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APPLICATION OF NEWTON’S LAW OF COOLING

CASE STUDY: ESTIMATION OF TIME OF DEATH

IN MURDER

BY

SAMUEL ASANTE (BSC. COMPUTER SCIENCE)

PG 6317711

A Thesis Submitted to the Department of Mathematics, Kwame Nkrumah University ofScience 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



ii

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 (20249016) …………… ………………

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



iii

DEDICATION

To my lovely wife, Cordelia and children

David, Eugenia and Desmond



iv

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.



v

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.



vi

CONTENTS

DECLARATION…………………………………………………………………………. ii

DEDICATION…………………………………………………………………………… iii

ACKNOWLEDGEMENT……………………………………………………………….. iv

ABSTRACT………………………………………………………………………………. v

CHAPTER 1……………………………………………………………………………… 1

INTRODUCTION………………………………………………………………………...1

1.1 Brief Introduction of Chapter 1………………………………………………...1

1.2 Background of the Study………………………………………………………...2

1.2.1 History of Newton‟s Law of Cooling………………………………………2

1.2.2 Newton‟s Law of Cooling…………………………………………………..3

1.2.3 Introduction to the Principle of Heat Transfer ……………………………...3

1.2.4 Profile of Study Area……………………………………………………….7

1.2.5 Some Basic Definitions……………………………………………………..9

1.3 Problem Statement……………………………………………………………..17

1.4 Main Objective…………………………………………………………………17

1.4.1 Specific Objectives………………………………………………………..17

1.5 Justification…………………………………………………………………….17

1.6 Methodology…………………………………………………………………...18

1.7 Organization of the Study……………………………………………………...18



vii

CHAPTER 2……………………………………………………………………………. 20

REVIEW OF RELATED STUDIES…………………………………………………….20

2.1 Review of Previous Studies…………………………………………………….20

CHAPTER 3…………………………………………………………………………….. 31

METHODOLOGY……………………………………………………………………… 31

3.1 Model Assumptions……………………………………………………………31

3.2 Newton‟s Model………………………………………………………………..32

3.2.1 Solution of Newton‟s Equation (Model)…………………………………..32

3.3 Rate of Post Mortem Cooling…………………………………………………..34

3.4 The following Assumptions are also made…………………………………….35

3.5 Software Development…………………………………………………………38

3.5.1 C Sharp (#) Programming Language……………………………………..40

3.5.2 Object-Oriented Programming…………………………………………….41

3.5.3 Object-Oriented Design…………………………………………………...42

3.5.4 Object-Oriented Analysis………………………………………………….42

3.5.5 Object……………………………………………………………………...43

3.5.6 Class………………………………………………………………………. 43

3.6 Programming Tools…………………………………………………………….44

3.6.1 Algorithm…………………………………………………………………. 44

3.6.2 Pseudocode……………………………………………………………….. 45



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3.6.3 Flowcharts…………………………………………………………………47

CHAPTER 4……………………………………………………………………………. 49

DATA ANALYSIS AND RESULTS……………………………………………………49

4.1 Source of Data…………………………………………………………………. 49

4.2 Parameter Estimation…………………………………………………………..50

4.3 Analysis of Results……………………………………………………………..51

4.4 Further Discussion of Results………………………………………………….57

CHAPTER 5……………………………………………………………………………. 60

CONCLUSION AND RECOMMENDATIONS………………………………………..60

5.1 Conclusion……………………………………………………………………... 60

5.2 Recommendation……………………………………………………………….60

REFERENCES………………………………………………………………………….. 61

APPENDIX A…………………………………………………………………………… 65

APPENDIX B…………………………………………………………………………… 71



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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



x

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



1

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



2

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

1.2.1

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).

http://www.eoht.info/page/Isaac+Newton

http://www.eoht.info/page/Pierre+Dulong

http://www.eoht.info/page/Alexis+Petit

http://www.eoht.info/page/Alexis+Petit

http://www.eoht.info/page/Pierre+Dulong

http://www.eoht.info/page/Isaac+Newton



3

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).

1.2.3 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



4

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

(http://www.efunda.com/formulae/heat_transfer/home/overview.cfm,Accessed:

14/11/2012).

1.2.3.1

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



5

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.

1.2.3.2

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.



6

1.2.3.3 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 72oF air temperature may feel

uncomfortably cold if the walls and ceiling are at 50oF. Conversely, they may feel

uncomfortably warm if the walls are 85oF. 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.

http://www.efunda.com/formulae/heat_transfer/conduction/overview_cond.cfm

http://www.efunda.com/formulae/heat_transfer/conduction/overview_cond.cfm



7

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

1921. 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.



