The theoretical aspects of the time dependent Z equation as a means of postmortem interval estimation using body temperature data only

It has been clearly demonstrated that the rectal cooling curve does not obey Newton's Law, which is exponential. The first success in modelling rectal cooling mathematically was achieved by Marshall and Hoare [1]. An amendment was made to the simple exponential curve which led to a good mathematical model, exhibiting the three main sections of rectal cooling, i.e. lag, linear and quasi-exponential. The resultant method of postmortem interval estimation required a knowledge of the body mass and height. The present study has led to a totally different amendment to Newton's Law, which provides a means of postmortem interval estimation from body temperature data only. The derivation of the method, with a background on Newton's Law follows.     

A Nonlinear Least Squares Approach to Time of Death Estimation Via Body Cooling

The problem of time of death (TOD) estimation by body cooling is revisited by proposing a nonlinear least squares approach that takes as input a series of temperature readings only. Using a reformulation of the Marshall–Hoare double exponential formula and a technique for reducing the dimension of the state space, an error function that depends on the two cooling rates is constructed, with the aim of minimizing this function. Standard nonlinear optimization methods that are used to minimize the bivariate error function require an initial guess for these unknown rates. Hence, a systematic procedure based on the given temperature data is also proposed to determine an initial estimate for the rates. Then, an explicit formula for the TOD is given. Results of numerical simulations using both theoretical and experimental data are presented, both yielding reasonable estimates. The proposed procedure does not require knowledge of the temperature at death nor the body mass. In fact, the method allows the estimation of the temperature at death once the cooling rates and the TOD have been calculated. The procedure requires at least three temperature readings, although more measured readings could improve the estimates. With the aid of computerized recording and thermocouple detectors, temperature readings spaced 10–15 min apart, for example, can be taken. The formulas can be straightforwardly programmed and installed on a hand‐held device for field use.     

Realization of an adaptive method of simulating the process of temperature change of a cadaver on a microcomputer

Programme based on mathematical model of the process of dead body temperature changing was developed for estimation of postmortem interval. Automatic retrieval of problem solution was performed on programmable microcalculators of "Electronica MK-61" type using adaptive approach. Diagnostical accuracy in case of dead body being preserved in permanent cooling conditions is +/- 3%.     

Cooling rates of the ear and brain in pig heads submerged in water: implications for postmortem interval estimation of cadavers found in still water

The state of the art for determining postmortem interval in submerged bodies reflects a serious lack of studies. The objectives of the present study were therefore to study cerebral and tympanic cooling in water and its relation to cooling in air, in a pig model. First of all, cerebral and tympanic cooling on a single head and on an entire body were compared and proven to be very similar in air and in water. Nine pairs of heads were then exposed to 9 temperature intervals from 0 degrees C to 20 degrees C. For every set temperature, one head was placed in water, the other in "ambient" air in a thermostatic chamber. Ear and brain temperature were simultaneously measured every 10 minutes during 8 hours. Results showed that both in air and in water, cooling curves were almost exponential, regardless of the site (ear or brain) or the environmental temperature. Cooling was always more rapid in water than in air. Cerebral and tympanic cooling always had a correlation coefficient of 0.98-0.99. Assuming that these cooling patterns are applicable to man, this research may provide a starting point for postmortem interval estimation in submerged cadavers.     

Rectal temperature time of death nomogram: sudden change of ambient temperature

Cooling experiments on dummies known as body-like cooling were performed with sudden decrease and increase of ambient temperature in the order of 15 degrees C. In the case of a sudden decrease of ambient temperature, a second 'temperature plateau' occurs which is shorter than the known plateau at the beginning of body cooling. The cooling curves can be described mathematically by a three-step procedure based on the two-exponential term of the 'nomogram method'. The second plateau at the beginning of the second cooling phase in sudden decreased ambient temperature requires a lower value of the constant A (1.15) compared with the known value (1.25) at the beginning of body cooling. In the case of a sudden increase of ambient temperature in the order of 15 degrees C no procedure could be found to model the cooling curves mathematically.     

Principles in construction of mathematical model for neonatal corpse cooling studies

The process of cooling of the body of a dead human was studied with consideration for the anatomical characteristics of adult and newborn human corpses. The body was simulated as a homogeneous elongated rotation ellipsoid and mathematical model was developed as function of relationship between body temperature and time. After appropriate characteristics in the mathematical model were established, cooling processes and effects of various factors on this process were analyzed for the final ellipsoid. The model suggests the possibility of amendments for environmental temperature and body parameters.     

