APPLICATION OF NEWTON S LAW OF COOLING CASE STUDY: ESTIMATION OF TIME OF DEATH IN MURDER SAMUEL ASANTE (BSC. COMPUTER SCIENCE) PG

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1 APPLICATION OF NEWTON S LAW OF COOLING CASE STUDY: ESTIMATION OF TIME OF DEATH IN MURDER BY SAMUEL ASANTE (BSC. COMPUTER SCIENCE) PG A Thesis Submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology, in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE: Industrial Mathematics College of Science/ Institute of Distance Learning. APRIL, 2013

2 DECLARATION I hereby declare that this submission is my own work towards the MSc. and that, to the best of my knowledge; it contains neither material previously published by another person nor material, which has been accepted for the award of any other degree of the University, except where the acknowledgement has been made in the text. SAMUEL ASANTE ( ) Student Name & ID No. Signature Date Certified by: Dr. E. OSEI-FRIMPONG. Supervisor Name Signature Date Certified by: PROF. S.K. AMPONSAH. Head of Dept. Name Signature Date Certified by: PROF. I.K. DONTWI. Dean-IDL Signature Date ii

3 DEDICATION To my lovely wife, Cordelia and children David, Eugenia and Desmond iii

4 ACKNOWLEDGEMENT First and foremost, I heartily express my sincerest gratitude to the Almighty God for seeing me through this thesis successfully. Next, I wish to express my profound gratitude to my honorable supervisor Dr. E. Osei- Frimpong for suggesting the thesis topic and also for his guidance, encouragement and contributions throughout this study. I am very much thankful for all his support and patience throughout the program. I would also like to thank all the lecturers of the courses I offered during the MSc Industrial Mathematics program. My special word of thanks goes to my wife, Cordelia, and children David, Eugenia and Desmond for their understanding, love and support throughout my study. Finally, my special gratitude goes to my mother, Ernestina Ansaah, and my dearest brother, Wilberforce Barfour Ansah (Okapi).Their support, love, encouragement and care have always been of great help. iv

5 ABSTRACT Determining the time of death of a person is a major responsibility of forensic investigation. This study presents a mathematical model based on Newton s law of cooling to determine the time of death of a person. Emphasis is put on the development of the method and computer software taking advantage of the decrease in body temperature. The model is based on four parameters: temperature of the body at time t, temperature of the surrounding area at time t, the weight of the body and the condition of the body. Computer software using Microsoft visual C sharp that allows first calculation of the time of death and graphical representation of the cooling of dead bodies is presented. The algorithms are based on a first-order linear differential equation formulated by Newton and modified with a two-exponential-model by Marshall and Hoare (1962). Several calculations are run by the program and the results are presented to allow comparism so that different parameters may be assumed and their effect on the length of the cooling period be assessed. Data from the following sources: Handbuch Gerichtliche Medizin, Volume 1 by Bernd Brinkmann and Burkhand Madea and ACTAMORPHOLOGIC, 2006; vol 3(2):51-54 Medical journal of Macedonian Association of Anatomists and Morphologists (MAAM) are used to test the computer solution. We conclude that the cooling model which includes the post mortem plateau gave good results and therefore the model can be applied to all cases. v

6 CONTENTS DECLARATION.ii DEDICATION iii ACKNOWLEDGEMENT..iv ABSTRACT.v CHAPTER 1 1 INTRODUCTION Brief Introduction of Chapter Background of the Study History of Newton s Law of Cooling Newton s Law of Cooling Introduction to the Principle of Heat Transfer Profile of Study Area Some Basic Definitions Problem Statement Main Objective Specific Objectives Justification Methodology Organization of the Study...18 vi

7 CHAPTER 2.20 REVIEW OF RELATED STUDIES Review of Previous Studies.20 CHAPTER METHODOLOGY Model Assumptions Newton s Model Solution of Newton s Equation (Model) Rate of Post Mortem Cooling The following Assumptions are also made Software Development C Sharp (#) Programming Language Object-Oriented Programming Object-Oriented Design Object-Oriented Analysis Object Class Programming Tools Algorithm Pseudocode..45 vii

8 3.6.3 Flowcharts 47 CHAPTER 4.49 DATA ANALYSIS AND RESULTS Source of Data Parameter Estimation Analysis of Results Further Discussion of Results.57 CHAPTER 5.60 CONCLUSION AND RECOMMENDATIONS Conclusion Recommendation.60 REFERENCES..61 APPENDIX A 65 APPENDIX B 71 viii

9 LIST OF FIGURES Figure 1.1 Sigmoidal shape of cooling curve. Mathematical description with the two exponential model by Marshall and Hoare...13 Figure 2.1: Henssge Nomogram for environmental temperature up to 230C(upper)..16 Figure 3.1: Flowchart of the Computer Program...48 Figure 4.1: A chart of estimated time and known time of death...57 Figure 5.1: A chart of estimated time and known time of death 58 ix

10 LIST OF TABLES Table 3.1 Definitions of Variables used in the Model..38 Table 4.1 Parameter estimates of Cooling Model..51 Table 4.2 Cases with known time of death (source: Handbuch Gerichtliche Medizin, Vol.1).52 Table 4.3 Cases with known time of death...53 Table 4.4 Summary of Temperature difference, estimated time of death and known time of death...59 x

11 CHAPTER 1 INTRODUCTION 1.1 Brief Introduction of Chapter 1 Establishing the time of death or the interval between the time of death and when the body is found cannot be determined with certainty. Unless death is witnessed, the exact time of death cannot be determined; however, sufficient information is often available to allow estimation of a range of time encompassing the actual moment of death. In general the short the postmortem interval, the narrower the estimated time ranges. Conversely, a longer postmortem interval entails a broader range estimate and after a greater chance for error. No single observation about a dead body is a reliable or accurate indicator of the postmortem interval. The most reliable estimate are based upon a combination of numerous observation made of the body and the scene of death. The combination of scene and body examination will give the investigation the best chance of reliably estimating when death occurred (Dix and Graham, 1999). An accurate estimate of time since death is an important aspect of every death investigation especially in suspicious death to link a suspect to the victim and establish the credibility of statements made by the expert witnesses (Amendt et al., 2007). Estimation of time since death is also one of the most important object of postmortem examination. Time passed since death continues to be a major problem for the forensic pathologist and its determination plays an important and vital issue in medico legal cases 1

