A new simulation-based model for calculating post-mortem intervals using developmental data for Lucilia sericata (Dipt.: Calliphoridae)
NNoname manuscript No. (will be inserted by the editor)
A new simulation-based model for calculating post-mortemintervals using developmental data for
Lucilia sericata (Dipt.: Calliphoridae)
Saskia Reibe · Philip v. Doetinchem · Burkhard Madea
Received: date / Accepted: date
Abstract
Homicide investigations often depend on the determination of a minimumpost-mortem interval (PMI min ) by forensic entomologists. The age of the most devel-oped insect larvae (mostly blow fly larvae) gives reasonably reliable information aboutthe minimum time a person has been dead. Methods such as isomegalen diagrams orADH calculations can have problems in their reliability, so we established in this studya new growth model to calculate the larval age of
Lucilia sericata (Meigen 1826). This isbased on the actual non-linear development of the blow fly and is designed to includeuncertainties, e.g. for temperature values from the crime scene. We used publisheddata for the development of
L. sericata to estimate non-linear functions describing thetemperature dependent behavior of each developmental state. For the new model it ismost important to determine the progress within one developmental state as correctlyas possible since this affects the accuracy of the PMI estimation by up to 75%. Wefound that PMI calculations based on one mean temperature value differ by up to 65%from PMIs based on an 12-hourly time temperature profile. Differences of 2 ◦ C in theestimation of the crime scene temperature result in a deviation in PMI calculation of15 - 30%.
Keywords
Forensic entomology · Growth modeling · Lucilia sericata · Developmentrates · Non-linear model
Insect development rates are applied not only in pest control management but alsoin forensic entomology (Greenberg, 1991). Several species of dipteran and coleopteranfamilies infest decaying material in order to breed offspring, and this includes the colo-nization of animal carcasses as well as dead bodies (Archer and Elger, 2003; Grassberger
S. Reibe and B. MadeaInstitute for Forensic Medicine, University of Bonn, Stiftsplatz 12, 53111 BonnTel.: +49-179-5181634Fax: +49-228-738339 E-mail: [email protected]. v. DoetinchemI. Physics Institute B, RWTH Aachen University, Sommerfeldstr. 14, 52074 Aachen a r X i v : . [ q - b i o . O T ] M a r and Frank, 2004; Lane, 1975). In homicide investigations, determination of the age oflarvae feeding on a corpse can indicate a minimal post-mortem interval (PMI min )(Goff, 1993). This is often important in forensic case work (Benecke, 1998).1.1 Life cycle of blow fliesThe general life cycle of blow flies includes four stages: egg stage, larval stage, pupalstage and imago stage (Tao, 1927). During the larval stage, three instars can be sep-arated: 1st, 2nd and 3rd instar, where the latter is divided due to behavioral changesin feeding and post-feeding larvae.Blow flies deposit egg clutches directly on the food substrate, such as a dead body(Smith and Wall, 1997), in a position where the eggs are protected and in a moistenvironment. This ensures a food supply for the hatching 1st instar larvae. The firstthree instars each undergo a moult to reach the next developmental stage; the stagescan be distinguished by the number of respiratory slits at the posterior end of thelarvae. The third instar stage lasts for longer than the first two and is divided in afeeding and a post-feeding phase. The latter is a preparation for pupation. Therefore,the larvae leave the food source to find a suitable place for pupation, emptying theirgut (Arnott and Turner, 2008). About one third of the pre-adult development time isspent in the post-feeding larval stage (Greenberg, 1991). Then pupation sets in andthe imago develops within the pupal case till eclosion (Lowne, 1890). This last stagepersists for about half of the time of the total development.The larval growth rate depends on its body temperature, which is directly influ-enced by environmental conditions as ambient temperature and the heat generatedby maggot aggregations (Slone and Gruner, 2007). Also, an important detail for PMIdetermination