8

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 (http://www.eservices.gov.gh

/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).



9

1.2.5 Some Basic Definitions

1.2.5.1

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.

1.2.5.2 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.



10

1.2.5.3 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.

(http://www.tampagov.net/dept_Police/about_us/Investigations_and_Support/Criminal_I

nvestigations/Forensic_Investigation/, Accessed:14/11/2012).

1.2.5.4

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

http://www.wisegeek.com/what-is-fraud.htm

http://www.wisegeek.com/what-is-fraud.htm



11

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.(http://www.pinow.com/investigations/forensic-

investigations,Accessed:14/11/2012)

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

http://www.pinow.com/investigations/forensic-investigations






12

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 (http://www.legalmetro.com/library/investigating-a-

crime-scene.html,Accessed:

14/11/2012).

1.2.5.5

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)

http://www.legalmetro.com/library/investigating-a-crime-scene.html






13

Figure 1.1 Sigmoidal shape of cooling curve. Mathematical description with

the two exponential model by Marshall and Hoare (Source: Henßgea and

Madea ,2004).

T0

Postmortal

temperature

Plateau Newton‟s law of

cooling

Ta

T - Ta

Tr - Ta

Temperature

Time (h)



14

1.2.5.6 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.

(http://en.wikipedia.org/wiki/Nomogram, 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 37.2oC.

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 23oC, Henssge

evolved two nomograms, the one for ambient temperatures above 23oC and the other for

ambient temperatures below 23oC. 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



15

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).



16

Figure 2.1: Henssge Nomogram for environmental temperature up to 230C (upper)

(Source: Henßgea and Madea, 2004)



17

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.

1.4.1

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



18

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



19

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.



20

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 200, 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 1974. 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 200 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

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21

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 00C yielded distinctly slighter temperature which were especially

marked at rectal temperatures up to approximately 110C 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





22

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 150C. 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









23

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 150

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,



24

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 finite-

element 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 medico-

legal 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

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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 12 – 36 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.



26

The resulting disturbance produced on the cooling curve is proved to obey a two-

exponential 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



27

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 15 – 16 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 10 – 12 hours. They tested the applicability of



28

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 1.1 – 1.2),

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







29

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 back-

calculation 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

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30

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.



31

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 Ta is assumed to remain constant.

Temperature of the body is the same as its surface temperature. That is we assume

uniform cooling.



32

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

3.2.1

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)



33

T(t) = +C (3.07)

At t = 0 , T(0 ) = To

To = + C ( 3.08)

C= To − ( 3.09)

( 3.10)

3.2 .2 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)



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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.



35

3.4 The following Assumptions are also made

In addition to the assumptions of the Newton‟s Model, the following are also

made.

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 37.20

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 in0C

Ta : Ambient temperature in 0C,

k and P are cooling rate constants and

C is also a constant

, T(0) =To ( 3.11)



36

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 =



37

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 :



38

Table 3.1 Definitions of Variables used in the Model

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

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

Ta Denotes ambient temperature

M Weight(Mass) of body

To Denotes rectal temperature at death ( t=0)

http://www.answers.com/topic/computer-1

http://www.answers.com/topic/data-processing

http://www.answers.com/topic/parallel-processing




http://www.answers.com/topic/data-processing

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39

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).

http://www.answers.com/topic/programming-language

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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



41

class; it is a stack-allocated type that can implement interfaces but does not support

inheritance (http://msdn.microsoft.com/en-s/library/aa287558%28v=vs.71%29.aspx,

Accessed:5/10/2012).

3.5.2 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

( http://www.uobabylon.edu.iq/uobColeges/ad_downloads/4_13173_344.pdf,

Accessed : 5/10,2012).

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42

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:http://www.mcs.vuw.ac.nz/research/design1/1996/submissions/17Owen_Astrach

an.htm, Accessed: 5/10/2012)

3.5.4 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.(http://www.mactech.com/articles/

frameworks/6_4/OO_Analysis_and_Design.html, Accessed: 5/10/2012)



43

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

(http://docs.oracle.com/cd/E28394_01/pt852pbh1/eng/

(psbooks/tpcd/chapter.htm?File=tpcd/htm/tp cd04.htm, Accessed:5/10/2012).

