Some of these may be manipulated by the formulator. Solutions are encountered frequently in pharmaceutical development, either as a dosage form in their own right or as a clinical trials material. Additionally, almost all drugs function in solution in the body. This chapter discusses the principles underlying the formation of solutions from solute and solvent and the factors that affect the rate and extent of the dissolution process. This process will be discussed particularly in the context of a solid dissolving in a liquid as this is the situation most likely to be encountered in the formation of a drug solution, either during manufacturing or during drug delivery.
Further properties of solutions are discussed in Chapter 3 and Because of the number of principles and properties that need to be considered, the contents of each of these chapters should only be regarded as introductions to the various topics.
The student is encouraged, therefore, to refer to the bibliography cited at the end of each chapter in order to augment the present contents.
It uses a large number of pharmaceutical examples to aid in the understanding of physicochemical principles. This chapter will begin by clarifying some of the key terms relevant to solutions. A solution may be defined as a mixture of two or more components that form a single phase that is homogeneous down to the molecular level.
The component that determines the phase of the solution is termed the solvent ; it usually but not necessarily constitutes the largest proportion of the system. The other component s are termed solute s and these are dispersed as molecules or ions throughout the solvent, i. The transfer of molecules or ions from a solid state into solution is known as dissolution. Fundamentally, this process is controlled by the relative affinity between the molecules of the solid substance and those of the solvent.
The extent to which the dissolution proceeds under a given set of experimental conditions is referred to as the solubility of the solute in the solvent. The solubility of a substance is the amount of it that passes into solution when equilibrium is established between the solute in solution and the excess undissolved substance. The solution that is obtained under these conditions is said to be saturated. A solution with a concentration less than that at equilibrium is said to be subsaturated.
Solutions with a concentration greater than equilibrium can be obtained in certain conditions; these are known as supersaturated solutions. Since the above definitions are general ones, they may be applied to all types of solution involving any of the three states of matter gas, liquid, solid dissolved in any of the three states of matter, i. However, when the two components forming a solution are either both gases or both liquids, then it is more usual to talk in terms of miscibility rather than solubility.
Other than the name, all principles are the same. One point to emphasize at this stage is that the rate of solution dissolution rate and amount which can be dissolved solubility are not the same and are not necessarily related. In practice, high drug solubility is usually associated with a high dissolution rate, but there are exceptions; an example is the commonly used film-coating material hydroxypropyl methylcellulose HPMC which is very water soluble yet takes many hours to hydrate and dissolve.
The majority of drugs and excipients are crystalline solids. Liquid, semi-solid and amorphous solid drugs and excipients do exist but these are in the minority.
For now, we will restrict our discussion to dissolution of crystalline solids into liquid solvents. Also, to simplify the discussion, it will be assumed that the drug is molecular in nature.
The same discussion applies to ionic drugs. Again, to avoid undue repetition in the explanations that follow, it can be assumed that most solid crystalline materials, whether drugs or excipients, will dissolve in a similar manner. The dissolution of a solid in a liquid may be regarded as being composed of two consecutive stages. First is an interfacial reaction that results in the liberation of solute molecules from the solid phase to the liquid phase.
This involves a phase change so that molecules of solid become molecules of solute in the solvent in which the crystal is dissolving. After this, the solute molecules must migrate through the boundary layers surrounding the crystal to the bulk of solution. These stages, and the associated solution concentration changes, are illustrated in Figure 2. Heating up a solvent gives the molecules more kinetic energy. The increased rapid motion means that the solvent molecules collide with the solute with greater frequency, and that the collisions occur with more force.
Both factors increase the rate at which the solute dissolves. As we will see in the next section, a temperature change not only affects the rate of dissolution, but also affects the amount of solute that dissolves. Rate of Dissolution Dissolution is the process by which a solute dissolves into a solvent and forms a solution. Surface Area The rate at which a solute dissolves depends upon the size of the solute particles. Agitation of the Solution Dissolving sugar in water will occur more quickly if the water is stirred.
