- Email: [email protected]
Determination of time-dependent skin temperature decrease rates in the case of abrupt changes of environmental temperature
Forensic Science International 113 (2000) 219–226
www.elsevier.com / locate / forsciint
Determination of time-dependent skin temperature decrease rates in the case of abrupt changes of environmental temperature a ¨ Gita Mall a , *, Michael Hubig b , Gundolf Beier a , Andreas Buttner , Wolfgang Eisenmenger a a
b
Institute of Legal Medicine, Ludwigs-Maximilians-University Munich, Frauenlobstr. 7 a, 80337 Munich, Germany German Remote Sensing Data Center ( DFD) at German Aerospace Research Center ( DLR), 82230 Weßling, Germany
Abstract The present study deals with the development of a method for determining time-dependent temperature decrease rates and its application to postmortem surface cooling. The study concentrates on evaluating skin cooling behavior since data on skin cooling in the forensic literature are scarce. Furthermore, all heat transfer mechanisms strongly depend on the temperature gradient between body surface and environment. One of the main problems in modelling postmortem cooling processes is the dependence on the environmental temperature. All models for postmortem rectal cooling essentially presuppose a constant environmental temperature. In medico-legal practice, the temperature of the surrounding of a corpse mostly varies; therefore, an approach for extending the models to variable environmental temperatures is desirable. It consists in ‘localizing’ them to infinitesimal small intervals of time. An extended model differential equation is obtained and solved explicitly. The approach developed is applied to the single-exponential Newtonian model of surface cooling producing the following differential equation: T 9S (t) 5 2 l(t)(T S (t) 2 T E (t)) (with T S (t) the surface / skin temperature, T E (t) the environmental temperature, l(t) the temperature decrease rate and T S9 (t) the actual change of skin temperature or first-order derivative of T S ). The differential equation directly provides an estimator: T S9 (t) l(t) 5 2 ]]]] T S (t) 2 T E (t) for the time-dependent temperature decrease rate. The estimator is applied to two skin cooling experiments with different types of abrupt changes of environmental temperature, peak-like and step-like; the values of the time-dependent temperature decrease rate function were calculated. By reinserting them, the measured surface temperature curve could be accurately reconstructed, indicating that the extended model is well suited for describing surface cooling in the case of abrupt changes of environmental temperature. 2000 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Skin temperature; Postmortem cooling; Temperature decrease rate
1. Introduction *Corresponding author. Tel.: 149-89-5160-5184; fax: 149-895160-5144. E-mail address: [email protected] (G. Mall).
The present study deals with the development of a method for determining time-dependent temperature decrease rates and its application to postmortem
0379-0738 / 00 / $ – see front matter 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S0379-0738( 00 )00209-7
220
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
surface cooling experiments with abruptly changing environmental temperature. The development of models for describing postmortem cooling processes plays an important role in the determination of the time since death in medico-legal practice. Since the core temperature, measured as rectal temperature, only slowly decreases towards the environmental temperature, it is suitable for estimating the duration of the postmortem cooling period. Various models for rectal cooling have been developed, among them models based on the single-exponential Newtonian approach [1], the model of Marshall and Hoare [2] with the coefficients of Henßge [3,4] and the model of Green and Wright [5,6]. All of the above models essentially presuppose a constant environmental temperature. In reality, the surrounding temperature of a corpse mostly varies and even under experimental conditions it is difficult to keep the environmental temperature exactly constant. It is therefore desirable to develop an approach for extending postmortem cooling models to variable environmental temperatures. The presented study concentrates on skin cooling since data for surface cooling experiments in the forensic literature are scarce [7–9]. Furthermore, all heat transfer mechanisms, which are important for physical modelling, essentially depend on the temperature gradient between skin and environment and not between core and environment. The aims of the present study were: 1. to develop an approach for extending postmortem cooling models to variable environmental temperatures and to apply it to the single-exponential Newtonian model; 2. to derive an estimator for the time-dependent temperature decrease rate from the extended Newtonian model; and 3. to present temperature time decrease rates of two skin cooling experiments with typical abrupt changes of the environmental temperature, peaklike and step-like.
