Use of Postmortem Temperature DecayResponse Surface Plots of Heat Transport inthe Human Eye to Predict Time of Death
ABSTRACT: A finite element heat transfer model of the human eye was previously constructed and applied to experimental postmortemtemperature decay curves collected in eyeballs of ten human bodies. The model was applied in the early postmortem period of 0–24 h underconditions of natural convection–radiation. Based upon this previous model, response surfaces for postmortem temperature decay were con-structed based upon variable ranges of the natural convective–radiation heat transfer coefficient from 7–13 W/m2 K, ambient temperatures of10–33°C, and times of 0–24 h. Mathematical equations to describe these response surfaces have been developed. This response surface methodis demonstrated for use by coroners/medical personnel to estimate time of death from recorded field temperature data collected over a 30-minperiod. Sensitivity of the model to small changes in the key variable of ambient temperature is explored. The response surface model is appliedto two cases of previously collected experimental eyeball temperature data. This response surface model method is only valid for constant sur-rounding temperatures, conditions of natural convection, no radiation effects, and postmortem times of 0–24 h.
KEYWORDS: forensic science, time of death, finite element model, human heat transport, response surface methodology, postmortemhuman temperature decay curve
In a previous paper (1), a finite element heat transfer modelwas proposed to estimate time of death in humans. This pro-posed model has advantages over other models in that it isapplied to the human eyeball, where morphologies are essentiallyconsistent across all adult human populations and temperatureplateau effects are minimal to nonexistent. Also, the model hasthe ability to adjust for any temperature plateau effects and ante-mortem hyperthermia/hypothermia conditions. The model wasapplied in the early postmortem period of 0–24 h. Twenty-fourhour was arbitrarily selected as the time limit of the modelbecause the temperature decay curve of the eyeball begins toflatten as it asymptotically approaches ambient room temperatureconditions. COMSOL Multiphysics� 4.0 (New England Execu-tive Park, Burlington, MA) finite element software was used tostudy heat transfer from the human eyeball to the surroundingair under conditions of natural convection/radiation. Experimen-tal postmortem temperature decay curves were collected in eye-balls of ten human bodies. The overall goal of this project wasto compare the temperature decay curve generated from the exe-cution of the COMSOL software model with that of experimen-tal postmortem decay curves. Model generated postmortemtemperature curves reflected experimental curves for ten caseswith coefficients of determination, r2, ranging from 0.9448–0.9953. These fits were especially satisfactory, given the fluctu-ating ambient temperature conditions experienced during the0–24-h period. However, as noted later, only two of the available
10 field cases were judged to be useful for the application of themethod outlined in this study.Response surface methodology (2,3) deals with the exploration
of response surfaces generated from mathematical functions.Describing response surfaces is a problem faced by experimentersin many technical fields, where in general, the response variableof interest is y and there is a set of predictor variables, x1, x2,…,xk. For example, y might be the viscosity of a polymer and x1, x2,and x3 might be the reaction time, the reactor temperature, andthe catalyst feed rate in the process. In some systems, based uponunderlying engineering, chemical, or physical principles, the nat-ure of the relationship between y and the x’s is exactly known. Inthis case, a model of the form y = g(x1, x2, x3) could be written.Based upon various values of the variables x1, x2, and x3, aresponse surface y could be generated and visually displayed. Inthe more common situation where the underlying mechanism iscomplex or is not fully understood, the experimenter mustapproximate the unknown function g with an appropriate empiri-cal model, y = g(x1, x2,…, xk) + e, where e represents the “error”in the system. This empirical model is called a response surfacemodel. In our current case of postmortem heat transfer from thehuman eye, the response surface y is the temperature in the centerof the eyeball. The predictor variables, x1, x2, and x3 are theambient temperature conditions surrounding the body, Tamb, thenatural convection/radiation (NCR) heat transfer coefficient, h,and at some time t after death.For example, a 2D plot of how the postmortem temperature