The rate of dissolution should noticeably increase as the temperature of the solution rises. Do not attempt these processes without training. Never use organic solvents near an open flame or source of ignition, as these solvents are highly flammable.
If you must use an organic solvent, wear safety glasses and do not place a container with the solvent directly on a heating surface. Instead, prepare a hot water bath by placing tap water into a beaker or large pan, and place the container with the solvent into the water bath. Never heat an organic solvent to boiling.
Updated April 24, Figure 6a shows the total number of undissolved particles over time. As the transition function scale parameter is increased, the total number of particles reaches its maximum value later, and its maximum value is lower.
The curve shapes in Fig. It shows that fragmentation events that produce the smallest particles relative to the original particle size generate the most surface area and do so at the highest rate, resulting in the fastest overall dissolution process.
Therefore, a lower transition function scale parameter results in a faster overall dissolution process. Finally, Fig. As the transition function logarithmic standard deviation is increased, the range of particle sizes produced in each fragmentation event increases. The effect of the specific fragmentation event on the overall dissolution process: range of new particle sizes. Simulated dissolutions were performed at transition function TF logarithmic standard deviations ranging from 0.
This parameter defines the width of the distribution of new particle sizes in logarithmic space, relative to the original particle size created during each fragmentation event. As the transition function logarithmic standard deviation is decreased, the amount of new surface area produced during each fragmentation event increases, directly increasing the overall rate of diffusive mass removal, and thus accelerating the entire dissolution process.
Figure 7a shows the total number of undissolved particles over time. As the transition function logarithmic standard deviation is increased, the total number of particles reaches its maximum value later, and its maximum value is lower. As the transition function logarithmic standard deviation is increased, the larger range of new particle sizes actually results in a smaller added surface area than an equivalent fragmentation event producing particles of more uniform size.
This results in a slower increase in total surface area, a smaller maximum surface area, and as shown in Fig. In order to better examine the idea of an external perturbing force influencing dissolution, we now explore an example from the field of pharmaceutical research. In the drug quality testing protocols prescribed by U.
Pharmacopeia, a dissolving pharmaceutical pill is subjected to a perturbing force in the form a stirring element, agitator, or flow apparatus in order to decrease overall dissolution time Furthermore, recent research has considered other methods of rapid solid dissolution, most notably the application of ultrasound pressure waves via a probe These pressure waves result in the formation and collapse of microbubbles of dissolved gas, a phenomenon known as ultrasonic cavitation.
This phenomenon results in a number of unique physical and chemical properties Most relevant to this work, cavitation has been shown to positively influence the dissolution process due both to energy deposition at the object surface and the improvement of flow characteristics in the solvent volume 20 , Furthermore, materials science research has shown that ultrasonic pressure waves eventually result in fatigue and fracture in many materials, so its influence on the fragmentation process here is hardly surprising An example of the effects of ultrasonic agitation on the dissolution of pharmaceutical tablets may be found in Supplementary Fig.
We began by fitting our model to experimental data in order to accurately express the diffusion and fragmentation characteristics of the model as functions of an applied perturbing force, in this case ultrasonic pressure waves emitted by a submerged ultrasound probe. The details of the fitting process and the resulting functions for the aforementioned parameters can be found in the supplementary information, specifically Supplementary Figs S4 and S5.
As detailed in equations 11 — 13 and Supplementary Figs S4 and S5 , both diffusion and fragmentation are affected by increases in ultrasound power. As the applied ultrasound power is increased, the mass transfer coefficient, k c , increases, causing the rate of diffusive mass removal to increase. As a result of these changes in the fragmentation parameters, the additional surface area generated by each fragmentation event is greater for higher applied ultrasound powers.
Increasing the applied ultrasound power affects both the fragmentation and diffusion characteristics of the dissolution process in such a way that the entire process is accelerated. The effect of applied ultrasound power on pharmaceutical pill dissolution.
