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 (17). However, for the following derivations, spherical geometry will be assumed. The surface area at any given time can then be expressed by the following:
S = 4TrJN0 (15)
where rt is the drug particle radius at any time t, and No is the number of drug particles 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, where p is the drug density, vo is the volume of one drug particle, X0 is the initial mass of drug, and r0 is the initial particle radius.
The previous equation can be solved for r0 and then made dynamic by replacing X0 and r0 with their respective time-dependent variables Xs and rt to yield. If dissolution is occurring in a closed system, such as the dissolution vessel, then the amount of dissolved drug Xd is given by
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 absorption was occurring.
Time (minutes)
Figure 1 Simulation showing the dissolution of solid drug (solid line) from Equation 19 with the concomitant appearance of dissolved drug (dashed line) from Equation 20.
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 favoured 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 totalled as follows:
XST = J2x*i (23)
XdT = J2x*i (24)
where Xs and Xd are the total amount of solid and dissolved drug from all particle size fractions, respectively, and n is the number of particle size fractions. In Equations 23 and 24, it should be noted that all particles, regardless of their size, are dissolving based on the same concentration gradient as the value of XdT is updated after each step of the numerical calculation and the same value for X^ is used for each particle size fraction ;’.
In summary, simulation of the dissolution of a polydisperse powder is accomplished by treating it as a collection of monosized fractions. At time zero, dissolution is the fastest because there is the most surface area and the concentration gradient is the greatest. Using the Runge-Kutta numerical method and Equations 21 and 22, the amount of drug that has dissolved from each particle size fraction is calculated, and after each step of the simulation, Equations 23 and 24 are used to sum up all the contributions from each particle size fraction. The total amount of dissolved drug from all fractions is then used during the next step of the numerical method so that each particle size fraction is dissolving against the same concentration gradient. Dissolution slows with time because the surface area and concentration gradient are getting smaller.
Typically, milled drug powders are distributed lognormally by mass about some geometric mean particle size. This means that one can find a collection of particles of similar size that are smaller than the mean particle size and another collection of particles of similar size that are larger than the mean particle size both collections of which are roughly equivalent in mass. However, since both collections are approximately equal in mass, the collection of smaller particles is made up of more particles and represents more surface area than the larger collection. As a result, the collection of smaller particles will dissolve faster and completely dissolve before the larger collection. Both the number and particle size distribution will change during the dissolution of a polydisperse powder, whereas only the particle size would change within each monosized fraction until complete dissolution was reached. At that point, the number within the fraction would become zero.
It is uncertain at what particle size one would be able to say that a particle is no longer a solid and that complete dissolution had occurred. However, particles calculated in the size range of molecular dimensions could probably be considered as completely dissolved. In computer simulation, without some statement as to when solid particles are completely dissolved, the calculated particle size will continue to decrease until the lower numerical limit of the computer system is reached. In agreement with the model described earlier for a polydisperse powder, it has been shown that during dissolution, the number of particles in a smaller particle size fraction decreased more rapidly relative to larger particle size fractions (18).
