Legal Medicine xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Legal Medicine journal homepage: www.elsevier.com/locate/legalmed
Temperature based forensic death time estimation: The standard model in experimental test M. Hubig a,⇑,1, H. Muggenthaler a,1, I. Sinicina b, G. Mall a a b
Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Germany Institute of Legal Medicine, Ludwig Maximilians-University Munich, Germany
a r t i c l e
i n f o
Article history: Received 28 January 2015 Received in revised form 17 April 2015 Accepted 12 May 2015 Available online xxxx Keywords: Temperature based death time estimation Henßge method Nomogram Cooling experiments Confidence interval probability check Bias
a b s t r a c t The determination of the time since death is essential to forensic homicide investigations since the time of death represents the presumed time of the offence. Erroneous death time estimates may lead to false acquittal or conviction of suspects. Since its introduction 30 years back, the nomogram method by Henßge has been established as the standard procedure of temperature-based death time determination in the early post-mortem period. The present study provides an independent investigation of the validity of its death time estimates and their corresponding 95%-confidence intervals. Comparison to post-mortem cooling curves recorded under controlled conditions of 84 suddenly deceased with known death times yielded the following results: (1) Violations of the predicted 95%-confidence intervals by the nomogram method were observed in 48 of 84 cases (57.1%). (2) The standard deviations computed from our experimental data considerably exceed those presupposed in the nomogram method for 95%-confidence interval derivation. (3) The nomogram method shows a clear trend to over-estimate the post-mortem interval in cases with high body mass and large surface area. Since in the light of our experiments the validity of the nomogram method seems to be problematic, death time estimates – and particularly their 95%-confidence interval limits – have to be interpreted carefully and should only be restrictively used as court evidence to support or refute alibis. Systematic overestimation of the post-mortem interval in bodies of high mass and large surface area must be taken into account. Ó 2015 Elsevier Ireland Ltd. All rights reserved.
1. Introduction Death time determination is essential to medico-legal investigations since the time of death represents the presumed time of the offence. Knowledge of the death time is required to check alibis of suspects. When used in court, erroneous death time estimates may lead to false acquittals or convictions. Temperature back-calculation based on the post-mortem cooling process provides the most accurate death time estimates in the early post-mortem period. The temperature of the deceased is measured at the crime scene commonly in the rectum. Fig. 1 illustrates the typically sigmoid shape of a post-mortem rectal temperature–time curve. ⇑ Corresponding author at: Biomechanics Department, Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Fürstengraben 23, 07743 Jena, Germany. Tel.: +49 3461 937913; fax: +49 3461 935552. E-mail address: [email protected] (M. Hubig). 1 The authors contributed equally.
The length of the post-mortem interval (PMI) can be determined from the position of the temperature–time measurement. The correctness of PMI determination depends on the correctness of the model cooling curve. If the curve is too flat, the PMI will be over-estimated. If the curve is too steep, the PMI will be under-estimated. The model developed by Henßge [1,2] is widely used for temperature-based death time determination. It is founded on the double-exponential model of post-mortem rectal cooling by Marshall and Hoare [3]:
T TA p Z eZt ept ¼ pZ T0 TA p Z where T0 is the core temperature at death time and TA stands for the ambient temperature at the scene. The original parameter definition, in which parameter Z depended on body mass and surface area, was simplified by Henßge:
http://dx.doi.org/10.1016/j.legalmed.2015.05.005 1344-6223/Ó 2015 Elsevier Ireland Ltd. All rights reserved.
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
2
M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
been published so far. In the light of the importance of the nomogram method as forensic standard procedure of death time determination, the aim of the present study was to investigate the precision of the method in 84 controlled postmortem cooling experiments with the bodies of suddenly deceased persons with known death times [11].
T
Tmeasured
2. Method and terminology
0
t^: death time estimate
PMI
Fig. 1. Temperature based death time determination. Graph: Rectal cooling model curve, Tmeasured: Measured rectal temperature, t ^ : Estimated death time by reverse diagram evaluation, PMI: Post-mortem interval.