12 because of the effect that forensic experts are often required to answer questions relating to the death in the courts of law (Kushwaha et al., 2010). 1.2 Background of the Study History of Newton s Law of Cooling This law of cooling is named after English physicist Isaac Newton who, in the late 17th century, conducted the first experiments on the nature of cooling. Specifically, noting that when the difference in temperature between the two bodies is small, approximately less than 10º C, that the rate of heat loss will be proportional to the temperature difference, Newton applied this principle to estimate the temperature of a red-hot iron ball, by observing the time which it took to cool from a red heat to a known temperature, and comparing this with the time taken to cool through a known range at ordinary temperatures (The Encyclopedia Britannica, 1910). According to this law, if the excess of the temperature of the body above its surroundings is observed at equal intervals of time, the observed values will form a geometrical progression with a common ratio. The inaccuracy of Newton s law become s considerable at high temperatures. The corrected Newton s law was formulated in 1817 by French Physicial chemist Pierre Dulong and physicist Alexis Petit who, experimenting through temperature ranges as high as 243º C, found that the quickness of cooling for a constant excess of temperature, increases in geometrical progression, when the temperature of the surrounding space increases in arithmetical progression ( Whewell,1866). 2

13 1.2.2 Newton s Law of Cooling The Newton's Law of Cooling model computes the temperature of an object of mass M as it is heated by a flame and cooled by the surrounding medium. The model assumes that the temperature T within the object is uniform. This lumped system approximation is valid if the rate of thermal energy transfer within the object is faster than the rate of thermal energy transfer at the surface. Newton s law of cooling states that the rate at which a warm body cools is approximately proportional to the difference between the temperature of the warm object and the temperature of its environment. Newton s law of cooling is generally limited to simple cases where the mode of energy transfer is convection, from a solid surface to a surrounding fluid in motion, and where the temperature difference is small, approximately less than 10º C (The Encyclopedia Britannica, 1910).When the medium into which the hot body is placed varies beyond a simple fluid, such as in the case of a gas, solid, or vacuum, etc., this becomes a residual effect requiring further analysis (Whewell, 1866) Introduction to the Principle of Heat Transfer Heat transfer is a science that studies the energy transfer between two bodies due to temperature difference. In the simplest of terms, the discipline of heat transfer is concerned with only two things: temperature, and the flow of heat. Temperature represents the amount of thermal energy available, whereas heat flow represents the movement of thermal energy from place to place. On a 3

14 microscopic scale, thermal energy is related to the kinetic energy of molecules. The greater a material s temperature, the greater the thermal agitation of its constituent molecules (manifested both in linear motion and vibrational modes). It is natural for regions containing greater molecular kinetic energy to pass this energy to regions with less kinetic energy. Several material properties serve to modulate the heat transferred between two regions at differing temperatures. Examples include thermal conductivities, specific heats, material densities, fluid velocities, fluid viscosities, surface emissivities, and more. Taken together, these properties serve to make the solution of many heat transfer problems an involved process. Heat transfer mechanisms can be grouped into three broad categories: Conduction, Convection and Radiation ( 14/11/2012) Conduction Conduction is the flow of heat through solids and liquids by vibration and collision of molecules and free electrons. The molecules of a given point of a system which are at higher temperature vibrate faster than the molecules of other points of the same system or of other systems- which are at lower temperature. The molecules with a higher movement collide with the less energized molecules and transfer part of their energy to the less energized molecules of the colder regions of the structure. For example, the heat transfer by conduction through the bodywork of a car. Metals are the best thermal conductors; while non-metals are poor thermal conductors. For heat to conduct from one object to 4

15 another, they must be in contact: break the contact and conduction ends (Nasif, 2009). Another example is, a spoon in a cup of hot soup becomes warmer because the heat from the soup is conducted along the spoon. Conduction is most effective in solids-but it can happen in fluids Convection Convection is the flow of heat through currents within a fluid (liquid or gas). Convection is the displacement of volumes of a substance in a liquid or gaseous phase. When a mass of a fluid is heated up, for example when it is in contact with a warmer surface, its molecules are carried away and scattered causing that the mass of that fluid becomes less dense. For this reason, the warmed mass will be displaced vertically and/or horizontally, while the colder and denser mass of fluid goes down (the low-kinetic-energy molecules displace the molecules in high-kinetic-energy states). Through this process, the molecules of the hot fluid transfer heat continuously toward the volumes of the colder fluid. For example, when heating up water on a stove, the volume of water at the bottom of the pot will be warmed up by conduction from the metallic bottom of the pot and its density decreases. Given that it gets lesser dense, it shifts upwards up to the surface of the volume of water and displaces the upper -colder and denser- mass of water downwards, to the bottom of the pot. Natural convection occurs when the flow of a liquid or gas is primarily due to density differences within the fluid due to heating or cooling of that fluid (Nasif, 2009). Forced convection occurs when the flow of fluid (liquid or gas) is primarily due to pressure differences. 5

16 Radiation Radiation is the transfer of heat from one object to another by means of electro-magnetic waves. Radiative heat transfer does not require that objects be in contact or that a fluid flow between those objects. Radiative heat transfer occurs in the void of space (that s how the sun warms us). People in a room at 72 o F air temperature may feel uncomfortably cold if the walls and ceiling are at 50 o F. Conversely, they may feel uncomfortably warm if the walls are 85 o F. Even though the air temperature is the same in both cases, the radiative cooling or warming of their bodies relative to the walls and ceiling will affect their comfort level (people sense heat loss or gain, not temperature). Radiation heat transfer is concerned with the exchange of thermal radiation energy between two or more bodies. Thermal radiation is defined as electromagnetic radiation in the wavelength range of 0.1 to 100 microns (which encompasses the visible light regime), and arises as a result of a temperature difference between 2 bodies. No medium need exist between the two bodies for heat transfer to take place as is needed by conduction and convection. Rather, the intermediaries are photons which travel at the speed of light. The heat transferred into or out of an object by thermal radiation is a function of several components. These include its surface reflectivity, emissivity, surface area, temperature, and geometric orientation with respect to other thermally participating objects. In turn, an object's surface reflectivity and emissivity is a function of its surface conditions (roughness, finish, etc.) and composition. 6

17 1.2.4 Profile of Study Area. The Gold Coast Police Force now called the Police Service was formed in 1894 but the real police work started in the then Gold Coast, now the Republic of Ghana, in the year The Ghana Police Service is organized on National basis, with a unified command under the Inspector-General of Police (IGP). The IGP, subject to the direction of the Minister of Interior, is responsible for exercising general day-to-day supervision over the operation and administration of the service. The Ghana Police Service has, since its inception been in the frontline of the criminal justice system of Ghana. It is clearly, the most visible arm of government as the symbol of law and order, to the people. Ghana Police Service is mandated by Article 200 of the 1992 constitution of the Republic of Ghana, and the Police Service Act 1970 (ACT 350). The constitution mandates the Service to operate on democratic policing principles. The Police Service Act 1970, Act350 spells out the core functions of the service as follows: To Protect life and Property, To prevent and detect crime, To apprehend and prosecute offenders, To maintain public order, to ensure a peaceful and safe environment to facilitate economic and social activities as a pre-requisite for making Ghana a Gateway to West Africa. 7

