Forensic Science, Medicine, and Pathology

, Volume 11, Issue 4, pp 530–537 | Cite as

Exploration of the R code-based mathematical model for PMI estimation using profiling of RNA degradation in rat brain tissue at different temperatures

Original Article


Precise estimation of postmortem interval (PMI) is crucial in some criminal cases. This study aims to find some optimal markers for PMI estimation and build a mathematical model that could be used in various temperature conditions. Different mRNA and microRNA markers in rat brain samples were detected using real-time fluorescent quantitative PCR at 12 time points within 144 h postmortem and at temperatures of 4, 15, 25, and 35 °C. Samples from 36 other rats were used to verify the animal mathematical model. Brain-specific mir-9 and mir-125b are effective endogenous control markers that are not affected by PMI up to 144 h postmortem under these temperatures, whereas the commonly used U6 is not a suitable endogenous control in this study. Among all the candidate markers, ΔCt (β-actin) has the best correlation coefficient with PMI and was used to build a new model using R software which can simultaneously manage both PMI and temperature parameters. This animal mathematical model is verified using samples from 36 other rats and shows increased accuracy for higher temperatures and longer PMI. In this study, β-actin was found to be an optimal marker to estimate PMI and some other markers were found to be suitable to act as endogenous controls. Additionally, we have used R code software to build a model of PMI estimation that could be used in various temperature conditions.


Postmortem interval (PMI) RNA Reverse transcription quantitative real-time PCR R code 


The precise estimation of postmortem interval (PMI) is a constant challenge in forensic science. Traditional methods used to estimate PMI in autopsy cases include livor mortis, rigor mortis, the digestive process of gastric contents and entomology, among others [1, 2]. However, none of these methods provides a complete solution to the question of PMI because the environment has many variables. In recent years different biomarkers such as proteins [3] and DNA [4] have been employed to build a mathematical model of PMI, demonstrating their great potential value in forensic cases. With crucial methodological advances in RNA extraction, reverse transcription, and the development of real-time quantitative polymerase chain reaction (RT-qPCR), many researchers have used RNA degradation to estimate PMI [5, 6, 7]. Various mRNA markers such as GAPDH, β-actin, and RPS-29 are considered to be appropriate endogenous control markers for biochemical research, while U6 snRNA is the most commonly used control marker in microRNA studies [5, 8]. However, despite their popularity, these molecules degrade over time, especially in harsh environments such as high temperatures, which reduces their efficacy as markers for PMI studies. Some studies have shown several RNA markers with good PMI correlation and that some microRNAs are sufficiently stable to be suitable as reference markers [6, 7].

In this study, the degradation rate of some RNA markers was explored under 4, 15, 25, and 35 °C to find the most suitable RNA markers for which ΔCt has the best correlation coefficient with prolonged PMI. This enables us to produce a robust mathematical model that can then be tested at various temperatures. R code software is a powerful tool for both model building and equation solving in such a complicated context.

Materials and methods

Animal samples

In total, 270 male and female Sprague–Dawley rats (body weight is 220 ± 20 g) were sacrificed by cervical dislocation and were randomly divided into a control group (PMI = 0 h) and four experimental groups, which were kept in a controlled environment chamber at 15 ± 1, 25 ± 1, and 35 ± 1°C respectively. Brain tissues were collected at 1, 3, 6, 12, 24, 36, 48, 72, 96, 120, and 144 h (n = 6) after death to develop mathematical models. To collect the tissue the brain was completely exposed by removing the scalp and skull. The anterior region of the brain was then carefully separated using ophthalmic scissors.

An additional 36 rats were sacrificed as described and randomly divided into three groups that were kept at 10 ± 1, 20 ± 1, and 30 ± 1 °C for validation of the models. Brain samples from these additional animals were collected at 10, 30, 50, and 100 h postmortem.

All samples were placed in RNA Latersolvent (Takara, Japan) immediately after collection. The animal experiments described in the present study were performed in accordance with the principles for the Care and Use of Laboratory Animals and were approved by the Science and Ethics Committee of Fudan University.

RNA isolation and integrity analysis

All equipment and bench surfaces that were used during this work were treated with RNase Away (Invitrogen, USA) prior to handling any of the samples to minimize the risk of RNA degradation by RNase during the experimental process. 80–500 mg of tissue from each sample was homogenized with 1 ml Trizol solvent (Invitrogen, USA) and 0.2 ml chloroform to cause protein denaturation. The supernatant was then decanted and mixed with 0.5 ml isopropanol before being stored at 20 °C for 1 h for precipitation. The mixture was then centrifuged at 10,000×g at 4 °C for 10 min. The resulting supernatant was discarded and the precipitate was washed with 75 % ethanol (3:1 DEPC-H2O). The washed samples were then centrifuged again and the supernatant was discarded. Finally the total RNA was dissolved in an appropriate volume of nuclease-free water to provide a solution with a concentration of 200–500 ng/μl. The concentration and purity of RNA were assessed by spectrophotometric analysis using NanoDrop 1000 (Erlangen, Germany). RNA integrity and the level of degradation were assessed by agarose gel electrophoresis.