Z ¼ 1:2815m0:625 0:0284 with m: body mass [kg]. p ¼ 5Z for ambient temperatures 623.2 °C. p ¼ 10Z for ambient temperatures P23.3 °C. The model parameters were fitted to curves of 41 post-mortem standard cooling experiments (body mass 9–112 kg, time elapsed between death and experiment 1–6 h, bodies naked in prone position on thermally indifferent ground, approximately constant ambient temperature around 9 °C in winter and 17.4 °C in summer with fluctuations of ±2 °C). Henßge introduced the body mass correction factor c based on 25 non-standard cooling experiments (dry and wet clothing, air movement). In case of insulating environmental conditions, a correction factor c > 1 is chosen to increase virtual body mass resulting in slowed down cooling. In case of non-insulating environmental conditions, a correction factor c < 1 is chosen to decrease virtual body mass and hence speed up cooling. Stipantis and Henßge [4] defined the normalized temperature Q: = (T TA)/(T0 TA) stating: ‘‘by normalizing to essential influencing factors systematic errors in death time back-calculation can be avoided’’. They derived 95%-confidence-intervals for three Q-ranges (Q1: early, Q2: medium, Q3: late cooling phase). The model can be applied at the crime scene using a nomogram or a special software (www.amasoft.de). Correction factors can be determined following the guidelines in current textbooks on forensic pathology [5–7]. As goes for all empirical models the validity of the Henßge-model strongly depends on the experimental sample and settings used for calibration. Only two studies so far investigated the precision of the nomogram method. Both studies are based on real forensic cases with a single rectal temperature measurement at the crime scene. In 1990 a multi-center study [8] collected 76 cases. In 46 cases the environmental conditions were relatively certain and true death time could be narrowed down to short time intervals based on the results of police investigations. In the other 30 cases environmental conditions were uncertain and the true death time spans relatively wide. A closer look at the 46 ‘reliable’ cases reveals that in 35 cases the time since death was below 10 h and therefore in the early cooling phase. In 2000 further 72 cases were presented [9]. In only 27 of these cases the true death time spans could be narrowed down to 0.5–1.0 h, in 5 cases to >1.0–2.0 h, in 7 cases to >2.0–5.0 h, in 6 cases to >5.0–10 h and in 8 cases to >10.0–25 h. In the remaining cases no reliable death time interval was available. Although implausible results by the nomogram method can be observed in practical case work [10], a study on the precision of the nomogram method on the basis of consecutive experimental rectal temperature measurements under controlled conditions has not
N = 84 postmortem cooling experiments an in bodies of recently and suddenly deceased with known times since death t n;k were performed under controlled conditions in a climatic chamber [11]. Rectal-temperature–time-measurements M n;k :¼ ðt n;k ; T n;k Þ were recorded every minute. Depending on the experiment duration the number K of measurements differed. Experimental boundary conditions were ambient temperature T A;n , initial body core temperature T 0;n , body mass mn and correction factor cn. The experiments were conducted in standard conditions, the correction factors cn – selected according to the general recommendations [5–7] – in the range of 1.0–1.4 had to account for dry clothing only. The Henßge-model cooling curve is computed for each cooling experiment an with temperature values T H n;k at time t n;k . Depending on its first recorded rectal temperature T n;0 an experimental case an is classified initially hyperthermic if
DT 0;n ¼ T 0;n T Hn ðtn;0 Þ > 0:5°C and initially hypothermic if DT 0;n ¼ T 0;n T Hn ðtn;0 Þ > 0:5°C. The experimental sample G0 of 84 cases is subdivided in subgroup G1 of 38 definitely normothermic cases and subgroup G2 of 46 potentially non-normothermic cases. G2 is further subdivided in subgroup G2A of 18 potentially hyperthermic cases and subgroup G2B of 28 potentially hypothermic cases. We adopted the solution of choosing arbitrary thresholds of ±0.5 °C + 37.2 °C to discriminate the potentially hypo-, hyper-, and normothermal group with three ideas in mind: (1) That the time difference between death and measurement start in case of hypothermia was relatively short and thus should not have led to a decline of the core temperature T under 37.2 °C at time of measurement. On the other hand we assumed, that hypothermia means a rise in the body core temperature of at least 1–2 °C. (2) The body core temperature in the group of normothermal subjects is subjected to statistical deviations. This fact, as well as the natural decline of body core temperature after death and before measurement begin forces the use of a time interval containing the regular core temperature of 37.2 °C. (3) In case of hypothermia the subjects should have had a initial core temperature of at least 37.2– 0.5 °C to be detectable. Certainly it is not possible to derive the thresholds from the soft criteria (1)–(3) so we had to introduce some arbitrariness at this point. This very fact as well as statistical deviations surely lead to some cases of misclassification and a blurring of the results. Since the results are highly significant we do not think that they would be changed by the misclassification of some cases. Death time back-calculation according to the Henßge-model is performed for each cooling experiment an and all rectal temperatures T n;k recorded during that experiment, resulting in K Henßge death time estimates t ^n;k , that can be compared to the (known) true death time tn;k . The difference between the estimated and the true death time is the error en;k :
en;k ¼ t ^n;k tn;k As t^n;k is a realization of a random variable t^ and t n;k a realization of the fixed value t, we can compute the expectation value E(e)
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
of the random variable e ¼ t^ t. The quantity E(e) is the bias b of the back-calculation: b ¼ Eðt ^ tÞ. For realizations t ^n;k and tn;k the bias bn is:
bn ¼ Ek t^n;k t n;k
T Hn;k T A;n ¼ T 0;n T A;n
The following 95%-confidence-interval radii r, which include selection errors of c of ±0.1 [6] were defined by Henßge for standard (c = 1.0) and non-standard (c – 1.0) conditions and for the different ranges Q1, Q2, Q3: Q1 (0.5 < Q 6 1.0) standard r = 2.8 h Q2 (0.3 < Q 6 0.5) standard r = 3.2 h Q3 (0.2 < Q 6 0.3) standard r = 4.5 h
non-standard r = 2.8 h. non-standard r = 4.5 h. non-standard r = 7.0 h.
