Mathematical modelling of drug release from a porous granule

Understanding drug release from pharmaceutical granules is vital to the development of targeted release profiles. A model describing diffusion and solubility-limited drug dissolu tion and release from a porous spherical granule of drug and excipient is considered. Ra dially varying porosity and initial concentration profiles which can arise in pharmaceutical granules are incorporated. A range of boundary-value and moving-boundary-value prob lems arise, depending on the relationship between the drug saturation concentration in the solvent medium and the initial drug concentration and porosity profiles. The model is derived in detail for the case where the initial drug concentration is greater than the drug saturation concentration in all parts of the granule. In this case, a moving bound ary forms at the granule surface and propagates inwards, separating an unextracted in ner core from a shell region which undergoes extraction via diffusion. The full model is non-dimensionalised and analysed using asymptotic methods and numerical solution. A leading-order model is derived by exploiting a small parameter corresponding to the ratio of the drug saturation concentration to the maximum initial concentration in the gran ule, allowing estimation of the time taken for the moving boundary to reach the granule centre. The behaviour of the full model is considered by solving it using a boundary immo bilisation method and the finite element method for a range of parameters and comparing to the leading-order model. Finally, the model outputs for the moving boundary position and normalised drug release are compared with experimental data from the literature

Mathematical modelling of drug release from a porous granule

Understanding drug release from pharmaceutical granules is vital to the development of targeted release profiles. A model describing diffusion and solubility-limited drug dissolu tion and release from a porous spherical granule of drug and excipient is considered. Ra dially varying porosity and initial concentration profiles which can arise in pharmaceutical granules are incorporated. A range of boundary-value and moving-boundary-value prob lems arise, depending on the relationship between the drug saturation concentration in the solvent medium and the initial drug concentration and porosity profiles. The model is derived in detail for the case where the initial drug concentration is greater than the drug saturation concentration in all parts of the granule. In this case, a moving bound ary forms at the granule surface and propagates inwards, separating an unextracted in ner core from a shell region which undergoes extraction via diffusion. The full model is non-dimensionalised and analysed using asymptotic methods and numerical solution. A leading-order model is derived by exploiting a small parameter corresponding to the ratio of the drug saturation concentration to the maximum initial concentration in the gran ule, allowing estimation of the time taken for the moving boundary to reach the granule centre. The behaviour of the full model is considered by solving it using a boundary immo bilisation method and the finite element method for a range of parameters and comparing to the leading-order model. Finally, the model outputs for the moving boundary position and normalised drug release are compared with experimental data from the literature

Asymptotic analysis of the dominant mechanisms in the coffee extraction process

Extraction of coffee solubles from roast and ground coffee is a highly complex process, depending on a large number of brewing parameters. We consider a recent, experimentally validated, model of coffee extraction, describing extraction from a coffee bed using a double porosity model, which includes dissolution and transport of coffee. It was shown that this model can accurately describe coffee extraction in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. Despite being based on some simplifying assumptions, this model can only be solved numerically. In this paper we consider asymptotic solutions of the model describing extraction from a packed coffee bed. Such solutions can explicitly relate coffee concentration to the process parameters. For an individual coffee grain, extraction is controlled by a rapid dissolution of coffee from the surface of the grain, in conjunction with a slower diffusion of coffee through the intragranular pore network to the grain surface. Extraction of coffee from the bed also depends on the speed of advection of coffee from the bed. We utilize the small parameter resulting from the ratio of the advection timescale to the grain diffusion timescale to construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared to numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters and so are useful to quickly fit extraction curves and investigate the influence of various process parameters on the extraction

Asymptotic analysis of the dominant mechanisms in the coffee extraction process

Extraction of coffee solubles from roast and ground coffee is a highly complex process, depending on a large number of brewing parameters. We consider a recent, experimentally validated, model of coffee extraction, describing extraction from a coffee bed using a double porosity model, which includes dissolution and transport of coffee. It was shown that this model can accurately describe coffee extraction in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. Despite being based on some simplifying assumptions, this model can only be solved numerically. In this paper we consider asymptotic solutions of the model describing extraction from a packed coffee bed. Such solutions can explicitly relate coffee concentration to the process parameters. For an individual coffee grain, extraction is controlled by a rapid dissolution of coffee from the surface of the grain, in conjunction with a slower diffusion of coffee through the intragranular pore network to the grain surface. Extraction of coffee from the bed also depends on the speed of advection of coffee from the bed. We utilize the small parameter resulting from the ratio of the advection timescale to the grain diffusion timescale to construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared to numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters and so are useful to quickly fit extraction curves and investigate the influence of various process parameters on the extraction

A moving-boundary model of dissolution from binary drug-excipient granules incorporating microstructure.