decay curve (PTDC) varies with the NCR heat transfer coeffi-cient, h, at a constant ambient temperature, Tamb = 20°C, isshown in Fig. 1. These curves were generated from execution ofthe software finite element model described in a previous paper
1University of Kentucky, 4810 Alben Barkley Dr., Paducah, KY 42001.Received 25 Sept. 2012; and in revised form 8 Jan. 2013; accepted
J Forensic Sci, March 2014, Vol. 59, No. 2doi: 10.1111/1556-4029.12333
Available online at: onlinelibrary.wiley.com
(1). All of these curves assume constant ambient temperatureconditions and a starting temperature in the body of 37°C. FromFig. 1, note at any specific time t, the eyeball temperature islower with increasing values of h. For example, at 10 h afterdeath, temperature in the eyeball is 24°C at a value of h = 7 W/m2 K and 21.9°C at a value of h = 14 W/m2 K. The engineer-ing units used to describe heat transfer are watts of energy persquare meter of heat transfer surface per temperature unit Kelvin,or W/m2 K. As explained earlier (1), the typical value for theNCR heat transfer coefficient for the human eyeball is 10 W/m2 K. This NCR heat transfer coefficient is comprised of both anatural convection and radiation heat transfer component andwill be referred to as a simple heat transfer coefficient, h. How-ever, with hirsute conditions (much head or facial hair) or of afleshy head, the heat transfer coefficient could lie within the7–10 W/m2 K range. On the other hand, with conditions of abald pate, an especially thin face, or slight conditions of forcedconvection, the heat transfer coefficient could be elevated and liewithin the 10–13 W/m2 K range. The heat transfer coefficient isa function of ambient temperature, radiation conditions, andsurface temperature of the eyeball. Surface temperature of theeyeball is also a complex function of conductive heat transfermechanisms within the eye and surrounding tissues.Using the finite element software model, 2D postmortem tem-
perature decay curves were generated for constant ambient tem-perature conditions of 10–30°C and heat transfer coefficients of7–14 W/m2 K over periods of 0–24 h. Initially, it was believedthat coroners/medical personnel might collect a portion (1–2 h)of the PTDC from recently deceased individuals in the field. As
demonstrated in an earlier paper (1), using the slope of the curveof the partial PTDC, it was believed the full PTDC could bereconstructed and time of death could be estimated. However,later, it was argued that the difference in slopes of variousPTDCs based upon temperature conditions, time, and values ofheat transfer coefficients were very slight and could not be easilyevaluated by using simple 2D graphs. Hence, a response surfacemethod was developed to address these shortcomings.Instead of relying upon individual measurements of curve
slope, it was decided to seek a comprehensive relationshipbetween all variables of NCR h, time, and ambient temperaturethat affect postmortem eyeball temperature. The 3D representa-tion of this relationship is called a response surface. Given aconstant ambient temperature, a single response surface can begenerated. Such a surface is shown in Fig. 2 for a constantambient temperature of 20°C. This response surface graphicallyillustrates a collection of all postmortem temperature decaycurves under a variety of external conditions. Note, the weakfunction of the heat transfer coefficient and its effect upon post-mortem eyeball temperature—eyeball temperature changes littleover varying values of heat transfer coefficient. On the otherhand, as to be expected, eyeball temperature is a strong functionof postmortem time.Choosing a single 2D curve (h = 10 W/m2 K) from Fig. 1, a
response surface for variable values of ambient room tempera-tures can be generated—this surface is visually displayed asFig. 3. For any selected value of the ambient room temperatureand time t, the postmortem temperature within the eyeball canbe found. From Fig. 3, note that for a constant ambient tempera-ture of 10°C, after 5 h the temperature within the eyeball is23.2°C and 10.5°C after 24 h. Alternatively, for a constant ambi-ent temperature of 30°C, after 5 h the temperature within the
FIG. 1––How variation in the the natural convection/radiation (NCR) heattransfer coefficient, h, affects the postmortem temperature decay curve. Thetop curve is for h = 7.0 W/m2 K. The next curve is for h = 7.5, and soforth. The lowermost curve is for h = 14.0.
FIG. 2––Postmortem eyeball temperature decay response surface as afunction of the natural convection/radiation (NCR) heat transfer coefficientand postmortem time. Ambient temperature conditions are assumed constantat Tamb = 20.0°C.