sents the classical approach for fitting surface temperature–time curves: T S (t) 5 (T S (0) 2 T E )e 2 l t 1 T E
where T S (t) represents the skin temperature, T S (0) the skin temperature at time of death, T E the environmental temperature and l the temperature decrease rate. The model is valid only for constant environmental temperature, T E 5 const. The classical Newtonian model is extended by ‘localizing’ it to small intervals of time [t, t 1 Dt] in which the environmental temperature T E as well as the temperature decrease rate l can be assumed to be constant. The ‘localized’ form of the classical Newtonian model, further referred to as ‘localized model’, is T S (t 1 Dt) 5 (T S (t) 2 T E (t))e 2 l Dt 1 T E (t)
T S (t 1 Dt) ¯ T S (t) 1 T 9S (t)Dt e 2 l Dt ¯ 1 2 lDt Thereby it is possible to approximate the value of the functions at time Dt. Inserting the Taylor series expansions in the ‘localized model’ (Eq. (2)) produces the following ‘localized’ differential equation: T 9S (t) 5 2 l(T S (t) 2 T E (t))
t
1E
2 L (t )
with u
E
L(u) 5 l(v) dv The single-exponential Newtonian model repre-
(3)
where T 9S (t) denotes the actual change of the skin temperature or the first-order derivative of the skin temperature T S (t). The differential equation can be solved by the so-called variation of constants; the solution represents the ‘extended Newtonian model’:
0
2.1. The extended Newtonian model
(2)
A Taylor series expansion of order one is applied to the function of the skin temperature as well as to the exponential function of the temperature decrease rate:
T S (t) 5 e 2. Material and method
(1)
0
l(u)T E (u)e L (u) du 1 T S (0)
2
(4)
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
where u and v denote the time variable in the integral expressions. This ‘extended Newtonian model’ is valid for non-constant environmental temperatures T E (t) and for non-constant temperature decrease rates l(t).
2.2. The temperature decrease rate l of the extended Newtonian model The differential equation (3) of the extended Newtonian model by a simple transformation provides an estimator for the temperature decrease rate l(t): T 9(t) l(t) 5 2 ]]]] T(t) 2 T E (t)
(5)
The estimator is valid for all cases where (T(t) 2 T E (t)) differs from zero. For numerical computations this restriction must be tightened: the absolute value of (T(t) 2 T E (t)) must be sufficiently different from zero.
2.3. Evaluation of experimental data 2.3.1. Experiments The method developed is applied to two cooling experiments, E1 and E2, with different types of abrupt changes of environmental temperature. 2.3.1.1. Experiment E1 Time of death, 10.10 a.m.; start of measurements, 11.50 a.m.; duration of measuring period, |21.4 h; skin temperature measured with surface sensor at uncovered forehead of male individual with body height 1.69 m and body mass 78.8 kg; environmental conditions, 2.75 h in closed cold room in still air at about 88C, then for |15 min in front of cold room in still air at |168C, then again in closed cold room (T E slightly increasing to about 10.58C). 2.3.1.2. Experiment E2 Time of death, 8.00 a.m.; start of measurements, 11.40 a.m.; duration of measuring period, 24.5 h; skin temperature measured with surface sensor at exposed forehead of male individual with body height 1.75 m and body weight 95 kg; environmental conditions, 3.75 h in closed room at almost constant
221
T E of about 248C, then kept in closed cold room in still air at about 98C (T E slightly increasing to about 12.58C). The environmental temperature was measured with a sensor |10 cm above the thigh region of the corpse. The temperature–time decrease curves of experiment E1 are presented in Fig. 1, and those of experiment E2 in Fig. 2.