All measurements M n;k recorded during the cooling experiment an are divided into three subgroups M Qn 1 , MQn 2 , M Qn 3 to be able to evaluate the Henßge-model in the three Q-ranges separately. The number of experiments an containing measurements M n;k in the Q-ranges Qs (s = 1,2 and 3) will be N s . We will use the index variable j instead of K to indicate that the measurement index belongs to one of the Q-ranges Qs (s = 1,2 and 3). Js(n) denotes the number of elements in MQn s . An experiment an is a case with a single violation of the 95%-confidence-interval if the 95%-CI ¼ ½t ^n;k r; t^n;k þ r centered around a single back-calculated Henßge death time estimate t ^n;k does not contain the true death time t n;k . An experiment an is a case with 5% violation of the 95%-confidence interval in Qs, if in the Q-range Qs (s = 1,2 and 3) the true death time tn;j is not contained in the 95%-CI ¼ ½t^n;k r; t^n;k þ r in more than 5% of the Henßge death time estimates t ^n;k back-calculated from all Js(n) time–temperature measurements Mn;j ¼ ðtn;j T n;j Þ in the set M Qn s . The variable v n;s takes the value v n;s ¼ 1 if an is a case with 5% violation of the 95%-CI in the range Qs and the value v n;s ¼ 0 else. An experiment an is a case with 5% violation of the 95%-confidence interval if in at least one of the Q-ranges Qs (s = 1, 2 and 3) there is a 5% violation of the 95%-confidence interval. The variable vn takes the value vn = 1 if v n;s ¼ 1 for any s = 1, 2 and 3 and the value vn = 0 if vn,1 = vn,2 = vn,3 = 0. The number ns of cases with more than 5% violations of the 95%-CI is calculated for each Q-range Qs (s = 1, 2, 3). In analogy, the number of cases with more than 5% violations of the lower 95%-CI limit is ns,lower, the number of cases with more than 5% violations of the upper 95%-CI limit is ns,upper. This can be expressed by the following formulae, where #{...} means the number of elements of the set {...}:
o n 9 # j2f1;:::;J ðnÞgj t^ t 0:05 J ðnÞ s : ; o 8 n 9 # j2f1;:::;J ðnÞgj t t^ 0:05 ns;lower ¼ # n 2 f1; . . . ; Ng J s ðnÞ : ; 8
0.05 strongly suggests rejection of the null hypothesis H0 : Q1 => n1/N1 = 37/84 = 0.440 > 0.05. Q2 => n2/N2 = 38/77 = 0.494 > 0.05. Q3 => n3/N3 = 28/63 = 0.444 > 0.05. The quotients are calculated for the subgroup G1 of definitely normothermic cases as well: Q1 => n1/N1 = 13/38 = 0.342 > 0.05. Q2 => n2/N2 = 16/35 = 0.457 > 0.05. Q3 => n3/N3 = 14/30 = 0.467 > 0.05. If the errors of the Henßge-model were random, they would have expectation value 0 and a symmetric probability density function in good approximation. Violations of the upper and of the lower limits of the 95%-CI would be observed in similar frequency. The quotient of the number of 5%-violations of the upper CI-limits and the number of 5%-violations of the lower CI-limits ns,lower/ns,upper in the Q-ranges Qs would be around 1. However, in the group G0 of all cases the quotients are: Q1 => n1,lower/n1,upper = 4/33 = 0.121 1. Q2 => n2,lower/n2,upper = 2/36 = 0.056 1.
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
Table 1 Bias bQi in temperature ranges Qi in group G0 and in subgroup G1. Bias bQ1 in Q-range Q1 [hours]
Bias bQ2 in Q-range Q2 [hours]
Bias bQ3 in Q-range Q3 [hours]
Sample G0 all cases
Sample G0 all cases
Subgroup G1 normothermic cases
Sample G0 all cases
Subgroup G1 normothermic cases
Subgroup G1 normothermic cases
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
1.45
2.66
1.11
1.57
3.93
6.16
3.25
3.84
4.29
6.10
4.86
5.53
Fig. 2. Frequency diagrams of biases bQ1 (early cooling phase), bQ2 (middle cooling phase), bQ3 (late cooling phase) in the groups G0 (all cases) and G1 (initially normothermic cases).
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx Table 2 Numbers Ns, ns, ns,upper, ns,lower of cases in (sub)groups G0, G1, G2A, G2B for temperature ranges Qi. NS
Q1 Q2 Q3
ns
ns,upper
ns,lower
G0
G1
G2A
G2B
G0
G1
G2A
G2B
G0
G1
G2A
G2B
G0
G1
G2A
G2B
84 77 63
38 35 30
18 16 11
28 26 22
37 38 28
13 16 14
11 10 7
13 12 7
33 36 27
13 16 14
7 8 6
13 12 7
4 2 1
0 0 0
4 2 1
0 0 0
Row 1: Number of cases in Q-range Qs, s = 1,2 and 3. Ns: number of cases in (sub-)samples. ns: number of cases in Qs with 95%-CI violation. ns,upper: number of cases in Qs with violation of upper 95%-CI limit (post-mortem interval over-estimated by Henßge-model). ns,lower: number of cases in Qs with violation of lower 95%-CI limit (post-mortem interval under-estimated by Henßge-model). Row 2: Sample and subgroups G0: all cases. G1: cases with definite initial normothermia. G2A: cases with potential initial hyperthermia. G2B: cases with potential initial hypothermia. Row 3: Frequencies in Q-range Q1 (early cooling). Row 4: Frequencies in Q-range Q2 (middle cooling). Row 5: Frequencies in Q-range Q3 (late cooling).