The use of absolute refractory period in the estimation of early postmortem interval

The estimation of the time since death (postmortem interval) is one of the most difficult problems in forensic pathology. Most methods currently employed use temperature-based algorithms intended to model the cooling of the body after death and thus estimate the postmortem interval. These methods are subject to considerable inaccuracy but their reliability can be improved if a range of other observed criteria such as lividity and rigor are also taken into consideration. The aim of the present study was to investigate the feasibility of using the absolute refractory period as an adjunct to the estimation of postmortem interval. The relationship between the `postmortem interval' and the `duration of absolute refractory period' was investigated using the rat sciatic nerve. A strong correlation between the duration of the absolute refractory period and the postmortem interval was observed. When both absolute refractory period and temperature were used in conjunction, the strength of this correlation was increased.     

Does a Draft Really Influence Postmortem Body Cooling?*

Estimation of the time of death (TOD) is a very important task of forensic pathologist, and measurement of body temperature is a method deemed to be most precise during the initial postmortem period. The study aimed at verification of the significance of airflow present in the room where a corpse is found on body cooling process and hence on determination of the TOD. The experiment was performed in pigs during which the postmortem temperature of the eye, muscles, and rectum was recorded--in still air and with generated draft in the room. The results showed that the moderate airflow present in the experimental conditions did not significantly affect the course of cooling of the investigated body sites. Despite moderate wind generated in the room, the air movement close to pigs' bodies was actually minimal. This allowed to conclude that to evaluate the TOD most precisely, one should first have reliable data on the actual velocity of air in the direct vicinity of the body rather than relying on subjective sensation of the draft and using various unnecessary corrective coefficients for TOD calculation.     

Multiple-probe thermography for estimating the postmortem interval: II. Practical versions of the Triple-Exponential Formulae (TEF) for estimating the time of death in the field

A simple, reliable, and relatively accurate method for estimating the time since death is described. It is based on the Triple-Exponential Formulae (TEF), which are devised for the first time in this study. The postmortem cooling rate of the brain, liver, and rectum in 117 forensic cases were investigated (Part I). The method can be used in the field as a computer program, reference graph, or reference chart-ruler. The program is simple and can easily be run by any user. There are six reference graphs representing the average brain, liver, and rectal cooling curves for naked and covered body groups. The ruler is designed for the rectal cooling curves for covered and naked bodies. This method requires one temperature measurement of the chosen body site and the environment. The postmortem interval is estimated as a probable value +/- a possible range of time estimates with a built-in confidence limit.     

A study of the shape of the post-mortem cooling curve in 117 forensic cases

The shape of the post-mortem cooling curve is described and its theoretical aspects are summarised. The concepts of the "infinite cylinder" and the "initial temperature plateau" are explained and their practical implications are discussed. Results of the study of the shape of post-mortem cooling curves for the brain, liver and the rectum of 117 cases are given. The bodies were monitored either naked or covered with blankets. For each case, temperatures of the three body sites and the environment were monitored soon after death and up to many hours post-mortem. Empirically derived three-exponential formulae were used in this study. The cooling curves for the three body sites were found to be of compound shapes and the slopes of the curves vary throughout the monitoring period. The "initial temperature plateau" was found on average in 22% of all cooling curves with the plateau incidence being significantly highest in the rectal curves (27% of rectal curves compared with 7% of brain and liver curves, P<0.1%). The effect of body sites, body build and covering of the torso on the occurrence of the plateau is assessed and discussed.     

Determination of the time of death according to rectal thermometry data using computer calculations

A mathematic model of estimation of time of death (TD) in newborns is proposed. This computer method of TD timing by changes of body temperature differs from earlier developed mathematical models by possibility to consider individual anthropometric and thermophysical characteristics of the body. It is more precise, shorter and easy to perform.     

Comment on the Marshall-Hoare-Henssge model for estimating the time since death

Timewise temperature variations in objects that are undergoing unsteady heating or cooling is a commonly encountered problem in the thermal sciences. One particular area of application is the cooling of a body post-death and the use of body temperatures to estimate the time of death. Here, a new approach based on the theory of transient heat transfer is formulated to allow efficient calculation of unsteady conduction problems. The theoretically derived unsteady temperature models are compared with experimentally based correlations (the Marshall-Hoare-Henssge model). The two approaches are found to agree very well. With this new theoretically based approach, timewise temperature variation can be calculated for both large and small Biot number transient problems.     

Historical aspects of the determination of the time of death

The origins of the determination of the time of death are presented with special reference to the signs of death, rigor mortis and body cooling, as well as to the supravital electrical response of skeletal muscle. The systematic method for future research resulting from these old experiments is presented, with the aim of making a more precise estimation of the time of death by standardization of the influencing factors.     

Determination of time-dependent skin temperature decrease rates in the case of abrupt changes of environmental temperature

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(S)'(t)=-lambda(t)(T(S)(t)-T(E)(t))(with T(S)(t) the surface/skin temperature, T(E)(t) the environmental temperature, lambda(t) the temperature decrease rate and T(S)'(t) the actual change of skin temperature or first-order derivative of T(S)). The differential equation directly provides an estimator:lambda(t)=-T(S)'(t)T(S)(t)-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.     