2.3.2. Numerical calculation of the estimator for l(t) Comparatively few measuring points lie within the peak- or step-dominated phases; a sequence of equidistance between points is generated by linear interpolation between two neighboring measuring points of the skin and the environmental temperature. This approximates the actual continuous course of the temperature curves. Mentioning tuples (t i , T S (t i ), T E (t i )) or their components refers to the interpolated values as well as to the measured values. The time distance between neighboring interpolated measuring points is Dt. An estimator for the time derivative T 9S (t i ) of the skin temperature is calculated for every point of time t i from the tuples (t i , T S (t i ), T E (t i )) according to T S (t i 11 ) 2 T S (t i 21 ) T 9S (t i ) 5 ]]]]]] Dt
(6)
The estimator for the temperature decrease rate l(t i ) is computed according to formula (5) for every point of time t i . The numerical consistency of the model is tested by reconstructing the skin temperature T S from the environmental temperature T E and the local temperature decrease rates l. A pointwise recursive forward computation starting at time t 0 is carried out: T S (t i 11 ): 5 l(t i )(T S (t i ) 2 T E (t i ))Dt 1 T S (t i )
(7)
3. Results The single-exponential Newtonian model is valid only for constant environmental temperature. By localizing it to infinitesimal small intervals of time a generalized Newtonian differential equation (3) was developed. An explicit solution (4) was obtained. It
222
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
Fig. 1. Skin temperature T S and environmental temperature T E in experiment E1.
Fig. 2. Skin temperature T S and environmental temperature T E in experiment E2.
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
represents a generalized version of the single-exponential model valid also for non-constant environmental temperatures and non-constant temperature decrease rates. The model could be applied to cooling processes with continuously changing environmental temperature; since abrupt changes of environmental temperature represent more difficult test situations, it was applied to two cooling experiments with a peak-like and a step-like course of environmental temperature. The extended differential equation directly provided an estimator (5) for the time-dependent temperature decrease rate l(t). The above estimator was used to compute the time-dependent temperature decrease rates l(t) (Figs. 3 and 4) in the two cooling experiments E1 and E2. The oscillation of the curve of the temperature decrease rates l(t) is an artefact caused by the quantization of the temperature measurements in steps of 0.18C. Especially in the later cooling phase, the amplitude of this oscillation considerably increases; this is due to the structure of the estimator, which assumes larger values if the difference between object and environmental temperature becomes smaller. The numerical consistency of the estimator was tested by a forward calculation reconstructing the skin temperature curve T S (t) and comparing it with the measured skin temperature curve.
223
As can be seen in Figs. 5 and 6, the results of the test were very good; the reconstructed and the measured curve completely overlap each other. This complete congruence demonstrates that the model is well suited even for cases with abruptly changing environmental temperature.
4. Discussion A common method for simplifying model approaches in physics consists of assuming the influencing quantities to be constant. This can be illustrated in the Newtonian model valid for constant environmental temperature and thermally thin objects only. The model was used by Rainy [1] for describing the postmortem cooling process. An important contribution has been put forward by Sellier [10], who solved the differential equation of heat transfer for a cooling cylinder, thereby explaining the differences between the cooling behavior of the core (sigmoid cooling curve with plateau) and of the surface (single-exponential). Improved models by Marshall and Hoare [2] and Henßge [3,4] and Green and Wright [5,6] are double exponential and recursively exponential, respectively, and therefore better fit the sigmoid shape of the rectal temperature
Fig. 3. Estimation of temperature decrease rate ( l) from data of experiment E1.
224
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
Fig. 4. Estimation of temperature decrease rate ( l) from data of experiment E2.
Fig. 5. Reconstruction of skin temperature curve (T S modelled) in comparison to measured curve (T S measured) for experiment E1.