SMART . HUMAN EYE TO PREDICT TIME OF DEATH 391
eyeball is 32.2°C and 30.1°C after 24 h. In a similar manner,other response surface curves can be generated for any selectedvalue of heat transfer coefficient, h.TableCurve 3D� 4.0 (Systat Software, Inc., San Jose, CA) soft-
ware was used to find the surface equation for each constant ambi-ent temperature condition. For example, at a constant ambienttemperature of 20.0°C, the best fit was a Chebyshev X, Y rationalorder 5/6 equation with 23 fitting parameters, yielding an r2 coeffi-cient determination value of 0.999977. Using this equation, timeof death (TOD) can be determined for any (x,y), where x = heattransfer coefficient, and y = eyeball temperature. These surfaceequations are lengthy and complex and are not provided here.Interested parties can contact the author for additional details.Response surfaces of postmortem temperature decay curves
are very powerful tools in estimating time of death (TOD) inhumans. When a coroner/medical personnel arrive at the sceneof a recently deceased individual, they can record the continuallydecreasing postmortem temperature of the body as a function oftime. This is a small portion of the overall postmortem tempera-ture decay curve (PTDC) that begins at death and asymptoticallyapproaches room temperature near 24 postmortem hours. Thistiny portion of the field-collected PTDC is represented as a darksurface in Fig. 3 at 17 h (1020 min) after TOD at an ambientroom temperature of 19°C. Imagine using this scant piece offield data and trying to match this small portion of the PTDC tothe overall response surface for postmortem temperature decayto estimate time of death. As explained below, this study pro-vides such a method.
Methods
As described in a previous paper (1), to estimate time ofdeath, the slope of the field postmortem temperature decay curve(PTDC) must be matched to the slope of the model PTDC. The
field PTDC is the curve collected by a coroner or medicalpersonnel by measuring temperature within the center of theeyeball of the deceased undisturbed individual over a 30-min per-iod at existing surrounding conditions (same physical location,same ambient temperature, etc.). The model PTDC is the familyof postmortem temperature decay curves within the eyeball gen-erated from execution of the finite element software model undervariable conditions of ambient room temperature, time, and NCRheat transfer coefficient. Theoretically, once the field PTDC issuccessfully matched to the slope of the model PTDC, the timeof death can be found by reverse engineering.This current work proposes to expand ideas from previous work
by demonstrating how the field PTDC can be matched to theappropriate postmortem temperature decay response surface. A typ-ical field PTDC is shown as the small dark curve in Fig. 3 recordedover a 30-min period (from 1002 to 1031 min) at constant ambienttemperature conditions of Tamb = 19°C and an NCR heat transfercoefficient of h = 10 W/m2 K. Of course, at the time of collectionof the field PTDC, it is not known that the 30-min data collectionperiod is 1002–1031 min since death. Only after matching theslope of the field PTDS to the slope of the model PTDS is itrevealed that the deceased individual expired some 16.7 h from theinitial start time of field collection temperature data.
Field Application
The COMSOL finite element software can be used to deter-mine TOD under most any set of conditions, including variableambient temperature conditions. To date, only results under con-ditions of natural convection/radiation heat transfer have beenpublished (1); however, other conditions of forced convection(wind) leading to elevated heat transfer coefficients, that is,>11 W/m2 K are being developed.It is desirable to create a simple tool for coroners/medical per-
sonnel to use in the field to determine time of death (TOD).With this idea in mind, the various equations to describe theresponse surfaces relating eyeball temperature to time and NCRh have been developed. These equations can be incorporated intoTableCurve software and used by field personnel to find TODunder a variety of circumstances. These circumstances are illus-trated in four cases shown below. These cases are presented inTables 1–4 and were selected to cover all possible situations thatmay be encountered by field personnel, that is, typical applica-tions, cases where a variable h is appropriate, cases where thereis not constant surrounding temperature conditions, and caseswhere temperature plateau effects may be prominent.A general overview of the data recorded in the various col-
umns of Tables 1–4 are provided:
• Column 1: Field time collected every 3 min (collected at aconstant ambient surrounding temperature).
