Exploring Mathematical Models for Calculating Blood Alcohol Concentration

Transcription

1 Exploring Mathematical Models for Calculating Blood Alcohol Concentration S. J. Kouba, M. B. M. Elgindi, R. W. Langer Department of Mathematics University of Wisconsin Eau Claire Eau Claire, WI ABSTRACT. A mathematical model for blood alcohol concentration (BAC) that takes into account gender, kidney function, liver function, the amount of alcohol consumed, body weight, and period of alcohol consumption is developed. Graphical results from a Visual Basic program implementing this model agree with the experimental data gathered. Results from this model can be used to inform the public about responsible use of alcohol. It will also allow a person to trace back BAC information to determine the BAC at a time an offense occurred. Finally, accurate determinations of the level and length of time the alcohol remains in the blood are important in studies related to the effects of alcohol on the brain cells and other body organs.. Introduction Blood alcohol concentration is affected by three different processes. These processes are absorption of alcohol into the bloodstream, metabolism of alcohol, and excretion of alcohol through breath and urine. Each of these processes is influenced by other factors. Absorption is influenced by the amount of food in the stomach, the rate the stomach empties into the intestines, and gender. Metabolism is influenced by the liver function, body weight, and the period of time over which alcohol is consumed. Finally, excretion is influenced by the amount of alcohol, gender, health, and the period of time over which alcohol is consumed. In order to make an

2 accurate model, we need to take all of these factors into account. Most of these factors are incorporated in our model. The effect of the amount of food consumed while drinking, however, still needs to be included into the model. This will be done in a future research project. Our model is based on the continuity equation, what enters the body is equal to what leaves the body. The amount of alcohol entering the body has to equal the amount of alcohol leaving the body through metabolism and excretion. 2. Mathematical Model for Blood Alcohol Concentration The following model consists of two differential equations for the blood alcohol concentration, the first for the period of alcohol consumption and the second for the period after the person has stopped drinking. We shall call these two phases the Drinking Phase and the Recovery Phase, respectively. Variables x = concentration of alcohol in the blood (BAC) u = amount of alcohol consumed / (b x amount of body water per given weight x s) b = gender ratio s = the length of the alcohol consumption period α = liver function parameter β = kidney function parameter υ = gender parameter t = time T = time when the alcohol concentration returns to zero m = time when the alcohol concentration reaches its peak value 2

3 M = peak alcohol concentration value During the Drinking Phase: Equations dx x u, t s, x () dt x (2.) This model is used to find the peak alcohol concentration, which we assume occurs at the end of the alcohol consumption period, i.e. we will assume that M = x (s). Note that this is a good approximation for large s. However, the peak of the alcohol concentration actually occurs slightly after the end of alcohol consumption i.e. m > s. During the Recovery Phase, we have the following model: dx dt x x, s t, x ( s) M (2.2) This model is used to determine the length of time it takes for the BAC to fall below the legal limit or some other prescribed level. 3. Results In our models we set α = and β =. The gender ratio b is different for males and females. This is because men and women have a different percentage of body fat. Since women have a higher percentage of body fat and alcohol is not absorbed by body fat, they have a smaller amount of body water in which the alcohol can be absorbed. This is like dropping the same amount of alcohol into a smaller pail of water. The alcohol concentration is higher in the smaller 3

4 pail. In [] McKim states that the percentage of body water for women is 66%, and for men the percentage is 75%. Accordingly, we took b = for men, and.88 for women. The gender parameter ν is also different for males and females since men and women absorb and metabolize alcohol differently. For men we took ν = 4 and.4 for women. With these values we compared our numerical results to the experimental data. The following figures are graphs of some results for the blood alcohol concentration found using an Excel/VBA program that we developed for approximating BAC based on the models (2.) and (2.2). We found that our approximations of the BAC compare well for both the male and female with the experimental BAC data presented in [2]. Our figures also demonstrate the effect of different factors on the BAC. For example, our results demonstrate how the gender, amount of alcohol consumed, weight, and health affect the BAC. Since the model s results agree well with experimental results, we know that we have a good model for the blood alcohol concentration. Blood Alcohol Concentration -Female Blood Alcohol Concentration -Male Figure Figure 2 Figures, 2 compare a female s BAC with a male s BAC while keeping all other factors the same. For these graphs we used a weight of 6 lbs, the amount of alcohol consumed as 5 4

5 drinks of the same type, the course of drinking as one hour, and we used the values for the liver and kidney function parameters stated above Figure 3 Figure 4 Figures 3, 4 above compare the BAC of two males with different weights while all other factors are kept the same. We set the amount of alcohol consumed as 4 drinks of the same type, the course of drinking as three hours, and we used the same kidney and liver function parameters as above. In Figure 3 we took the weight to be 5 lbs, while in Figure 4 we took it to be 25 lbs. Blood Alcohol Concentration-Female Blood Alcohol Concentration-Female Figure 5 Figure 6 5

6 Figures 5, 6 compare the BAC of two females consuming different amounts of the same of alcohol while all other factors remain the same. In both cases we took a weight of 4 lbs, the course of drinking as four hours and the same liver and kidney function parameters. Figure 5 assumes 2 drinks while Figure 6 assumes 5 drinks Figure 7 Figure 8 Figures 7, 8 compare the BAC of two males with different liver function parameters. In both cases we took a weight of 8 lbs, the course of drinking as three hours, 3 drinks of the same type, and the same kidney function parameter. Figure 7 is for a healthy the liver while Figure 8 is for an unhealthy liver. 4. Conclusions In this paper we developed a model for describing blood alcohol concentration that takes into account several factors including gender, liver function, kidney function, amount of alcohol consumed, body weight, and the length of the drinking period. This model agrees well with the experimental data from [2]. Because it agrees well with experimental data, it could be used in many different settings. It could be used to trace back from a current level of blood alcohol 6

7 concentration at the time when an alleged offense had occurred. It could also be used by educators to inform the public about how alcohol affects the body. Finally, it could be used in biological studies concerning the alcohol effects on the brain and other body organs functions. ACKNOWLEDGMENT. This research was supported by the National Science Foundation Grant /92974 (REU site program SUREPAM) and the University of Wisconsin Eau Claire s Center for Alcohol Studies and Education. The authors are grateful to Dr. Mahmoud Ahmed of Marshfield Clinic for helping them understand the physical aspects of the project. REFERENCES [] MCKIM, W A., Tutorial on Calculating Blood Alcohol Levels Memorial University of Newfoundland. < [2] PAVLIC, M., GRUBWIESER, P., LIBISELLER, K., and RABL, W. "Elimination Rates of Breath Alcohol." Science Direct (26). 7