Table 1.

Governing equations of physical phenomena involving transport of drugs and physiological fluids.

Physical Phenomenon	Mathematical Relation	Definitions	Example Applications
Discrete particle dynamics at atomistic scale	Newton’s 2. Law of Motion $m_i\frac{d^2{\overrightarrow{r}}_i}{d{t}^2}={\overrightarrow{F}}_i$	mi: mass of the ith particle ${\overrightarrow{r}}_i$: position of the ith particle ${\overrightarrow{F}}_i$: external force on ith particle due to intermolecular potentials.	• Aspherical modeling of atoms and coarse-grained particles (MD)(17) • NP-plasma membrane interaction (MD) (16) • NP targeting kinetics (DPD)(29)
Discrete particle dynamics in the presence of time-scale separation	Langevin Equation $m_i\frac{d^2{\overrightarrow{r}}_i}{d{t}^2}=\xi \frac{d{\overrightarrow{r}}_i}{dt}+{\overrightarrow{F}}_{B,i}$	${\overrightarrow{F}}_{B,i}$: random Brownian force exerted on the ith particle ξ: friction coefficient	• Mobility and diffusivity analysis of charged particles (BD) (26) • Increasing speed of brute-force MD while retaining meaningful details (milestoning) (MD) (28) • Estimation of effective diffusivity within porous ECM (61,64)
Continuum fluid dynamics	Navier Stokes Equations $\rho \left(\frac{\partial \overrightarrow{v}}{\partial t}+\left(\overrightarrow{v}\bullet \nabla \right)\overrightarrow{v}\right)=-\nabla p+\mu {\nabla }^2\overrightarrow{v}$	ρ: density of the fluid $\overrightarrow{v}$: velocity field of the fluid p: pressure field of the fluid μ: dynamic viscosity of the fluid	• Analysis of NP distribution within micro-vessels (Coarse-grained MD and IMEFEM)(60) • Flow of interstitial fluid within 3D fiber network (FEM) (61,63)
Continuum fluid dynamics in porous media	Darcy’s Law $\overrightarrow{v}=-K\nabla p$	K: hydraulic conductivity	• Estimation of hydraulic conductivity of porous ECM (FEM) (61,63)
Spatiotemporal distribution of species in continuum	Advection-Diffusion Equation $\frac{\partial C}{\partial t}=-\overrightarrow{v}\bullet f\nabla \mathrm{C}+\nabla \bullet \left(D_{eff}\nabla \mathrm{C}\right)$	C: species concentration Deff: effective diffusivity f: retardation coefficient	• Intra-tissue particle distribution and penetration (BD)(65) • Biotransport in arterial blood clots (FEM) (40)
Dynamics of species transport (accumulation and clearance) in physiological compartments	Compartmental Mass Balance Equations   $\dot{m}={\dot{m}}_{in}-{\dot{m}}_{out}-{\dot{m}}_{elimination}$ Flow-Limited Model $V\frac{dC}{dt}=\sum Q_{in}{C}_{in}-\sum Q_{out}{C}_{out}-R_{cl}V$ Interface-Limited Model    $\frac{dC}{dt}=P{C}_{in}-\frac{P}{K_p}C-R_{cl}$	V: compartment volume C: species concentration within the compartment. Qin, Qout and Cin, Cout: volumetric flow rates into/out of the compartment and corresponding concentrations in fluids Kp: partition coefficient P: interface transport coefficient Rcl: rate of elimination (clearance) per unit volume	• Adaptation of adult PBPK model to analyse paediatric plasma profiles (Interface-Limited PBPK)(47) • Alleviation of vascular barriers using focused ultrasound combined with microbubbles (Flow-Limited PBPK) (53)