Increasing the applied ultrasound power changes the fragmentation characteristics such that the amount of new surface area created during each fragmentation event increases. Additionally, increasing the applied ultrasound power also results in increased flow rates and other changes that result in increased diffusion.
Both of these changes tend to accelerate the overall dissolution process as the applied ultrasound power is increased. As shown in Fig. In these cases, the maximum number of particles is still reached faster as the ultrasound power increases, but the maximum number of particles itself decreases. A possible explanation for this occurrence is that at high ultrasound power, the resulting increase in surface area is greater than that at lower applied ultrasound powers because the fragmentation events create a smaller number of larger particles but do so with a larger fraction of the original particles at high ultrasound powers compared with a larger number of smaller particles from a smaller fraction of the original particles at lower ultrasound powers.
This hypothesis is corroborated by Fig. We now demonstrate the design capabilities of this model by optimizing a hypothetical battery-powered ultrasound-assisted dissolution device. Such a device, given a limited total amount of expendable energy, is restricted in the power and duration of ultrasound agitation it can provide.
In addition, this device can operate constantly, or in a pulsatile manner. By simulating this device operating at varying ultrasound powers, pulse frequencies, and duty cycles, we endeavour to determine the optimal settings for each variable given a limited amount of usable energy.
Figure 9a shows the effect of changing the power applied during each pulse. All of the curves in this plot exhibit the same basic shape, wherein ultrasound-induced fragmentation creates new surface area and accelerates the overall rate of dissolution until the total available energy is exhausted and dissolution continues without any fragmentation. Moreover, the results suggest that even in situations with limited total available energy, the highest applied power results in the fastest dissolution, even though it exhausts the available energy the fastest.
Ultrasound power pulse optimisation study. Next, Fig. The curves in this plot still exhibit the same exhaustive behaviour observed in Fig. Though higher frequencies appear smooth at the displayed time scales, the application or absence of ultrasound power is quite evident for lower frequencies, which only pulse a few times throughout the course of the simulation.
Most notably, though the total dissolved volume behaves differently for each pulse frequency while ultrasound power is applied, they exhibit similar behaviour after the available energy is exhausted, and they result in very close amounts of total volume dissolved. This suggests that within certain time frames, the total applied energy is more important to the speed of the overall process than the pattern by which it is applied.
That said, after the non-pulsed case, the pulse patterns which dissolve the most mass in the allotted time are those with lower frequencies. The curves in this plot again exhibit the same exhaustive behaviour observed in Fig.
In keeping with the other results detailed in this figure, the constant power case a duty cycle of 1 results in the fastest total dissolution process and the no-power case a duty cycle of 0 results in the slowest. The duty cycles between the two extremes obey this trend, with higher duty cycles resulting in faster overall dissolution. Again, the modality by which the most energy is imparted to the system in the shortest time is the one that results in the fastest dissolution.
Taking into account the possible ultrasound powers and pulsing modalities, the results of this optimisation study suggest that by employing the highest possible ultrasound power, the lowest possible pulse frequency no pulsing if possible , and the highest possible duty cycle again, resulting in constant power if possible , the fastest overall dissolution process will be achieved.
In all cases, applying all available energy in the shortest possible timespan appears to produce the fastest dissolution. We have developed and explored a novel partial differential equation model of dissolution governed by two interacting phenomena: surface area dependent diffusive mass removal and physical fragmentation.
This model adds to existing literature by describing the time evolution of particle size distributions as dissolving particles are subjected to both phenomena. While surface area-dependent mass removal by diffusion is required for the total dissolution of an object, both surface area-dependent diffusion and physical fragmentation have profound effects on the resulting particle size distribution and on the bulk dissolution rate at large.
Characterizing the fragmentation process is essential, because in most cases, the chemical composition and diffusion characteristics of the solute and solvent are not subject to change. In these cases, physical fragmentation is the only independently controllable process.