• Column 2: Corresponding collected field eye temperatures, °C.• Column 3: Selected values of the NCR heat transfer coeffi-
cient, h, to be tested (7–9 for fleshy head or much facial hair,9–11 for typical applications, or 10–12 for bald pate or slightconvective conditions).
• Column 4: Measurement of the slope of the field data curveand the software model curve for corresponding values of theNCR heat transfer coefficient, h.
• Column 5: Percent differences between the slopes of the fielddata curve and the software model curve. The lowest percentdifference will indicate which value of h is preferred.
FIG. 3––Postmortem eyeball temperature decay response surface as afunction of ambient temperature conditions and postmortem time. The natu-ral convection/radiation (NCR) heat transfer coefficient was assumed con-stant at h = 10 W/m2 K.
392 JOURNAL OF FORENSIC SCIENCES
TABLE1––Typicalap
plication.
12
34
56
78
910
11
FieldData
@T a
mb=20
.0(m
in)
Eye
Tem
perature
(oC)
h(W
/m2K)
FieldData/
Model
Equ
ation
Slope/Slope
%Difference
FieldData/Model
Equ
ation
2ndDerivative/2ndDerivative
%Difference
TPE
(min)
FieldData/
Model
Equation
2ndDerivative/
2ndDerivative
%difference
TOD
(min
priorto
initial
temperature
data)
026
.11
9�1
.727e�
2 /�1
.598e�
2�7
.65.180e
�5 /4.82
4e�5
+6.9
326
.05
10�1
.727e�
2 /�1
.714e�
2�0
.85.180e
�5 /5.69
2e�5
+9.9
0n/a
n/a
249
626
.00
11�1
.727e�
2 /�1
.814e�
2+5.0
5.180e
�5 /6.52
1e�5
+25
.99
25.95
1225
.90
1525
.84
1825
.79
2125
.74
2425
.69
2725
.64
3025
.59
TOD,tim
eof
death;
TPE
,temperature
plateaueffects.
TABLE2––Variableh.
12
34
56
78
910
11
FieldData
@T a
mb=22
.0(m
in)
Eye
Tem
perature
(oC)
h(W
/m2K)
FieldData/Mod
elEqu
ation
Slope/Slope
%Difference
FieldData/Model
Equ
ation
2ndDerivative/2ndDerivative
%Difference
TPE
(min)
FieldData/Model
Equ
ation
2ndDerivative/2ndDerivative
%Difference
TOD
(min
prior
toinitial
temperature
data)
024
.97
9�8
.455e�
3 /�7
.588
e�3
�10.3
2.84
9e�5 /5.207e
�6
�81.7
324
.95
10�8
.455e�
3 /�8
.137
e�3
�3.76
2.84
9e�5 /2.367e
�5
�16.9
624
.92
11�8
.455e�
3 /�8
.609
e�3
+1.8
2.84
9e�5 /2.609e
�5
�8.4
0n/a
n/a
422
924
.89
1224
.87
1524
.84
1824
.82
2124
.79
2424
.77
2724
.74
3024
.72
TOD,tim
eof
death;
TPE
,temperature
plateaueffects.
SMART . HUMAN EYE TO PREDICT TIME OF DEATH 393
TABLE3––Variableam
bienttemperature.
12
34
56
78
910
11
FieldData
@T a
mb=20
.0(m
in)
Eye
Tem
perature
(oC)
h(W
/m2K)
FieldData/Mod
elEqu
ation
Slope/Slope
%Difference
FieldData/Model
Equation
2ndDerivative/2ndDerivative
%difference
TPE
(min)
FieldData/Model
Equ
ation
2ndDerivative/2ndDerivative
%Difference
TOD
(min
priorto
initial
temperature
data)
021
.20
9�4
.867e�
3 /�2
.878
e�3
�40.9
521
.17
10�4
.867e�
3 /�3
.076
e�3
�36.8
n/a
n/a
n/a
n/a
n/a
Invalid
mod
el10
21.15
11�4
.867e�
3 /�3
.276
e�3
�32.7
1521
.12
2021
.10
2521
.08
3021
.05
3521
.03
4021
.00
TOD,tim
eof
death;
TPE
,temperature
plateaueffects.