Validation of a new method of providing case-specific time-of-death estimates using cadaver temperatures

The near ubiquitous presence of numerical simulation has made case-specific calculations of body temperatures following death possible so that accurate calculations of body temperatures can provide valuable information for estimating the time of death and can aid in forensic investigations. Here, a computational approach is described that has been validated against multiple, independent, and experimental investigations. The approach only requires one subjective input parameter (the heat transfer coefficient). A simple algorithm serves as a guidepost to the selection of this parameter. The algorithm incorporates clothing and the space in which the body is housed. Heat transfer coefficients that range from h = 2 W/m²/°C for bodies that are heavily clothed to h = 9 W/m²/°C for bodies that are nude (in air). The method also requires setting of ambient temperature conditions (ambient temperature)—however, that input is often available. The paucity of inputs makes this technique remarkably easy to employ. The new method is also able to calculate cadaver cooling rates for situations where the cadaver is in a timewise or spatially changing thermal environment (diurnal temperature variations, bodies partially submerged in water, changes to weather, insolation, etc.). Results from the present calculations are compared with a large body of measurements from the literature; it was found that the predictions and measurements were in excellent agreement, regardless of the ambient temperature conditions and the nature of the clothing of the body. This new calculation approach can be used with reasonable accuracy for determining cadaver cooling and time since death.     

Estimation of the post mortem period by multiple-site temperature measurements and the use of a new algorithm

The estimation of the post mortem period, in cases where death occurred under suspicious circumstances, is usually attempted using temperature measurements taken at a single body site. Early investigations of the validity of such an approach use the abdominal skin surface, the axilla and the rectum as measurement sites (B.H. Knight, Forensic Sci. Int., 36 (1987) 47-55). However, it has recently become more common to use the rectum alone, though the ear and the nasal passages have also been utilized. Whatever site is employed, the estimates are frequently found to be inaccurate. There are several fundamental reasons for these inaccuracies, the most prominent being the unknown variation in the ambient temperature between the time of death and the commencement of measurements, and the unknown body temperature at the time of death. This paper proposes a method of overcoming the above difficulties by taking a series of measurements concurrently at a number of body sites, a technique used by several previous workers (B.H. Knight, Forensic Sci. Int., 36 (1987) 47-55). Initial investigations have shown that an improved estimation of post mortem period is obtainable by the application of a suitable decision-making algorithm.     

Factors influencing the precision of estimating the postmortem interval using the triple-exponential formulae (TEF). Part II. A study of the effect of body temperature at the moment of death on the postmortem brain, liver and rectal cooling in 117 forensic cases

The temperatures of three body sites, namely, the brain, liver and the rectum as well as the temperature of the environment were continuously monitored, every 5-10 min, in 117 forensic cases commencing soon after death and in most cases, within 45 min postmortem. The body temperature at the moment of death was empirically determined by a computer-based extrapolation method. Thus, temperature data for the first 3h of each body site were fitted to single-exponential equations and the fitted curve was extrapolated backwards to obtain the intercept on the Y-axis (the temperature axis). The effect of body temperature at the moment of death on postmortem cooling rate is examined and factors influencing body temperature at death are discussed. Forensic fatalities associated with hyper and hypothermia are reviewed briefly.     

The role of intestinal bacterial heat production in confounding postmortem temperature measurements

To assess the influence of anaerobic bacterial heat production in human stools as a confounding factor in postmortem rectal temperature measurements, in vitro experiments were carried out with human stools incubated at 37 degrees C for 6 h and at decreasing temperatures simulating a postmortem body cooling. Although a statistical significant heat production was observed, it was not relevant enough to explain a postmortem temperature plateau or a substantial rise in the postmortem body temperature. The experiments suggest that stools merely reflect the environmental thermal changes rather than producing bias and confounding by a bacterial heat production.     

A diagnostic program for estimating the time of death using microcalculators MK-61 and B3-34

The programme design is based on the process of changes in dead body temperature. Specific features of the process of changes in temperature values of a dead body are characterized by a parameter which involves process sample obtained by temperature measurements in diagnostical area of a cadaver at the accident scene during 1 hour. Programme provides for estimation of postmortem interval within the first 60 h after death occurrence. Diagnostical deviations don't exceed +/- 3% as compared to the real time of death.     

Influence of electricity on post-mortal body temperature

In a case of suicidal application of electricity differences between the rectal temperature of the body and the suspected time of death were observed. In order to answer the question whether an electric current from hand to hand over >30 min led to a rise in body temperature FEM-based computer simulations and animal experiments were carried out. Both resulted in a warming of the soft parts in the arm without warming the body core. Thus a temperature-based estimation of the time since death can also be used in cases with electricity as the cause of death. Besides, in the animal experiment we found a spontaneous rise in the body core temperature even without application of electricity which may be a reason for the typical temperature plateau after death.