Results and discussion

Ethanol elimination with particular consideration of low BAC values

The hourly rates of blood alcohol concentration (BAC) decrease across the different studies and across the various sections of the concentration–time curves are summarised in Table 1.

Interval | Wheat beer [‰/h] | Wheat beer + Eezup! [‰/h] | Apple wine [‰/h] | Grappa [‰/h] | Vodka [‰/h]
cmax – 0‰ | 0.143 | 0.150 | 0.162 | 0.168 | 0.159
0.3–0.0‰ | 0.111 | 0.126 | 0.142 | 0.143 | 0.125
0.6–0.3‰ | 0.162 | 0.175 | 0.174 | 0.182 | 0.184
0.9–0.6‰ | 0.110 | 0.109 | 0.163 | 0.143 | 0.139
cmax – 0.6‰ | 0.105 | 0.104 | 0.155 | 0.135 | 0.133
cmax – 0.9‰ | 0.078 | 0.102 | 0.121 | 0.154 | 0.069

The overall hourly decline in blood alcohol concentrations across the full range [cmax–0‰] was 0.156‰/h, which is consistent with the average elimination rate reported in the literature [8].

Analysis of the 75 data sets in the range below the 0.15‰ threshold (linear regressions of the subsection [0.15–0‰]) yielded a mean decline in BAC of 0.083‰/h (Supplementary Table S1). This value lies below the rate of 0.1‰/h commonly regarded as the minimum elimination rate and is clearly lower than the elimination rate observed across the entire BAC-time curve [cmax–0‰] (Supplementary Table S2).

Closer inspection of the calculated elimination rates (Supplementary Table S2) initially reveals that elimination velocities were broadly comparable across all regression ranges of the BAC-time curves, irrespective of the type of alcoholic beverage. Wheat beer (Fig. 1) showed a decline of 0.11‰/h; the same beverage combined with “Eezup!” (Fig. 2) exhibited a nearly unchanged rate of 0.12‰/h. The arithmetic means for both self-produced (Fig. 3_)_ and commercially available apple wine (Fig. 4), as well as for grappa (Fig. 5), amounted to 0.15‰/h, while vodka (Fig. 6) showed a rate of 0.13‰/h. Apple wine and grappa yielded the highest average elimination rates, whereas wheat beer showed the lowest.

Fig. 1: Blood alcohol concentration-time curve (wheat beer)

Fig. 2: Blood alcohol concentration-time curve (wheat beer + Eezup!)

Fig. 3: Blood alcohol concentration-time curve (apple wine, homemade)

Fig. 4: Blood alcohol concentration-time curve (apple wine, commercial)

Fig. 5: Blood alcohol concentration-time curve (grappa)

Fig. 6: Blood alcohol concentration-time curve (vodka)

In the interval of particular interest, between 0.3 and 0‰, the overall mean of all elimination rates was approximately 0.13‰/h. Elimination rates appeared consistent within this interval and were of the same order of magnitude as those observed in higher intervals (e.g. cmax–0.6‰). This was confirmed by the ANOVA including all five groups (F = 1.64, p = 0.204 > 0.05), indicating no statistically significant differences between the groups, which may therefore be considered statistically homogeneous. As expected, this conclusion is further supported by the low overall dispersion of all values (standard deviation = 0.0309, corresponding to ~ 24% of the mean; range = 0.1149). These findings support continued mathematical application of a linear relationship in this segment of the BAC curve.

In the higher ranges of the BAC-time curves, elimination rates originating from cmax appeared lower than the average across the entire curve, which can be explained by the transition from the absorption phase to the elimination phase. Thus, mean elimination rates in the ranges 0.9–0.6‰ and cmax–0.6‰ were approximately 0.13‰/h, while only the subsection 0.6–0.3‰ showed a slightly higher mean of approximately 0.18‰/h. This demonstrates a degree of consistency that is reliable for established back-calculation rules. In principle, all observed average elimination rates exceeded 0.1‰/h. Taken together, the modal value and median of approximately 0.13‰/h may therefore be regarded as the most suitable descriptive parameter for ethanol elimination velocity overall, particularly given that this value consistently represents both the arithmetic mean across all subdivided segments of the BAC curve and the cross-sectional mean across all beverage types.

Overall, with respect to the research question, it can be concluded that ethanol elimination rates in the range 0.3–0‰ (mean ≈ -0.129‰/h), which corresponds to measured initial BACs below 0.15‰, did not fall significantly below those of other intervals of the elimination curve (mean x̄ ≈ -0.1375‰/h, SD = 0.0248‰/h, minimum = -0.1755‰/h, maximum = -0.1048‰/h). To verify this, a Grubbs test (G test) for identification of outliers in a small normally distributed sample was applied. In the present case, no outlier was detected, as G < Gcrit (Grubbs statistic G = 0.327; critical value Gcrit ≈ 1.89 for α = 0.05, n = 6).