is that each species has its own temperature dependent growth rate.1.2 Methods for PMI determinationIn forensic case work, two different methods are frequently used to calculate a PMI.The first uses isomegalen or isomorphen diagrams, by which the lengths or the devel-opmental stage of the larvae are combined as a function of time and mean ambienttemperature in a single diagram (Grassberger and Reiter, 2001). According to its orig-inators, this method is optimal only if the body and therefore the larvae were notundergoing fluctuating temperatures, e.g. in an enclosed environment where the tem-perature was nearly constant.The second method of calculating a PMI estimates the Accumulated Degree Daysor Hours (ADD or ADH). ADH values represent a certain number of ”energy hours”that are necessary for the development of insect larvae. The degree day or hour conceptassumes that the developmental rate is proportional to the temperature within a certainspecies-specific temperature range (overview in (Higley and Haskell, 2009)). However,the relationship of temperature and development rate (reciprocal of development time)is typically curvilinear at high and low temperatures and linear only in between.The formula for calculating ADH is given byADH = T · ( Θ − Θ ) (1) where T is the development time, Θ is the ambient temperature, and the minimumdevelopmental threshold temperature Θ is a species-specific value, the so called devel-opment zero, which is the x-intercept, i.e., an extrapolation of the linear approximationof the reciprocal of the developmental time. This value has no biological meaning, it isthe mathematical consequence of using a linear regression analysis (Higley and Haskell,2009).One basic condition for using the ADH method is that the ADH value for com-pleting a developmental stage stays constant within certain temperature thresholds.For example a developmental duration for finishing a certain stage of 14 days at 25 ◦ Cresults in 238 ADD when a base temperature of 8 ◦ C is assumed. A developmental du-ration of 19 days at 21 ◦ C results in 231 ADD, both ADD-values are in the same range.We analyzed a published data-set for the development of
Lucilia sericata (Meigen 1826)(Grassberger and Reiter, 2001) and calculated the corresponding ADH values for thesedata. Fig. 1 shows the calculated ADH values for a base temperature of Θ =8 ◦ C (ascalculated by a linear regression analysis for the used data-set). In the figure we see anew effect: for the younger and also shorter developmental phases the ADH values arenearly constant over the complete range of temperatures, but for the post-feeding andthe pupal stages the ADH values are strongly temperature dependent.In general, the ADH method seems to give good results only when the larvae ofinterest have been exposed to temperatures similar to those used in generating thereference value applied in the PMI calculation (Anderson, 2001). Moreover, the tem-perature range in which the development rate is actually linear is not wide enoughto cover all temperatures during a typical summer in Germany (see also examples forMay/June 2008 in Fig. 3). Furthermore, neither developmental durations nor base tem-peratures for development have been calculated for species originating from Germany.The method must therefore be used carefully.Furthermore, it is highly problematic that uncertainties for temperature measure-ments from a crime scene cannot be taken into account by either of the commonlyused methods for PMI determination. It is difficult to determine the actual temper-ature controlling the larvae at a real crime scene. Since temperature is the variablethat most influences development, it is crucial to consider it as accurately as possible.The standard procedure is to use temperatures of the nearest weather station for thedesired time frame and correct them by applying a regression starting from tempera-tures measured at the crime scene, when taking the larvae as evidence (Archer, 2004).The corrected values still contain uncertainties that cannot be accounted for by themethods currently used for PMI determination. No information exists for either modelabout the quality of the method or the error intervals of the calculated PMIs.
