Mathematically Modelling the Post-Mortem Interval (PMI) | Sample IB Math Internal Assessment | Real-Life Application of Logarithms

Introduction

Post Mortem Interval (PMI) is referred as the time that has elapsed after a person has expired^[1]. PMI is a very reliable source of information that criminologists can use in order to solve or investigate a particular crime case. There are number of ways to estimate PMI, such as looking at the changes in the intestinal content, stage of invasion, blow fly larval development and so on^[2]. These methods give more accurate estimation of PMI as it is useful in cases with both short PMI and long PMI^[1]. But other methods such as Henssge-nomogram and the electrical or mechanical stimulation of skeletal muscles are only useful for short PMI because at one point of time the corps will attain the ambient temperature (room temperature) and stops cooling^[1]. This method purely relies on the temperature factor, the cooling of the corpse. This method of death time estimation is based on the cooling of the corpse. From my prior knowledge I knew that the cooling of any object follows an exponential decrease. And so, it was clear that post mortem interval could be mathematically modeled using exponential functions. Through this model I aim at arriving at an equation which will help in estimating the time of death.

This method of estimating PMI is no more in action due to its limitations mentioned above. However, it is very fascinating to know how man has been able to translate a real life issue into a mathematical language and further mathematically model the problem and find a solution for it. Yet another interesting part which fascinates me the most is the exponential function and the real world. Right from the very common radioactive decay, charging and discharging of capacitors, analyzing the population growth, bacterial growth, predicting cost and revenue in economics, there are number of applications of exponential function to solve real life problems. Initially I didn’t have any idea how to go about with this because there were number of applications based on exponent functions and majority of them were commonly know and discussed ones. However, my objective of this exploration was to apply the concept that was learnt in class to a real life scenario and to show how mathematics as a language expressed in terms of equations, symbols and numbers has got significance in understanding the world.

Data Collection

To obtain the data and graph for the study, I had to gather secondary data from authenticated journals since I had no scope of conducting the experiment at lab. The gathered data shown below is obtained from ‘Journal of Criminal Law and Criminology’^[4]. This data represents the change in rectal temperature of a corpse with respect to time. This data was recorded on 28^th January 1953 and it is a very old data.

          TABLE 1

Rectal Temperatures at (°F)
10 am	11 am	12 pm	1 pm	2 pm	3 pm	4 pm	5 pm	6 pm	7 pm	8 pm
97.7	96.8	95.8	94.8	94.1	93.3	92.7	91.9	91.1	90.4	89.9

                TABLE 2

Room Temperature in Laboratory(°F)
Min.	Max.	Avg.
79.0	82.0	80.3

From the data shown in TABLE 1, it’s understood that the temperature of the corpse was recorded at consecutive hours and the temperature is expressed in terms of Fahrenheit (°F). But for my convenience, I will be using Celsius (°C) to express the temperature. The conversion of Fahrenheit to Celsius can be done using the equation:

                  TABLE 3

Rectal Temperature of the corpse (°C)
Time (Hours)	0	1	2	3	4	5	6	7	8	9	10
Temperature (°C)	36.5	36.0	35.4	34.9	34.5	34.1	33.7	33.3	32.8	32.4	32.2

Average room temperature: 26.8°C

Methodology

Graph 1

This graph represents the cooling of a corpse according to the data provided in TABLE 3. I used logger pro to produce the graph with x-axis representing the time t in hours and y-axis representing temperature in °C. Curve fitting was added into the graph in order to get a proper visualization of the data since the available wasn’t enough for further analysis.

It was surprising to see that the plotted graph was very similar to the curve fit with RMSE (Root-Mean-Square-Error) value of just 0.06327°C. RMSE basically means the deviation of values from the true value (in this case the best fit). The applied curve fit was of the general equation:

Here, B represents the horizontal asymptote of the function, which I assume it to be a value close to the ambient temperature. Plugging in the obtained values of A, C and B from the graph, the curve can be expressed in the form of an exponential function:

Using the above equation, the time at which the corpse attains the ambient temperature can be determined by solving for t. The ambient temperature is considered to be the average room temperature where the data collection was taken place and according to TABLE 2, the average room temperature is 26.8°C. Plugging this into the equation, we get:

This shows us that, approximately after 39 hours the corpse will attain the ambient temperature and further there won’t be any major reduction in the body temperature. Since we know that the function of the curve is: , we can plot a graph of this equation and visualize it since Graph 1 is only a small part of the entire exponential curve.