RNA markers

Nine RNA markers were chosen in this study. β-actin and GAPDH are housekeeping genes which are commonly used as endogenous control markers. The ribosomal protein (RP) S29 mRNA (RPS29), was shown to be stable for normalizing in the first 12 h under 25 °C [5]. 5SrRNA, 18SrRNA, and U6 are frequently used as control markers in microRNA studies [9, 10]. Tissue specific and abundant mature microRNAs (miRs) were chosen in this research because they were considered to be less susceptible to degradation owing to their small size of about 22 base pairs. miR-9 and miR-125b were chosen because their expression was specific to brain tissues according to the miR base database [11]. Let-7a was also chosen because it was an abundant biomarker in all tissues [12].

The primers corresponding to these markers were designed to span at least one exon/exon boundary (except for miRNA and U6) to ensure the amplification and detection of cDNA alone. The details of RNA markers and the primer are given in Table 1.
Table 1

Primers used to amplify RNA markers by RT-qPCR

Primer name

Sequence 5′–3′

Sequence 5′–3′

Product size (bp)

Accession number























Uni-miR qPCR primer





Uni-miR qPCR primer





Uni-miR qPCR primer





Uni-miR qPCR primer





Uni-miR qPCR primer



The symbol ‘‘~’’ denotes the approximate length of the PCR product

Real-time quantitative polymerase chain reaction (RT-qPCR)

Total RNA was reverse transcribed using a PrimeScript RT reagent Kit with gDNA Eraser (Perfect Real Time) (Takara, Japan) according to the manufacturer’s protocol. At the same time, 500 ng total RNA was reverse transcribed by adding a polyA tail using a One Step PrimeScript miRNA cDNA Synthesis Kit (Perfect Real Time) (Takara, Japan) according to the manufacturer’s protocol for microRNA analysis. The cDNA product was then diluted by a ratio of 1:10 for further use and stored at −20 °C for RT-qPCR.

Real-time PCR was performed in an ABI Prism 7500 fluorescence quantitative PCR instrument (Applied Biosystems, USA). An amplification mixture was prepared using the SYBR Premix Ex Taq kit (Takara, Japan) according to the manufacturer’s protocol. The reactions were performed in a total volume of 20 μl containing 2 μl cDNA, 10 μl SYBR premix, 0.4 μl dye, 0.4 μl forward primers, 0.4 μl reverse primers and 6.8 μl RNase-free dH2O. A standard curve was similarly produced in a 20 μl volume with 2 μl cDNA diluted 5–8 times. The cycling parameters were 30 s at 95 °C followed by 40 cycles of 5 s at 95 °C and 34 s at 64 °C. Reactions were prepared in duplicate for each sample and for each of the 3 assays. Copies of the specific endogenous markers were quantified and presented as the mean cycle threshold (Ct) values detected with sequence-detection system software v2.3 using a threshold value of 0.2 (Applied Biosystems, USA). All RT-qPCR experiments were performed according to the Quantitative Real-Time PCR Experiments (MIQE) guidelines [13].

Mathematical model and statistical method

ΔCt represents the differences in Ct values between target biomarkers and control biomarkers. The bivariate cubic curve fit was explored to show the trends over time for the Ct and ΔCt values. Curve estimation analysis of ΔCt values, temperature, and PMIs of rat samples was performed to derive an optimal mathematical model using R software (v3.0.1), a tool for statistical computing and graphics. With R software, a quadratic equation was built with three unknowns (temperature, PMI, and ΔCt). First, the equation was composed by including the terms t, PMI, PMI × t, and PMI2. The animal data was used to fit this type of equation in the R software, and statistical significance (P < 0.05) was applied to compose the final equation. This was then transformed into a three-dimensional visual statistical model with Matlab7.0 software.