We analyzed developmental data for
L. sericata at different temperatures and fittedan individual exponential function for each developmental stage. Data used as input tothe model were published by Grassberger and Reiter (2001) and represent the minimaltime in hours to complete each larval phase (egg stage = stage 0, 1st instar = stage 1,2nd instar = stage 2, 3rd instar feeding = stage 3, 3rd instar post-feeding = stage 4 andpupal stage = stage 5) until eclosion of the adult blow fly. The used data-set is one ofthe rare sets which covers a lot of temperatures and the resulting growth curve seemsto represent growth behavior well (see original paper). Unfortunately, Grassberger and
Reiter do not give any error values for their measurements, so we assumed an error forthe developmental times of about 1 hour. These authors used 250 g of raw beef liverin plastic jars, and placed 100 eggs on the food substrate. The jars were placed in aprecision incubator. At each temperature regime the procedure was repeated 10 times.Every 4 hours, four of the most developed maggots were removed from the plastic jars,killed in boiling water, and preserved in alcohol (Adams and Hall, 2003) and then theirstage of development was determined.2.1 Data fitOur new larval growth model is based on the data shown in Fig. 2, in which the durationof each developmental stage was measured as a function of temperature (Grassbergerand Reiter, 2001). These data points were fitted with an exponential function of theform: T α ( Θ ) = a α · exp ( − τ α · Θ ) + T ,α (2)where T α is the duration of one developmental stage α as a function of temperature Θ . The parameters fitted for the different stages are shown in Table 1. The param-eter τ α defines how strongly the time interval depends on temperature; the higherthe parameter in Table 1, the steeper is the gradient of the fitted curve. T ,α repre-sents the minimum time interval required for finishing a certain developmental stageand a α provides the absolute normalization. The developmental stages of the mag-gots were determined every ∆T = 4 h, such that time measurement errors are set to σ T = ∆T / √
12 following an uniform distribution. It is assumed that the maggot bodytemperature is known to an accuracy of 3 % in order to take into account uncertaintiesabout differences between ambient and maggot body temperature. The parameters a α , τ α and T ,α were determined by minimizing the sum of error squares. As seenin Fig. 2, the exponential function accurately models the behavior during all devel-opmental stages and will be used below. In all stages the developmental duration attemperatures below 24 ◦ C starts to rise exponentially. Fig. 1 shows the calculated ADHvalues corresponding to eq. (1) (data points). In addition, the figure shows the functionADH α ( Θ ) = T α ( Θ ) · ( Θ − Θ ) (lines). T α ( Θ ) is calculated by eq. (2) with the previouslyfitted parameters (Table 1). Again, the functions give a reasonable description of thedata. Nevertheless, the model is an empirical one, based on the observations of thedata points generated by Grassberger and Reiter (2001).2.2 PMI calculationFor European and especially German temperatures, calculation of the total develop-mental duration must allow for non-linear temperature behavior in order to ensureaccuracy. The basic idea underlying a new approach in PMI determination is to fol-low an ambient time-temperature profile Θ ( t ) backwards in time starting from thetime point t F at which the maggots of interest were collected. The idea of backwardscalculation is obviously similar to the ADH method, but in the new model the im-portant improvement is the way of calculating the larval age. The latter is calculatedsuccessively during certain time steps using the fitted functions (introduced in Fig. 2) corresponding to the current developmental stage. In each stage α the relative devel-opmental progress is P α (values 0 - 1) where 0 is the beginning and 1 is the finishingpoint of each developmental stage; e.g. a maggot in the middle of the post-feedingstage is P = 0 .
5, at the end of the post-feeding stage it is P = 0 . t α, spent in each individual stage is calculated by solving therelation: P α = (cid:90) t α, t α +1 , d tT α ( Θ ( t F − t )) . (3)where dt/T ( Θ ( t )) is the infinitesimal relative development.The calculation starts with the developmental stage of the maggot at the timeof collection, summing the developmental progress of each stage backwards until thebeginning of the egg stage is reached. The calculation for each collection stage uses t α +1 , = t F . The total development time t or post-mortem interval (PMI) is thengiven by t = (cid:88) α t α, . (4)For the new model a program was written in C++ using Root (http://root.cern.ch/).This program includes all mentioned mathematical steps and produces the figuresshown here as output. For each new PMI calculation, the corresponding temperatureprofile can be inserted and individually chosen uncertainties can be included.2.3 Consideration of uncertainties by Monte-Carlo simulationTo explore the uncertainties in the total developmental duration, a Monte-Carlo sim-ulation was applied, which is commonly used for simulations in life sciences (Manssonet al., 2005). It is a method for calculating one final uncertainty after considering allstatistically independent uncertainties that influence e.g. the larval age. The meanPMI with corresponding standard variation is calculated n times taking into accountand varying all uncertainties described in the following. First, the developmental pro-files T α ( Θ ) have uncertainties due to the measurement procedure. Second, the time-temperature profile from the collection scene is not known precisely and must be ap-proximated using temperature values from nearby weather stations. The variations areintroduced for each model as follows:Development profile: The mean duration values of the temperature-time data are ran-domly smeared with a uniform distribution with corresponding error σ T ; for themaggot body temperature σ Θ,b a Gaussian distribution is used. New fits with thefunction in eq. (2) are performed for each stage.Time-temperature profile: Deviations between the temperature profile at the collectionscene and the nearest weather station are accounted for by Gaussian smearing oftime t and temperature Θ , with the corresponding errors σ t and σ Θ as widthfor each data point. σ Θ can be inserted in the model‘s calculation individuallydependent on the differences between the temperatures at finding place and weatherstation.We calculated the PMI for a mock crime scene with the following parameters: theerror of the measurement of the original data σ T = 4 / √
12 h, the errors of the data ofthe weather station σ t = 1 h and σ Θ = 2 ◦ C, the difference of the ambient temperature and larval body temperature σ Θ,b = 3 % for 10.000 models for a fixed collection stageprogress of P α = 0 .