TABLE4––Evidenceof
TPE.
12
34
56
78
910
11
FieldData
@T a
mb=19
.0(m
in)
Eye
Tem
perature
(oC)
h(W
/m2K)
FieldData/Model
Equation
Slope/Slope
%Difference
FieldData/Model
equatio
n2ndDerivative/2ndDerivative
%Difference
TPE
(min)
FieldData/Mod
elEqu
ation
2ndDerivative/2ndDerivative
%Difference
TOD
(min
prior
toinitial
temperature
data)
022
.34
9�9
.061e�
3 /�8
.523e�
3�5
.93
22.31
10�9
.061e�
3 /�9
.140e�
3+0.9
2.331e
�5 /2.663e
�5
+14
.260
2.663e
�5 /2.66
3e�5
0.0
479+60
=53
96
22.28
11�9
.061e�
3 /�9
.680e�
3+6.8
922
.26
1222
.23
1522
.20
1822
.17
2122
.15
2422
.12
2722
.09
3022
.07
TOD,tim
eof
death;
TPE
,temperature
plateaueffects.
394 JOURNAL OF FORENSIC SCIENCES
• Column 6: The second derivative of the field data curve andthe software model curve for corresponding values of the pre-ferred NCR heat transfer coefficient, h.
• Column 7: Percent differences between the second derivativeof the field data curve and the second derivative of the soft-ware model curve. These differences are used to test for thepresence of any temperature plateau effects (TPE).
• Column 8: Indicates the minutes of TPE. If the differencebetween the second derivative of the field and model curvesis less than or equal to 10% at the previously selected valueof h in column 7, then no TPE is indicated. Therefore, thecolumn value is 0 min (see column 8 of Table 1). If the dif-ference between the second derivative of the field and modelcurve is greater than 10%, then TPE is indicated. The fieldcurve must be indexed to the right by adding additional min-utes to allow for TPE, typically 5–60 min. See column 8 ofTable 4 where 60 min has been added to the field curve.
• Column 9: If the field curve has been adjusted for additionalTPE minutes, the second derivative of the field curve andmodel curve are computed again.
• Column 10: The percent difference between the second deriv-ative of the field curve and model curve should now be zero(see column 10 of Table 4). If it is not zero, then the valueof the TPE minutes added to the field curve in column 8must be adjusted to give a zero difference between the sec-ond derivative of the field curve and model curve. This is atrial and error procedure.
• Column 11: Now, the field curve is congruent to the modelcurve, and the model curve can be reverse engineered toyield TOD. This final value of TOD will include any TPE, ifpreviously indicated.
Situation 1. Typical Application—Field data are collected fromthe eyeball of a deceased individual according to the procedurepreviously described (1). The coroner must use a thermometerwith precision of 0.01°C, which are readily available in today’smarket. Thermometers with resolution of only 0.1°C will notcapture the small differences in the slopes of the postmortemtemperature decay curves. Eyeball temperatures are recordedevery 3 min over a 30-min time period, as shown in columns 1and 2 of Table 1. A constant ambient temperature is assumed inthe use of this model. If the body has been moved from anotherlocation, if forced convection is suspected (wind or strongHVAC air currents blowing across the face), if the body hasbeen subjected to strong radiation effects (outside exposure tothe sun), or if large swings in room temperature has occurreddue to a cycling HVAC system, this model does not apply.Typically, a value of 10 W/m2 K is the NCR h used to
describe natural convective/radiation heat transfer in the humaneyeball (1). However, this model will accommodate smallchanges in the NCR h due to slight forced convection effects,which elevate h, or the presence of large quantities of hair orflesh on the head, which tend to reduce h. This sensitivity willbe demonstrated in Situation 2. The sensitivity is explored overvalues of 9–11 NCR h, as shown in column 3 of Table 1.Once the field temperatures have been collected, the data are
incorporated into a graphing program, such as SigmaPlot�
(Systat Software, Inc.), where linear regression is performed onthe curve. This is called the field curve. Over a 30-min time per-iod, the portion of the postmortem temperature decay curve(PTDC) is very nearly linear. Using the equation of the fieldPTDC for a constant ambient temperature (20°C in this case),TableCurve software is used to estimate corresponding times for