A further 36 rat samples were used for verification of the mathematical model. For each case, the known temperature and ΔCt values were inserted into the equation of the animal model. Then the R software calculated the value of PMI at intervals of 0.1 until the smallest deviation between the corresponding output ΔCt value and the known ΔCt value was found, indicating the optimal PMI as our estimated PMI. Model quality was judged by the error rate as shown:
$$ {\text{Error}}\;{\text{rate}} = \left[ {\left( {{\text{estimated}}\;{\text{PMI}} - {\text{real}}\;{\text{PMI}}} \right)/{\text{real}}\;{\text{PMI}}} \right] \times 100\;\% . $$

The R code for the calculating procedure and the Matlab7.0 code for the visual transformation are presented in Supplemental Material 1.

The statistical significance of the resulting data was analyzed with GraphPad v5.0 (GraphPad software Inc., USA), Excel 2010 (Microsoft, USA), with α = 0.05.


RNA extraction and integrity

Total RNA was successfully extracted from all rats. RNA yield significantly correlates with PMI at 15, 25, and 35 °C, where RNA yield declines with PMI at 25 and 35 °C (Fig. 1). The 260/280 ratios of all samples ranged from 1.9–2.1 after purification. Electrophoresis analysis shows that RNA integrity degrades with PMI, particularly at high temperatures (Fig. 2). The degradation rate of RNA integrity is faster in muscle than in brain compared to another study which was conducted at 25 °C [6]. Brain is supposed to be less affected by exogenous RNase due to protection from the skull [14].
Fig. 1

RNA yield significantly correlates with PMI at 15 °C (R2 = 0.1602, P<0.05), 25 °C (R2 = 0.4959, P<0.05), and 35 °C (R2 = 0.6217, P<0.05); RNA yield declines with PMI at 25 and 35 °C

Fig. 2

Electrophoretic analysis was carried out to detect the postmortem change in total RNA. RNA integrity degrades with PMI, particularly at high temperatures. The control group (0 h) shows three clear bands, which begin to fade with PMI and become blurry after 96 h at 25 °C or after 48 h at 35 °C

Correlation between ΔCt value and PMI in the rat model

Amplifications were successfully performed for all candidate biomarkers though RT-qPCR. The tests used to validate the specificity of the RT-qPCR (Supplemental Material 2) included amplification plots, dissociation curves, standard curves, and PCR efficiency of candidate biomarkers [13, 15].

Ct and ΔCt values of RNA markers were subjected to bivariate cubic curve fit analysis with GraphPad software. The Ct values of nine candidate markers varied more with prolonged PMI. Interestingly, U6 was found to degrade with PMI particularly at higher temperatures, demonstrating that it is not a suitable endogenous control in this study (Fig. 3). After calculations with geNorm software, miR-125b and miR-9 were chosen as control markers because of their high stability (Supplemental Material 3). They also have the lowest standard deviation value across all the RNA markers (Fig. 4; Table 2). Data were normalized according to the ΔCt model using the following formula:
$${\Delta }{\text{Ct}} = {\text{Ct}}\left( {{\text{target}}\;{\text{gene}}} \right)-\left( {{\text{Ct}}\;{\text{miR}}{\text{-}}125{\text{b}} \times {\text{Ct}}\;{\text{miR}}{\text{-}}9} \right)^{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} } $$
Fig. 3

Ct (U6) apparently increases with PMI particularly at high temperatures, demonstrating that U6 was not suitable as an endogenous control in this study

Fig. 4

Ct (miR-9) and Ct (miR-125b) have the minimum SD (standard deviation) value among all the RNA markers that were chose (Table 4) and show the highest stability by geNorm software (Supplemental Material 3). The curve fit between these Ct values and increased PMI in rat samples is presented here. The changes in Ct value are minimal within 144 h even at 35 °C when compared with the other markers, making them the best choice for endogenous control markers

Table 2

Standard deviation of Ct values at different temperatures


4 °C

15 °C

25 °C

35 °C

All samples























































SD (standard deviation) value of the Ct (RNA) in the corresponding group is calculated. For example, SD (β-actin) = 1.0352 means the SD value of all the Ct (β-actin) from 0 h to 144 h at 4 °C is 1.0352. All samples means the Ct (β-actin) of the 270 rat samples. Results shows that Ct (miR-9)and Ct (miR-125b) have the minimum SD valuein all the RNA markers

The R2 values of ΔCt calculations are presented in Table 3. Our results show that ΔCt (β-actin) significantly correlates with PMI in all of the temperature groups, and also has the highest average R2 value with PMI (Fig. 5; Table 3). Therefore, it was subsequently used as a control marker to explore our new mathematical model.
Table 3

Correlation between ∆Ct and PMI at different temperatures


4 °C

15 °C

25 °C

35 °C


∆Ct (β-actin)

0.8925 (<0.0001)

0.9490 (<0.0001)

0.9363 (<0.0001)