18 As per the new motto of the service, TO PROTECT AND SERVE WITH HONOUR the Ghana Police Service is committed to protect and serve all residents in their communities, using democratic policing principles, and appropriate technology to protect life and property, and personal dignity. The vision of the Ghana Police Service is to be a World Class Police Service capable of delivering planned, democratic, protective, and peaceful services up to standards of international best practice. The Ghana Police Service is divided into twelve administrative regions, namely, Accra, Tema, Ashanti, Eastern, Brong Ahafo, Volta, Western, Central, Northern, Upper East, Upper West, and Railways, Ports and Harbour Regions ( /GPS/SitePages/GPS-Home.aspx). The Criminal Investigations Department (CID) is the criminal investigation arm of the Ghana Police Service. The CID is mandated to ensure a proactive and professional approach to the prevention and detection of crime, protection of life and property and the apprehension and prosecution of offenders. The CID oversees criminal intelligence gathering, training of CID detectives and criminal investigation assignments of the Ghana Police Service. Personnel of the CID are employed as specialists in connection with various aspects of crime detection and investigation (C.I.D., 2012). 8

19 1.2.5 Some Basic Definitions Homicide A homicide is a crime where a person kills someone. There are 3 different types of homicide. There's justifiable where you're trying to protect someone and the person dies. Excusable is when the person is killed by unlawful acts that you try to stop. Criminal homicide is when one person kills another. Included among homicides are murder and manslaughter, but not all homicides are a crime, particularly when there is a lack of criminal intent. Non-criminal homicides include killing in self-defense, a misadventure like a hunting accident or automobile wreck without a violation of law like reckless driving, or legal (government) execution. Suicide is a homicide, but in most cases there is no one to prosecute if the suicide is successful. Assisting or attempting suicide can be a crime Post-Mortem Interval The time elapsed from death until discovery and medical examination of the body. If the time in question is not known, a number of medical or scientific techniques are used to determine it. This also can refer to the stage of decomposition of the body. 9

20 Forensic Investigation The word forensic simply means applying scientific applications or techniques to investigate a crime. A forensic investigation is the practice of lawfully establishing evidence and facts that are to be presented in a court of law. The term is used for nearly all investigations, ranging from cases of financial fraud to murder. The Forensic Investigation Unit provides expertise at crime scenes through the use of photography, identification, evidence collection, processing, and preservation. Technicians also perform latent fingerprint comparisons and provide expert court testimony. There are actually a number of different techniques for forensic investigation. There are four types of technicians in the unit: Crime Scene Technician, Latent Fingerprint Specialist, Photo Lab Technician, and Office Support Specialist. In addition to skills required by their specialization, each technician must have a exceptional understanding of law enforcement techniques and needs to perform their assigned tasks in a way that assists future courtroom proceedings. ( nvestigations/forensic_investigation/, Accessed:14/11/2012) Crime Scene Investigation A crime scene is any physical scene, anywhere, that may provide potential evidence to an investigator. It may include a person s body, any type of building, vehicles, and places in the open air or objects found at those locations. Crime scene examination therefore 10

21 refers to an examination where forensic or scientific techniques are used to preserve and gather physical evidence of a crime. Crime Scene Investigation gather fingerprints, blood, bodily fluids, and other evidence found at the crime scene in order to solve a crime or even determine whether a crime has taken place. Investigating a crime scene can be a very long, tedious process that should be completed by a trained professional with excellent attention to detail. It not only encompasses collection of evidence, but also the necessary analysis to ensure that the evidence is credible and relevant.( Investigating a crime scene is one of the most important parts of any criminal investigation. If done right, it can actually be the key to solving a crime. Crime Scene Investigators, or CSIs, use special methods and equipment for investigating a crime scene. These methods include using certain type of equipment, special investigation methods, and most importantly, preserving the integrity of a crime scene so that nothing gets moved or disturbed. Investigating how a crime occurred can offer a lot of insight into why the crime occurred at all. Since evidence gathered at a crime scene is what puts a criminal in jail, crime scenes are very important. There are many steps that have to be taken when conducting a criminal investigation and investigating a crime scene. Firstly, detectives have to try and figure out why and how a crime was committed. They examine a crime scene looking for information or clues such as fingerprints, weapons, and DNA. They investigate the victims history to determine why someone would want to harm them. After they have formed a hypothesis, they try to find proof that somebody committed a crime so that they can arrest the suspects. They look at both the motive and 11

22 the actual evidence of the crime and try to see if their hypothesis makes sense. The suspects then enter the criminal justice system where they are tried using the evidence collected at the crime scene. Crime scene investigators have special equipment that they use to investigate crime scenes. This equipment is packed into crime scene kits. They include fingerprint powders designed to find fingerprint on surfaces, brushes, magnetic power wands, and placards for marking important evidence, tape, scissors, tweezers, tape, pencils, and pens. They also include chemicals for detecting bodily fluids like luminal and black lights. Bodily fluids can be the most valuable piece of evidence because it can yield DNA, the holy grail of proof ( 14/11/2012) Temperature Plateau When temperature is plotted against time for the cooling of a human body, there is an initial maintenance of body temperature which may last for some hours. That period is what Harry Rainy in 1868 referred to as Plateau and this is shown in Figure 1.1. A plateau is a period after death where the body does not cool at all and the body temperature may even rise a bit (Rainy, 1968) 12

23 Temperature ( o C) T 0 T 0 - Ta Postmortal temperature Plateau Newton s law of cooling Tr - Ta Ta Time (h) Figure 1.1 Sigmoidal shape of cooling curve. Mathematical description with the two exponential model by Marshall and Hoare (Source: Henßgea and Madea,2004). 13

24 Henssge Nomogram Method A nomogram also called a nomograph is a graphical calculating chart, a two-dimensional diagram designed to allow the approximate graphical computation of a function. ( Accessed:14/11/2012). Henssge's nomogram is based upon a formula which approximates the sigmoid shaped cooling curve. It requires the measurement of deep rectal temperature and assumes a normal temperature at death of 37.2 o C. This formula has two exponential terms within it. The first constant describes the post mortem plateau and the second constant expresses the exponential drop of the temperature after the plateau according to Newton's law of cooling. In an individual case, the constant expressing the exponential drop of the temperature after the plateau is simply calculated from the body weight. The first constant which describes the post mortem temperature plateau was found to be significantly related to the second constant in that, bodies with a low rate of cooling also had a longer plateau phase than bodies with a high rate of cooling. Using previously published data which establishes that the relative length of the post mortem temperature plateau depends upon the environmental temperature but is nonlinear and pronounced in environmental temperatures above 23 o C, Henssge evolved two nomograms, the one for ambient temperatures above 23 o C and the other for ambient temperatures below 23 o C. Figure 1.2 below shows an example of its use. Within each of these two nomograms there is a differing allowance for the effect of environmental temperature on the rate of cooling as well as an allowance for the effect of body weight.in order to determine the possible time of death for each individual case, a line is drawn that links the rectal and environment temperatures. Through the cross-point 14

