Higuchi equation

The Higuchi equation , also known as the square root law according to Higuchi , describes the release of a substance evenly suspended in an insoluble matrix by diffusion . Takeru Higuchi published it in 1961 to characterize the release of medicinal substances from suspension ointments. It can also be used to describe the release from other matrices, such as solid drug forms formed from polymers in which the drug is embedded.

In its simplified ("classical") form, the Higuchi equation reads:

{\ displaystyle {\ frac {M_ {t}} {A}} = {\ sqrt {2 \ C_ {0} \ D \ C_ {s} \ t}}}

This is the amount of drug released at a time , the effective diffusion surface, the diffusion coefficient , the initial concentration in the carrier matrix and the saturation concentration in the matrix material. ${\ displaystyle M_ {t}}$ ${\ displaystyle t}$ ${\ displaystyle A}$ ${\ displaystyle D}$ ${\ displaystyle C_ {0}}$ ${\ displaystyle C_ {S}}$

For semi-solid and solid suspension drug forms, the Higuchi equation thus establishes a simple mathematical relationship between the amount of drug released and the time, to the square root of which it is directly proportional ("square root law").

Derivation

Higuchi's approach is based on the consideration that in the course of the release from an insoluble carrier matrix, the diffusion path through the matrix to be covered by the substance does not remain constant, but increases steadily, since the matrix becomes increasingly depleted in substance towards the edge. As the diffusion path increases, the concentration gradient and, consequently, the rate of release decreases . ${\ displaystyle h}$

Higuchi assumed the following relationship:

{\ displaystyle d \ left ({\ frac {M_ {t}} {A}} \ right) = \ left (C_ {0} - {\ frac {C_ {s}} {2}} \ right) ie}

The mathematical solution for the diffusion path leads, after insertion into Fick's First Law, to the Higuchi equation in the following form: ${\ displaystyle h}$

{\ displaystyle {\ frac {M_ {t}} {A}} = {\ sqrt {D (2C_ {0} -C_ {s}) C_ {s} \ t}}}

The prerequisites for the applicability of the Higuchi equation are that the drug is readily water-soluble and the substance concentration in the acceptor medium is very low (“sink condition”), while the matrix is insoluble in water and non-degradable and has a pore-free, planar surface. The suspended substance must be very fine-grained so that the particle diameter is significantly smaller than the layer thickness of the dosage form. Furthermore, the amount of substance initially incorporated in the matrix must be greater than its saturation concentration in the material of the matrix ( ). As soon as this last condition is no longer given, i.e. the reservoir is exhausted, the equation no longer applies. ${\ displaystyle C_ {0}> C_ {s}}$

For the frequently occurring case that the amount of substance initially incorporated in the matrix is even very much greater than the saturation concentration ( ), the Higuchi equation is simplified to the form mentioned at the beginning. ${\ displaystyle C_ {0} \ gg C_ {s}}$

variants

Based on his first equation, Higuchi developed different variants. Another describes the release from porous matrices. The equation includes the porosity of the carrier matrix (i.e. the volume fraction created by pores and capillaries) and the tortuosity of the capillaries: ${\ displaystyle \ epsilon}$ ${\ displaystyle \ tau}$

{\ displaystyle {\ frac {M_ {t}} {A}} = {\ sqrt {\ frac {D \ \ epsilon \ (2C_ {0} - \ epsilon \ C_ {s}) C_ {s} \ t} {\ tau}}}}

swell

Individual evidence

^ Voigt, Rudolf: Voigt Pharmaceutical technology for study and work . 12th, completely revised edition. Deutscher Apotheker Verlag, Stuttgart 2015, ISBN 978-3-7692-6194-3 .
↑ T. Higuchi: Rate of release of medicaments from ointment bases containing drugs in suspension. In: J Pharm Sci . 1961 50: 874-875. PMID 13907269 ; doi : 10.1002 / jps.2600501018 .
↑ J. Siepmann, NA Peppas: Higuchi equation: Derivation, applications, use and misuse In: Int J Pharm . Vol. 418, 2011, pp. 6–2. DOI: 10.1016 / j.ijpharm.2011.03.051 .

literature

Claus-Dieter Herzfeldt, Jörg Kreuter: Basics of the drug form theory. Springer Berlin Heidelberg, 1999, ISBN 978364258448-0 , p. 400.
AN Martin, J. Swarbrick and A. Cammarata, eds. u. completely revised by H. Stricker: Physical Pharmacy. Physico-chemical principles applied pharmaceutically. Martin Swarbrick Cammarata. WVG Stuttgart, 1987, p. 309.
P. Costa, JM Sousa Lobo: Modeling and comparison of dissolution profiles . In: European Journal of Pharmaceutical Sciences 13 (2001), pp. 123-133.

[1] Voigt, Rudolf: Voigt Pharmaceutical technology for study and work . 12th, completely revised edition. Deutscher Apotheker Verlag, Stuttgart 2015, ISBN 978-3-7692-6194-3 .

[2] T. Higuchi: Rate of release of medicaments from ointment bases containing drugs in suspension. In: J Pharm Sci . 1961 50: 874-875. PMID 13907269 ; doi : 10.1002 / jps.2600501018 .

[3] J. Siepmann, NA Peppas: Higuchi equation: Derivation, applications, use and misuse In: Int J Pharm . Vol. 418, 2011, pp. 6–2. DOI: 10.1016 / j.ijpharm.2011.03.051 .