Table 3.

A few mathematical models and equations which are often used for controlled-release and kinetic-release characteristics of the AI from the host into the solution.

No	Mathematical Model	Equations	Denotations	Source(s)
1	Zeroth-order	C = C0 − K0 t	C = Amount of drug released C0 = Initial amount of drug in solution K0 = Zeroth-order rate constant t = Time	[55]
2	First-order	dC/dt = −KC	K = First order rate constant	[55]
3	Second-order	1 − (Mt/M0)/t = kt−1/2 + b		[58]
4	Hixson–Crowell	C01/3 − Ct1/3 = Kt	Ct = Amount of drug released in time, t C0 = Initial amount of drug in table K = Rate constant	[51,55]
5	Fick’s first law	J = −Df dc/dx	J = Amount of substance passing perpendicularly through a unit of surface area per unit of time Df = Diffusion coefficient dc/dx = Concentration gradient	[56]
6	Fick’s second law	$\frac{\partial \mathsf{\phi }}{\partial \mathrm{t}}=\mathrm{D} \frac{\partial 2 \mathsf{\phi }}{\partial \mathrm{x}2}$	φ = Concentration in mol/m3 φ = φ(x,t) is a function that depends on location x and time t D = Diffusion coefficient in m2/s	[56]
7	Korsmeyer–Peppas	F = (Mt/M) = Km tn	F = Fraction of drug release time Mt = Amount of drug release time M = Total amount of drug dosage Km = Kinetic constant n = Diffusion or release exponent t = Time	[55]
8	Wiebull	$$\begin{gathered} f\left(x, \lambda , k\right) \\ =\left\{\begin{array}{c}\frac{k}{\lambda }{\left.\left( \frac{x}{\lambda }\right.\right)}^{k-1}{e}^{-\left(\frac{x}{\lambda }\right)k} x \ge 0,\\ \\ 0 x<0\end{array}\right. \end{gathered}$$	K < 1 = Failure rate decreases over time K = 1 = Failure rate is constant over time k > 1 = Failure rate increases over time	[57]
9	Higuchi	Q = KH t 1/2	Q = Cumulative amount of drug released at the time, t KH = Higuchi constant t = Time	[52]
10	Baker–Lonsdale	F1 = 3/2(1 − (1 − Ct/C∞)2/3) Ct/C∞ = kt	Ct = Drug release amount at time, t C∞ = Amount of drug released K = Release constant	[53,55]
11	Hopfenberg	Qt/Q∞ = 1 – (K0t/C0a0)	Qt = Amount of drug released in time, t Q∞ = Amount of drug dissolved when the dosage form is exhausted C0 = Initial concentration of the drug A0 = Initial radius sphere for a slab	[54]