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
225
Fig. 6. Reconstruction of skin temperature curve (T S modelled) in comparison to measured curve (T S measured) for experiment E2.
decrease; they still presuppose a constant environmental temperature. In medico-legal practice, the temperature of the surroundings of a corpse mostly varies; even under experimental conditions it is difficult to keep the environment at an exactly constant temperature. Slight, continuous changes of the environmental temperature will most probably not influence the decrease of the core temperature, while they have a noticeable effect on the course of the surface temperature. Abrupt, peak-like or steplike, changes of the environmental temperature significantly alter the decrease of the skin temperature and also influence rectal cooling. Althaus and Henßge [11] in this respect recently published an investigation on the coefficients of the double-exponential model in cases of abrupt rise or fall of the ambient temperature. The present study provides an approach (‘localization’) for extending the above models to variable environmental temperatures. The localization approach was applied to the single-exponential Newtonian model which is suitable for describing the exponential shape of the skin temperature decrease curves. Data on skin cooling behavior in the forensic literature are scarce. Lyle and Cleveland [7] only
measured the cooling of the uncovered skin of the forehead and published a normed curve. Simonsen et al. [8] measured the temperature of the axillary skin, which is not representative. Newitt and Green [9] concentrated on the method of thermography and do not mention cooling data. Knowledge of skin cooling is essential for understanding and investigating the postmortem cooling process according to the contribution of the different heat transfer mechanisms. Radiation, convection and conduction strongly depend on the temperature gradient between body surface and environment and not between core and environment. To be able to compute the heat loss and to model the postmortem cooling process physically, it is necessary to quantify skin cooling by determining the skin temperature decrease rate. Since the body surface is extremely sensitive to even slight changes of the environmental temperature, the quantification has to take into account non-constant environmental temperatures and non-constant temperature decrease rates. We were able to derive an estimator for time-variable temperature decrease rates. It was thereby possible to quantify skin cooling in two experiments with typical abrupt changes of the environmental temperature. The numerical con-
226
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
sistency of the presented method for quantifying the temperature decrease of the skin was tested successfully by accurately reconstructing the measured skin temperature curves.
Acknowledgements The authors would like to thank Ms Mona Eckl for carrying out the measurements.
References [1] H. Rainy, On the cooling of dead bodies as indicating the length of time that has elapsed since death, Glasgow Med. J. (New Series) 1 (1869) 323–330; cited in: C. Henßge, B. Madea, Methoden zur Bestimmung der Todeszeit an Leich¨ ¨ en, Schmidt-Romhild-Verlag, Lubeck, 1988, pp. 142–144. [2] T.K. Marshall, F.E. Hoare, Estimating the time of death, J. Forensic. Sci. 7 (1962) 56–81, 189–211, 211–221. ¨ ¨ [3] C. Henßge, Die Prazision von Todeszeitschatzungen durch ¨ die mathematische Beschreibung der rektalen Leichenabkuhlung, Z. Rechtsmed. 83 (1979) 49–67.
¨ [4] C. Henßge, Todeszeitschatzungen durch die mathematische ¨ Beschreibung der rektalen Leichenabkuhlung unter ver¨ schiedenen Abkuhlbedingungen, Z. Rechtsmed. 87 (1981) 147–178. [5] M.A. Green, J.C. Wright, Postmortem interval estimation from body temperature data only, Forensic Sci. Int. 28 (1985) 35–46. [6] M.A. Green, J.C. Wright, The theoretical aspects of the time dependent Z-equation as a means of postmortem interval estimation using body temperature data only, Forensic Sci. Int. 28 (1985) 53–62. [7] H.P. Lyle, F.P. Cleveland, Determination of the time of death by body heat loss, J. Forensic Sci. 1 (1956) 11–24. [8] J. Simonsen, J. Voigt, N. Jeppesen, Determination of the time of death by continuous post-mortem temperature measurements, Med. Sci. Law 17 (1977) 112–122. [9] C. Newitt, M.A. Green, A thermographic study of surface cooling of cadavers, J. Forensic Sci. Soc. 19 (1979) 179– 181. [10] K. Sellier, Determination of the time of death by extrapolation of the temperature decrease curve, Acta Med. Leg. Soc. 11 (1958) 179–302. [11] C. Althaus, C. Henßge, Rectal temperature time of death nomogram: sudden change of ambient temperature, Forensic Sci. Int. 99 (1999) 171–178.