each field temperature value. These field temperatures and esti-mated times are plotted in graphing software, and linear regres-sion is performed on the curve. This is called the model curve.Slopes of the field curve at NCR h values 9–11 are comparedwith corresponding slopes of the model curve. The slopes of thefield data and model curve are recorded in column 4 of Table 1.Differences between the two slopes are compared, as shown incolumn 5 of Table 1. The least difference in slopes will indicatewhich value of NCR h is appropriate for estimating TOD. In thisexample, a value of NCR h = 10 W/m2 k is the best choice withonly a – 0.8% difference between the slopes of the field curveversus the slope of the model curve.Next, the PTDC must be checked for the presence of any tem-
perature plateau effects (TPE). These effects sometime occur dueto antemortem conditions including fever, drugs, inflammation,etc. that create elevated/reduced temperatures in the human body.Also, in some cases, the temperature can remain flat over a post-mortem period of time after death, sometimes even up to asmuch as 2 h (1,4).Though the field data can be linearized with graphing soft-
ware, it does have a slight curvature to it and can also be fittedwith a second-order polynomial for the purposes of estimatingthe 2nd derivative. The 2nd derivative of the field curve andmodel curve are determined and recorded, as indicated in col-umn 6 of Table 1. Again, their differences are compared, asshown in column 7. Small differences <10% are acceptablewithin the inherent error of the field data. Small differences indi-cate no substantial TPE as shown in column 8. Once no TPE isindicated, the model curve can be judged to closely approximatethe field curve. In this case, as shown in column 11, the modelcurve indicates 249 min as TOD. Situation 4 (shown below) willillustrate how to adjust the TOD to accommodate temperatureplateau effects.
Situation 2. Variable NCR h—In some cases, the value of theNCR heat transfer coefficient may vary from the typical value of10 W/m2 K. Following the same procedure as in Situation 1,with the exception that Tamb = 22.0°C, slopes of the field curveare compared with those of the model curve. As shown in col-umn 5 of Table 2, smaller differences in slopes are reported foran NCR h = 11 W/m2 K. This may indicate the presence ofsome slight forced convection effects, or perhaps the individualhad a bald head. The model needs to be exercised again for avalue of NCR h = 12 W/m2 K, to compare differences in slopesof field and model curves (not performed here). However, in thissituation, slope differences were judged to be acceptable and thePTDC for h = 11 W/m2 K was selected. No TPE was indicatedbecause differences in the 2nd derivatives was <10% as indi-cated in column 7 of Table 2, and a TOD = 422 min was esti-mated (see column 11).
Situation 3. Variable Ambient Temperature—Unfortunately,this is a common situation. The PTDC is extremely sensitive tosurrounding ambient temperature conditions, as shown in thesensitivity analysis below. In a previous paper (1), there weredifferent degrees of variability of ambient temperature conditionsin deceased individuals (see cases 1–10). Some of this variabilitywas due to the fact that individuals died in their homes or carsat a given surrounding temperature and were transported to themorgue some 2–3 h later where the bodies were exposed tocycling HVAC temperature swings. If the coroner/medicalpersonnel encounters a deceased individual in a room withessentially constant surrounding temperature conditions, good
SMART . HUMAN EYE TO PREDICT TIME OF DEATH 395
estimates of TOD will be expected with the method described inthis present study.In Situation 3, field temperatures were collected every 5 min
over a 40-min time period with actual ambient temperaturesvarying from 21.5–19.0°C. These field temperatures are actualexperimental temperatures from a deceased individual case 1 (1).To demonstrate use of the present method, a constant ambienttemperature of 20.0°C was assumed (see columns 1 and 2 ofTable 3). Following the same procedure used in Situation 1,there are very large differences between the slopes of the fieldcurves and the model curves for all values of NCR h (see col-umn 5 of Table 3). These values are unacceptable, and the pres-ent model is invalid and cannot be used to get reliable estimatesfor TOD. Aside: the COMSOL finite element model can itselfbe used to accommodate variable ambient temperature condi-tions, and the TOD was estimated in this case to be 1000 min(16.7 h). However, the simplified procedure as presented in thisstudy cannot be used in conditions of variable surrounding ambi-ent temperature.