25 obtained by the oblique line, a line is drawn and afterwards, taking into consideration the body weight expressed in kg, the possible time of death expressed in hours is read on the nomogram (Pounder, 1995). 15

26 Figure 2.1: Henssge Nomogram for environmental temperature up to 230C (upper) (Source: Henßgea and Madea, 2004) 16

27 1.3 Problem Statement Determining the time of death is important in both criminal and civil cases. In criminal cases, it can set the time of the murder, free a suspect or suggest a possible suspect. In civil cases, the time of death might determine who inherits property or whether an insurance policy was in force ( 2001 by CRC Press LLC). Unfortunately, there is no system in Ghana where crime investigators rely on to estimate time of death when there is murder. Investigators use manual and inaccurate methods to sometimes estimate the time. 1.4 Main Objective The main objective of the study is to determine the time of death in murder with a modified Newton s law of cooling by Marshall and Hoare Specific Objectives To use analytical approach to find the general solution of the model. Develop computer software to estimate the time of death. 1.5 Justification The number of murder cases in the country has risen by 35% between January and June 2012, according to the latest half year report on crime by the Ghana Police Service. The 17

28 Police have been unable to solve several of these murders. Determining time of death is extremely important in a death investigation as it focuses the investigation into the correct time frame. In Ghana, Investigators sometimes do not use scientific method in solving most of these murders. Crime Investigators sometimes estimate it manually and it takes longer time. It is against this backdrop that this research is being carried out to find an appropriate system for estimating the time of death in murder. The study is very important because it can help Police Investigators to determine the time of death and also give an important piece of information in some coroner's cases, especially those that involve criminal or insurance investigations. 1.6 Methodology An Analytical approach to the solution of first-order linear differential equation formulated by Newton and modified by Marshall and Hoare is first solved explicitly to find the general solution. This task is achieved by assuming the existence of an integrating factor. An algorithm which solves the equation is implemented. The algorithm is further developed into computer codes by using Microsoft visual C sharp, a modern, object-oriented, and type-safe programming language to estimate the time of death. 1.7 Organization of the Study The thesis consists of five Chapters. In chapter 1, we considered the introduction, background of the study, problem statement and objectives of the study. The 18

29 justifications, methodology of the study and thesis organization were also put forward. Chapter 2 presents the relevant and adequate literature on the problem at hand. Chapter 3 is devoted for the research methodology of the study. In Chapter 4, we shall put forward data collection and analysis. Also we will come up with software analysis and development for the determination of time of death in a murder case. Chapter 5 which is the final chapter of the study considers the conclusion and recommendation of the study. 19

30 CHAPTER 2 REVIEW OF RELATED STUDIES Several researchers have carried out work on the estimation of time of death of humans. In this Chapter, some of these related studies are discussed. 2.1 Review of Previous Studies In Henssge et al. (1984) the authors subdivided into three groups twenty-nine corpses. The bodies were suspended undressed in a tub holding 1,000l in nearly still water of temperatures approximately 20 0, 10 0 and 0 0 C. The rectal temperature was measured, normally until the 33rd hour postmortem. Time of death was calculated by means of the mathematical analytical two-exponential formula suggested by Marshall and Hoare in 1962, in the version used by Brown and Marshall in The adapting parameters of the formula were standardized according to the principle of Henssge in 1979 and in 1981 and related to standardization by adjusting factors to body weight stated for standard values of cooling. After termination of the postmortem temperature plateau, it was found that undressed corpses suspended in water of temperatures of approximately 20 0 C and 10 0 C cool as quickly as undressed corpses of half the body mass in calm air of the same temperatures. As to the duration of the postmortem temperature plateau in water suspension time from the time of death, it may only be indirectly concluded that it is linked to the subsequent speed of cooling in the same way which is well known in the 20

31 case of air cooling. Statistical standard values were given concerning the differences between the computed and the real times of death. Unexpectedly, the experiments in water at approximately 0 0 C yielded distinctly slighter temperature which were especially marked at rectal temperatures up to approximately 11 0 C in corpses of great body mass and small body surface in proportion to and equally, without regard to body mass. As an explanation of this, a decrease in the thermal conductivity of the subcutaneous adipose tissue in connection with a decrease in tissue temperature was then discussed. Again in 2000, the authors used the temperature-based nomogram method for estimation of the time period since death at the scene of death as the primary method within a compound method in 72 consecutive cases The situation and cooling conditions inspected and evaluated by the forensic pathologist at the scene were described as far as necessary to enable handling of the method. A comparison of the estimated period since death with the period determined by the police investigations demonstrated the reliability of the method. There were no contradictions in any of the 60 cases between the period of death estimated by this method and that determined by the police investigations. The criminal investigations were effectively supported in the earliest stages in 11 cases despite the fact that the period estimated was of considerable duration. Also, Henssge (2006) presented that the main principle of the determination of the time since death was the calculation of a measurable date along a time-dependent curve back to the start point. Characteristics of the curve (e.g. the slope) and the start point were influenced by internal and external, ante mortem and postmortem conditions. These influencing factors were taken into consideration quantitatively in order to improve the precision of death time estimation. It does not make any sense to study the postmortem 21

32 time course of any analyte without considering influencing factors and giving statistical parameters of the variability. Comparison of different methods required an investigation of the same postmortem interval. For practical purposes, the author concluded that the amount of literature on estimating the time since death has a reverse correlation with its importance in practice. Lynnerup (1993) presented a simple BASIC computer program that enables solving of Marshall and Hoare's equation for the postmortem cooling of bodies.in his presentation the author stated that in the 1960s, Marshall and Hoare presented a "Standard Cooling Curve" based on their mathematical analyses on the postmortem cooling of bodies. Although fairly accurate under standard conditions, the "curve" or formula was based on the assumption that the ambience temperature is constant and that the temperature at death is known. Also, Marshall and Hoare's formula expressed the temperature as a function of time, and not vice versa, the latter being the problem most often encountered by forensic scientists. The author proposed that by having a computer program that solves the equation, giving the length of the cooling period in response to a certain rectal temperature, and which allows easy comparison of multiple solutions, the uncertainties related to ambience temperature and temperature at death can be quantified, substantiating estimations of time of death. Althaus and Henssge (1999) performed cooling experiments on dummies known as body-like cooling with sudden decrease and increase of ambient temperature in the order of 15 0 C. In the case of a sudden decrease of ambient temperature, a second temperature plateau occurred which is shorter than the known plateau at the beginning of body cooling. The cooling curves were described mathematically by a three-step procedure 22

