of the body from the scene of death to the place where it was found. The histograms of errors of the calculated cooling times (Fig. 6) demonstrate no systematic error and a distribution well within the known confidence limits of the method [7]. References
Second step. From Eq. (12) it results: DT ¼ 16:9 ð1 0:542Þ ¼ 7:7 C
(15)
Third step. From Eq. (13) it results: T 0r ¼ 21:8 7:7 ¼ 14:1 C
(16)
Fourth step. From Eq. (1), after replacing Tr with T 0r and assuming a constant ambient temperature Ta1, it results: 14:1 4:2 ¼ 1:25 expðB tÞ 0:25 expð5 B tÞ (17) 37:7 4:2 which yields t = 20 h. The same value is obtained by the Nomogram. In this example the estimated value coincides with the effective cooling time. Q¼
3. Discussion It is well known [7] that the results of dummy experiments can be transferred on body cooling. Therefore, the argument exposed in Section 2, based on the experiments on dummies reported in Ref. [5], may provide
[1] C. Henßge, B. Madea, Estimation of the time since death in the early postmortem period, Forensic Sci. Int. 144 (2004) 167–175. [2] C. Henßge, Die Pra¨zision von Todeszeitscha¨tzungen durch die mathematische Beschreibung der rektalen Leichenabku¨hlung, Z. Rechtsmed. 83 (1979) 49–67. [3] C. Henßge, Death time estimation in case work. I. The rectal temperature time of death nomogram, Forensic Sci. Int. 38 (1988) 209– 236. [4] T.K. Marshall, F.E. Hoare, Estimating the time of death. I. The rectal cooling after death and its mathematical expression, J. Forensic Sci. 7 (1962) 56–81. [5] L. Althaus, Zur Todeszeitbestimmung aus der Leichenabku¨hlung bei sprunghaft wechselnder Umgebungstemperatur, Master’s Thesis, Universita¨t Essen, 1997. [6] L. Althaus, C. Henßge, Rectal temperature time of death nomogram: sudden change of ambient temperature, Forensic Sci. Int. 99 (1999) 171–178. [7] C. Henßge, Themperature-based methods II, in: B. Knight (Ed.), The Estimation of the Time Since Death in the Early Postmortem Period, Arnold, London, 2002, pp. 43–102.