PBPK modeling

The tutorial approaches model design as a short chain of yes-or-no questions about the drug formulation. Does the nanoparticle dissolve in aqueous fluid, or does it stay intact as a colloid? Is it large and hydrophilic, so crossing membranes is the slow step, or small and lipophilic, so blood flow is the bottleneck? Should macrophage uptake be expected? The answers pick between a perfusion-limited (flow-limited) and a permeability-limited (membrane-limited) layout, and decide whether dissolution and macrophage subcompartments are needed ^[1]. Answering these questions before writing equations keeps the later parameter fitting and validation grounded.

The paper illustrates four layouts that come out of the decision flow. In (a), a conventional non-nanoparticle drug uses a flow-limited structure: each tissue is a single well-mixed compartment fed by arterial blood and drained to venous blood. In (b), a colloidal nanoparticle that does not dissolve uses a membrane-limited structure, splitting each tissue into a capillary and a tissue subcompartment separated by a permeability term. In (c), the same membrane-limited layout adds phagocytizing-cell (PC) subcompartments inside tissues with high macrophage density. In (d), a dissolvable nanoparticle uses a flow-limited structure where particles release free drug at rate k_rel into the surrounding compartment ^[1].

Each tissue compartment is described by a mass-balance ODE built from the law of mass action. For a baseline flow-limited tissue, compartment 2 fed by compartment 1 obeys Eq. (1), where Q is blood flow, C₁ and C₂ are concentrations, and k_p is the partition coefficient (a distribution coefficient for colloidal nanoparticles). Tissues that clear drug add an excretion term dM_ex/dt collecting renal and biliary clearance, giving Eq. (2). For colloidal nanoparticles, a permeability factor X/(1+X) gates flow across the capillary–tissue membrane, giving Eq. (5). For dissolvable nanoparticles, a first-order release term k_rel · V_t · C couples the particle and dissolved-drug equations with opposite signs on each side, giving Eq. (4) ^[1].

Because colloidal nanoparticles do not dissolve, they are subject to immune clearance. The tutorial treats macrophage uptake as a small two-step subcompartment with absorption rate k_ab and desorption rate k_de, placed inside tissues with high macrophage density such as the liver (Kupffer cells). The governing equation for those tissues picks up an extra term, (k_de · M_{2,m}) − (k_ab · V_2 · C_2), where M_{2,m} is the amount currently held in macrophages, giving Eq. (6). This term is what allows the model to capture the long tissue residence times that colloidal nanoparticles exhibit in vivo ^[1].

Inputs split into three groups. Physicochemical properties, such as partition coefficient, molecular weight, and solubility, depend only on the drug. Drug–biological properties, such as fraction unbound, Michaelis–Menten constants, and permeability, depend on both the drug and the organism. Anatomic and physiological properties, such as organ volumes, surface areas, blood flows, and expression levels, come from the organism. Many are taken from literature or databases; the rest are fitted to in vivo biodistribution data. Nanoparticle-specific additions include the release constant, maximum uptake rate in phagocytic cells, Hill coefficient, and desorption constant. Sensitivity analysis helps decide which parameters are worth the effort to estimate. For instance, whether the dissolution rate of a protein-corona-coated nanoparticle actually moves the output. Validation is performed on held-out data, usually by comparing log₁₀(predicted) against log₁₀(measured) tissue concentrations via R² ^[1].