Q3 => n3,lower/n3,upper = 1/27 = 0.037 1. In the definitely normothermic subgroup G1 the quotients are: Q1 => n1,lower/n1,upper = 0/13 = 0.000 1. Q2 => n2,lower/n2,upper = 0/16 = 0.000 1. Q3 => n3,lower/n3,upper = 0/14 = 0.000 1. 3.3. Influencing variables The group G0 with all N = 84 cases is divided into two subgroups depending on the value of the violation variable vn indicating a 5% violation of the 95%-confidence intervals in any Q-range Qs (vn = 1) or not (vn = 0). The body mass index BMI, the body mass m, the body height l, the body surface area BSA, the ambient temperature TA, the correction factor c and the time delay d between death and experiment are tested as potential influencing variables. The variables are tested for normal distribution in each of the groups G0[v = 0] and G0[v = 1]. Since especially in the group G0[v = 1] many variables were not normally distributed, nonparametric tests were applied. Table 3 presents the results of the Median and Kruskal– Wallis tests. The Median-test is known for its low test power, so differences between the results of the Median- and the Kruskal– Wallis test are to be expected. 3.4. Linear regression analysis Fig. 3 demonstrates linear regression analysis with the biases bQs as dependent and the most significant influencing variable body surface area BSA as independent variable. The regression analysis is performed in the sample G0 of all cases and in the subgroup G1 of definitely normothermic cases and for all Q-ranges. Since some variables bQi do not have a normal pdf, the common parametric regression test could not be performed. We thus report the R2 values: Q1 and G0: R2 = 0.302; Q2 and G0: R2 = 0.444; Q3 and G0: R2 = 0.258; Q1 and G1: R2 = 0.489; Q2 and G1: R2 = 0.461; Q3 and G1: R2 = 0.299. 4. Discussion The Henßge-model or nomogram method can be applied already at the crime scene meeting the needs of police investigators. Due to the body mass correction factor the method can also be applied to non-standard cooling scenarios frequently encountered in medico-legal practice. It has thus become the forensic standard procedure of temperature-based death time
determination in many countries. The model pursues an empirical approach and its validity depends on the experimental sample and settings. Apart from the original calibration experiments validation studies [8,9] concentrated on real cases with uncertain death times and environmental conditions, in which only single temperature– time measurements were taken. The model also has obvious problems. Its parameters depend on the body mass only and not on the constitution. It is unlikely that the body mass correction factor alone can compensate the complex influences of different boundary conditions on the cooling process. The present study – for the first time – provides a systematic and quantitative approach to evaluate the Henßge-model, based on rectal cooling curves of 84 suddenly deceased, with known death times, promptly transferred to a climatic chamber, where the ambient temperature of the death scene was kept constant [11]. Clothing and covering were left unchanged. The bodies were positioned on a metal tray as in the experiments by Henßge, thus a change of bearing compared to the death scene could not be avoided. If the Henßge-model was correct, deviations of the model-predicted from the experimental cooling curves would have to be attributed to random errors of measurement or parameter selection. Throughout the cooling process (Q1-early, Q2-middle, Q3-late cooling phase) the bias had positive mean values of 1.45–4.29 h in the overall and of 1.11–4.86 h in the initially normothermic sub-sample. The standard deviations ranged from 2.66 to 6.10 h in the overall and from 1.57 to 5.53 h in the normothermic sub-sample and exceeded the standard deviations used by Henßge (1.3–3.4 h) to estimate the width of his 95%-confidence-intervals. As the bias did not display a normal distribution in Q1 and Q2, the validity of the symmetric 95%-confidence intervals is at least questionable. In 48 of 84 cases (57.1%) the 95%-confidence intervals did not contain the true death time in more than permitted 5% in at least one of the Q-ranges. Furthermore, violations of the upper 95%-confidence-interval limits (over-estimation of the PMI) were predominant. Bias statistics as well as frequencies of upper and/or lower 95%-confidence-interval violations prove a systematic error in the Henßge-model. Body surface area BSA and body mass are significant influencing variables. Linear regression analysis displays a clear trend towards overestimation of the PMI with increasing body surface area and body mass. In the Henßge-model, a ‘normal’ initial rectal temperature of 37.2 °C is assumed. A ready objection to our experiments could be the cases with initial hypo-/hyperthermia. Our sample consisted of suddenly deceased who were neither severely ill nor
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
Table 3 Results of Median-test and Kruskal-Wallis-test in comparison of medians in (sub)groups G0[v = 0] vs G0[v = 1] and G1[v = 0] vs G1[v = 1] of variables BMI, mass, height, BSA, TA, c, delay. Cases