5. The results are shown in Fig. 3 (upper part) which is a directoutput of the new program that calculates the PMI. The lower time axis defines theprogress of the temperature profile forward in time, representing the time frame ofinterest. The temperature profile used here (black line) is taken from the minimum andmaximum temperatures in May and June 2008 measured at Cologne/Bonn airport. Theright end of the diagram marks a fictional time point of maggot collection and thereforethe starting point for PMI calculation. The upper time axis depicts the PMI backwardsin time starting from the moment of maggot collection. For each developmental stagethe PMI was calculated by following a linear interpolation between the maximum andminimum temperatures. The histograms illustrate the PMI distribution for each stageand show a clear single peak structure. The arrows on the top show the 1-standarddeviation interval for each stage around the mean PMI value, and range between 0.1and 1.2 days (depending on the stage). Since no data points below temperatures
Θ < ◦ C were measured, the functions T α ( Θ ) were extrapolated to lower temperatures.As expected, the PMI and the corresponding standard deviation increase with higherdevelopmental duration (see arrows above histogram).Since the exact progress within the developmental stage at collection time is mostof the time also unknown, a third uncertainty is introduced:Stage progress: The developmental stage at collection time was determined only tointeger precision, so that it is assumed the exact progress is an uniformly distributedvalue between 0 and 1. Consequently, the starting value for the PMI calculation P α at time t F is randomly and uniformly chosen within the interval [0,1] for eachmodel.Fig. 3 (lower part) shows the PMI calculation for the same parameters as before,but without setting the progress of the development for each stage to a fixed value.The 1-standard deviation values increase by 0.3 to 3.3 days. The resulting uncertaintyin the progress of the stage contributes about 75 % to the total PMI error interval.In addition, the histograms show deviations from a clear single peak structure, e.g.for the pupal stage, implying that the PMI probabilities for 21 days and 26 days arenearly the same. To use the new model, the crucial parameter is therefore the correctdetermination of the progress of the developmental stage of maggots collected from acorpse.2.4 Estimation of temperature at the location of maggot collectionThe impact of correct temperature determination at the maggot collection scene isshown in Fig. 4. The data points represent the mean PMIs with an error bar of 1standard deviation as a function of collection stage for three different temperatureprofiles. The triangles show the PMIs for the original temperature profile as measuredat Cologne/Bonn airport. The bullets (squares) show the results for the same profile butsubtracted (and added) by 2 ◦ C. As expected, the PMIs and the corresponding standarddeviations of the lower (higher) temperature profile increase (decrease) relative to thenominal profile. These differences in temperature of 2 ◦ C give rise to an effect of 15 -30 %. That implies that a miscalculation of the temperature at the crime scene of 2 ◦ Cwill result in a miscalculation of the PMI by 15-30 %. The later the stage, the greaterthe deviation from the actual PMI. t , t F ] are compared with PMIs from our model using the threetemperature profiles introduced previously. The calculated mean temperatures wereas follows (calculated for the time frame till completion of each stage): stage 0 =18 ◦ C, stage 1 = 19 ◦ C, stage 2 = 20 ◦ C, stage 3 = 19 ◦ C, stage 