3.5.6

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

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(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

3.6.1

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: http://www.unf.edu/~broggio/cop2221/2221pseu.htm, Accessed: 5/10/2012).

http://www.unf.edu/~broggio/cop2221/2221pseu.htm





45

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: http://users.csc.calpoly.edu/~jdalbey/SWE/pdl_std.html, Accessed:5/10/2012)

Steps

Initialize counter to zero

Initialize rectal temperature, ambient temperature and Initial temperature to zero

Initialize Time of death to 0.001.

Define other variables like rate constants k, p, xValue, yValue and TotalValue.

Input Rectal temperature T

Input Ambient temperature Ta

Input Initial temperature To

Input Weight of body M

Select body condition as “Condition”

Calculate Body surface area as BSA=0.1173*W*100.6466

Calculate size Factor as SF=0.8*BSA*10000/W

Calculate rate constant k from k = 0.0006125 *SF – 0.05373



46

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 0.001 to time of death

end if

increase the counter by 1

end while

print „Time of death is equal to „,Time

end program



47

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: http://www.edrawsoft.com/Flowchart-Definition.php, 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

http://www.edrawsoft.com/images/examples/Process-Flowchart.png

http://www.edrawsoft.com/images/examples/Process-Flowchart.png



48

Flowchart of the Model

Figure 3.1: Flowchart of the Computer Program.

Input Initial Temp To

Set t= 0.001

Start

Select condition

If condition==”clothed”, P =0.3

If condition== “naked “, P=0.4

1. Input Rectal Temp T

2. Input Room Temp Ta

3. Input Initial Temp To

Set diff= T*0.00001

BSA=0.1173*W*100.6466

SF =0.8*BSF*10000/w

K=0.0006125-0.5373

Count = 0

yes

No

yes

No

while count > 0

x=Ta +(To-Ta)e –kt

y=(k/k-p)(To-Ta)(e –pt

- e-kt

)

Total= x+y

Set Value= T -Total

If Value <= diff

t= t + 0.001

Count=Count+1

Set Time = t

Out ut Time

Stop



49

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 * 0.0006125 –

0.05375 (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).



50

4.2 Parameter Estimation

The parameters of the model are estimated based on the following information

(Lynnerup, 1993).

BSA = 0.1173 * W * 100.6466

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 * 0.0006125 – 0.05375

For clothed body, p =0.3 and for naked body p= 0.4

The above parameter estimates are summarized in table 4.1 below.



51

Table 4.1 Parameter estimates of Cooling Model

Parameter Value Source

k SF * 0.0006125 – 0.05375 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.



52

Table 4.2 Cases with known time of death (source: Handbuch Gerichtliche

Medizin, Vol.1)

C A

E

A M

B I E N T T E M P E R A T U R E

( T O C )

M E A N

A M B I E N T

T E

M P E R A T U R E

R E

C T A L T E M P E R A T U R E

B O

D Y W E I G H T ( k g )

K N

O W N T I M E O F D E A T H

C L

O T H E D

( + )

E S

T I M A T E D T I M E S I N C E

D E

A T H O F T H E

M

O D E L

115.5 / 16.5 16.00 33.1 62 5.80 + 5.69

222.50 / 23.0 22.75 32.8 57 7.70 + 7.77

318.0 / 19.0 18.75 33.5 96 11.30 + 6.98

425.0 / 27.3 26.15 33.3 78 11.60 + 10.04

526.4 / 27.4 26.90 33.0 116 29.10 + 14.05

6 1.50 / 1.50 1.50 16.7 78 16.60 + 16.64

74.50 / 6.0 5.25 28.3 69 8.40 + 7.81

816.7 / 18.2 17.45 26.3 71 16.90 + 15.14

915.7 / 17.2 16.45 25.4 56 13.90 + 14.08



53

Table 4.3 Cases with known time of death (source: ACTA MORPHOLOGIC, 2006

Vol. 3(2):51-54)

C A S E

A G E

S E X

K N O W N T I M E O F D E A T H

R E C T A L T E M P E R A T U R E

A M B I E N T

T E M P E R A T U R E

B O D Y W E I G H T ( k g )

B O D Y H E I G H T ( c m )

C L O T H E D ( + )

N A K E D

( - )