Situation 4. Evidence of TPE—Field temperatures were col-lected every 3 min over a 30-min time period under constanttemperature conditions of 19°C, as recorded in columns 1 and 2of Table 4. Following the same procedure as in Situation 1,there appears to be a large difference between the 2nd deriva-tives of the field and model curves, as shown in column 7 ofTable 4. This may indicate the presence of temperature plateaueffects.An arbitrary number of minutes are added to the field curve.
If present, TPE’s are typically 5–60 min. Plot the new time ver-sus the field temperature data. Take the 2nd derivative of thenew model curve and compare it to the 2nd derivative of the ori-ginal field data curve. When both 2nd derivatives are the same,the appropriate TPE has been identified. Repeat the procedure ofadding minutes until these 2nd derivatives are the same, asshown in column 9 of Table 4. The final TOD will be the num-ber of TPE minutes added to the original model generated time.In this case, 479 min + 60 min TPE = 539 min.
Sensitivity Analysis
From the previous sensitivity studies (5), it was revealed thattime of death estimates is most sensitive to parameter estimatesof initial body temperature and ambient surrounding temperatureconditions. Variations in other parameters, such as geometry,mass, heat capacity, and thermal conductivity, are less importantin estimating time of death from core body temperature data, butare even less important in estimating time of death from eyeballtemperature data. As previously demonstrated (1), the eyeballmodel can be adjusted to accommodate changes in assumed ini-tial eyeball temperature and core body conditions at time ofdeath. Therefore, the remaining single parameter that will influ-ence estimated TOD is the temperature of surrounding ambientconditions. This sensitivity is demonstrated in the followingFigures.Figure 4 illustrates the sensitivity of the model to errors in the
assumption of constant ambient temperature. If the ambient tem-perature is underestimated by as much as 10% from an assumedambient temperature of 28°C (h = 10 W/m2 K), then after 20 hfrom TOD, there will be 2.75 degrees error in the eyeball modeltemperature. Similarly, for an assumed ambient temperature of20°C (h = 10 W/m2 K), then after 20 h from TOD, there willbe 1.9 degrees error in the eyeball model temperature. Finally,
for an assumed ambient temperature of 12°C (h = 10 W/m2 K),then after 20 h from TOD, there will be 1.3 degrees error in theeyeball model temperature. The small gray dotted lines indicatethe sensitivity of the dark curves to different assumed values ofh (8 and 12 W/m2 K). As previously shown, the model is notvery sensitive to small changes in the value of the NCR heattransfer coefficient.Figure 5 illustrates how well the model conforms to the exper-
imental data if the ambient temperature is underestimated oroverestimated from the actual surrounding temperature. Forexample, given an actual ambient temperature of 28°C, if theambient temperature is underestimated as much as 10%, the cor-relation coefficient between the model and the experimental datais reduced from a perfect 1.0 to a poor correlation of 0.54. If10% overestimated, correlation is reduced even further to almost0.2. Other temperature curves of Tamb = 20°C and 12°C areshown. In Fig. 5, also again, the small gray dotted lines indicatethe sensitivity of the dark curves to different assumed values ofh (8 and 12 W/m2 K).
Application to Previous Experimental Data
In previous work (1), 10 experimental cases were offered todemonstrate use of a finite element software model to estimatetime of death. It was desired to apply equations of the responsesurface models to this experimental data to demonstrate accu-racy. However, the response surface model as presented in thisstudy is based upon constant ambient temperature conditions.The experimental cases do not strictly fulfill this criteria due to:
FIG. 4––Sensitivity analysis. Effect of % change of assumed ambient tem-perature from actual ambient temperature upon postmortem eyeball tempera-ture. Dark curves are based upon assumed value of h = 10 W/m2 K. Graydotted lines indicate assumed values of h = 8 and 12 W/m2 K.