www.elsevier.com / locate / forsciint
Determination of time-dependent skin temperature decrease rates in the case of abrupt changes of environmental temperature a ¨ Gita Mall a , *, Michael Hubig b , Gundolf Beier a , Andreas Buttner , Wolfgang Eisenmenger a a
b
Institute of Legal Medicine, Ludwigs-Maximilians-University Munich, Frauenlobstr. 7 a, 80337 Munich, Germany German Remote Sensing Data Center ( DFD) at German Aerospace Research Center ( DLR), 82230 Weßling, Germany
Abstract The present study deals with the development of a method for determining time-dependent temperature decrease rates and its application to postmortem surface cooling. The study concentrates on evaluating skin cooling behavior since data on skin cooling in the forensic literature are scarce. Furthermore, all heat transfer mechanisms strongly depend on the temperature gradient between body surface and environment. One of the main problems in modelling postmortem cooling processes is the dependence on the environmental temperature. All models for postmortem rectal cooling essentially presuppose a constant environmental temperature. In medico-legal practice, the temperature of the surrounding of a corpse mostly varies; therefore, an approach for extending the models to variable environmental temperatures is desirable. It consists in ‘localizing’ them to infinitesimal small intervals of time. An extended model differential equation is obtained and solved explicitly. The approach developed is applied to the single-exponential Newtonian model of surface cooling producing the following differential equation: T 9S (t) 5 2 l(t)(T S (t) 2 T E (t)) (with T S (t) the surface / skin temperature, T E (t) the environmental temperature, l(t) the temperature decrease rate and T S9 (t) the actual change of skin temperature or first-order derivative of T S ). The differential equation directly provides an estimator: T S9 (t) l(t) 5 2 ]]]] T S (t) 2 T E (t) for the time-dependent temperature decrease rate. The estimator is applied to two skin cooling experiments with different types of abrupt changes of environmental temperature, peak-like and step-like; the values of the time-dependent temperature decrease rate function were calculated. By reinserting them, the measured surface temperature curve could be accurately reconstructed, indicating that the extended model is well suited for describing surface cooling in the case of abrupt changes of environmental temperature. 2000 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Skin temperature; Postmortem cooling; Temperature decrease rate
1. Introduction *Corresponding author. Tel.: 149-89-5160-5184; fax: 149-895160-5144. E-mail address: [email protected] (G. Mall).
The present study deals with the development of a method for determining time-dependent temperature decrease rates and its application to postmortem
0379-0738 / 00 / $ – see front matter 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S0379-0738( 00 )00209-7
220
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
surface cooling experiments with abruptly changing environmental temperature. The development of models for describing postmortem cooling processes plays an important role in the determination of the time since death in medico-legal practice. Since the core temperature, measured as rectal temperature, only slowly decreases towards the environmental temperature, it is suitable for estimating the duration of the postmortem cooling period. Various models for rectal cooling have been developed, among them models based on the single-exponential Newtonian approach [1], the model of Marshall and Hoare [2] with the coefficients of Henßge [3,4] and the model of Green and Wright [5,6]. All of the above models essentially presuppose a constant environmental temperature. In reality, the surrounding temperature of a corpse mostly varies and even under experimental conditions it is difficult to keep the environmental temperature exactly constant. It is therefore desirable to develop an approach for extending postmortem cooling models to variable environmental temperatures. The presented study concentrates on skin cooling since data for surface cooling experiments in the forensic literature are scarce [7–9]. Furthermore, all heat transfer mechanisms, which are important for physical modelling, essentially depend on the temperature gradient between skin and environment and not between core and environment. The aims of the present study were: 1. to develop an approach for extending postmortem cooling models to variable environmental temperatures and to apply it to the single-exponential Newtonian model; 2. to derive an estimator for the time-dependent temperature decrease rate from the extended Newtonian model; and 3. to present temperature time decrease rates of two skin cooling experiments with typical abrupt changes of the environmental temperature, peaklike and step-like.
