Mass of Solid Drug

Xs is the mass of solid drug remaining at any time t, D is the drug diffusion coefficient, S is the drug surface area, h is the hydrodynamic diffusion layer thick- ness, Cs is the drug solubility, Xd is the mass of dissolved drug at any time t, and V is the volume of fluid in which the drug is dissolving. In trying to solve the earlier equation, it should be noted that the drug surface area would not remain constant as drug dissolves after being released from typical dosage forms. The amount of dissolved drug would also not be constant. As a result, the rate of dissolution is continually changing. Dissolution testing is typi- cally done under sink conditions; therefore, the Xd/V term is small compared to Cs so that former can be ignored. However, in trying to establish a mechanistically based in vitro/in vivo correlation, the assumption that sink conditions would exist in the GI tract is an especially bad one for poorly soluble drugs. Also, as will be shown, testing dissolution under sink conditions is not necessary and can make instrumental analysis of dissolution more difficult. What is required is a numerical solution of the Noyes–Whitney equation to make the application of the theory as general as possible by eliminating the need to make frequently bad assumptions in order to solve the equation analytically. The following approach has been previously described (15,16). If one assumes that a drug particle has certain geometry, then surface area can be expressed in terms of drug mass if the drug density is known. The simplest geometry to use is spherical, although other geometries could be used. However, for the following derivations, spherical geometry will be assumed. The surface area at any given time can then be expressed by the following. where rt is the drug particle radius at any time t, and N0 is the number of drug par- ticles present initially. It will be shown later how one could handle a polydisperse drug powder, but for now, it will be assumed that all drug particles are exactly the same size and that they will all dissolve at the same rate. If this were the case, then the number of drug particles would not change with time until they completely dissolved at which time the number of particles would be zero. The number of drug particles present initially can be calculated by dividing the initial mass of drug or dose by the mass of one drug particle. Figure 1 shows the numerical calculation of XS and Xd with time based on Equations 19 and 20, respectively. Because the simulation is for a closed system, the two curves representing XS and Xd are symmetric. This would not be the case if one were to simulate drug dissolving in the GI tract while drug absorp- tion was occurring. Equations 19 and 20 are only able to handle a single particle size. To expand the application to polydisperse powders, it will be assumed that a poly- disperse powder can be simulated as a collection of monodisperse powder In solving these equations numerically using the Runge–Kutta method, the values of XSi and Xdi are calculated at each step of the numerical method, the size of which can be selected as a trade-off between accuracy favored by smaller step sizes versus speed of calculation for larger step sizes. A typical step size would be approximately one second. After each step, the amount of solid and dissolved drug from each particle size fraction i would be totaled as follows.