We generated a mathematical model describing the behavior of a mixture of sensitive and resistant bacterial populations in a mouse-thigh infection model under differing amounts of antimicrobial pressure. The mass-balance equations (Equations 1-5 parallel first-order inhomogeneous differential equations) for the combined pharmacokinetic-pharmacodynamic model that describe the exposure-response relationships for the sensitive and resistant subpopulations are described below. Equations 6 and 7 correspond to symbols inserted into equations 4 and 5.

Equation 1

dX₁/dt = -K_a • X₁

Equation 2

dX₂/dt = K_a • X₁ - [(SCL/V_c) + K₂₃] • X₂ + K₃₂ • X₃

Equation 3

dX₃/dt = K₂₃ • X₂ - K₃₂ • X₃

Equation 4

dN_S/dt = K_gmax-S • N_S • E - K_kmax-S • M • N_S

Equation 5

dN_R/dt = K_gmax-R • N_R • E - K_kmax-R • N_R • M

Equation 6

E = 1 - (N_R + N_S)/POPMAX

Equation 7

M = (X₂/V_c)^H/[(X₂/V_c)^H + EC₅₀^H]

Equations 1-3 describe levofloxacin pharmacokinetics in the mouse (a standard two-compartment open model with first-order input and elimination). These equations were fitted to the pharmacokinetic data obtained from a separate cohort of infected mice. Mean population-parameter estimates obtained from the pharmacokinetic analysis were then fixed in Equations 1-3 for the pharmacodynamic analysis. X₁ is the amount of drug in the absorption compartment (i.e., the intraperitoneal space), X₂ is the amount of drug in the central compartment (i.e., serum), and X₃ is the amount of drug in the peripheral compartment (i.e., tissues and organs, including the thigh muscles). K_a (h^-1) is the first-order absorption rate constant for the diffusion of drug from the peritoneal cavity into the central (serum) compartment. SCL (l/h) is the rate of clearance of drug from serum (central compartment). V_c is the volume of the central compartment. K₂₃ (h^-1) and K₃₂ (h^-1) are the first-order transfer constants between the central and peripheral compartments.

Equations 4 and 5 describe the rates of change of the sensitive and resistant subpopulations, respectively, over time. The model equations for describing the rate of change of the numbers of organisms in the sensitive and resistant bacterial subpopulations were developed based on the in vivo observation that bacteria at the site of infection are in logarithmic growth phase in the absence of drug and exhibit an exponential density-limited growth rate (Equation 6). There is one equation to describe the sensitive population (Equation 4) and one to describe the resistant bacterial subpopulation (Equation 5). In each, first-order growth was assumed, up to a density limit. Each population has an independent growth rate constant (K_gmax-S for sensitive and K_gmax-R for resistant). As the organisms approach maximal bacterial density, they approach stationary phase. This is accomplished mathematically by multiplication of the first-order growth terms by E (a logistic growth term; Equation 6). The maximal bacterial density (POPMAX) is identified as part of the estimation process. Most of the information used in identifying this parameter is derived from the bacterial growth in the control group.

Equations 4 and 5 also allow the antibacterial effect of the different drug doses administered to be modeled. For both sensitive and resistant populations, there is an independent effect of the drug dose on the two populations, one mediated through Equation 4 (sensitive population) and one through Equation 5 (resistant population). There is a maximal kill rate that the drug can induce for each population (K_kmax-S and K_kmax-R). The killing effect of the drug was modeled as a saturable Michaelis-Menten-type kinetic event (M; Equation 7) that relates the kill rate to serum drug concentration, where H is the slope constant and EC₅₀(mg/l) is the drug concentration at which the bacterial kill rate is half-maximal. The drug effect observed is a balance between growth and death induced by the drug concentrations achieved.

Other model forms were evaluated for Equations 4 and 5 that had growth rate constants that were dependent on drug concentration. Evaluation of these larger models did not demonstrate improved performance of the model. Consequently, under the rule of parsimony, the simpler model was employed.

Measured outputs were changes in total-population density (total = N_S + N_R) and resistant-mutant density (resistant = N_R), where the sensitive (S) and resistant (R) subpopulations were enumerated on drug-free plates and plates containing 3 × MIC of the drug, respectively.