4 = 16 ◦ C, stage5 = 17 ◦ C. The PMI values based on the temperature profile and those based on amean temperature value agree to within about 5 % for the high temperature value(original profile + 2 ◦ C) in all stages. The deviation between mean temperature andthe original temperature profile exceeds the 10 % level starting at the 3rd instar feedingstage, and increases to 25 % in the pupal stage. This effect becomes even larger for thelow temperature profile (original profile subtracted by 2 ◦ C). Starting from the 2ndinstar stage, the deviation increases from about 10 % up to about 65 % for the pupalstage. This means that use of mean temperature values overestimates the influence oflow temperatures and underestimates periods of high temperatures. The effect shouldbe larger if the mean temperature during the development is lower still, e.g. in springor fall. In general, more data points are needed for the developmental duration at lowtemperature ranges to provide more reliable statements.2.6 Does the model work in a real case?We calculated the PMI in a real case where the actual PMI was known due to aconfession of the offender. At the end of August 2007 the victim was killed in earlymorning and was found 4 days later also in the morning on a grassland. This leads toa PMI of approximately 96 hours. The victim was stabbed to death and had severalwounds which would act as attractant to the blow flies. It can be assumed that blowflies started ovipositing early after death occurred (Reibe and Madea, 2010). Autopsywas performed directly after the corpse was recovered and several 2nd instar larvae of
L. sericata were collected. The largest larvae measured 6.1 mm. Hourly temperaturevalues were taken from a weather station 10 km away. The mean temperature was 16 ◦ C. Using Grassberger and Reiter‘s isomegalen diagram for a larvae measuring 6 mmand a mean temperature of 16 ◦ C results in a time interval of 3.2 days plus 30 hours(larval development time plus egg period). In total a PMI of 107 hours is indicated.This would shift the time of oviposition to nighttime, which is a highly unlikely event(Amendt et al., 2008). The same data can be used to calculate the ADH value for
L. sericata for reaching 6 mm in order to calculate the PMI not based on the meantemperature but on hourly data. As mentioned earlier, a regression analysis of the dataset reveals a base temperature of 8 ◦ C. The corresponding ADH value is therefore 856,based on the equation: ADH = 107(16 −
8) (5)Subtracting the hourly ADH values, estimated by the temperature values from theweather station and the base temperature, from the starting value of 856 results in aPMI of 101 hours.To use the new model for calculating larval age in the real case, information aboutthe progress of the 2nd instar larval stage was required. In the original work of Grass-berger and Reiter (2001, Figure 1) a figure is included showing the growth of the larvae and also the time points for each moult. According to this figure, the 2nd instar stagesets in after the larvae have reached a size of approximately 4 mm and ends when thelarvae have reached a size of approximately 8mm. As the largest larvae we collectedmeasured 6 mm, we chose P=0.5 as progress for the larval stage. We included thehourly temperature profile and chose a temperature error of 1 ◦ C. The result of thecalculation was a PMI of 99 hours (SD = 3 hours).These calculations of a PMI in a real case show that all three methods give rea-sonable results. Furthermore, it becomes obvious that the new model is a possiblealternative for the existing methods with the benefit of directly providing a standarddeviation for the calculation.