Accurate mechanistic in vitro dissolution models can deliver insight into drug release behaviour and guide formulation development. Drug release profiles from drug-excipient granules can be impacted by variation of porosity and drug load within granules, which may arise from inherent variability in granulation processes. Here, we analyse and validate a recent model of drug release from a single spherical granule with a matrix of insoluble excipient, incorporating radial variation of porosity and drug load. The model is presented and specialised to the case where the initial drug load is large compared to the capacity of the granule's pores at solubility. In this limit, the model reduces to a single ordinary differential equation describing depletion of a shrinking, drug-saturated core. Model validation is performed using drug release data from the literature for a granule system consisting of acetaminophen and microcrystalline cellulose. A new extended model to describe dissolution from a polydisperse collection of granules is derived. The performance is compared to single particle models using equivalent spherical diameters. The developed model provides a new tool to explore the dissolution parameter space for these systems and for considering the impact of radial variation of granule porosity and drug load arising from manufacturing processes

Coffee extraction kinetics in a well mixed system

The extraction of coffee solubles from roasted and ground coffee is a complex operation, the understanding of which is key to the brewing of high quality coffee. This complexity stems from the fact that brewing of coffee is achieved through a wide variety of techniques each of which depends on a large number of process variables. In this paper, we consider a recent, experimentally validated model of coffee extraction, which describes extraction from a coffee bed using a double porosity model. The model incorporates dissolution and transport of coffee in the coffee bed. The model was shown to accurately describe extraction of coffee solubles from grains in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. The full model equations can only be solved numerically. In this work we consider asymptotic solutions, based on the dominant mechanisms, in the case of coffee extraction from a dilute suspension of coffee grains. Extraction in this well mixed system, can be described by a set of ordinary differential equations. This allows analysis of the extraction kinetics from the coffee grains independent of transport processes associated with flow through packed coffee beds. Coffee extraction for an individual grain is controlled by two processes: a rapid dissolution of coffee from the grain surfaces in conjunction with a much slower diffusion of coffee through the tortuous intragranular pore network to the grain surfaces. Utilising a small parameter resulting from the ratio of these two timescales, we construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared with numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters, so the solutions facilitate quick investigation of the influence of various process parameters on the coffee extraction curves

Coffee extraction kinetics in a well mixed system

The extraction of coffee solubles from roasted and ground coffee is a complex operation, the understanding of which is key to the brewing of high quality coffee. This complexity stems from the fact that brewing of coffee is achieved through a wide variety of techniques each of which depends on a large number of process variables. In this paper, we consider a recent, experimentally validated model of coffee extraction, which describes extraction from a coffee bed using a double porosity model. The model incorporates dissolution and transport of coffee in the coffee bed. The model was shown to accurately describe extraction of coffee solubles from grains in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. The full model equations can only be solved numerically. In this work we consider asymptotic solutions, based on the dominant mechanisms, in the case of coffee extraction from a dilute suspension of coffee grains. Extraction in this well mixed system, can be described by a set of ordinary differential equations. This allows analysis of the extraction kinetics from the coffee grains independent of transport processes associated with flow through packed coffee beds. Coffee extraction for an individual grain is controlled by two processes: a rapid dissolution of coffee from the grain surfaces in conjunction with a much slower diffusion of coffee through the tortuous intragranular pore network to the grain surfaces. Utilising a small parameter resulting from the ratio of these two timescales, we construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared with numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters, so the solutions facilitate quick investigation of the influence of various process parameters on the coffee extraction curves

Time evolution of soluble coffee.

Time sequence of the remaining soluble coffee concentrations in the coffee bed using the CFD model for the fine coffee grind.</p

Espresso control chart with extraction uniformity.

Coffee brewing control chart for espresso strength preferences. The espresso strength categories are adopted from ref. [10].</p

Conical case computational domain.

Conical case computational domain with included boundary conditions.</p