E S T I M A T E D T I M E S I N C E

D E A T H O F T H E

M O D E L

1 51 M 4 34.9 17 65 168 + 3.94

2 54 F 4 34.4 22 65 160 + 5.44

3 54 M 4 35.1 19.3 78 180 + 4.29

4 26 F 4 34.7 22.5 75 166 - 4.85

5 16 M 4 35.9 22.4 65 175 - 2.96

6 49 M 5 33 24 75 176 - 7.94

7 35 M 5 36.5 21 80 175 - 2.07

8 38 M 6 32.8 22.4 80 174 - 7.70

9 20 F 6 32.2 22 58 165 - 7.35



54

10 55 M 6 34 21 75 172 - 5.41

11 58 M 6 33.2 16.5 60 158 + 5.47

12 69 M 6 33 20 78 173 - 6.47

13 52 M 6 34.1 24.5 70 166 + 6.86

14 44 F 7 32 21 57 160 - 7.15

15 35 M 7 30.6 16 80 175 + 8.79

16 53 M 7 32.5 21.3 80 172 + 8.42

17 75 M 7 33 24 78 176 + 8.86

18 76 M 7 33.9 24 82 175 + 7.44

19 60 M 7 33.8 24 84 178 + 7.69

20 59 M 7 33 22.5 70 164 + 7.76

21 44 M 7 35.4 30 75 177 + 7.17

22 62 M 8 32 21 72 168 - 7.86

23 38 M 8 32.4 21.2 55 159 - 6.66



55

24 60 M 8 32.6 21 85 180 - 7.60

25 29 M 9 32 24 50 159 - 8.16

26 57 M 10 31.6 19 78 174 + 8.61

27 40 M 10 30.9 23 80 180 + 12.14

28 42 M 12 27.2 15.7 75 173 + 12.39

29 44 M 13 29.6 24 80 178 + 16.23

30 74 M 13 28.8 21 83 180 - 13.64

31 54 M 13 28.1 16.5 80 179 - 11.14

32 53 M 14 28.4 18 75 172 - 11.29

33 67 F 14 27.2 20.6 54 151 - 13.42

34 32 F 14 27.5 20 45 166 - 11.42

35 36 M 15 27 23 75 174 + 21.55

36 60 M 15 25.9 24.4 76 175 - 33.07

37 53 M 15 24.1 23.6 75 180 - 49.05



56

38 37 M 15 24.5 20 80 181 + 23.35

39 32 M 16 24 21 73 166 + 27.07

40 37 M 17 23.5 23 95 187 + 58.30

41 25 M 19 23 19 80 184 + 25.91

42 25 M 19 23.3 22 95 180 + 41.52

43 20 M 19 23 22 85 184 + 45.06

44 24 M 20 22.7 17 75 180 + 21.53

45 75 M 20 22.5 17 70 173 + 21.30

46 34 M 20 22.5 17 76 175 + 22.18

47 31 F 21 22 20.6 65 155 - 33.76

48 44 M 22 21.6 21 73 175 + 49.26

49 24 M 24 21.3 20 80 178 + 41.50

50 58 M 24 21.8 21 60 163 - 39.67



57

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.

Figure 4.1: A chart of estimated time and known time of death

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9

Known Time of Death 5.8 7.7 11.3 11.6 29.1 16.6 8.4 16.9 13.9

Estimated Time of Death of

Model5.69 7.77 6.98 10.04 14.05 16.64 7.81 15.14 14.08

T i m e o f D e a t h ( H r s )

Cases

Column Chart of Time of Death



58

Figure 4.2: A chart of estimated time and known time of death.

The results however, show some level of discrepancies from the known time of death. In

Figure 4.2, the chart shows some level of discrepancies from case numbers 34 to 43 and

from 47 to 50. This happened as a result of temperature difference between the

environmental (ambient) and rectal temperatures. When the post mortem period increased

above 15hrs, the difference between the ambient temperature and the rectal temperature

were very small. From Table 4.3, case numbers 44, 45 and 46 gave a very small

difference between the known time of death and the estimated time of death because the

temperature difference between the ambient and rectal temperatures were big.

0

10

20

30

40

50

60

70

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

T i m e o f D e a t h ( h r s (

Cases

Column chart of time of death

Known Time of Death Estimated Time of Death of Model



59

Table 4.4 summarizes the temperature difference between the ambient and rectal

temperatures, the known time of death and the estimated time of death obtained from the

computer software.