33 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 required a lower value of the constant compared with the known value at the beginning of body cooling. In the case of a sudden increase of ambient temperature in the order of 15 0 C, the authors could not find a procedure to model the cooling curves mathematically. In Guy (2000) the author used single external auditory canal (EAC) temperatures from cases of suspicious deaths to verify the hypothesis that a single EAC temperature can be used to estimate a time since death (TSD). Two different types of thermometers were used (infrared and alcohol-in-glass) to record ambient and body temperatures, which were in turn applied to previously published algorithms without the use of corrective factors to estimate the TSD. In addition, 18 anatomical, environmental, and daily activity factors were investigated as to whether they may influence the temperature within the EAC and thus require the introduction of a corrective factor into an algorithm, other than one used to take into account the difference between the rectal and EAC temperature during life and after death. Of the ones examined, only head position, wind speed, daily circadian rhythm, drinking of hot drinks, and possibly mental thought were shown to influence temperature, but the difference was so small that the introduction of a corrective factor into an algorithm was considered unnecessary. Al-Lousi et al. (2001) described a simple, reliable, and relatively accurate method for estimating the time since death. The method was based on the Triple-Exponential Formulae (TEF), which the authors devised for the first time in their study. The postmortem cooling rate of the brain, liver, and rectum in 117 forensic cases were investigated. The method was used in the field as a computer program, reference graph, 23

34 or reference chart-ruler. There were six reference graphs representing the average brain, liver, and rectal cooling curves for naked and covered body groups. The ruler was designed for the rectal cooling curves for covered and naked bodies. This method required one temperature measurement of the chosen body site and the environment. The postmortem interval was estimated as a probable value. Hartorg and Lotens (2004) analyzed two murder cases in which available methods did not provide a sufficiently reliable estimate of the postmortem time. In both cases a study was performed to verify the statements of suspects. The authors developed a finiteelement computer model that simulates a human torso and its clothing. With this model, changes to the body and the environment were modeled; this was very relevant in one of the cases, as the body had been in the presence of a small fire. In both cases it was possible to falsify the statements of the suspects by improving the accuracy of the postmortem time estimate. The estimated postmortem time in both cases was within the range of Henssge's model. The standard deviation of the postmortem time estimate was 35 minutes in the first case and 45 minutes in the second case, compared to 168 minutes in Henssge's model. In conclusion, the authors noted that the model as presented can have additional value for improving the accuracy of the postmortem time estimate. In Mall et al. (2005) the authors presented that the temperature-oriented death time determination is based on mathematical model curves of postmortem rectal cooling. All mathematical models require knowledge of the environmental conditions. In medicolegal practice homicide is sometimes not immediately suspected at the death scene but afterwards during external examination of the body. The environmental temperature at the death scene remains unknown or can only be roughly reconstructed. In such cases the 24

35 question arises whether it is possible to estimate the time since death from rectal temperature data alone recorded over a longer time span. The authors theoretically deduced formulae which were independent of the initial and environmental temperatures and thus proved that the information needed for death time estimation was contained in the rectal temperature data. Since the environmental temperature at the death scene may differ from that during the temperature recording, an additional factor was used. This is that the body core was thermally well isolated from the environment and that the rectal temperature decreased after a sudden change of environmental temperature continued for some time at a rate similar to that before the sudden change. The study further provided a curve-fitting procedure for such scenarios. The procedure was tested in rectal cooling data of from 35 corpses using the most commonly applied model of Henssge. In all cases the time of death was exactly known. After admission to the medico-legal institute the bodies were kept at a constant environmental temperature for h and the rectal temperatures were recorded continuously. The curve-fitting procedure led to valid estimates of the time since death in all experiments despite the unknown environmental conditions before admission to the institute. The estimation bias was investigated statistically. The 95% confidence intervals amounted to ±4 h, which seems reasonable compared to the 95% confidence intervals of the Henssge model with known environmental temperature. They concluded that the presented method may be of use for determining the time since death even in cases in which the environmental temperature and rectal temperature at the death scene have unintentionally not been recorded. Bisegna et al. (2007) presented a procedure for the postmortem interval estimation in the presence of a rapid increase of ambient temperature occurred during the cooling phase. 25

36 The resulting disturbance produced on the cooling curve is proved to obey a twoexponential law and is removed from the theoretical or modified body temperature, which enables the estimation of the time since death by means of the standard Nomogram method. Verica et al. (2007) studied estimation of time since death in the field of forensic medicine and analyzed some of the existing methods, compared obtained results to determine which method gives more precise results of the estimation of time since death. The authors presented the analysis of 50 cases autopsied at the Institute of Forensic Medicine and Criminology in Skopje, with known time of death. Rectal temperature was taken with digital thermometer. Simultaneously, environment temperature was measured as well as the body weight; it was recorded whether the body was covered or naked. In order to estimate time since death, following methods were applied: Method I, Method II, Al-Alousi and Anderson and Henssge- nomogram. Comparison of the known time of death with the time obtained by the applied methods showed a discrepancy of few hours. Comparison of results obtained by application of the above stated methods showed that the Henssge-nomogram gives less discrepancy from the true time of death. In Kalizan and Hauser (2007) a systematic two-stage study was conducted in pigs to verify the models of postmortem body temperature decrease currently employed in forensic medicine. During their investigations, temperature recordings were performed in four body sites (eyeballs, orbit soft tissues, muscles and rectums). The results of their study supported the possible use of the eyeball and also the orbit soft tissues as temperature measuring sites at the early phase after death; they have narrowed the significance of rectum temperature measurements to the late stage of postmortem body 26

37 temperature decrease, shown insignificant correlations between the body weight and the temperature decrease rate constant and illustrated the functional increase of the time of death estimation error as the body cools, expressed in the distinct tendency to overestimate the calculated time of death as compared to the actual one. In the second stage of their experiment, a lack of a plateau phase was demonstrated, at least from 30 min post mortem. It was also found that in the very early post mortem period, the kinetics of cooling of all the body sites studied was better described by the two-exponential model than the single exponential one. Their study also showed that the weak airflow present in the experimental conditions did not practically affect the course of cooling of the investigated body sites. Eyeball temperature measurements with an infra-red laser thermometer performed during the experiment proved to be of no use for determination of the time of death. The experiments allowed for defining the so far unreported value of physiological temperature of pig eyeball as 38 degrees C. In Karhunen et al. (2008) the authors presented a paper on time of death of victims found in cold water environment. Here they presented their experience on two cases with known post-mortem times. A 14-year-old girl (rectal temperature 15.5 C) was found assaulted and drowned after a rainy cold night (+5 C) in wet clothing (four layers) at the bottom of a shallow ditch, lying in non-flowing water. The post-mortem time turned out to be h. Four days later, at the same time in the morning, after a cold (±0 C) night, a young man (rectal temperature 10.8 C) was found drowned in a shallow cold drain (+4 C) wearing similar clothing (four layers) and being exposed to almost similar environmental and weather conditions, except of flow (7.7 l/s or 0.3 m/s) in the drain. The post-mortem time was deduced to be hours. They tested the applicability of 27