Sample G0 (N = 84)
Variable
BMI
Mass
Height
BSA
TA
c
Delay
BMI
Mass
Height
BSA
TA
c
Delay
Median-test P
0.19
0.002
0.001
Contents lists available at ScienceDirect
Legal Medicine journal homepage: www.elsevier.com/locate/legalmed
Temperature based forensic death time estimation: The standard model in experimental test M. Hubig a,⇑,1, H. Muggenthaler a,1, I. Sinicina b, G. Mall a a b
Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Germany Institute of Legal Medicine, Ludwig Maximilians-University Munich, Germany
a r t i c l e
i n f o
Article history: Received 28 January 2015 Received in revised form 17 April 2015 Accepted 12 May 2015 Available online xxxx Keywords: Temperature based death time estimation Henßge method Nomogram Cooling experiments Confidence interval probability check Bias
a b s t r a c t The determination of the time since death is essential to forensic homicide investigations since the time of death represents the presumed time of the offence. Erroneous death time estimates may lead to false acquittal or conviction of suspects. Since its introduction 30 years back, the nomogram method by Henßge has been established as the standard procedure of temperature-based death time determination in the early post-mortem period. The present study provides an independent investigation of the validity of its death time estimates and their corresponding 95%-confidence intervals. Comparison to post-mortem cooling curves recorded under controlled conditions of 84 suddenly deceased with known death times yielded the following results: (1) Violations of the predicted 95%-confidence intervals by the nomogram method were observed in 48 of 84 cases (57.1%). (2) The standard deviations computed from our experimental data considerably exceed those presupposed in the nomogram method for 95%-confidence interval derivation. (3) The nomogram method shows a clear trend to over-estimate the post-mortem interval in cases with high body mass and large surface area. Since in the light of our experiments the validity of the nomogram method seems to be problematic, death time estimates – and particularly their 95%-confidence interval limits – have to be interpreted carefully and should only be restrictively used as court evidence to support or refute alibis. Systematic overestimation of the post-mortem interval in bodies of high mass and large surface area must be taken into account. Ó 2015 Elsevier Ireland Ltd. All rights reserved.
1. Introduction Death time determination is essential to medico-legal investigations since the time of death represents the presumed time of the offence. Knowledge of the death time is required to check alibis of suspects. When used in court, erroneous death time estimates may lead to false acquittals or convictions. Temperature back-calculation based on the post-mortem cooling process provides the most accurate death time estimates in the early post-mortem period. The temperature of the deceased is measured at the crime scene commonly in the rectum. Fig. 1 illustrates the typically sigmoid shape of a post-mortem rectal temperature–time curve. ⇑ Corresponding author at: Biomechanics Department, Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Fürstengraben 23, 07743 Jena, Germany. Tel.: +49 3461 937913; fax: +49 3461 935552. E-mail address: [email protected] (M. Hubig). 1 The authors contributed equally.
The length of the post-mortem interval (PMI) can be determined from the position of the temperature–time measurement. The correctness of PMI determination depends on the correctness of the model cooling curve. If the curve is too flat, the PMI will be over-estimated. If the curve is too steep, the PMI will be under-estimated. The model developed by Henßge [1,2] is widely used for temperature-based death time determination. It is founded on the double-exponential model of post-mortem rectal cooling by Marshall and Hoare [3]:
T TA p Z eZt ept ¼ pZ T0 TA p Z where T0 is the core temperature at death time and TA stands for the ambient temperature at the scene. The original parameter definition, in which parameter Z depended on body mass and surface area, was simplified by Henßge:
http://dx.doi.org/10.1016/j.legalmed.2015.05.005 1344-6223/Ó 2015 Elsevier Ireland Ltd. All rights reserved.
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
2
M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
been published so far. In the light of the importance of the nomogram method as forensic standard procedure of death time determination, the aim of the present study was to investigate the precision of the method in 84 controlled postmortem cooling experiments with the bodies of suddenly deceased persons with known death times [11].
T
Tmeasured
2. Method and terminology
0
t^: death time estimate
PMI
Fig. 1. Temperature based death time determination. Graph: Rectal cooling model curve, Tmeasured: Measured rectal temperature, t ^ : Estimated death time by reverse diagram evaluation, PMI: Post-mortem interval.