0.8421 (<0.0001)



0.7569 (0.0002)

0.1214 (0.2671)

0.5721 (0.0044)

0.5368 (0.0067)


∆Ct (18SrRNA)

0.3523 (0.0419)

0.8019 (<0.0001)

0.7468 (0.0003)

0.8248 (<0.0001)


∆Ct (RPS29)

0.1146 (0.2819)

0.6693 (0.0011)

0.2302 (0.1145)

0.8932 (<0.0001)


∆Ct (U6snRNA)

0.0001 (0.9890)

0.7346 (0.0004)

0.8686 (<0.0001)

0.9560 (<0.0001)


∆Ct (let7a)

0.0561 (0.4587)

0.3027 (0.0638)

0.0005 (0.9437)

0.1868 (0.1606)


Pearson correlations between the ∆Ct and PMI are calculated by GraphPad software

∆Ct = Ct (target gene) − (Ct miR-125b × Ct miR-9)1/2

Results shows that ∆Ct (β-actin) has the highest average R2 and is used to explore our mathematical model by R software

Fig. 5

Results for the ΔCt values of β-actin show significant and consistent degradation with increased PMI, especially at higher temperatures, making it the most reliable and valuable marker for PMI estimation

Building a visual three-dimensional statistically animal model

The result of the R software shows that terms t, PMI, PMI × t, and PMI2 are statistically significant (P < 0.05). These terms are combined in the final equation. The quadratic equation was based on animal data with three unknowns; the visual model is shown in Fig. 6. The t value is the temperature of each sample. The coefficient of each term and the intercept are calculated using R software.
Fig. 6

The cubic curve fit of ΔCt (β-actin) and prolonged PMI under different temperature conditions was taken and transformed into four equations. The fit between the curve generated by the R software and the animal data results from the quadratic equation with three unknowns, which uses the temperature and ΔCt (β-actin) as parameters. The equation was then transformed into a three-dimensional visual mathematical model using Matlab7.0 software

Verification of animal samples

Comparison of the animal model to the calculated data shows that the mathematical model explored here should accurately estimate the PMI of the Sprague–Dawley rat at different temperatures, not only at the temperatures we chose to use. In fact, the model actually shows increased accuracy for higher temperatures and longer PMI (Table 4).
Table 4

Verification of the mathematical model

RealPMI (h)

T (°C)

∆Ct (β-actin)

EstPMI (h)

Average ± SD

Error rate (%)









11.2 ± 5.87










23.53 ± 3.72










58.43 ± 4.8










97.27 ± 21.29










12.47 ± 8.29










20.97 ± 6.71










53.63 ± 24.27










94.87 ± 18.3










14.3 ± 6.77










30.37 ± 11.64










51.37 ± 12.16










110.47 ± 6.86


36 other rats were used to verify our mathematical model. Model quality was judged by the estimated PMI ± SD and error rate. Error rate = [(estimated PMI − real PMI)/real PMI] × 100 %. Results shows that the model can estimate the PMI of the rat at different temperatures within 35 °C, with reasonable error. The model actually shows increased accuracy for higher temperatures and longer PMI


PMI is defined as the time interval between physiological death and the examination of the deceased. Traditional methods of PMI estimation using postmortem phenomena are often imprecise or experimental. However, innovative research is underway to estimate PMI based on algor mortis using new technology and physical sciences software for enhanced accuracy [16, 17]. Retrospective study of postmortem phenomena, such as re-establishment of rigor mortis and mechanically stimulated idiomuscular contraction, also present further evidence that can be commonly used in forensic practice.

In recent years, the development of technology to detect biological markers after death has enhanced accuracy in PMI estimation. RNA is an extremely labile molecule that is prone to damage from either intrinsic factors such as enzymatic degradation by RNases, or external factors such as light, humidity, or high temperatures [18]. RNA is thought to degrade very quickly after death and was considered unsuitable for estimating PMI unless the degradation rate could be quantitatively measured. This is now possible with RT-qPCR. From semi quantification by RT-PCR to absolute quantification by qRT-PCR, recent research has explored postmortem RNA degradation to assess its potential value in indicating PMI [5, 19, 20, 21]. Some studies have explored the differences of intragroup RIN [22] and others use ΔCt values to help estimate PMI [5]. However, most of the experimental work uses rats or pigs for long-term research projects [23] and such studies are often conducted without consideration of parameters like temperature. Although autopsy samples can be analyzed reliably using qRT-PCR according to Inoue [21], using human postmortem tissues for mRNA expression studies is challenging because of the high biological variability between cases [24]. The lack of human data to verify the animal model also makes it controversial for practical forensic applications.

