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.
The volume-dependent fragmentation rate can be a constant or a function of the particle volume V. In this case, we enforce that the smallest particles cannot themselves fragment, but simply dissolve away over time, by constructing g V in the form. This is done to minimize the round-off error associated with the fragmentation of very small particles close to the minimum simulated particle size.
The transition function describes each fragmentation event. In this framework, we employ what we have found to be the simplest and most versatile transition function for our purposes, a normalized lognormal distribution,. The diffusive mass removal model and the fragmentation model defined in equations 2 and 5 are combined to to obtain a single governing equation,.
After examining the data and the behaviour of the model, three parameters were found to largely determine the overall system behaviour. When these parameters were fit properly to experimental data, our model was able to robustly reproduce the average behaviour of the physical system. Supplementary Figs S4 and S5 detail the fitting process by which our modelling framework was fit to experimentally collected dissolution data, and the following functional forms were developed for each of the three driving parameters:.
This is an Open Access article which permits unrestricted noncommercial use, provided the original work is properly cited. This article has been cited by other articles in PMC. Keywords: particle size, solubility, dissolution, nanocrystal, bioavailability, coenzyme Q Introduction Since Sucker et al produced nanoparticles in the s, 1 nanonization has attracted much attention, especially for improving the bioavailability of poorly soluble drugs. Preparation of naked coenzyme Q10 nanocrystal and coarse suspensions A precipitation method was developed for producing suspensions of naked coenzyme Q 10 nanocrystals.
Characterization of nanocrystal suspensions Particle size distribution and zeta potential The particle size distribution of the naked nanocrystals was determined by dynamic light scattering using a PSS Nicomp ZLS equipped with a He-Ne laser source at Transmission electron microscopy The samples of different-sized nanocrystal suspensions were diluted with distilled water, pipetted onto collodion film-coated copper grids with a mesh size of , dried using absorbent paper, stained with phosphorus tungsten acid for 4 minutes, and finally dried under ambient conditions.
Differential scanning calorimetry Thermographs of the nanocrystals and bulk drugs in suspension were obtained using a differential scanning calorimetry instrument 1 STARe; Mettler Toledo, Schwerzenbach, Switzerland. Preparation of dissolution media Three types of dissolution medium were utilized in this work to investigate the size effect in a series of in vitro dissolution and solubility experiments.
Solubility measurement Shaking behavior The kinetic solubility of bulk coenzyme Q 10 with different amounts of drug content was examined in dissolution medium A, and two types of shaking behavior were investigated. Solubility of CoQ 10 nanocrystals and bulk drugs The equilibrium solubility values for the four coenzyme Q 10 nanocrystals and bulk drugs in the three types of dissolution medium were determined using the dilution method.
Bioavailability studies in beagle dogs Twelve healthy beagle dogs body weight Pharmacokinetic analysis Standard pharmacokinetic parameters for coenzyme Q 10 were obtained from the plasma concentration-time curves.
Results and discussion Size distribution and zeta potential of coenzyme Q10 nanocrystals in suspension As shown in Figure 1 , suspensions of the bulk drug and the naked coenzyme Q 10 nanocrystals of different sizes had different appearances.
Open in a separate window. Figure 1. Figure 2. Abbreviation: PI, polydispersity index. Evaluation of morphology and crystal form Transmission electron micrographs clearly show that the nanocrystals were rounded Figure 3 , which is crucial for investigation of the relationship between size and solubility proposed by the classical Ostwald—Freundlich equation. Figure 3. Figure 4. Figure 5.
Solubility For poorly soluble coenzyme Q 10 , the aqueous solutions provided solubility and dissolution results which were too low for measurement, so the nonionic surfactant Tween 20 was used, in accordance with an already reported method.
Investigation of shaking behavior In the literature, solubility is always measured by adding an excess amount of drug to the dissolution medium, while shaking continuously in a water bath and sampling at a fixed time such as 24 hours. Figure 6. Solubility of coenzyme Q 10 nanocrystals and bulk drug Solubility experiments are usually conducted using drugs in the solid state. Figure 7. Figure 8. In vitro dissolution study In vitro dissolution of coenzyme Q 10 from the suspensions containing 80 nm, nm, nm, or nm naked nanocrystals and bulk drugs was also investigated in the three types of dissolution medium.
Figure 9. Bioavailability study in dogs All serum samples were analyzed following the HPLC method described earlier. Figure Conclusion Four naked stable coenzyme Q 10 nanocrystals of different size were prepared and characterized to generate a drug model for investigating the effects of particle size. Footnotes Disclosure The authors report no conflicts of interest in this work. References 1. Hydrosols-alternatives for the parenteral application of poorly water soluble drugs.
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A new understanding of the relationship between solubility and particle size. J Solution Chem. Kipp JE. The role of solid nanoparticle technology in the parenteral delivery of poorly water-soluble drugs.
The rate of solution of solid substances in their own solutions. J Am Chem Soc. Dokoumetzidis A, Macheras P. A century of dissolution research: from Noyes and Whitney to the Biopharmaceutics Classification System. Solubility and dissolution enhancement: an overview. J Pharm Res. Nanosuspension for improving the bioavailability of a poorly soluble drug and screening of stabilizing agents to inhibit crystal growth. Preparation and evaluation of nanosuspensions for enhancing the dissolution of poorly soluble drugs.
Nanosuspensions as the most promising approach in nanoparticulate drug delivery systems. Rabinow BE. Nanosuspensions in drug delivery. Nat Rev Drug Discov. Drug nanoparticles by antisolvent precipitation: mixing energy versus surfactant stabilization. The solubility of a substance fundamentally depends on the solvent used, as well as temperature and pressure.
The solubility of a substance in a particular solvent is measured by the concentration of the saturated solution. In contrast, the solubility of gases increases as the partial pressure of the gas above a solution increases. An increase in pressure and an increase in temperature in this reaction results in greater solubility.
An increase in pressure results in more gas particles entering the liquid in order to decrease the partial pressure.
Therefore, the solubility would increase. Increasing the temperature will therefore increase the solubility of the solute. An example of a solute whose solubility increases with greater temperature is ammonium nitrate, which can be used in first-aid cold packs. Effect of the temperature on solubility i. At high temperature the solubility of a solution is high so it is able to dissolve more solute, but when it is cooled, the solubility of the solution decreases and due to which the solute separate out as solid.
Therefore, the solubility concentration increases with an increase in temperature. If the process is exothermic heat given off. A temperature rise will decrease the solubility by shifting the equilibrium to the left.
The solubility of Na2SO4 increases up to In organic compounds, the presence of polarity, especially hydrogen bonding, usually leads to a higher melting point. The melting points of polar substances are higher than the melting points of nonpolar substances with similar sizes. Polarity is important because it determines whether a molecule is hydrophilic from the Greek for water-loving or hydrophobic from the Greek for water-fearing or water-averse.
Molecules with high polarity are hydrophilic, and mix well with other polar compounds such as water. When the temperature of an object increases, the average kinetic energy of its particles increases.
Therefore, the thermal energy of an object increases as its temperature increases. Therefore the kinetic energy will be the highest when the temperature is the highest. Temperature is directly proportional to the average translational kinetic energy of molecules in an ideal gas. Solid particles have the least amount of energy, and gas particles have the greatest amount of energy. Radiation can occur not only from solid surfaces but also from liquids and gases.
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