396 JOURNAL OF FORENSIC SCIENCES
(i) large differences in ambient temperature conditions betweenwhere the individual expired and conditions within the morgue,(ii) large fluctuations in the morgue temperature due to cyclingHVAC controls, and (iii) the presence of significant temperatureplateau effects (Cases 1, 4, and 5). For example, the Case 10individual expired in surrounding conditions of 15°C for threehours prior to being brought to the morgue, where typical sur-rounding temperatures in the morgue were near 22°C.Cases 3 and 9 were useful cases to apply the response surface
model. A constant ambient temperature condition of 21.5°C andan NCR h = 10 W/m2 K was assumed in both cases. This is inspite of the facts that the Case 3 individual had been deceasedfor 205 min under ambient temperature conditions of 21°C andCase 9 had been deceased for 130 min under ambient tempera-ture conditions of 20°C, prior to being brought into the morgue.As shown in Fig. 6, there is good agreement between theresponse surface model and the two experimental curves. TheCase 3 model curve was only applied from 0 to 600 min post-mortem due to significant changes in the ambient temperatureconditions within the morgue after 600 min. Case 9 was appliedover the full duration of 0–1250 min (20.8 h).
Conclusions
In conclusion, the response surface model method as pre-sented in this study is believed to be a practical accurate methodfor use by coroners/medical personnel for determining time ofdeath in humans between 0–24 postmortem hours under constantambient surrounding temperature conditions and natural convec-tion/radiation heat transfer. To assist field personnel, the lengthy,
complex equations describing the response surfaces for all casesof NCR h values of 7–13 W/m2 K, ambient temperatures across10–33°C, and times of 0–24 h are available from the author.The method was successfully applied to two cases of experi-
mental field data (Cases 3 and 9 (ref. 1)), but additional data areneeded to truly validate the method.Application of the response surface model method as pre-
sented in this study is limited to constant surrounding tempera-ture conditions, conditions of natural convection, no evidence ofradiation effects, and postmortem time periods of 0–24 h.Experimental work is continuing to further validate the model
and to expand use of the response surface model for use in con-ditions of forced convection, where wind or HVAC currentsmay be blowing across the face of the deceased individual.
Nomenclature
h natural convective/radiation heat transfer coefficient,W/m2 K
HVAC heating, ventilation, and air conditioning
NCR natural convection/radiation
PTDC postmortem temperature decay curve
r2 statistical correlation coefficient of determination
T temperature, °C
Tamb ambient temperature, °C
TOD time of death, min
t time, s
FIG. 5––Sensitivity analysis. Effect of % change of assumed ambient tem-perature from actual ambient temperature upon correlation coefficientbetween experimental results and model results. Dark curves are based uponassumed value of h = 10 W/m2 K. Gray dotted lines indicate assumed valuesof h = 8 and 12 W/m2 K.
FIG. 6––Response surface model applied to Cases 3 and 9 of experimentaldata, ref. (1). Cases were selected because they offered approximate constantsurrounding temperature conditions.
SMART . HUMAN EYE TO PREDICT TIME OF DEATH 397
References
1. Smart JL, Kaliszan MR. Use of a finite element model of heat transport inthe human eye to predict time of seath. J Forensic Sci 2013;58(S1):s69–77.
2. Box GEP, Draper NR. Empirical model-building and response surfaces.New York, NY: John Wiley & Sons, 1987.
3. Myers RH, Montgomery DC, Anderson-Cook CM. Response surfacemethodology. New York, NY: John Wiley & Sons, 2009.
4. Smart JL, Kaliszan MR. The postmortem temperature plateau effect andits role in the estimation of time of death in humans. Leg Med2012;14:55–62.
5. Smart JL. Estimation of time of death with a Fourier series unsteady stateheat transfer model. J Forensic Sci 2010;55:1481–7.
Additional information and reprint requests:Jimmy L. Smart, Ph.D., P.E.University of Kentucky4810 Alben Barkley DrivePaducah, KY 42002E-mail: [email protected]