The new model improves the larval age calculation in specific ways. It can be usedin non-linear parts of the temperature dependent development, and includes individu-ally defined uncertainties for a temperature profile determined retrospectively from thenearest weather station. In the new model the temperature profile plus the determina-tion of the larval stage are translated into a mean PMI as well as a standard deviation.PMI calculation using mean temperatures, however, can lead to severe deviations fromthe real PMI.So far, the main uncertainty arises from the fact that the developmental stage isdetermined only on a 1 - 6 scale (egg, 1st instar, 2nd instar, 3rd instar feeding, 3rd instarpost-feeding and pupae). As shown above, 75 % of the uncertainties in the model dependon the exact determination of the developmental progress, and additional length values,as shown for the PMI calculation in the real case, will propably increase its accuracyleading to more accurate PMI calculations. Moreover, the next step is to produce owngrowth data with known error values to refine the inclusion of uncertainties that areonly rough estimates at the present time and to improve the till now only empiricalmodel.Nevertheless, the new PMI calculation program is suitable for use in forensic casework as a general tool for PMI determination. Scientists from every country or climaticregion can incorporate their own growth values for different species and ensure a highaccuracy in PMI determination.
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RH Porter, LondonMansson RA, Frey JG, Essex JW and Welsh AH (2005) Prediction of properties fromsimulations: a re-examination with modern statistical methods. J Chem Inf Model45(6):1791–1803Reibe S and Madea B (2010) How promptly do blow flies colonise fresh carcasses? Astudy comparing indoor vs. outdoor locations. For Sci Int 195:52–57Slone D and Gruner S (2007) Thermoregulation in larval aggregations of carrion-feedingblow flies (Diptera: Calliphoridae). J Med Entomol 44(3):516–523Smith KE and Wall R (1997) The use of carrion as breeding sites by the blowfly Luciliasericata and other Calliphoridae. Med Vet Entomol 11(1):38–44Tao S (1927) A Comparative Study of the early Larval Stages of some common Flies.Am J Epidemiol 7:735 – 761 [deg C] Q temperature 15 20 25 30 35 ADH eggsfirstsecondthirdpost-feedpupa
Fig. 1
Calculated ADH values for the development of
L. sericata using eq. (1) (data points)and fitted functions (lines) calculated using eq. (2) and estimated parametres (Table 1). Orig-inal data by Grassberger and Reiter 2001.
Table 1
Fitted parameters for the development-time-function T α ( Θ ). α Stage a α [h] τ α [ ◦ C − ] T ,α [h]0 eggs 1 . · . · . · . · . · . · [deg C] Q temperature 15 20 25 30 35 t i m e T [ h ] )+ 3.74 Q eggs: 128.85 exp(-0.10 )+ 8.32 Q first: 1005.39 exp(-0.20 )+10.73 Q second: 1103.19 exp(-0.19 )+24.99 Q third: 2339.95 exp(-0.22 )+84.87 Q post-feed: 266887.22 exp(-0.46 )+119.36 Q pupa: 7447190.98 exp(-0.57 Fig. 2
Developmental data of
L. sericata with fitted functions (eq. (2)). Original data byGrassberger and Reiter 2001.2 time [h]0 200 400 600 800 1000 t e m p e r a t u r e [ d e g C ] eggsfirstsecondthirdpost-feedpupa time [h]0 200 400 600 800 1000 t e m p e r a t u r e [ d e g C ] eggsfirstsecondthirdpost-feedpupa Fig. 3 α . The arrows at top show 1-standard deviationintervals around the mean PMIs for each stage. Upper part:
PMI calculation with parameters σ T = 4 / √
12 h, σ t = 1 h, σ Θ = 2 ◦ C, σ Θ,b = 3 % for 10.000 models for a fixed collection stageprogress of P α = 0 . Lower part:
PMI calculation as before but the progression of the stageis not fixed.3 a stage 0 1 2 3 4 5 P M I [ d ays ] - 2 deg C Q as measured Q + 2 deg C Q Fig. 4
Calculated PMIs and SDs for each stage for a temperature profile of May/June 2008measured at Cologne/Bonn airport in comparison to profiles ± ◦ C. The later the stage thegreater the SD and the differences in the calculated PMI for differences of 2 ◦ C.4 a stage 0 1 2 3 4 5 m od e l / P M I m ea n P M I - 2 deg C Q as measured Q + 2 deg C Q Fig. 5
Deviations of calculated PMIs when using the mean temperature and the actual 12-hourly temperature profile of May/June 2008 from Cologne/Bonn airport. Additionally, theresults are shown for the temperature profiles ± ◦◦