sents the classical approach for fitting surface temperature–time curves: T S (t) 5 (T S (0) 2 T E )e 2 l t 1 T E
where T S (t) represents the skin temperature, T S (0) the skin temperature at time of death, T E the environmental temperature and l the temperature decrease rate. The model is valid only for constant environmental temperature, T E 5 const. The classical Newtonian model is extended by ‘localizing’ it to small intervals of time [t, t 1 Dt] in which the environmental temperature T E as well as the temperature decrease rate l can be assumed to be constant. The ‘localized’ form of the classical Newtonian model, further referred to as ‘localized model’, is T S (t 1 Dt) 5 (T S (t) 2 T E (t))e 2 l Dt 1 T E (t)
T S (t 1 Dt) ¯ T S (t) 1 T 9S (t)Dt e 2 l Dt ¯ 1 2 lDt Thereby it is possible to approximate the value of the functions at time Dt. Inserting the Taylor series expansions in the ‘localized model’ (Eq. (2)) produces the following ‘localized’ differential equation: T 9S (t) 5 2 l(T S (t) 2 T E (t))
t
1E
2 L (t )
with u
E
L(u) 5 l(v) dv The single-exponential Newtonian model repre-
(3)
where T 9S (t) denotes the actual change of the skin temperature or the first-order derivative of the skin temperature T S (t). The differential equation can be solved by the so-called variation of constants; the solution represents the ‘extended Newtonian model’:
0
2.1. The extended Newtonian model
(2)
A Taylor series expansion of order one is applied to the function of the skin temperature as well as to the exponential function of the temperature decrease rate:
T S (t) 5 e 2. Material and method
(1)
0
l(u)T E (u)e L (u) du 1 T S (0)
2
(4)
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
where u and v denote the time variable in the integral expressions. This ‘extended Newtonian model’ is valid for non-constant environmental temperatures T E (t) and for non-constant temperature decrease rates l(t).
2.2. The temperature decrease rate l of the extended Newtonian model The differential equation (3) of the extended Newtonian model by a simple transformation provides an estimator for the temperature decrease rate l(t): T 9(t) l(t) 5 2 ]]]] T(t) 2 T E (t)
(5)
The estimator is valid for all cases where (T(t) 2 T E (t)) differs from zero. For numerical computations this restriction must be tightened: the absolute value of (T(t) 2 T E (t)) must be sufficiently different from zero.
2.3. Evaluation of experimental data 2.3.1. Experiments The method developed is applied to two cooling experiments, E1 and E2, with different types of abrupt changes of environmental temperature. 2.3.1.1. Experiment E1 Time of death, 10.10 a.m.; start of measurements, 11.50 a.m.; duration of measuring period, |21.4 h; skin temperature measured with surface sensor at uncovered forehead of male individual with body height 1.69 m and body mass 78.8 kg; environmental conditions, 2.75 h in closed cold room in still air at about 88C, then for |15 min in front of cold room in still air at |168C, then again in closed cold room (T E slightly increasing to about 10.58C). 2.3.1.2. Experiment E2 Time of death, 8.00 a.m.; start of measurements, 11.40 a.m.; duration of measuring period, 24.5 h; skin temperature measured with surface sensor at exposed forehead of male individual with body height 1.75 m and body weight 95 kg; environmental conditions, 3.75 h in closed room at almost constant
221
T E of about 248C, then kept in closed cold room in still air at about 98C (T E slightly increasing to about 12.58C). The environmental temperature was measured with a sensor |10 cm above the thigh region of the corpse. The temperature–time decrease curves of experiment E1 are presented in Fig. 1, and those of experiment E2 in Fig. 2.