For absolute quantification of RNA transcript levels in molecular research, the mRNA transcript level is always normalized by using housekeeping genes as an internal control [25]. However, even housekeeping genes degrade with prolonged PMI, complicating their use as internal control markers. More stable RNA markers are still needed for this purpose.

MicroRNA (miR) refers to a class of small non-coding single-stranded RNA, only 21–25 nucleotides long, thought to be highly conserved in the evolution of genomes and relatively stable in comparison to mRNA. Studies using human adipose tissues and cells indicate that microRNAs could be considered suitable candidates for endogenous controls [26].

Animal-based PMI research shows that heart-abundant miR-1 is stable within 7 days after death [6]. In this study, we selected brain-specific miR-9 and miR-125b to explore their potential as internal standards. Results show that they maintain high stability over 6 days even under high temperature. Both miR-9 and miR-125b were chosen as endogenous control markers to normalize the target markers for which the relationship between PMI and ΔCt values was analyzed.

The small nuclear RNA, U6 snRNA, only exists in the eukaryotic nucleus and is highly stable because of its short hairpin structure and lack of nuclease [27]. However, it begins to degrade after 96 h, even at temperatures under 15 °C, and more rapidly under higher temperatures (Fig. 3). This may be because U6 becomes directly exposed to nucleases without the protection of ribonucleoproteins, which degrade quickly [7]. Similarly, some studies also demonstrate that U6 is not suitable as an endogenous control in some research [28, 29].

β-actin, known as a housekeeping gene, is commonly used as reference in biomedical research because of its multiple sources and homology. Many researchers have used its degradation rate to estimate PMI [5, 7]. Similarly, our results for the ΔCt values of β-actin show significant and consistent degradation with increased PMI, especially at higher temperatures, making it the most reliable and valuable marker for PMI estimation. Through testing of four temperature groups, a multi-parametric mathematical model was established and was found to be effective in the verification of data from rat samples. However, research using human data is scarce but still necessary to extend this field of research further in the future.

The statistical method of linear regression is often used in the study of PMI estimation in legal medicine. However, its use in estimating PMI is questionable because the linear assumptions lack a means for verification. Very little research has explored alternative ways to improve the estimation, such as using R code-based PMICALC software [30]. Similarly, the degradation of RNA doesn’t always follow a linear path, especially with longer PMI, and the profile of RNA degradation varies with temperature. To take temperature into consideration, R software was chosen because it provides a method to build the model with three unknowns and makes it easy to solve the equation using the iterative method.

In summary, we have created a standardized animal model for PMI estimation, showing the influence of temperature and allowing verification. However, the postmortem RNA degradation in human tissues is not only dependent on PMI but also other factors, and particularly environmental conditions, cause of death and the individual’s circumstances. It is necessary to next apply this model to human samples, and assess the other compounding factors. Further studies will also need to find more effective biomarkers to be able to accurately determine PMI with this methodology.

Key points

  1. 1.

    Brain-specific mir-9 and mir-125b are effective endogenous control markers that are not affected by PMI up to 144 h at temperatures between 10 ± 1 and 35 ± 1 °C.

  2. 2.

    U6 degrades with PMI, particularly at higher temperatures, demonstrating that it was not a suitable endogenous control for this study.

  3. 3.

    ΔCt (β-actin) has the best correlation coefficient with PMI among the markers we chose, demonstrating that β-actin is an optimal RNA marker to estimate PMI.

  4. 4.

    Our animal model of PMI estimation using R software could be used in various temperature conditions. It is a powerful tool for both model building and equation solving in such a complicated context.


Supplementary material

12024_2015_9703_MOESM1_ESM.docx (134 kb)
Supplementary material 1 (DOCX 134 kb)
12024_2015_9703_MOESM2_ESM.docx (207 kb)
Supplementary material 2 (DOCX 206 kb)
12024_2015_9703_MOESM3_ESM.docx (3.4 mb)
Supplementary material 3 (DOCX 3512 kb)


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jianlong Ma
    • 1
  • Hui Pan
    • 1
  • Yan Zeng
    • 1
  • Yehui Lv
    • 1
  • Heng Zhang
    • 1
  • Aimin Xue
    • 1
  • Jieqing Jiang
    • 1
  • Kaijun Ma
    • 2
  • Long Chen
    • 1
  1. 1.Department of Forensic MedicineShanghai Medical School of Fudan UniversityShanghaiPeople’s Republic of China
  2. 2.Forensic Lab, Criminal Science and Technology InstituteShanghai Public Security BureauShanghaiPeople’s Republic of China

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