Interpretation and presentation of alcohol results
Widmark’s equation
A question that is often asked as part of a DUI investigation is ‘How much alcohol is the suspect likely to have consumed to have produced the blood or breath alcohol determined by testing?’ or alternatively, ‘Is the subject’s recollection of how much alcohol they drank conversant with the blood or breath alcohol value determined?’
During the early part of the 20th century, E. M. P. Widmark, a Swedish physician, did much of the foundational research regarding alcohol pharmacokinetics in the human body (Widmark 1981). In addition, he developed an algebraic equation that allowed any one of seven variables to be estimated given the other six. According to Widmark’s equation (11.1), the amount of alcohol consumed (N) is a function of these several variables:
N=f(W, r, Ct, b, t, z) (11.1)
where N is the amount of alcohol consumed, W is the body mass, r is the volume of distribution (a constant), Ct is the BAC, b is the alcohol elimination rate, t is the time since the first drink and z is the volume of alcohol per drink. These variables are related according to equation (11.2):
where N is the number of drinks consumed, W is the body mass in ounces (or kg), r is the volume of distribution (a constant relating the distribution of water in the body in L/kg), Ct is the BAC in kg/L, β is the alcohol elimination rate in kg/L/h, t is the time since the first drink in hours, z is the volume of alcohol per drink in fluid ounces (or L) and 0.8 is the density of ethanol (i.e. 0.8 ounces per fluid ounce or 0.79 g/mL). It should be noted that the units used to measure body mass and mass of alcohol in the drink (i.e. 0.8 X volume of alcohol in the drink) must be the same. A different value is used for the volume of distribution for males and females. Women generally have a higher proportion of fat relative to water compared to men and hence on a mass basis men have a higher percentage of water per unit mass. As alcohol is water soluble and therefore partitions preferentially into the aqueous phase, the volume of distribution is greater in men than women. The generally accepted value of r for men is 0.68L/kg and for women is 0.55L/kg. The examples shown in Figure 1 illustrate typical uses of the Widmark equation in DUI cases. Table 11.4 illustrates typical ranges of values for the alcohol content of different bever ages, which can help to define ‘a drink’ in terms of its percentage alcohol content by volume. Figure 2 illustrates a typical Widmark alcohol concentration–time curve for an individual who consumed alcohol on an empty stomach. The rapid rise to the peak concentration, typical of empty-stomach consumption, is observed along with the classic linear elimination portion of the curve. Widmark’s equation was applied to this individual by computing the estimated BrAC (see below) as a function of time. A linear estimation curve is observed below that of the subject’s actual BrAC curve. For this individual, Widmark’s method (using the assumed r and b parameters) underestimated the actual BrAC. Experimental plots such as this are useful to assess the validity of the Widmark estimation methods but cannot normally be applied in DUI cases because BAC or BrAC values are typically taken at a single point in time and hence the full pharmacokinetic dataset required is not available.
Figure 1 Use of the Widmark equation.
Figure 2 Comparison of actual BrAC with the Widmark estimate.