Table 4.4 Summary of Temperature difference, estimated time of death and

known time of death

Temperature Difference

(0C)

Known time of Death Estimated time of death

17.9 4 3.94

16.7 6 5.47

13.0 6 6.47

12.6 10 8.61

10.4 14 11.29

7.8 13 13.64

4.5 15 23.35

3.0 16 27.07

2.5 21 33.76

1.5 19 41.52

1.0 19 45.06

0.6 22 49.26



60

CHAPTER 5

CONCLUSION AND RECOMMENDATIONS

5.1

Conclusion

Estimation of time of death by using the model show that accuracy of results largely

depends on the length of postmortem period. From the analysis, it was observed that the

shorter the postmortem period, the more accurate is the estimate of time of death .The

inaccuracy of the model become‟s considerable at low temperature difference between

the rectal and ambient temperatures. Consequently, the longer the post mortem interval,

the wider is the range of estimate as to when death probably occurred. We conclude that

the Marshall and Hoare double exponential model is the most appropriate model for

estimating the time of death in murder.

5.2

Recommendation

Variables such as posture of the body, site of reading of the postmortem body

temperature, emaciation and micro-environment (rain, humidity etc) that were not

considered in this thesis for the determination of time of death could be study to improve

the current model to account for the long post mortem period and to accurately estimate

time of death. This improvement can be used for scientific crime investigation.



61

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65

APPENDIX A

The controls that are referred to in the code

Object

type

Name Description

TextBox txttimeA read only text box that displays time

of death

TextBox txtrectalA text box that accepts a rectal

temperature

TextBox txtmass

A text box that accepts the mass of body

TextBox txtambientA text box that accepts ambient

temperature

TextBox txtInitialA text box that accepts initial

temperature

TextBox txtLogPathA read only text box that displays the

path the records are kept

Button btnOKCalculates the time of death when

clicked

Button btnSaveSaves the records when clicked

Button btnExitCloses the form when clicked

Button btnNextClears the textboxes when clicked



66

Object type Label

Object Property Value

Label Name lblTimeofDeath

AutoSize True

Font Times New Roman, 12F

Location Point(132, 188)

Size (124, 23)

Text Time of Death

Label Name lblPath

AutoSize True



Size (47, 23)

Text Path

Label Name lblmass

AutoSize True



Size (125, 23)

Text Mass of body

Label Name lblRectalTemperature

AutoSize True



Size (172, 23)



67

Text Rectal Temperature

Label Name lblInitialTemperature

AutoSize True



Size (168, 23)

Text Initial Temperature

Label Name lblAmbientTemperature

AutoSize True



Size (190, 23)

Text Ambient Temperature

Object type Textbox


Textbox Name txtLogPath

Enabled False


Fontstyle Regular

Location Point(268,227)

Multiline True

Size (155,27)

Text c:\\Test\Death Time.txt

Textbox Name txttime

Enabled False



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Fontstyle Bold


Multiline True

Size (155,23)

ForeColor Blue

Textbox Name txtambient


Fontstyle Bold


Multiline True

Size (155,23)

Textbox Name txtInitial


Fontstyle Regular


Multiline True

Size (155,23)

Textbox Name txtRectal


Fontstyle Regular


Multiline True

Size (155,23)

Textbox Name txtmass




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Fontstyle Regular


Multiline True

Size (155,23)

Object type Button


Exit Name btnExit

Location Point(342, 331)Size (54,23)

Text Exit

UseVisualStyle

BackColor

True

Save Name btnSave


Size (64,23)

Text Save

UseVisualStyle

BackColor

True

Next Name btnNext


Size (64,23)

Text Next

UseVisualStyle

BackColor

True



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OK Name btnOK


Size (64,23)

Text Ok

UseVisualStyle

BackColor

True



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APPENDIX B

Program Structure

using System;

using System.Collections.Generic;

using System.ComponentModel;

using System.Data;

using System.Drawing;

using System.Linq;

using System.Text;

using System.Windows.Forms;

using System.IO;

using System.Numeric;

using ZedGraph;



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namespace THESIS_2013

{

public partial class frmMainApp : Form

{

// Initialising variables

Int64 MAXCOUNT = Int64.MaxValue;

int Count;

double Ta = 0,Time = 0, To = 37.2, xValue = 0.0,yValue = 0.0,

TotalValue = 0.0;

double p = 0, k = 0, t = 0.001, T = 0, W = 0, BSA = 0, SF = 0;

string Option;

public frmMainApp()

{

InitializeComponent();

}

private void btnexit_Click(object sender, EventArgs e)

{

this.Close();

}

private void btnok_Click(object sender, EventArgs e)

{

//double difference = Math.Abs(T * .00001);

/******************************************************



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* This method calculates the time of death in murder

* based on Newton's law of cooling with modifications

* The rectal temperature of the body,Initial rectal temperature

* mass of the body and the rate constants are accepted

* as inputs ant the time is determined.