No statistically significant differences were detected between elimination behaviours of the different types of alcohol. The Grubbs significance test (α = 0.05, two-sided) yielded a mean x̄ ≈ -0.1337‰/h, SD = 0.0170‰/h, minimum = -0.1513‰/h, maximum = -0.1132‰/h, Grubbs statistic G = 1.2046, and critical value Gcrit = 1.7150; since G < Gcrit, no significant outlier was present.

Across the various BAC intervals and beverage types, ethanol metabolism rates remained largely consistent, with no substantial differences between elimination at higher and lower blood ethanol levels that would be statistically significant or relevant for back-calculation in favour of the defendant (mean x̄ ≈ -0.1375‰/h, SD = 0.0248‰/h, minimum = -0.1755‰/h, maximum = -0.1048‰/h, G = 1.5311, Gcrit = 1.8871; no outliers detected).

It is recommended, as a precautionary measure, to set the cut-off for permissible retrograde calculation of the minimum blood alcohol concentration at 0.06‰. This threshold was not selected as a mere analytical limit of detection or quantification; rather, it is intrinsically linked to the structure and applicability range of the present model. Specifically, 0.06‰ represents the lowest BAC range for which a sufficient number of post-absorptive data points were available and for which stable regression-based elimination behaviour and model-based time corrections could be consistently demonstrated. The simulation-based approach, including Monte Carlo modelling, was explicitly developed and validated within this concentration domain (0.06–0.15‰), where elimination kinetics remained sufficiently predictable and the resulting correction factors showed robust behaviour. At the same time, this threshold lies clearly above the analytical detection and quantification limits of the applied forensic blood alcohol analysis, while remaining sufficiently low to capture legally relevant borderline constellations. In this concentration range, analytical imprecision and further factors may exert a non-negligible influence on the measured value, thereby limiting the robustness of any retrospective calculation, such that the additional uncertainty introduced by back-calculation outweighs the evidential value of the result. In addition, at very low blood alcohol concentrations, ethanol metabolism increasingly approaches first-order kinetics as enzymatic saturation diminishes, further reducing the predictability of elimination behaviour. Reported endogenous ethanol concentrations in human blood typically range between approximately 0.01 and 0.05‰, depending on methodological approach and reference source [9, 10]. Taken together, these considerations justify a conservative lower limit of 0.06‰ for retrograde calculation of minimum BAC as a data-supported and model-consistent lower boundary – one that balances empirical robustness, biological plausibility, analytical limitations, and forensic conservatism, rather than reflecting a purely technical analytical limit, thereby providing an appropriate safeguard against overinterpretation of analytically and physiologically unstable low-concentration findings.

Consideration of BAC values below 0.15‰ for back-calculation

The descriptive statistical parameters of BAC elimination below 0.15‰ are shown in Table 2.

Mean BAC decline [‰/h] | n
Study |  | 
Wheat beer | 0.075 | 15
Wheat beer + Eezup! | 0.080 | 11
Apple wine | 0.105 | 18
Grappa | 0.072 | 9
Vodka | 0.084 | 22
Total |  | 75
Arithmetic mean x̄ | 0.083‰/h | 
Median | 0.080‰/h | 
Standard deviation (SD) | 0.030‰/h | 
Variance | < 0.001(‰/h)² | 
Minimum | 0.028‰/h | 
Maximum | 0.200‰/h | 
95% confidence interval (CI) | [0.076; 0.091]‰/h | 
Standard error (SE) | ± 0.004‰/h | 
Margin of error (95%) = z x SE ; z = 1,96 | ± 0.007‰/h | 

Using a Student’s t test with 74 degrees of freedom, it was assessed whether the mean elimination rate below 0.15‰ differs significantly from the conventionally conservative value of 0.1‰/h. The test evaluates the deviation of the arithmetic mean (x̄ = 0.083‰/h) from the reference value (µ₀ = 0.1‰/h) in units of the standard error:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{c}t\left(df\left(Degrees\;of\;Freedom\right)=n-1\right)\\=t\left(74\right)=\left(\overline x-\mu_0\right)/SE\approx-4;p<0.00001\end{array}$$\end{document}

Thus, the deviation is highly significant, amounting to almost five standard deviations, with the true mean lying 4.8 times the standard error away from the reference value of 0.1‰/h.

