Various aspects of mathematical modeling and statistical analysis are used daily in health care and by the pharmaceutical industry. Administering an effective dose of antibiotics is key when treating certain types of infections. In this era of antibiotic resistance, receiving the correct dosage is particularly important; mathematics can play an important role in this process.
Figure 1. Schematic of the model.
Mathematicians often develop a specific type of model—called a physiologically-based pharmacokinetic (PBPK) model—to investigate the uptake, distribution, metabolism, and elimination of a drug. PBPK modeling incorporates known physiological parameters, such as body weight, organ volumes, and blood flow rates in particular tissues, as well as information about urine and feces excretion. Researchers then use differential equations to examine the movement of an antibiotic through the body and determine the concentrations of the drug in both the blood and specific organs and tissues.
Our work involves the development of a PBPK model for the antibiotic ertapenem; Figure 1 displays a schematic of the model . Ertapenem, an antibiotic commonly used to treat community-acquired and mixed infections, is part of the class of antimicrobials called carbapenems [5, 6]. Carbapenems have the widest spectrum of antimicrobial activity against both gram-positive and gram-negative bacteria [2, 7]. Unlike other carbapenems which have a half-life of approximately one hour and must be administered several times a day, ertapenem has a half-life of approximately four hours and can thus be administered just once a day. This antibiotic is usually administered intravenously over a thirty-minute period to individuals who are hospitalized due to infection.
Ertapenem is highly bound to human plasma proteins. Only the free, unbound portion of the drug actually saturates the tissues and can be excreted . Moreover, because only the free (or unbound) concentration of the drug is considered to be medicinally effective, we chose to examine both the total concentration and the free concentration in the blood.
To simulate this model, we need values for all represented parameters; some of these are known from the literature and based on experimentation, but others are unknown. Fortunately, there exists data from clinical trials. By optimizing a cost function, we can estimate values of these unknown parameters to fit the simulated solution of the model to the data. In our particular model, we compared blood concentrations of ertapenem determined by the model to data from published clinical trials for males of normal height and normal weight. Figure 2 shows the model’s fit to the data for the total concentration of ertapenem in the blood; the Nix data was used in the optimization and the Merck data was used for validation [3, 6].
Figure 2. Total concentration of ertapenem in blood [3, 6].
Because of the way ertapenem is absorbed, the amount of fat in the body can affect how the drug binds, how quickly it passes through the body, and thus how effective it is. Our work has focused on determining if body height and body weight, which govern the body mass index (BMI) for an individual, should play a role in antibiotic dosage.
Table 1. Classifications of BMI.
BMI is the ratio of weight (in kilograms) to the square of the height (in meters). We considered five classifications of BMI (see Table 1). For each classification, we analyzed the area under the curve of the total concentration in the blood as well as the area under the curve of the free concentration in the blood, the peak total concentration in the blood, the minimum free concentration in the blood, and the amount in the fat. Figure 3 illustrates the area under the curve of the total concentration of ertapenem in the blood for adult males and females. Although the results are similar for males and females, we note that females have a higher range of concentrations. We also observe that there is a lower area under the total concentration curve for both males and females with higher BMI; this could lead to potentially less effective antibiotics. Additionally, those with higher BMI have a higher minimum free concentration, and less chance of developing resistant bacteria. For individuals with lower BMI, there is a higher area under the total concentration curve and a lower minimum free concentration, which could lead to a greater probability of undesired side effects and possible development of antibiotic resistance. People with lower BMI also had higher peak concentrations than those with higher BMI. Additionally, females had substantially higher peak concentrations than males.
Figure 3. Area under the total concentration curve of ertapenem in the blood. 3a. Distribution of area under the curve (AUC) for males; the solid black vertical line represents the AUC for a normal height, normal weight male. 3b. Distribution of AUC for females.
It is clear that both BMI and sex play significant roles in the distribution and elimination of ertapenem. Researchers should continue examining such aspects to work toward more effective dosing of the drug, which could significantly impact your health.
 Clewell, H.J. III, Reddy, M.B., Lave, T., & Andersen, M.E. (2008). Physiologically based pharmacokinetic modeling. In S.C. Gad (Ed.), Preclinical Development Handbook: ADME Biopharmaceutical Properties (pp 1167-1127).Hoboken, NJ: John Wiley & Sons, Inc.
 Fuchs, P.C., Barry, A.L., & Brown, S.D. (2001). In vitro activities of ertapenem (mk-0826) against clinical bacterial isolates from 11 North American medical centers. Antimicrobial Agents and Chemotherapy, 45(6), 1915-1918.
 Merck & Co. Inc. (2012). Highlights of prescribing information, Invanz®. Ertapenem for injection.
 Joyner, M.L., Manning, C.C., Forbes, W., Maiden, M., & Nikas, A.N. (2016). A physiologically-based pharmacokinetic model for the antibiotic ertapenem. Mathematical Biosciences and Engineering, 13(1), 119–133.
 Majumdar, A.K., Musson, D.G., Birk, K.L., Kitchen, C.J., Holland, S., McCrea, J.,…,Rogers, J.D. (2002). Pharmacokinetics of ertapenem in healthy young volunteers. American Society for Microbiology, 46(11), 3506-3511.
 Nix, D., Majumdar, A., & DiNubile, M. (2004). Pharmacokinetics and pharmacodynamics of ertapenem: an overview for clinicians. Journal of Antimicrobial Chemotherapy, 53(2), ii23-ii28.
 Shah, P.M., & Isaacs, R.D. (2003). Ertapenem, the first of a new group of carbapenems. Journal of Antimicrobial Chemotherapy, 52, 538-542.
|| Cammey Cole Manning is a professor of mathematics and head of the Mathematics and Computer Science Department at Meredith College. She earned her B.S. in mathematics and computer science at Duke University and completed her Ph.D. in applied mathematics with a concentration in computational mathematics at North Carolina State University. She has been actively involved with SIAM, the Association for Women in Mathematics, and the Statistical and Applied Mathematical Sciences Institute, particularly in mentoring and planning professional development programs and workshops for undergraduates, graduate students, and early-career individuals.
|| Michele Joyner is an associate professor of applied mathematics in the Mathematics & Statistics Department at East Tennessee State University. She earned her B.S. in mathematics at the Georgia Institute of Technology and completed her Ph.D. in applied mathematics with a concentration in computational mathematics at North Carolina State University. She has actively been involved in directing students interested in industrial mathematics through a research course in collaboration with many local companies in East Tennessee. She is also a member of SIAM, the Mathematical Association of America, and the Association for Women in Mathematics.