This control is most readily facilitated through the application of an external perturbing force, such as mechanical stirring of the solvent, physical impact with the body to be dissolved, or as modelled above, the application of ultrasound. Through our simulations, we observed that the fragmentation process has a strong effect on the kinetics of the overall dissolution process. In all cases, the most rapid increase in surface area through fragmentation or otherwise will always result in the fastest overall dissolution of an object.
Furthermore, we noticed that with one notable exception, in all cases the dissolution process reaches a critical point at which the total surface area stops increasing and begins to decrease as surface area increase by fragmentation is matched and overtaken by surface area decrease due to particle size decrease and disappearance due to diffusive mass removal. We discovered that the faster this critical point is reached, the faster the overall dissolution process proceeds.
The exception to this rule occurs when there is either no fragmentation or very little such that the initial rate of surface area increase is never able to overcome the decrease in surface area due to diffusive mass removal. Finally, we demonstrated the capabilities of this model by performing an optimisation simulation in which the optimum ultrasound power and pulse pattern were determined for a simple battery-powered dissolution device. This suggests potential applications for this model in the design of devices incorporating controlled dissolution of solids in liquid solutes.
Future challenges in this field include the incorporation of spatial effects to account for heterogeneities in both the solute and particle distribution, which we assume to be small for the ultrasound power dissolution due to rapid mixing. The incorporation of spatial effects, including the tracking of fluid flow and particle transport, would allow for the removal of the assumption of a well-mixed solution and all for the modelling of slowly stirred dissolution cases where heterogeneities in the solute and particle distribution play a larger role; recent literature has demonstrated that the Lattice Boltzmann method can be applied to these types of problems 24 , However, the addition of spatial effects would increase the mathematical complexity and computational cost of the model.
In this case, the stochastic aspects of the fragmentation process cannot be neglected. Furthermore, while the experiments used to validate our model show low variability in the overall dissolution curves, this is not necessarily the case for other dissolution experiments. It would be interesting to reinterpret our fragmentation distribution from one describing the evolution of a particle distribution to one describing a probability distribution for the chance of a given particle to fragment into a certain distribution of resulting particles.
Such a reformulation would allow our model to account for this variability within the overall dissolution process and more accurately simulate these more complex dissolution regimes. Our model takes into account the two physical processes contributing to solid dissolution in a liquid solvent, represented by two sub-models: the surface-area-dependent, concentration gradient-based diffusive mass removal and the fragmentation of all undissolved particles.
Both processes can be driven by an external perturbing force, examples of which include mechanical agitation, solvent flow, and ultrasonic pressure waves.
This model idealizes the each particle as a mass of arbitrary shape submerged in a fluid solvent. At all solute-solvent interfaces, mass is constantly being removed by diffusion. The rate of of this diffusive mass removal is described by the Nernst-Brunner equation equation 1.
Importantly, this model relies on the assumption that the agitated liquid is well mixed, as the dissolved solute concentration is calculated as an average value for the entire fluid volume. Throughout this process, particles of all sizes are tracked as a distribution N V , t of the number of undissolved particles of each volume, V at each time, t.
At its simplest, S V is of the form. As a validation step, this model was compared against the analytically-solved Nernst-Brunner equation equation 1 , generating the results shown in Supplementary Fig. Given the same initial conditions and parameters, our model predicted the volume loss over time for a single dissolving particle as the Nernst-Brunner equation to within 1. The fragmentation model considers how particles break apart during the dissolution process.
A fragmentation event of a single particle involves the breakdown of the particle into smaller particles of varying sizes. Each fragmentation event is defined by two mathematical constructs: the fragmentation rate and the transition function.
The volume-dependent fragmentation rate, a parameter denoted by g in equation 5 , represents the rate at which each undissolved particle of volume V fragments into new, smaller particles. The evolution of the particle-size distribution function N V , t is thus described by the integral equation.
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