Graph 2 now gives more visual clarity about the exponential nature of the cooling of a corpse. It can be seen that it follows an exponential decrease and thus the power of exponential being negative. This graph validates the fact that the body will stop cooling when it attains the ambient temperature. In the graph, it can be observed that between 20°C and 30°C on y-axis, the curve becomes a straight line indicating that the body has reached the ambient temperature of around 28.6°C and it cannot decrease way beyond the ambient temperature. The (e.q.1) shows

24.94°C as the horizontal asymptote carrying an uncertainty of ±1.95. This supports my assumption that the asymptote will be a value closer to the ambient temperature.

 Let’s look upon to a different scenario where in there is no ‘Logger pro’ software determining the function of the graph plotted by providing with appropriate curve fits. It is known that the average body temperature of a normal living human being is around 37°C and from the obtained secondary data (shown in TABLE 3), it can be inferred that the person has just recently expired since the initial rectal temperature reading (36.5°C) is almost near to the average human body temperature (37°C). Taking this into consideration, I will be modeling a hypothetical situation and finding time elapsed since the death of a man.

Situation: An unknown man’s dead body was happened to be found in an abandoned place at 12 pm. The police officers were informed about this by the people who saw the lying dead body. Within in no time an ambulance arrived and the body was taken to a nearby hospital. The man’s body was taken for further medical examination to learn how the person has actually died and when did he die. The rectal temperature of the body was recorded as 34.1°C at 12.30 pm. The rectal temperature was recorded at every hour for 5 hours and the data is shown below. The average room temperature was 26.8°C. (Note that the values shown below are obtained from TABLE 3).          

                  TABLE 4

Time (hrs)	0	1	2	3	4	5
Temp (°C)	34.1	33.7	33.3	32.8	32.4	32.2

 

From the data set provided, let’s try to determine the time when the person died. We don’t have sufficient data set to plot a graph and check whether the cooling of the corpse has an exponential nature. Therefore, we assume the decrease in temperature to have an exponential nature and at one point of time the body stops cooling i.e.it has attained the room temperature.

 

 Assuming the cooling will follow an exponential nature, further assumptions can be made that:

·   The power of the exponential will carry a negative value since it’s an exponential decrease

·   According to the law of nature, the temperature of the body cannot go down beyond the ambient temperature 26.8°C to a greater extent. Therefore, the horizontal asymptote of this function can be defined as y=26.8.

Based on my 1^st assumption, I can mathematically write it as:

Where, T represents temperature and t representing time.

Based on my 2^nd assumption, the graph will be translated 26.8 units up on y-axis because we know that the corpse can’t cool down beyond the ambient temperature and so the graph should converge at y = 26.8. Adding the translation constant T_a which represents the ambient temperature, we get:

Still, we see that the equation is not complete because at t = 0, the temperature T should be equal to 34.1°C. But, when t = 0 is plugged in into the (e.q.2), the temperature is, T = 27.8°C. Also, graphically we can tell it as incorrect. Plotting (e.q.2) on a graphic calculator gives us a curve:

Therefore I assumed multiplying the function e^-t from (e.q.2) by 34.1 would give the y-intercept 34.1 at x = 0. But I understood that the vertical translation constant T_b (ambient temperature, horizontal asymptote) would change my y value. This made me realize that, in order to get y-intercept as 34.1 at x = 0, the graph should undergo both vertical and horizontal stretch or compression

In order for this, (e.q.2) should be re-written as:

The general equation can be written as:

Here, the k represents the vertical stretch or compression along with the y values and ‘a’ represents the horizontal stretch or compression along the x-axis.

So, using (e.q.4) let’s solve for k and a, then determine the translation constant along the y-axis and the horizontal translation constant along x-axis.

Now, values should be plugged into (e.q.5). We can algebraically solve the equation by producing 2 different equations and solving them. The values should be taken from the data set provided on TABLE 4.