This indicates that, although the average elimination rate below 0.15‰ (0.083‰/h) remains close to that of other intervals in the re-analysis using back-calculation simulation scenarios, the Student’s t test demonstrates a statistically significant deviation from the conservative value of 0.1‰/h. This finding justifies the need for alternative calculation models for the interval ≤ 0.15‰.

When evaluating new approaches for forensic back-calculation rules, it must be considered that, below 0.15‰, both the mean ethanol elimination rate of 0.083‰/h and the upper bound of the 95% confidence interval lie below the elimination rate of 0.1‰/h otherwise used as a minimum. In the conservative model, this value is applied; however, when starting from low BAC values, this may lead to a slight overestimation of elimination and, in individual cases, to incorrect retrograde BAC values. Since elimination rates exceed 0.1‰/h in all other segments of the curve, any disadvantage is considered minor.

To avoid this error, additional back-calculation rules were evaluated, either by using lower elimination rates or by compensating for the temporarily reduced elimination via shortening of the back-calculation time interval (time deduction).

In the realistic model, the mean elimination rate (r) of 0.083‰/h is applied, and a time adjustment to be subtracted (Δtreal) is calculated according to the following equation:

General formula for back-calculation deduction:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\triangle t_{real}=\left(\left(0.1-r\right)/0.1\right)\times60$$\end{document}

The upper and lower bounds of the 95% confidence interval were used as limits for the realistic range of elimination rates during back-calculation:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CI_{95\%}=\overline x+z\left(SD/\surd\right)$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CI_{95\%\;real}=0.8342\pm1.96\left(0.02992/\surd75\right)=0.08342\pm0.00677$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CI_{95\%\;real}=\left[0.7665;0.09019\right]$$\end{document}

For the realistic model, the standard deviation of the mean elimination rate was ± 0.02992‰/h; the 95% confidence interval was [0.07665; 0.09019]‰/h, and the margin of error was ± 0.00677‰/h (margin of error = (CI95% upper limit − CI95% lower limit)/2 = (0.09019‰/h − 0.07665‰/h)/2 = 0.00677‰/h). On this basis, the following realistic time deductions (Δt__real) can be derived for retrograde calculation given an initial measured BAC within the respective concentration interval (see the first column of Table 3):

BAC range [‰] | Elimination rate r [‰/h] | Calculation(Δtreal = ((0.1 − r)/0.1) × 60) | ∆treal = realistic time reduction [min]
0.06–0.09 | 0.07665 | (0.1–0.07665)/0.1 × 60 | 13.99
0.10–0.12 | 0.08342 | (0.1–0.08342)/0.1 × 60 | 9.95
0.13–0.15 | 0.09019 | (0.1–0.09019)/0.1 × 60 | 5.89

Applying this model would entail subtracting approximately 14 min from the time interval between blood sampling and the incident for back-calculation of the minimum BAC when the measured BAC lies within [0.06–0.09‰]; a deduction of 10 min for BACs within [0.10–0.12‰]; and a deduction of 6 min when starting from [0.13–0.15‰]. This rule is intended to compensate for slower elimination at low BAC values and to prevent overestimation of BACretrograde without unduly restricting the applicability of retrograde calculation. It provides acceptable reliability, with a low margin of error and proportionate time deductions (see Fig. 7).

Fig. 7: Single-point analysis of all measurement values including retrograde extrapolation simulation: conventional conservative model vs. realistic model vs. extreme model

To validate this retrograde rule (“realistic model”), a Monte Carlo simulation with 1,000 iterations per BAC interval (as specified in the first column of the table) was performed. Inter-individual variability was incorporated by drawing, for each iteration, an elimination rate r from a normally distributed random distribution with mean x̄ = 0.08342‰/h and empirically determined standard deviation (SD) of 0.02992‰/h (n = 75). Using the standardised formula (see General formula for back-calculation deduction), the corresponding realistic back-calculation time deduction Δtreal was computed as described above.

Subsequently, for each of the three categories of initial BAC values ([0.06–0.09‰], [0.10–0.12‰], [0.13–0.15‰]), the corresponding empirical elimination rate r was applied as described in the Methods. The resulting Δtreal values were statistically evaluated (Table 4).

BAC range [‰] | Elimina-tion rate r [‰/h] | Mean ∆treal [min] | SD ∆treal [min] | Minimum ∆treal [min] | Maximum ∆treal [min] | 95% CI [min]
0.06–0.09 | 0.07665 | 13.98 | 2.04 | 7.66 | 20.68 | [10.10; 17.84]
0.10–0.12 | 0.08342 | 9.89 | 2.06 | 3.43 | 16.83 | [5.97; 14.01]
0.13–0.15 | 0.09019 | 5.82 | 2.10 | -1.39 | 11.51 | [1.53; 9.84]

The Monte Carlo simulation results showed dispersion consistent with the analytically calculated time deductions and confirmed the methodological robustness and validity of the derived time deductions Δtreal across all three interval ranges. Moreover, the narrow confidence intervals provide a well-founded means of accounting for individual variability in retrograde BAC estimation.