38 five practical methods to estimate time of death and found Henssge s temperature time of death nomogram method with correction factors as the most versatile and gave also most accurate results, although there was limited data on choosing of correction factors. In the first case, the right correction factor was close to 1.0 (recommended ), suggesting that wet clothing acted like dry clothing in slowing down body cooling. In the second case, the right correction factor was between 0.3 and 0.5, similar to the recommended 0.35 for naked bodies in flowing water. In Hubig et al. (2011) the authors investigated the influence of variations in the environmental temperature, initial body core temperature, core temperature and time on the standard deviation of the most established model commonly used in forensic practice which was developed by Henssge to estimate the time since death. Two different approaches were used for calculating the standard deviation: the law of error propagation and the Monte Carlo method. Errors in environmental temperature measurements as well as deviations of the initial rectal temperature were identified as major sources of inaccuracies in model based death time estimation. Michael et al. (2011) mentioned that model-based methods play an important role in temperature-based death time determination. The most prominent method uses Marshall and Hoare's double exponential model with Henssge's parameter determination. The formulae contain body mass as the only non-temperature parameter. The authors said that Henssge's method is well established since it can be adapted to non-standard cooling situations varying the parameter body mass by multiplying it with the corrective factor. The authors investigated the influence of measurement errors of body mass M as well as variations of the corrective factor on the error of the Marshall and Hoare-Henssge death 28

39 time estimator. A formula for the relative error of death time estimator as a function of the relative error of body mass was derived. Simple approximations of order 1 and 0 nevertheless yielded acceptable results validated by Monte Carlo simulations. They also provided the rule of thumb according to which the quotient of the standard deviations of the estimated death time and the standard deviations of the body mass was equal to the quotient of the estimated death time and the body mass. Additionally, formulae and their approximations were derived to quantify the influence of Henssge's body mass corrective factor on death time estimation. In a range of body masses between 50 and 150 kg, the relative variation of the body mass corrective factor is approximately equal to the relative variation of the death time. This formula was applied and compared to computations and to experimental cooling data with good results. In Mergenthaler et al. (2012), the authors noted that the most common method used in determining the estimated time since death in the early post-mortem phase is backcalculation based on rectal temperature decrease. Cooling experiments are essential for model generation and validation. Post-mortem temperature models are necessary to perform back-calculations. Thus far, cooling experiments have not been performed under controlled environmental conditions. In their study they provided data on 84 post-mortem cooling experiments under strictly controlled environmental conditions. For a period of 5 years, starting in 2003, deceased persons with a known time of death and known environmental conditions at the death scene were transferred to a climatic chamber for the process of body cooling. The environmental temperature was programmed to the death scene temperature and kept constant throughout the process of body cooling. Rectal and ambient temperatures were measured every minute. Relevant case-specific 29

40 information was summarized in a FileMaker database. The database serves as a reference tool for the comparison of real cases in forensic routine and to check the plausibility of model-derived estimates. Smart and Kalizan (2012) examined evidence to seek an explanation of the possible causes or contributing factors to the temperature plateau phenomenon and its influence on time of death estimation. The concept of the temperature plateau effect was reviewed, and investigation was conducted into its possible prediction under post mortem conditions. The authors concluded that the appearance of a temperature plateau effect in postmortem body core temperature decay curves is currently random and cannot be predicted. This unpredictability is based upon the inter-individual differences in core body temperature, hyperthermia, use of drugs, trauma, etc. and biomarker concentrations (electrolytes, thyroxine, etc.) at ante mortem times, which will ultimately affect the shape of the postmortem temperature decay curve. According to the authors, studies indicated that the temperature plateau effect is diminished or even absent in the head tissues, including eye and ear. The possibility of precise estimation of the time of death in the early post mortem period based on eye temperature measurements was also commented. 30

41 CHAPTER 3 METHODOLOGY In this chapter, an ordinary first order differential equation model for time estimation of death is formulated. Due to the complex nature of the model, a computer program written in Visual C # is employed to solve the model. The inputs to the program includes: the surrounding temperature, rectal temperature of the body, mass of the body and the condition of the body. The output is the time of death. The detailed listing and examples of the input and output are presented in Chapter four. The general structure of the computer program is described by the flow chart shown in figure 3.1. Some important variables in the text and program are listed in Appendix A. The complete computer program is presented in Appendix B. Some key terms and few theories that are relevant to this thesis are also presented. 3.1 Model Assumptions Temperature of the surrounding T a is assumed to remain constant. Temperature of the body is the same as its surface temperature. That is we assume uniform cooling. 31

42 3.2 Newton s Model Newton s law of cooling based on the assumptions above and which is also defined in chapter one is stated mathematically as (3.01) where T : Temperature of the cooling object at time t t : time in hours since the first reading : Temperature of surrounding medium (Ambient temperature) k : Constant of proportionality Solution of Newton s Equation (Model) ( 3.02) (3.03) Multiplying both sides by the integrating factor (3.04) Integrating both sides with respect to t dt = dt (3.05) = (3.06) 32

43 T(t) = +C (3.07) At t = 0, T(0 ) = T o T o = + C ( 3.08) C= T o ( 3.09) ( 3.10) Modified Newton s Model Rainy (1868) discovered that a period after death called plateau exist where the body does not cool at all and the body temperature may even rise and further stated that bodies recently dead are not found to cool in conformity with Newton s law of cooling. Rainy modified Newton s model by determining a minimum and maximum of time within which the time of death will be included. The model could not estimate the maximum time because it was difficult to fix it. In 1962, Marshall and Hoare established the fact that the basic assumption of Newton s law of cooling was invalid when applied to the very early of the cooling of a deceased human body. Marshall and Hoare modified Newton s model by adding an exponential term that represents the postmortem plateau. The model contains two exponential parts. The first represents the postmortem plateau and the second constant shows the exponential drop of time after the plateau according to Newton s law of cooling (Leinbach, 2010) 33

44 3.3 Rate of Post Mortem Cooling According to Pounder (1995), University of Dundee, the linear rate of post mortem cooling is affected by environmental factors and other than the environmental temperature and the body temperature at the time of death. These include: The size of the body. The greater the surface area of the body relative to its mass, the more rapid will be its cooling. Consequently, the heavier the physique and the greater the obesity of the body, the slower will be the heat loss. The exposed surface area of the body radiating heat to the environment will vary with the body position. Clothing and coverings. These insulate the body from the environment and therefore cooling is slower. Cooling of a naked body is half as fast as when clothed. Movement and humidity of the air. Air movement accelerates cooling by promoting convection and even the slightest sustained air movement is significant. Cooling is said to be more rapid in a humid rather than dry atmosphere because moist air is a better conductor of heat. The humidity of the atmosphere will affect cooling by evaporation where the body or its clothing is wet. Immersion in water. For a given environmental temperature, cooling in still water is about twice as fast as in air, and in flowing water, about three times as fast. Clearly the will cool more rapidly in cold water than warm water. 34