Z ¼ 1:2815m0:625 0:0284 with m: body mass [kg]. p ¼ 5Z for ambient temperatures 623.2 °C. p ¼ 10Z for ambient temperatures P23.3 °C. The model parameters were fitted to curves of 41 post-mortem standard cooling experiments (body mass 9–112 kg, time elapsed between death and experiment 1–6 h, bodies naked in prone position on thermally indifferent ground, approximately constant ambient temperature around 9 °C in winter and 17.4 °C in summer with fluctuations of ±2 °C). Henßge introduced the body mass correction factor c based on 25 non-standard cooling experiments (dry and wet clothing, air movement). In case of insulating environmental conditions, a correction factor c > 1 is chosen to increase virtual body mass resulting in slowed down cooling. In case of non-insulating environmental conditions, a correction factor c < 1 is chosen to decrease virtual body mass and hence speed up cooling. Stipantis and Henßge [4] defined the normalized temperature Q: = (T TA)/(T0 TA) stating: ‘‘by normalizing to essential influencing factors systematic errors in death time back-calculation can be avoided’’. They derived 95%-confidence-intervals for three Q-ranges (Q1: early, Q2: medium, Q3: late cooling phase). The model can be applied at the crime scene using a nomogram or a special software (www.amasoft.de). Correction factors can be determined following the guidelines in current textbooks on forensic pathology [5–7]. As goes for all empirical models the validity of the Henßge-model strongly depends on the experimental sample and settings used for calibration. Only two studies so far investigated the precision of the nomogram method. Both studies are based on real forensic cases with a single rectal temperature measurement at the crime scene. In 1990 a multi-center study [8] collected 76 cases. In 46 cases the environmental conditions were relatively certain and true death time could be narrowed down to short time intervals based on the results of police investigations. In the other 30 cases environmental conditions were uncertain and the true death time spans relatively wide. A closer look at the 46 ‘reliable’ cases reveals that in 35 cases the time since death was below 10 h and therefore in the early cooling phase. In 2000 further 72 cases were presented [9]. In only 27 of these cases the true death time spans could be narrowed down to 0.5–1.0 h, in 5 cases to >1.0–2.0 h, in 7 cases to >2.0–5.0 h, in 6 cases to >5.0–10 h and in 8 cases to >10.0–25 h. In the remaining cases no reliable death time interval was available. Although implausible results by the nomogram method can be observed in practical case work [10], a study on the precision of the nomogram method on the basis of consecutive experimental rectal temperature measurements under controlled conditions has not
N = 84 postmortem cooling experiments an in bodies of recently and suddenly deceased with known times since death t n;k were performed under controlled conditions in a climatic chamber [11]. Rectal-temperature–time-measurements M n;k :¼ ðt n;k ; T n;k Þ were recorded every minute. Depending on the experiment duration the number K of measurements differed. Experimental boundary conditions were ambient temperature T A;n , initial body core temperature T 0;n , body mass mn and correction factor cn. The experiments were conducted in standard conditions, the correction factors cn – selected according to the general recommendations [5–7] – in the range of 1.0–1.4 had to account for dry clothing only. The Henßge-model cooling curve is computed for each cooling experiment an with temperature values T H n;k at time t n;k . Depending on its first recorded rectal temperature T n;0 an experimental case an is classified initially hyperthermic if
DT 0;n ¼ T 0;n T Hn ðtn;0 Þ > 0:5°C and initially hypothermic if DT 0;n ¼ T 0;n T Hn ðtn;0 Þ > 0:5°C. The experimental sample G0 of 84 cases is subdivided in subgroup G1 of 38 definitely normothermic cases and subgroup G2 of 46 potentially non-normothermic cases. G2 is further subdivided in subgroup G2A of 18 potentially hyperthermic cases and subgroup G2B of 28 potentially hypothermic cases. We adopted the solution of choosing arbitrary thresholds of ±0.5 °C + 37.2 °C to discriminate the potentially hypo-, hyper-, and normothermal group with three ideas in mind: (1) That the time difference between death and measurement start in case of hypothermia was relatively short and thus should not have led to a decline of the core temperature T under 37.2 °C at time of measurement. On the other hand we assumed, that hypothermia means a rise in the body core temperature of at least 1–2 °C. (2) The body core temperature in the group of normothermal subjects is subjected to statistical deviations. This fact, as well as the natural decline of body core temperature after death and before measurement begin forces the use of a time interval containing the regular core temperature of 37.2 °C. (3) In case of hypothermia the subjects should have had a initial core temperature of at least 37.2– 0.5 °C to be detectable. Certainly it is not possible to derive the thresholds from the soft criteria (1)–(3) so we had to introduce some arbitrariness at this point. This very fact as well as statistical deviations surely lead to some cases of misclassification and a blurring of the results. Since the results are highly significant we do not think that they would be changed by the misclassification of some cases. Death time back-calculation according to the Henßge-model is performed for each cooling experiment an and all rectal temperatures T n;k recorded during that experiment, resulting in K Henßge death time estimates t ^n;k , that can be compared to the (known) true death time tn;k . The difference between the estimated and the true death time is the error en;k :
en;k ¼ t ^n;k tn;k As t^n;k is a realization of a random variable t^ and t n;k a realization of the fixed value t, we can compute the expectation value E(e)
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
of the random variable e ¼ t^ t. The quantity E(e) is the bias b of the back-calculation: b ¼ Eðt ^ tÞ. For realizations t ^n;k and tn;k the bias bn is:
bn ¼ Ek t^n;k t n;k
T Hn;k T A;n ¼ T 0;n T A;n
The following 95%-confidence-interval radii r, which include selection errors of c of ±0.1 [6] were defined by Henßge for standard (c = 1.0) and non-standard (c – 1.0) conditions and for the different ranges Q1, Q2, Q3: Q1 (0.5 < Q 6 1.0) standard r = 2.8 h Q2 (0.3 < Q 6 0.5) standard r = 3.2 h Q3 (0.2 < Q 6 0.3) standard r = 4.5 h
non-standard r = 2.8 h. non-standard r = 4.5 h. non-standard r = 7.0 h.