* ***************************************************/

try

{

if (IsValidData())

{

Ta = Convert.ToDouble(txtAmbient.Text);

W = Convert.ToDouble(txtweight.Text);

T = Convert.ToDouble(txtRectal.Text);

To = Convert.ToDouble(txtInitial.Text);

Option=Convert.ToString(cbocondition.Text);

if (Option =="Clothed Body")

{

p = 0.3;

}

else if (Option == "Naked Body")

{

p = 0.4;



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}

// Count = 0;

BSA = 0.1173 * Math.Pow(W, 0.6466) ;

SF = 0.8 * BSA * 10000 / W;

k= 0.0006125 * SF - 0.05373;

Count = 1;

// while (Count < MAXCOUNT)

while (Count > 0)

{

xValue = Ta + (To - Ta) * Math.Exp(-k * t);

yValue = (k / (k - p)) * (To - Ta) * (Math.Exp(-p * t) - Math.Exp(-k * t));

double difference = Math.Abs(T * 0.0001);

TotalValue = xValue + yValue;

if (Math.Abs(T - TotalValue) <= difference)

{

Time = t;

break;

}

else

{

t = t + 0.001;

}

Count++;



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}

txttime.Text = Time.ToString("F2") + "hrs";

}

}

catch (FormatException)

{

MessageBox.Show("Invalid Numeric Format.Please check all entries", "Entry Error");

}

catch (OverflowException)

{

MessageBox.Show("Overflow Error.Please enter accurate values", "Entry Error");

}

catch (Exception ex)

{

MessageBox.Show(ex.Message, ex.GetType().ToString());

}

}

public bool IsValidData( )

{

// validate the Ambient temperature textbox

if (!IsPresent(txtAmbient, "Ambient Temperature"))

return false;



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if (!IsDouble(txtAmbient, "Ambient Temperature"))

return false

if (!IsWithinRange(txtAmbient, "Ambient Temperature", 0))

return false;

// validate the mass of body textbox

if (!IsPresent(txtweight, "Mass of Body"))

return false;

if (!IsDouble(txtweight, "Mass of Body"))

return false;

if (!IsWithinRange(txtweight, "Mass of Body", 0))

return false;

// validate the Rectal temperature textbox

if (!IsPresent(txtRectal, "Rectal Temperature"))

return false;

if (!IsDouble(txtRectal, "Rectal Temperature"))

return false;

if (!IsWithinRange(txtRectal, "Rectal Temperature", 0))

return false;

// validate the Initial temperature textbox

if (!IsPresent(txtInitial, "Initial Temperature"))

return false;

if (!IsDouble(txtInitial, "Initial Temperature"))

return false;



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if (!IsWithinRange(txtInitial, "Initial Temperature", 0))

return false;

}

public bool IsPresent(TextBox textbox, string name)

{

if (textbox.Text == "")

{

MessageBox.Show(name + " is a required fiel", "Entry Error");

textbox.Focus();

return false;

}

return true;

}

public bool IsDouble(TextBox textbox, string name)

{

try

{

Convert.ToDouble(textbox.Text);

return true;

}

catch (FormatException)

{

MessageBox.Show(name + "must be a decimal value", "Error Entry");



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textbox.Focus( );

return false;

}

}

public bool IsWithinRange(TextBox textbox, string name, double min)

{

double number = Convert.ToDouble(textbox.Text);

if (number < min)

{

MessageBox.Show(name + " must not be less than zero or negative", "Error Entry");

textbox.Focus();

return false;

}

return true;

}

private void btnSave_Click(object sender, EventArgs e)

{

this.Text = Logger.LogWrite(txtLogPath.Text, txtRectal.Text, txtInitial.Text,

txtAmbient.Text, txtweight.Text, txttime.Text);

}

private void showRecordToolStripMenuItem_Click(object sender, eventArgs e)

{

frmDisplayResults mydata = new frmDisplayResults();



mydata.Show();

}

private void exitToolStripMenuItem_Click(object sender, EventArgs e)

{

this.Close(); this.Close();

}

private void btnNext_Click(object sender, EventArgs e)

{

txtAmbient.Clear();

txtInitial.Clear();

txtweight.Clear();

txtRectal.Clear();

txttime.Clear();

cbocondition.Text = "";

txtAmbient.Focus();

}

private void txtAmbient_TextChanged(object sender, EventArgs e)

{

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