The negative minimum observed in the range [0.13–0.15‰] is purely statistical in nature and results from isolated random draws with r > 0.1‰/h. In forensic practice, Δtreal would instead be set to 0 in such cases to avoid adding an artificial increase in the back-calculation interval.

Overall, these findings confirm the applicability of the retrograde model for measured BAC values < 0.15‰.

The realistic model was not intended as the final forensic recommendation, but as an intermediate derivative model to assess whether a differentiated, data-driven correction of the back-calculation interval is methodologically plausible. It demonstrates that the reduced elimination rate below 0.15‰ can be translated into proportionate time deductions that vary according to the measured BAC range. However, because forensic back-calculation of a minimum BAC must avoid any relevant risk of overestimation in disfavour of the subject, the realistic model primarily serves as a plausibility and validation step. The final practical recommendation is therefore based on the extreme model, which deliberately prioritises maximum forensic safety over precision.

The extreme model, which excludes any disadvantage to the affected individual due to even a slight overestimation of the back-calculated BAC, is based on the lowest elimination rate identified (0.02859‰/h). This was derived from the terminal low-concentration segment of the curve with the lowest elimination rate. Inspection of the corresponding BAC profile showed regular elimination behaviour above 0.15‰, with rates within the expected range. The markedly lower value in the terminal segment is attributable to minor fluctuations and a flattened regression. It therefore reflects late-phase variability rather than a biologically implausible elimination pattern. Accordingly, the extreme back-calculation deduction Δtextreme is given as follows (Table 5):

BAC range [‰] | Most conservative elimination rate r [‰/h] | Calculation(∆textreme = ((0.1 – r)/0.1) × 60) | Extreme deduction ∆textreme [min]
0.06–0.15 | 0.02859 | (0.1–0.02859)/0.1 × 60 | 42.85

For instance, if blood sampling was performed 5 h after the incident and the measured BAC was 0.06‰, the uncorrected conservative model would yield a retrograde BAC of 0.56‰ (0.06‰ + 0.1‰/h × 5 h). Under the realistic model, the applicable time deduction for the range 0.06–0.09‰ would be approximately 14 min, resulting in an effective back-calculation interval of 4 h 46 min and a retrograde BAC of approximately 0.54‰. Under the extreme model, the uniform deduction of 43 min would reduce the effective interval to 4 h 17 min, yielding approximately 0.49‰. This example illustrates that the realistic model provides a data-driven estimate of the expected correction, whereas the extreme model applies the maximum empirically justified deduction in favour of the subject.

Applying this one-off deduction from the back-calculation interval (Δtextreme) guarantees, based on the present data set, that not a single back-calculation result overestimates a true BAC, thereby ensuring an interpretation in favour of the subject. The most conservative elimination rate (r = 0.02859‰/h) represents the slowest elimination behaviour actually observed in the 75 valid curves analysed.

Consequently, a uniform time deduction of 43 min for retrograde calculation of the minimum BAC provides maximum forensic safety (see Figs. 7, 8 and 9).

Fig. 8: Example – Blood alcohol concentration-time curve with fitted regression lines

Fig. 9: Example – Retrograde calculation using the conventional conservative model versus the extreme model, starting from 0.06‰ over 5 h

As the extreme model is based on the lowest observed elimination rate and does not incorporate dispersion, a repeated Monte Carlo simulation to validate its results is not required. Although this model achieves maximum forensic safety, it will be overly restrictive in the vast majority of cases due to the large time deduction and may yield unrealistically low values of BACretrograde. On the other hand, cases in which retrograde calculation would otherwise have been omitted below 0.15‰ are likely to involve scenarios requiring longer back-calculation intervals. Accordingly, the extreme model may open up additional possibilities for evaluation in such constellations.

In all models, back-calculation is performed using an hourly elimination rate of 0.1‰/h; however, in the realistic and extreme models, the corresponding subtractions from the relevant back-calculation intervals must additionally be applied.

The assumption of a linear relationship adopted for reasons of practicality appears justified on the basis of the present data sets. Accordingly, retrograde calculation of blood alcohol concentrations should also be scientifically comprehensible and defensible for measured values < 0.15‰, particularly when incorporating a time correction component.