All measurements M n;k recorded during the cooling experiment an are divided into three subgroups M Qn 1 , MQn 2 , M Qn 3 to be able to evaluate the Henßge-model in the three Q-ranges separately. The number of experiments an containing measurements M n;k in the Q-ranges Qs (s = 1,2 and 3) will be N s . We will use the index variable j instead of K to indicate that the measurement index belongs to one of the Q-ranges Qs (s = 1,2 and 3). Js(n) denotes the number of elements in MQn s . An experiment an is a case with a single violation of the 95%-confidence-interval if the 95%-CI ¼ ½t ^n;k r; t^n;k þ r centered around a single back-calculated Henßge death time estimate t ^n;k does not contain the true death time t n;k . An experiment an is a case with 5% violation of the 95%-confidence interval in Qs, if in the Q-range Qs (s = 1,2 and 3) the true death time tn;j is not contained in the 95%-CI ¼ ½t^n;k r; t^n;k þ r in more than 5% of the Henßge death time estimates t ^n;k back-calculated from all Js(n) time–temperature measurements Mn;j ¼ ðtn;j T n;j Þ in the set M Qn s . The variable v n;s takes the value v n;s ¼ 1 if an is a case with 5% violation of the 95%-CI in the range Qs and the value v n;s ¼ 0 else. An experiment an is a case with 5% violation of the 95%-confidence interval if in at least one of the Q-ranges Qs (s = 1, 2 and 3) there is a 5% violation of the 95%-confidence interval. The variable vn takes the value vn = 1 if v n;s ¼ 1 for any s = 1, 2 and 3 and the value vn = 0 if vn,1 = vn,2 = vn,3 = 0. The number ns of cases with more than 5% violations of the 95%-CI is calculated for each Q-range Qs (s = 1, 2, 3). In analogy, the number of cases with more than 5% violations of the lower 95%-CI limit is ns,lower, the number of cases with more than 5% violations of the upper 95%-CI limit is ns,upper. This can be expressed by the following formulae, where #{...} means the number of elements of the set {...}:
o n 9 # j2f1;:::;J ðnÞgj t^ t 0:05 J ðnÞ s : ; o 8 n 9 # j2f1;:::;J ðnÞgj t t^ 0:05 ns;lower ¼ # n 2 f1; . . . ; Ng J s ðnÞ : ; 8
0.05 strongly suggests rejection of the null hypothesis H0 : Q1 => n1/N1 = 37/84 = 0.440 > 0.05. Q2 => n2/N2 = 38/77 = 0.494 > 0.05. Q3 => n3/N3 = 28/63 = 0.444 > 0.05. The quotients are calculated for the subgroup G1 of definitely normothermic cases as well: Q1 => n1/N1 = 13/38 = 0.342 > 0.05. Q2 => n2/N2 = 16/35 = 0.457 > 0.05. Q3 => n3/N3 = 14/30 = 0.467 > 0.05. If the errors of the Henßge-model were random, they would have expectation value 0 and a symmetric probability density function in good approximation. Violations of the upper and of the lower limits of the 95%-CI would be observed in similar frequency. The quotient of the number of 5%-violations of the upper CI-limits and the number of 5%-violations of the lower CI-limits ns,lower/ns,upper in the Q-ranges Qs would be around 1. However, in the group G0 of all cases the quotients are: Q1 => n1,lower/n1,upper = 4/33 = 0.121 1. Q2 => n2,lower/n2,upper = 2/36 = 0.056 1.
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
Table 1 Bias bQi in temperature ranges Qi in group G0 and in subgroup G1. Bias bQ1 in Q-range Q1 [hours]
Bias bQ2 in Q-range Q2 [hours]
Bias bQ3 in Q-range Q3 [hours]
Sample G0 all cases
Sample G0 all cases
Subgroup G1 normothermic cases
Sample G0 all cases
Subgroup G1 normothermic cases
Subgroup G1 normothermic cases
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
1.45
2.66
1.11
1.57
3.93
6.16
3.25
3.84
4.29
6.10
4.86
5.53
Fig. 2. Frequency diagrams of biases bQ1 (early cooling phase), bQ2 (middle cooling phase), bQ3 (late cooling phase) in the groups G0 (all cases) and G1 (initially normothermic cases).
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx Table 2 Numbers Ns, ns, ns,upper, ns,lower of cases in (sub)groups G0, G1, G2A, G2B for temperature ranges Qi. NS
Q1 Q2 Q3
ns
ns,upper
ns,lower
G0
G1
G2A
G2B
G0
G1
G2A
G2B
G0
G1
G2A
G2B
G0
G1
G2A
G2B
84 77 63
38 35 30
18 16 11
28 26 22
37 38 28
13 16 14
11 10 7
13 12 7
33 36 27
13 16 14
7 8 6
13 12 7
4 2 1
0 0 0
4 2 1
0 0 0
Row 1: Number of cases in Q-range Qs, s = 1,2 and 3. Ns: number of cases in (sub-)samples. ns: number of cases in Qs with 95%-CI violation. ns,upper: number of cases in Qs with violation of upper 95%-CI limit (post-mortem interval over-estimated by Henßge-model). ns,lower: number of cases in Qs with violation of lower 95%-CI limit (post-mortem interval under-estimated by Henßge-model). Row 2: Sample and subgroups G0: all cases. G1: cases with definite initial normothermia. G2A: cases with potential initial hyperthermia. G2B: cases with potential initial hypothermia. Row 3: Frequencies in Q-range Q1 (early cooling). Row 4: Frequencies in Q-range Q2 (middle cooling). Row 5: Frequencies in Q-range Q3 (late cooling).