2.3.2. Numerical calculation of the estimator for l(t) Comparatively few measuring points lie within the peak- or step-dominated phases; a sequence of equidistance between points is generated by linear interpolation between two neighboring measuring points of the skin and the environmental temperature. This approximates the actual continuous course of the temperature curves. Mentioning tuples (t i , T S (t i ), T E (t i )) or their components refers to the interpolated values as well as to the measured values. The time distance between neighboring interpolated measuring points is Dt. An estimator for the time derivative T 9S (t i ) of the skin temperature is calculated for every point of time t i from the tuples (t i , T S (t i ), T E (t i )) according to T S (t i 11 ) 2 T S (t i 21 ) T 9S (t i ) 5 ]]]]]] Dt
(6)
The estimator for the temperature decrease rate l(t i ) is computed according to formula (5) for every point of time t i . The numerical consistency of the model is tested by reconstructing the skin temperature T S from the environmental temperature T E and the local temperature decrease rates l. A pointwise recursive forward computation starting at time t 0 is carried out: T S (t i 11 ): 5 l(t i )(T S (t i ) 2 T E (t i ))Dt 1 T S (t i )
(7)
3. Results The single-exponential Newtonian model is valid only for constant environmental temperature. By localizing it to infinitesimal small intervals of time a generalized Newtonian differential equation (3) was developed. An explicit solution (4) was obtained. It
222
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
Fig. 1. Skin temperature T S and environmental temperature T E in experiment E1.
Fig. 2. Skin temperature T S and environmental temperature T E in experiment E2.
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
represents a generalized version of the single-exponential model valid also for non-constant environmental temperatures and non-constant temperature decrease rates. The model could be applied to cooling processes with continuously changing environmental temperature; since abrupt changes of environmental temperature represent more difficult test situations, it was applied to two cooling experiments with a peak-like and a step-like course of environmental temperature. The extended differential equation directly provided an estimator (5) for the time-dependent temperature decrease rate l(t). The above estimator was used to compute the time-dependent temperature decrease rates l(t) (Figs. 3 and 4) in the two cooling experiments E1 and E2. The oscillation of the curve of the temperature decrease rates l(t) is an artefact caused by the quantization of the temperature measurements in steps of 0.18C. Especially in the later cooling phase, the amplitude of this oscillation considerably increases; this is due to the structure of the estimator, which assumes larger values if the difference between object and environmental temperature becomes smaller. The numerical consistency of the estimator was tested by a forward calculation reconstructing the skin temperature curve T S (t) and comparing it with the measured skin temperature curve.
223
As can be seen in Figs. 5 and 6, the results of the test were very good; the reconstructed and the measured curve completely overlap each other. This complete congruence demonstrates that the model is well suited even for cases with abruptly changing environmental temperature.
4. Discussion A common method for simplifying model approaches in physics consists of assuming the influencing quantities to be constant. This can be illustrated in the Newtonian model valid for constant environmental temperature and thermally thin objects only. The model was used by Rainy [1] for describing the postmortem cooling process. An important contribution has been put forward by Sellier [10], who solved the differential equation of heat transfer for a cooling cylinder, thereby explaining the differences between the cooling behavior of the core (sigmoid cooling curve with plateau) and of the surface (single-exponential). Improved models by Marshall and Hoare [2] and Henßge [3,4] and Green and Wright [5,6] are double exponential and recursively exponential, respectively, and therefore better fit the sigmoid shape of the rectal temperature
Fig. 3. Estimation of temperature decrease rate ( l) from data of experiment E1.
224
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
Fig. 4. Estimation of temperature decrease rate ( l) from data of experiment E2.
Fig. 5. Reconstruction of skin temperature curve (T S modelled) in comparison to measured curve (T S measured) for experiment E1.
