# Dosage of Antibiotics: Should It Be the Same for Everyone?

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.

Our work involves the development of a PBPK model for the antibiotic ertapenem; Figure 1 displays a schematic of the model [4]. 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 [1]. 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.

**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.

**References**

[1] 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.

*Antimicrobial Agents and Chemotherapy, 45*(6), 1915-1918.

*Highlights of prescribing information, Invanz®*. Ertapenem for injection.

*Mathematical Biosciences and Engineering, 13*(1), 119–133.

*American Society for Microbiology, 46*(11), 3506-3511.

*Journal of Antimicrobial Chemotherapy, 53*(2), ii23-ii28.

*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. |