45 3.4 The following Assumptions are also made made. In addition to the assumptions of the Newton s Model, the following are also No strong radiation(e.g. sun, heater, cooling system) No uncertain severe changes of the cooling condition during the period between the time of death and examination.(the place of death must be the same as where the body was found. The Ambient temperature is maintained at C. Marshall and Hoare observed that the cooling of a body is being influenced initially by a phenomenon whose effect decays with time.they approximated it with an exponential function (Leinbach, 2010). The expression for the rate of cooling of a deceased body according to Marshall and Hoare is where t : time T: Body temperature in 0 C T a : Ambient temperature in 0C, k and P are cooling rate constants and C is also a constant, T(0) =T o ( 3.11) 35

46 The first order differential equation has an integrating factor u(t) = Multiplying through by the integration factor u(t) = we have, (3.12) Therefore, Integrating both sides T = (3.13) T = (3.14) Multiplying equation (3.14) by we have T = 36

47 The solution of the linear differential equation then becomes T(t) = (3.15) At T(0) = = A = ( 3.16) Substitute equation (3.16) into equation (3.15) T(t) = T(t) = T(t) = ) but C = k( ) Therefore the final solution reduces to T(t) = ) (3.17) with the parameters defined as follows : 37

48 Table 3.1 Definitions of Variables used in the Model Parameters Definitions k Rate constant known as the cooling factor p Rate constant for the Plateau C Constant T Denotes rectal temperature at any time T a Denotes ambient temperature M Weight(Mass) of body T o Denotes rectal temperature at death ( t=0) 3.5 Software Development Computer program, a series of instructions that a computer can interpret and execute; programs are also called software to distinguish them from hardware, the physical equipment used in data processing. These programming instructions cause the computer to perform arithmetic and logical operations or comparisons (and then take some additional action based on the comparison) or to input or output data in a desired sequence. In conventional computing the operations are executed sequentially; in parallel processing the operations are allocated among multiple processors, which execute them 38

49 concurrently and share the results. Programs are often written as a series of subroutines, which can be used in more than one program or at more than one point in the same program. Systems programs are those that control the operation of the computer. Chief among these is the operating system-also called the control program, executive, or supervisor-which schedules the execution of other programs, allocates system resources, and controls input and output operations. Processing programs are those whose execution is controlled by the operating system. Language translators decode source programs, written in a programming language, and produce object programs, which are in machine language and can be understood by the computer. These include assemblers, which translate symbolic languages that have a one-to-one relationship with machine language; compilers, which translate an algorithmic or procedural-language program into a machine-language program to be executed at a later time; and interpreters, which translate source-language statements into object-language statements for immediate execution. Other processing programs are service or utility programs, such as those that "dump" computer memory to external storage for safekeeping and those that enable the programmer to "trace" program execution, and application programs, which perform business and scientific functions, such as payroll processing, accounts payable and receivable posting, word processing, and simulation of environmental conditions ( Maddix and Morgan,1989). 39

50 3.5.1 C Sharp (#) Programming Language Microsoft C# (pronounced C sharp) is a new programming language designed for building a wide range of enterprise applications that run on the Dot NET Framework. An evolution of Microsoft C and Microsoft C++, C# is simple, modern, type safe, and object oriented. C# code is compiled as managed code, which means it benefits from the services of the common language runtime. These services include language interoperability, garbage collection, enhanced security, and improved versioning support. C# is introduced as Visual C# in the Visual Studio Dot NET suite. Support for Visual C# includes project templates, designers, property pages, code wizards, an object model, and other features of the development environment. The library for Visual C# programming is the Dot NET Framework. C# is an elegant and type-safe object-oriented language that enables developers to build a variety of secure and robust applications that run on the.net Framework. You can use C# to create traditional Windows client applications, XML Web services, distributed components, client-server applications, database applications, and much, much more. Visual C# provides an advanced code editor, convenient user interface designers, integrated debugger, and many other tools to make it easier to develop applications based on version 4.0 of the C# language and version 4.0 of the Dot NET Framework. As an object-oriented language, C# supports the concepts of encapsulation, inheritance, and polymorphism. All variables and methods, including the Main method, the application's entry point, are encapsulated within class definitions. A class may inherit directly from one parent class, but it may implement any number of interfaces. Methods that override virtual methods in a parent class require the override keyword as a way to avoid accidental redefinition. In C#, a struct is like a lightweight 40

51 class; it is a stack-allocated type that can implement interfaces but does not support inheritance ( Accessed:5/10/2012) Object-Oriented Programming Object-oriented programming is a method of implementation in which programs are organized as cooperative collections of objects, each of which represents an instance of some class, and whose classes are all members of a hierarchy of classes united via inheritance relationships. There are three important parts to this definition: Object-oriented programming uses objects, not algorithms, as its fundamental logical building blocks. Each object is an instance of some class. Classes may be related to one another via inheritance relationships. A program may appear to be object-oriented, but if any of these elements is missing, it is not an object-oriented program. Specifically, programming without inheritance is distinctly not object oriented; that would merely be programming with abstract data types ( Accessed : 5/10,2012). 41

52 3.5.3 Object-Oriented Design Object-oriented design is a method of design encompassing the process of object oriented decomposition and a notation for depicting both logical and physical as well as static and dynamic models of the system under design. There are two important parts to this definition: object-oriented design leads to an object-oriented decomposition and uses different notations to express different models of the logical (class and object structure) and physical (module and process architecture) design of a system, in addition to the static and dynamic aspects of the system (source: an.htm, Accessed: 5/10/2012) Object-Oriented Analysis Object-oriented analysis is a method of analysis that examines requirements from the perspective of the classes and objects found in the vocabulary of the problem domain. Basically, the products of object-oriented analysis serve as the models from which we may start an object-oriented design; the products of object-oriented design can then be used as blueprints for completely implementing a system using object-oriented programming methods.( frameworks/6_4/oo_analysis_and_design.html, Accessed: 5/10/2012) 42

53 3.5.5 Object An object represents a unique instance of a data structure defined by the template provided by its class. Each object has its own values for the variables belonging to its class and responds to methods defined by that class. After an object has been created (instantiated) from a class, you can change its properties. A property is an attribute of an object. Properties define: Object characteristics, such as name or value. The state of an object such as deleted or changed. Some properties are read-only and cannot be set, such as Name or Author. Other properties can be set, such as Value or Label. Objects are different from other data structures. They include code (in the form of methods), not just static data. A method is a procedure or routine, associated with one or more classes, that acts on an object ( (psbooks/tpcd/chapter.htm?file=tpcd/htm/tp cd04.htm, Accessed:5/10/2012) Class A class is a set of objects that share a common structure and a common behavior. Every object is associated with a class. For example, all the objects that capture information about employees could fall into a class called Employee, because there are attributes 43

54 (e.g., ID number, First name, Last name address, birth date, phone, and sex) and methods (e.g., calculate Salary, employee status, and Pension Scheme) that all employees share. A class therefore defines the properties of an object and the methods used to control the object s behavior. 3.6 Programming Tools Algorithm An algorithm is a procedure for solving a problem in terms of the actions to be executed and the order in which those actions are to be executed. An algorithm is merely the sequence of steps taken to solve a problem. The steps are normally sequence, selection, iteration, and a case-type statement (source: Accessed: 5/10/2012). 44

