A statistical mechanical model for drug release: relations between release parameters and porosity M´arcio Sampaio Gomes-Filho a , Marco Aur´ elio Alves Barbosa b , Fernando Albuquerque Oliveira a a Instituto de F´ ısica, Universidade de Bras´ ılia, Bras´ ılia-DF, Brazil b ProgramadeP´os-Gradua¸ c˜aoemCiˆ encia de Materiais, Faculdade UnB Planaltina, Universidade de Bras´ ılia, Planaltina-DF, Brazil Abstract A lattice gas model is proposed for investigating the release of drug molecules on devices with semi-permeable, porous membranes in two and three dimensions. The kinetic of this model was obtained through the analytical solution of the three-dimension diffusion equation for systems without membrane and with Monte Carlo simulations. Pharmaceutical data from drug release is usually adjusted to the Weibull function, exp[−(t/τ ) b ], also known as stretched exponential, and the dependence of adjusted parameters b and τ is usually associated, in the pharmaceutical literature, with physical mechanisms dominating the drug dynamics inside the capsule. The relation of parameters τ and b with porosity λ are found to satisfy, a simple linear relation for between τ and λ −1 , which can be explained through simple physically based arguments, and a scaling relation between b and λ, with the scaling coefficient proportional to the system dimension. Keywords: drug release, Weibull distribution function, capsule membrane, porosity 1. Introduction The advances in the synthesis technology of porous ma- terials allowed the development of new matrices (mono- lithic) and membrane pharmaceutical devices where size, shape and pore distribution can be fine controlled during the fabrication process [1, 2, 3, 4]. Understanding the con- nection between drug release rates and the characteristics of the device has an enormous potential for improving the treatment of various diseases. Mathematical modeling of drug release usually involves finding the proper form of a diffusion equation consider- ing the essential physical phenomena occurring as particles diffuse through the capsule device [5, 6, 7]. In the phar- maceutical literature it is also a common procedure to fit drug release data to semi-empirical functions and use this information to obtain insights onto the processes through which the drug is released. Since many factors can con- tribute to determine the final drug release, this procedure can be subject to ambiguous interpretations that could make the data analysis even more confusing [8]. Thus, a more systematic approach for understanding the relation between release data and physical processes occurring in- side the capsule is desired. We work in this direction by simulating minimalist lattice models devised to describe both drug and physical device (or capsule) and investi- gate the relation between drug release patterns and the system porosity through the semi-empirical parameters of the Weibull function. Email addresses: aureliobarbosa@unb.br (Marco Aur´ elio Alves Barbosa), fao@fis.unb.br (Fernando Albuquerque Oliveira) In this work a lattice gas model is proposed for inves- tigating the release of drug molecules encapsulated on de- vices with semi-permeable, porous membranes in two and three dimensions, following a previous work on 1D and 2D systems [9]. Release patterns were obtained through ana- lytical solution of the three-dimension diffusion equation, for systems without membrane, and Monte Carlo simula- tions (MC), for systems with porous membrane, and ad- justed to the Weilbull function, exp[−(t/τ ) b ], which is also known as stretched exponential. The dependence of the characteristic time τ with the membrane content, defined as the inverse power of porosity, ζ = λ −1 , was found to satisfy linear relation, that is justified using reasonable physical arguments. The parameter b was found to satisfy a scaling relation with ζ from a regime without membrane for up to 90% of membrane coverage. This article is organized as follows, in the next section we present the Weibull distribution and discuss previous investigations about its semi-empirical parameters using statistical mechanical models, the current model and the simulations protocol are introduced in Section 2, while our results and discussions are presented in Section 3. An an- alytical solution for the diffusion equation of a continuous system similar to our 3D lattice model is presented in Ap- pendix A. 1.1. Weibull distribution The Weibull distribution function was originally pro- posed by Waloddi Weibull in 1951 as the ‘simplest possible’ empirical function that could be used to adjust non-linear experimental data from complex systems [10]. Distribu- tions on systems as diverse as electric bulb duration, life Preprint submitted to Physica A October 22, 2019 arXiv:1910.08839v1 [cond-mat.soft] 19 Oct 2019