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
225
Fig. 6. Reconstruction of skin temperature curve (T S modelled) in comparison to measured curve (T S measured) for experiment E2.
decrease; they still presuppose a constant environmental temperature. In medico-legal practice, the temperature of the surroundings of a corpse mostly varies; even under experimental conditions it is difficult to keep the environment at an exactly constant temperature. Slight, continuous changes of the environmental temperature will most probably not influence the decrease of the core temperature, while they have a noticeable effect on the course of the surface temperature. Abrupt, peak-like or steplike, changes of the environmental temperature significantly alter the decrease of the skin temperature and also influence rectal cooling. Althaus and Henßge [11] in this respect recently published an investigation on the coefficients of the double-exponential model in cases of abrupt rise or fall of the ambient temperature. The present study provides an approach (‘localization’) for extending the above models to variable environmental temperatures. The localization approach was applied to the single-exponential Newtonian model which is suitable for describing the exponential shape of the skin temperature decrease curves. Data on skin cooling behavior in the forensic literature are scarce. Lyle and Cleveland [7] only
measured the cooling of the uncovered skin of the forehead and published a normed curve. Simonsen et al. [8] measured the temperature of the axillary skin, which is not representative. Newitt and Green [9] concentrated on the method of thermography and do not mention cooling data. Knowledge of skin cooling is essential for understanding and investigating the postmortem cooling process according to the contribution of the different heat transfer mechanisms. Radiation, convection and conduction strongly depend on the temperature gradient between body surface and environment and not between core and environment. To be able to compute the heat loss and to model the postmortem cooling process physically, it is necessary to quantify skin cooling by determining the skin temperature decrease rate. Since the body surface is extremely sensitive to even slight changes of the environmental temperature, the quantification has to take into account non-constant environmental temperatures and non-constant temperature decrease rates. We were able to derive an estimator for time-variable temperature decrease rates. It was thereby possible to quantify skin cooling in two experiments with typical abrupt changes of the environmental temperature. The numerical con-
226
G. Mall et al. / Forensic Science International 113 (2000) 219 – 226
sistency of the presented method for quantifying the temperature decrease of the skin was tested successfully by accurately reconstructing the measured skin temperature curves.
Acknowledgements The authors would like to thank Ms Mona Eckl for carrying out the measurements.
References [1] H. Rainy, On the cooling of dead bodies as indicating the length of time that has elapsed since death, Glasgow Med. J. (New Series) 1 (1869) 323–330; cited in: C. Henßge, B. Madea, Methoden zur Bestimmung der Todeszeit an Leich¨ ¨ en, Schmidt-Romhild-Verlag, Lubeck, 1988, pp. 142–144. [2] T.K. Marshall, F.E. Hoare, Estimating the time of death, J. Forensic. Sci. 7 (1962) 56–81, 189–211, 211–221. ¨ ¨ [3] C. Henßge, Die Prazision von Todeszeitschatzungen durch ¨ die mathematische Beschreibung der rektalen Leichenabkuhlung, Z. Rechtsmed. 83 (1979) 49–67.
¨ [4] C. Henßge, Todeszeitschatzungen durch die mathematische ¨ Beschreibung der rektalen Leichenabkuhlung unter ver¨ schiedenen Abkuhlbedingungen, Z. Rechtsmed. 87 (1981) 147–178. [5] M.A. Green, J.C. Wright, Postmortem interval estimation from body temperature data only, Forensic Sci. Int. 28 (1985) 35–46. [6] M.A. Green, J.C. Wright, The theoretical aspects of the time dependent Z-equation as a means of postmortem interval estimation using body temperature data only, Forensic Sci. Int. 28 (1985) 53–62. [7] H.P. Lyle, F.P. Cleveland, Determination of the time of death by body heat loss, J. Forensic Sci. 1 (1956) 11–24. [8] J. Simonsen, J. Voigt, N. Jeppesen, Determination of the time of death by continuous post-mortem temperature measurements, Med. Sci. Law 17 (1977) 112–122. [9] C. Newitt, M.A. Green, A thermographic study of surface cooling of cadavers, J. Forensic Sci. Soc. 19 (1979) 179– 181. [10] K. Sellier, Determination of the time of death by extrapolation of the temperature decrease curve, Acta Med. Leg. Soc. 11 (1958) 179–302. [11] C. Althaus, C. Henßge, Rectal temperature time of death nomogram: sudden change of ambient temperature, Forensic Sci. Int. 99 (1999) 171–178.