55 3.6.2 Pseudocode Pseudocode is a kind of structured English for describing algorithms. It allows the designer to focus on the logic of the algorithm without being distracted by details of language syntax. At the same time, the Pseudocode needs to be complete. It describes the entire logic of the algorithm so that implementation becomes a rote mechanical task of translating line by line into source code (source: Accessed:5/10/2012) Steps Initialize counter to zero Initialize rectal temperature, ambient temperature and Initial temperature to zero Initialize Time of death to Define other variables like rate constants k, p, xvalue, yvalue and TotalValue. Input Rectal temperature T Input Ambient temperature T a Input Initial temperature T o Input Weight of body M Select body condition as Condition Calculate Body surface area as BSA=0.1173*W* Calculate size Factor as SF=0.8*BSA*10000/W Calculate rate constant k from k = *SF

56 If Condition is equal to Clothed body then calculate p=0.3 else if Condition is equal to Naked body then calculate p= 0.4 end if Set maximum counter to MAXIMUM_VALUE while counter is less than MAXIMUM_VALUE calculate xvalue = calculate yvalue = ) calculate TotalValue=xValue + yvalue subtract TotalValue from T if the difference is less than threshold then set time of death to Time exit from loop else add to time of death end if increase the counter by 1 end while print Time of death is equal to,time end program 46

57 3.6.3 Flowcharts A flow Chart is a diagrammatic representation that illustrates the sequence of operations to be performed to get the solution of a problem. Flow charts are generally drawn in the early stages of formulating computer solutions. Flowcharts facilitate communication between programmers and business people. These flowcharts play a vital role in the programming of a problem and are quite helpful in understanding the logic of complicated and lengthy problems. Once the flowchart is drawn, it becomes easy to write the program in any high level language. Often we see how flowcharts are helpful in explaining the program to others. Hence, it is correct to say that a flowchart is a must for the better documentation of a complex program (source: Accessed: 5/10/2012). In this thesis, we shall concern ourselves with the program flow chart, which describes what operations and in what sequence are required to solve the Newton s cooling model. The flow chart is shown below 47

58 Flowchart of the Model Start Input Initial Temp T o Set t= Select condition If condition== clothed, P =0.3 If condition== naked, P= Input Rectal Temp T 2. Input Room Temp Ta 3. Input Initial Temp To Set diff= T* BSA=0.1173*W* SF =0.8*BSF*10000/w K= Count = 0 yes while count > 0 No x=t a +(T o -T a )e kt y=(k/k-p)(t o -T a )(e pt - e -kt ) Total= x+y Set Value= T -Total If Value <= diff yes No t= t Set Time = t Output Time Count=Count+1 Figure 3.1: Flowchart of the Computer Program. Stop 48

59 CHAPTER 4 DATA ANALYSIS AND RESULTS In this chapter, a computer program that solves the model by iteration is presented. Firstly, the program accepts the ambient temperature, weight of the body, rectal temperature and initial temperature as inputs. The program also takes clothing into account when solving the model. The body surface area (BSA) is calculated based on the formulae provided by Livingston and Lee (2001).The rate constant k termed cooling factor is calculated using Marshall s linear relationship k = size factor * (Lynnerup,1993). The program also request if the body is naked or clothed and base on the answer given, the value for the rate constant for the plateau p is chosen. Once the program accepts all the inputs along with the calculated parameters, the time since death is determined. 4.1 Source of Data Data for analyzing the model were obtained from the following sources in the literature (ACTA MORPHOLOGIC,2006; vol 3(2):51-54 Medical journal of Macedonian Association of Anatomists and Morphologists (MAAM), Handbuch Gerichtliche Medizin, Volume 1 by Bernd Brinkmann, and Burkhand Madea). 49

60 4.2 Parameter Estimation The parameters of the model are estimated based on the following information (Lynnerup, 1993). BSA = * W * where BSA: Body surface Area W : weight of the body. SF = 0.8 * BSA (m2) / W (kg) where SF: Size Factor of the body. The parameter k which is the rate constant of the Newton s model is given by k = SF * For clothed body, p =0.3 and for naked body p= 0.4 The above parameter estimates are summarized in table 4.1 below. 50

61 Table 4.1 Parameter estimates of Cooling Model Parameter Value Source k SF * Lynnerup (1993) Journal of Forensic sciences P 0.3 for Clothed body 0.4 for Naked body Lynnerup (1993) Journal of Forensic sciences 4.3 Analysis of Results Analysis of 59 cases obtained from the following sources in the literature: ACTA MORPHOLOGIC, 2006; vol 3(2):51-54, Medical journal of Macedonian Association of Anatomists and Morphologists (MAAM) and Handbuch Gerichtliche Medizin, Volume 1 by Bernd Brinkmann, and Burkhand Madea with Known time of death is presented. Rectal temperature and environmental temperature were measured as well as the body weight. The clothing condition (naked or clothed) was also recorded. Comparison is made between the known time of death with the time obtained from the computer software. Tables 4.2 and 4.3 below summarize all the 59 cases together with results obtained by the computer program. 51

62 CAE AMBIENT TEMPERATURE (T O C) MEAN AMBIENT TEMPERATURE RECTAL TEMPERATURE O BODY WEIGHT(kg) KNOWN TIME OF DEATH CLOTHED ( +) ESTIMATED TIME SINCE DEATH OF THE MODEL Table 4.2 Cases with known time of death (source: Handbuch Gerichtliche Medizin, Vol.1) / / / / / / / / /

63 CASE AGE SEX KNOWN TIME OF DEATH RECTAL TEMPERATURE (T O C) AMBIENT TEMPERATURE BODY WEIGHT(kg) BODY HEIGHT(cm) CLOTHED ( +) NAKED ( - ) ESTIMATED TIME SINCE DEATH OF THE MODEL Table 4.3 Cases with known time of death (source: ACTA MORPHOLOGIC, 2006 Vol. 3(2):51-54) 1 51 M F M F M M M M F

64 10 55 M M M M F M M M M M M M M M

65 24 60 M M M M M M M M M F F M M M

66 38 37 M M M M M M M M M F M M M

67 Time of Death(Hrs) 4.4 Further Discussion of Results The developed model of cooling is validated against known data provided in Tables 4.2 and 4.3 above. Results obtained by the computer program of the model show that there is no significant difference between the known time of death and the estimated time of death of the model. This is illustrated in Figures 4.1 and 4.2 below. From the charts in Figures 4.1 and 4.2, it is possible to see that with a total of 7 and 37 cases respectively, there was either no discrepancy or very small difference in time between the known time of death and the estimated time of death of the model Known Time of Death Estimated Time of Death of Model Column Chart of Time of Death Cases Figure 4.1: A chart of estimated time and known time of death 57