Table 1.

General descriptions of the common simulation methods applied for drug-delivery studies.

Scales	Length and Time Scales [74]	Descriptions	Formulations
Quantum scale	~10−10 m and ~10−12 s	The nuclei and electrons are the particles of interest at this scale and quantum mechanics (QM) methods are used to model their state by solving the Schrödinger wave equation [74].	$-\frac{h^2}{8{\pi }^{2}m}{\nabla }^2\mathsf{\varphi }{\left(r\right)}_k+U\left(r\right)\mathsf{\varphi }{\left(r\right)}_k=E_k\mathsf{\varphi }{\left(r\right)}_k,$ $\mathsf{\varphi }{\left(r\right)}_k:\mathrm{Wave}\mathrm{equation}$ Ek: energy eigenstate $U\left(r\right):$ Potential h: Planck constant r: coordinates vector m: mass
Atomistic scale	~10−9 m, ~10−9–10−6 s	The Monte Carlo (MC) technique is a stochastic method that uses random numbers to generate a sample population of the system from which one can calculate the properties of interest [74,75].	A new configuration can be produced by arbitrarily or systematically moving one atom from position i$\to j$ and can be accepted if $∆H=H\left(j\right)-H\left(i\right)<0$ If $∆H>0$ the move is accepted only with a certain probability $p_{i\to j}$ which is given by $p_{i\to j}\propto \exp \left(-\frac{∆H}{k_{B}T}\right)$. According to Metropolis et al. [76], one can determine the new configuration according to the following rule: $\mathsf{\xi }\le \exp \left(-\frac{∆H}{k_{B}T}\right),\mathrm{the}\mathrm{move}\mathrm{is}\mathrm{accepted};$ $\mathsf{\xi }>\exp \left(-\frac{∆H}{k_{B}T}\right),\mathrm{the}\mathrm{move}\mathrm{is}\mathrm{not}\mathrm{accepted}.$ H: Hamiltonian kB: the Boltzmann constant $\mathsf{\xi }$: a random number between 0 and 1
		The Molecular dynamics (MD) simulation technique allows one to predict the time evolution of a system of interacting particles (e.g., atoms, molecules, granules, etc.) and estimate the relevant physical properties [75].	The simulation of a many-body system would require the formulation and solution of equations of motion of all constituting particles, which for a particle i is $m_i\frac{d^2{\mathit{r}}_i}{d{t}^2}={\mathit{f}}_i$, mi: the particle mass ri: the particle position vector. fi: the force acting on the ith particle The interaction potentials describe in detail how the particles in a system interact with each other, i.e., how the potential energy of a system depends on the particle coordinates. Some of the most common simulations use AMBER [77], GROMOS [78] CHARMM [79] and OPLS [80].
		Molecular mechanics (MM) is a simulation technique to minimize large molecular structures such as DNA, RNA, proteins and their complexes, in which atoms are treated as masses, and bonds as springs with appropriate force constants. For minimizations calculations, the positions of the atoms within a molecule must be systematically or randomly moved and the energy recalculated with the goal of finding a lower energy and hence more stable molecule [81,82].	Similar to MD simulation, MM is based on Newton’s equation of motion. The interactions between the particles in the system can be described via the force-field potentials applied in MD simulations [82].
Mesoscopic scale	~10−6 m, ~10−6–10−3 s	Coarse-grained molecular dynamics (CGMD) methods overcome length and time scale limitations of atomistic simulations though coarse-graining large molecules by several connected beads [13].	Commonly used forcefields in CGMD are: Weeks–Chandler–Andersen potential, COS potential and Finite Extensible Elastic (FENE) bond potential [83]. MARTINI forcefileds [70].
		The Dissipative particle dynamics (DPD) method is also a mesoscopic simulation technique which can correctly account for the hydrodynamic interactions by considering water molecules explicitly. In DPD simulations, a cluster of atoms are represented by one bead and its dynamics is governed by Newton’s equation of motion [13,74,75].	Beads i and j interact through simple pairwise force consisting of a conservative force (FCi j), a dissipative force (FDi j), and a random force (FRi j). The total force applied on each bead i due to bead j is given as a sum of these three terms [71] $F=F_{ij}^C+F_{ij}^D+F_{ij}^R$