Q3 => n3,lower/n3,upper = 1/27 = 0.037 1. In the definitely normothermic subgroup G1 the quotients are: Q1 => n1,lower/n1,upper = 0/13 = 0.000 1. Q2 => n2,lower/n2,upper = 0/16 = 0.000 1. Q3 => n3,lower/n3,upper = 0/14 = 0.000 1. 3.3. Influencing variables The group G0 with all N = 84 cases is divided into two subgroups depending on the value of the violation variable vn indicating a 5% violation of the 95%-confidence intervals in any Q-range Qs (vn = 1) or not (vn = 0). The body mass index BMI, the body mass m, the body height l, the body surface area BSA, the ambient temperature TA, the correction factor c and the time delay d between death and experiment are tested as potential influencing variables. The variables are tested for normal distribution in each of the groups G0[v = 0] and G0[v = 1]. Since especially in the group G0[v = 1] many variables were not normally distributed, nonparametric tests were applied. Table 3 presents the results of the Median and Kruskal– Wallis tests. The Median-test is known for its low test power, so differences between the results of the Median- and the Kruskal– Wallis test are to be expected. 3.4. Linear regression analysis Fig. 3 demonstrates linear regression analysis with the biases bQs as dependent and the most significant influencing variable body surface area BSA as independent variable. The regression analysis is performed in the sample G0 of all cases and in the subgroup G1 of definitely normothermic cases and for all Q-ranges. Since some variables bQi do not have a normal pdf, the common parametric regression test could not be performed. We thus report the R2 values: Q1 and G0: R2 = 0.302; Q2 and G0: R2 = 0.444; Q3 and G0: R2 = 0.258; Q1 and G1: R2 = 0.489; Q2 and G1: R2 = 0.461; Q3 and G1: R2 = 0.299. 4. Discussion The Henßge-model or nomogram method can be applied already at the crime scene meeting the needs of police investigators. Due to the body mass correction factor the method can also be applied to non-standard cooling scenarios frequently encountered in medico-legal practice. It has thus become the forensic standard procedure of temperature-based death time
determination in many countries. The model pursues an empirical approach and its validity depends on the experimental sample and settings. Apart from the original calibration experiments validation studies [8,9] concentrated on real cases with uncertain death times and environmental conditions, in which only single temperature– time measurements were taken. The model also has obvious problems. Its parameters depend on the body mass only and not on the constitution. It is unlikely that the body mass correction factor alone can compensate the complex influences of different boundary conditions on the cooling process. The present study – for the first time – provides a systematic and quantitative approach to evaluate the Henßge-model, based on rectal cooling curves of 84 suddenly deceased, with known death times, promptly transferred to a climatic chamber, where the ambient temperature of the death scene was kept constant [11]. Clothing and covering were left unchanged. The bodies were positioned on a metal tray as in the experiments by Henßge, thus a change of bearing compared to the death scene could not be avoided. If the Henßge-model was correct, deviations of the model-predicted from the experimental cooling curves would have to be attributed to random errors of measurement or parameter selection. Throughout the cooling process (Q1-early, Q2-middle, Q3-late cooling phase) the bias had positive mean values of 1.45–4.29 h in the overall and of 1.11–4.86 h in the initially normothermic sub-sample. The standard deviations ranged from 2.66 to 6.10 h in the overall and from 1.57 to 5.53 h in the normothermic sub-sample and exceeded the standard deviations used by Henßge (1.3–3.4 h) to estimate the width of his 95%-confidence-intervals. As the bias did not display a normal distribution in Q1 and Q2, the validity of the symmetric 95%-confidence intervals is at least questionable. In 48 of 84 cases (57.1%) the 95%-confidence intervals did not contain the true death time in more than permitted 5% in at least one of the Q-ranges. Furthermore, violations of the upper 95%-confidence-interval limits (over-estimation of the PMI) were predominant. Bias statistics as well as frequencies of upper and/or lower 95%-confidence-interval violations prove a systematic error in the Henßge-model. Body surface area BSA and body mass are significant influencing variables. Linear regression analysis displays a clear trend towards overestimation of the PMI with increasing body surface area and body mass. In the Henßge-model, a ‘normal’ initial rectal temperature of 37.2 °C is assumed. A ready objection to our experiments could be the cases with initial hypo-/hyperthermia. Our sample consisted of suddenly deceased who were neither severely ill nor
Please cite this article in press as: Hubig M et al. Temperature based forensic death time estimation: The standard model in experimental test. Leg Med (2015), http://dx.doi.org/10.1016/j.legalmed.2015.05.005
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M. Hubig et al. / Legal Medicine xxx (2015) xxx–xxx
Table 3 Results of Median-test and Kruskal-Wallis-test in comparison of medians in (sub)groups G0[v = 0] vs G0[v = 1] and G1[v = 0] vs G1[v = 1] of variables BMI, mass, height, BSA, TA, c, delay. Cases
Sample G0 (N = 84)
Variable
BMI
Mass
Height
BSA
TA
c
Delay
BMI
Mass
Height
BSA
TA
c
Delay
Median-test P
0.19
0.002
0.001
