Mathematical Modeling of Biofilm Eradication Using Optimal Control

We propose and analyze a model for antibiotic resistance transfer in a bacterial biofilm and examine antibiotic dosing strategies that are effective in bacterial elimination. In particular, we consider a 1-D model of a biofilm with susceptible, persistor and resistant bacteria. Resistance can be transferred to the susceptible bacteria via horizontal gene transfer (HGT), specifically via conjugation. We analyze some basic properties of the model, determine the conditions for existence of disinfection and coexistence states, including boundary equilibria and their stability. Numerical simulations are performed to explore different modeling scenarios and support our theoretical findings. Different antibiotic dosing strategies are then studied, starting with a continuous dosing; here we note that high doses of antibiotic are needed for bacterial elimination. We then consider periodic dosing, and again observe that insufficient levels of antibiotic per dose may lead to treatment failure. Finally, using an extended version of Pontryagin’s maximum principle we determine efficient antibiotic dosing protocols, which ensure bacterial elimination while keeping the total dosing low; we note that this involves a tapered dosing which reinforces results presented in other clinical and modeling studies. We study the optimal dosing for different parameter values and note that the optimal dosing schedule is qualitatively robust.

💡 Research Summary

The paper presents a comprehensive mathematical framework for studying antibiotic treatment of bacterial biofilms that contain three phenotypic sub‑populations: susceptible (Bs), persister (Bp), and resistant (Br) cells. The authors formulate a one‑dimensional moving‑boundary PDE system that couples the volume fractions of these bacterial groups with the concentrations of a limiting nutrient (C) and an antibiotic agent (A). The biofilm thickness L(t) evolves according to an advective growth term derived from the total biomass production and a quadratic detachment term, thereby capturing both growth and erosion processes.

The governing equations incorporate biologically motivated nonlinearities: growth follows Michaelis–Menten kinetics (fs(C), fr(C)), horizontal gene transfer (HGT) from susceptible to resistant cells is modeled by a term μ Br Bs, and antibiotic‑induced killing is represented by linear or Hill‑type functions kd A for susceptible cells and gp(A) for persisters. Persisters arise from susceptible cells at a rate kl fs(C) and revert back at a rate kr(A) that depends on the presence of antibiotic. Resistant cells experience a fitness cost q and may revert to susceptibility via mis‑segregation. Antibiotic diffusion is described by a standard diffusion term DA∂²A/∂x² together with consumption terms pi(A)Bi that are larger for susceptible cells (νs>νr>νp). Nutrient diffusion and consumption are treated analogously.

The authors first analyze the model without control (u(t)=0). By setting time derivatives to zero they identify four classes of equilibria: (i) the disease‑free (all bacterial fractions zero), (ii) boundary equilibria where only one phenotypic class survives, and (iii) interior coexistence equilibria where all three phenotypes persist. Linearization around each equilibrium and application of the Routh–Hurwitz criteria yield explicit conditions on parameters such as the antibiotic killing rate kd, the HGT rate μ, the fitness cost q, and the nutrient supply that determine stability. In particular, a sufficiently high kd or a large dosing intensity guarantees global stability of the disease‑free equilibrium, whereas low kd combined with high μ leads to a stable coexistence state, indicating treatment failure.

Next, the paper investigates prescribed dosing strategies. Continuous high‑dose therapy rapidly drives the system toward the disease‑free state but may be clinically infeasible due to toxicity. Periodic (on‑off) dosing is explored by varying the dose period τ and the on‑dose fraction α. Numerical simulations reveal a sharp threshold: short τ or low α allow resistant cells to dominate, while long τ with adequate α can suppress persisters and enable eradication. These results corroborate earlier experimental observations that intermittent dosing can be more effective than continuous low‑dose regimens when a fitness cost is present.

The central contribution is the formulation and solution of an optimal control problem. The objective functional minimizes the total administered antibiotic ∫₀ᵀ u(t) dt while enforcing a terminal constraint that the total viable biomass (∫₀ᴸ(T) (Bs+ Bp+ Br) dx) stays below a clinically relevant threshold. By extending Pontryagin’s Maximum Principle to the PDE‑ODE coupled system with a moving boundary, the authors derive the Hamiltonian, adjoint equations for the state variables, and the optimality condition for the control. The resulting optimal control law is of bang‑bang type with a singular arc, but numerical implementation (using a forward–backward sweep method) shows that the optimal schedule follows a “tapered” pattern: an initial high dose that quickly reduces susceptible cells, followed by a gradual decline that maintains pressure on persisters while limiting total drug exposure. Sensitivity analyses across a range of μ, DA, and nutrient inflow demonstrate that the tapered shape is robust; the exact timing and magnitude adjust modestly but the qualitative structure persists.

The discussion acknowledges modeling simplifications (one‑dimensional geometry, homogeneous diffusion coefficients, neglect of immune response) and suggests extensions such as multi‑species biofilms, combination therapy, and stochastic HGT. Nonetheless, the work provides a rare analytical treatment of optimal antibiotic dosing in a spatially explicit biofilm context, linking equilibrium analysis, numerical experiments, and control theory. It offers actionable insight: clinicians may achieve eradication with lower total antibiotic use by front‑loading the dose and then tapering, especially when resistance incurs a fitness penalty. This bridges theoretical biofilm modeling with practical antimicrobial stewardship.