Scientific method as envisaged by eminent scientists is fundamental to the investigation of new knowledge based upon evidence. Scientists use hypothesis and logic to prepare explanations for physical phenomenon in the form of equations. It is an undeniable fact that most pharmacy curriculums/courses avoid being confronted with any form of mathematics or even scientific equations. And yet, according to the mathematician Stanley Gudder, "The essence of mathematics is not to make simple things complicated, but to make complicated things simple". One of the most important controlled release equations is called "Higuchi equation". The Higuchi equation has helped define the mathematical perspective of controlled release drug delivery systems since the era of development of sustained release dosage forms.
The release of a drug from a drug delivery system (DDS) involves both dissolution and diffusion. Several mathematical equations models describe drug dissolution and/or release from DDS. In the modern era of controlled-release oral formulations, 'Higuchi equation' has become influential kinetic equation in its own right, as evidenced by employing drug dissolution studies that are recognized as an important element in drug delivery development. Today the Higuchi equation is considered one of the widely used and the most well-known controlled-release equation.
T. Higuchi developed an equation for the release of a drug molecule from creams and ointments in 1960 (Higuchi, T., J. Soc. Cosmetic Chem., 1960, 11:85-97). He applied this equation to diffusion of solid drugs dispersed in homogenous dosage system. In 1961, Higuchi published a mathematical equation, the so called 'Classical Higuchi equation' in the Journal of Pharmaceutical Sciences to describe the release rate of drugs from planar system having homogenous matrix (Higuchi, T., J. Pharm. Sci., 1961, 50:874-875). The classical basic Higuchi equation is: (1), where Q is the cumulative amount of drug released in time t per unit area, Co is the initial drug concentration, Cs is the drug solubility in the matrix and D is the diffusion coefficient of the drug molecule in the matrix. The above equation (1) can be expressed in more simple form as: (2), where k is equal to , known as Higuchi dissolution constant and treated sometimes in a different manner by different researchers. Thus the cumulative release of drug released is proportional to the square root of time. The rate of release can be increased by increasing the drug's solubility (Cs) in the polymer matrix and vice versa. Therefore, the simple Higuchi model according to equations 1 and 2 will result a linear Q versus t0.5 plot having gradient, or slope, equal to k and we say the matrix follows t0.5 kinetics (i.e. 'root t'). It is important to note that a few assumptions are made in this Higuchi model. These assumptions are (i) the initial drug concentration in the system is much higher than the matrix solubility; (ii) perfect sink conditions are maintained; (iii) the diffusivity of the drug is constant and (iv) the swelling of the polymer is negligible. The sink conditions are achieved by ensuring the concentration of the released drug in the release medium never reaches more than 10 per cent of its saturation solubility. The concentration profiles of a drug initially suspended in an ointment dosage form are represented in Fig.1. The solid line represents the variation of drug concentration after exposure of the ointment to perfect sink conditions for a certain time t. There is a sharp discontinuity and otherwise linear concentration profiles at distance h from the surface. To distances higher than h, the concentration gradient is essentially constant, provided Co Cs. At a time t, the amount of drug release by the system corresponds to the shaded area in the figure. After an additional time interval, ?t, the new concentration profile of the drug is given by the broken line. Under these conditions, Higuchi derived the simple relationship between the release of the drug and square root of time.

Fig. 1. Drug theoretical concentration profile of an ointment containing suspended drug in direct contact with a perfect sink release medium

Other forms of Higuchi equation
Higuchi developed also other models, such as spherical homogeneous and planar or spherical heterogeneous systems. In a scenario when the matrix is not homogenous but instead contains granular materials (i.e. granular matrix), the release of solid drug involves the simultaneous penetration of the surrounding liquid, the dissolution of drug, and leaching out of the drug solution through the pores. Higuchi developed an alternate model for drug release from a planar heterogeneous matrix system, where a factor to account for the tortuosity of the system and describes the porosity of the matrix (Higuchi, T., J. Pharm. Sci., 1963, 52:1145-1149). The second alternate form of Higuchi equation is: (3), in which is the matrix porosity and is the tortuosity of the capillary system, both parameters being dimensionless quantities. Equation 1 differs from equation 3 only in the addition of porosity ( ) and tortuosity ( ). Porosity is the fraction of matrix that exists as pores and channels. Tortuosity is due to increase in path length of diffusion resulting from branching and bending of pores. The classical Higuchi equation was developed to study the diffusion-controlled release of dispersed drugs from non-swellable and non-biodegradable matrices. Since then, equation has not only adapted into alternate forms, to account for changing matrix concepts, but also used by various researchers to interpret their experimental drug release data. Today the Higuchi model has a large application in polymeric system and is a common practice to analyze experimental drug release data to get a rough estimate of underlying release mechanisms. Because assessing drug release/dissolution from solid dosage forms is a very important element in dosage form design, the Higuchi release model continues to be best describing release mechanisms.

The author is Reader in Pharmacy,
Annamalai University, Annmalainagar 608 002, (TN)
Email: cdl_scbasak@sancharnet.in