Population Density Modulates Drug Inhibition and Gives Rise to Potential Bistability of Treatment Outcomes for Bacterial Infections
A. Main panel: Theoretical (solid and dashed lines) and numerical (shaded region) phase diagrams indicate treatment outcomes in PK/PD model as a function of initial cell density (ranging from 0 to the carrying capacity, C) and initial antibiotic concentration D0. Solid red lines, stable fixed points of population density (theory). Dashed red lines, unstable fixed points (theory). The curved dashed red line is the phase boundary (separatrix) indicating the critical density above which a population will survive. A region of growth bistability, where treatment can lead to success or failure depending on initial cell density, exists for antibiotic concentrations K0γ(0) ≤ D0 ≤ K0γ(C), where K0 is the MIC and γ(n) is a nonlinear function that depends on the maximum drug kill rate (gmin), the Hill coefficient (h), the drug decay rate (kd), and the dosing period T (S1 Text). Shaded regions indicate treatment failure in numerical solutions of the PK/PD model. Upper right inset: numerical solution of PK/PD equations for five different initial densities (indicated by red and black squares on the phase diagram). Lower inset: temporal dynamics of antibiotic concentration. For numerical phase diagram and simulations, gmin = -0.05 and ε = 0.9. Simulations in insets correspond to D0 = 200 (in units of MIC, K0). B. Phase diagrams from both theory (solid and dashed lines) and numerical simulations (shaded region) for increasing maximum kill rates (gmin = -0.25, -1, -2 from top to bottom) and populations densities on the order of 108 cells/mL (corresponding to the OD ranges measured here). gmin is measured in units of gmax; biologically, gmax≈1 hr-1 for bacteria, so one can also view these units as inverse hours. C. Initial dose of antibiotic (units of MIC, K0) required to clear infections of density OD = 0 (dashed red), OD = 0.4 (blue line), and OD = 0.8 (red line) for different maximum kill rates for the case with no density dependence (ε = 0, left), modest density dependence (ε = 0.5, middle), and strong density dependence (ε = 0.9, right). In all panels, the Hill coefficient h = 2, kd = ½, and T = 8, corresponding to a treatment period of 8 hours and a natural drug decay rate of ½ hr-1. Qualitatively similar results are found for other parameters (Figure G of S1 Text).