 

This shows that the produced function matches with data set of TABLE 4. So, let’s estimate the time when the person had actually died. We know that the average human body temperature is around 37°C. Let’s substitute T= 36.5 (such that we can reflect back onto e.q.1 and check to what extent the calculating values matches with these values) and find at what value for x does the value of y becomes 36.5°C in (e.q.8)

This shows us that the man was dead approximately 5 hours before the body was found. That is, the person died around 7 in the morning. Comparing this answer with TABLE 3 also confirms the answer to be correct. Using (e.q.8) and (e.q.1) let’s find the temperature value (function value) of the body for 10 hours by substituting values for t from 0-10 and compare them with the original data set provided in TABLE 3.

Graph 6

TABLE 5

Time (hrs)	0	1	2	3	4	5	6	7	8	9	10
Difference between the temperature values obtained from e.q.1 and Expt.data	0	0	-0.1	-0.1	-0.1	0	+1	+1	-1	-1	0
Difference between the temperature values obtained from e.q.8 and Expt.data	-0.2	-0.1	-0.1	-0.1	0	0	0	0	-0.1	-0.1	0

Conclusion

Graphically analysing, both (e.q.1) and (e.q.8) gave out identical graphs without any major difference (Graph 7) except for the y-intercept being different. It was good to see that the logically produced equation (e.q.8) suited with (e.q.1) produced by the software by applying a best fit curve. However, a reference graph of the experimental data could have been used for comparison and check to what extend did the graphs produced from (e.q.1) and (e.q.8) deviate from the original graph. But sufficient data wasn’t available to plot such a graph because cooling of a corpse is a very slow process and so, it takes lots of time to record sufficient data in order to produce such a graph for comparison. Yet another limiting factor is that, these dead bodies must not be kept out for a long period of time as the human organs starts to decompose and produces virtually intolerable smell and so they are refrigerated. However, with the available data of 10 hours, (e.q.1), (e.q.8) and the experimental values are plotted on the same axes for comparison. It is observed that there isn’t any major deviation of values from the experimental value. Determining the mean error for each equation shown on TABLE 5, it can be deduced that (e.q.1) lies more closer to the experimental value than (e.q.8) signifying that values of (e.q.1) being more accurate.

The justification for why (e.q.1) gave out comparatively a bit more accurate data is because, (e.q.1) was produced using a best fit curve approach and so the best fit line would have taken account of the slight variations in the temperature. According to the original data, the average room temperature was recorded as 26.8°C and so the vertical translation constant, T_b in the equation should have been 26.8. But then we find in (e.q.1) that the value for T_b is given as 24.94°C. This shows that the rate in change in temperature shown in TABLE 3 was not due to the average ambient temperature being 26.8°C alone but due to 24.94°C with an uncertainty of ±1.95°C. From this, it can be understood that ambient temperature came around in the range 26.8°C - 23°C. In case of the hypothetical situation, it was determined that the man was happened to be dead 5 hours before his body was found. But in reality, this value needn’t be true as the ambient temperature needn’t be the same every time since the time the person has expired and temperature is most likely to be lesser than 26.8°C in the morning, so at that point of time, the rate at which the body cools will also vary from what we have generalized. This shows the discrepancy involved in this particular mathematical modeling. In order to overcome this, multiple parameters must be included into the modeling which makes the equation more and more complex.

However, I was able to solve the hypothetical scenario using this model even though there existed slight anomalies in the data produced. So rather than ’determining’ post-mortem interval, it would be accurate to say, ‘estimating’ the post-mortem interval. After all, this has made me more curious to explore real life problems and how the mathematic concepts learnt in class can be applied solve real life situations.

Bibliography

[1]      H. T. Gelderman, L. Boer, T. Naujocks, A. C. M. Ijzermans, and W. L. J. M. Duijst, “The development of a post-mortem interval estimation for human remains found on land in the Netherlands,” Int. J. Legal Med., vol. 132, no. 3, pp. 863–873, 2018.

[2]      R. Sharma, R. Kumar Garg, and J. R. Gaur, “Various methods for the estimation of the post mortem interval from Calliphoridae: A review,” Egypt. J. Forensic Sci., vol. 5, no. 1, pp. 1–12, 2015.

[3]      C. Henßge and B. Madea, “Estimation of the time since death in the early post-mortem period,” Forensic Sci. Int., vol. 144, no. 2–3, pp. 167–175, 2004.

 [4]     G. Webster and N. Kathirgamatamby, “Post-Mortem Temperature and the Time of Death,” vol. 46, no. 4, 1956.