Method for predicting the solubility of a molecule in a polymer at a given temperature

Abstract

The invention provides an improved method to predict the solubility of a drug or other molecule in a solid polymer or other matrix at any temperature. The instant invention provides a method to determine the difference in specific enthalpy, specific entropy and specific Gibbs energy between a solid solution and the unmixed components, as well as a method to use those data to predict the solubility of a drug or other molecule in a solid polymer or other matrix. The method uses known thermodynamics equations and thermal analysis data, such as obtained from DSC (differential scanning calorimetry) at temperatures that are lower than the temperature at which the solubility is predicted. The method allows prediction of the drug-in-polymer solubilities without the use of elevated temperatures, but still avoids impractically long experiments. The instant invention can predict the solubility at many temperatures, but is particularly useful in the pharmaceutical sciences to predict the solubility of a drug in a polymer at typical storage temperatures, which are typically near room temperature or below.

Claims

1. A method for determining the difference in specific enthalpies at a given temperature T and pressure due to heat content differences between a solid solution and unmixed components of the same composition, comprising a) providing unmixed components of a substance and a polymer in a known weight ratio and at a given temperature; b) providing a solid solution mixture of said substance in said polymer in the same weight ratio at said temperature; c) obtaining the specific heats of said unmixed components using DSC or other appropriate method, from the lowest temperature for which data is used T.sub.min to said temperature T or higher; d) obtaining the specific heat of said solid solution using DSC or other appropriate method, from the lowest temperature for which data is used T.sub.min to said temperature T or higher; e) taking the difference between the specific heats of said solid solution and said unmixed components over the experimental range from T.sub.min to said temperature T or higher to construct a function C.sub.P; f) providing an approximating function c that fits said experimental C.sub.P data; g) calculating values of c for temperatures ranging from absolute zero to T.sub.min; h) determining the difference in specific enthalpy due to the heat content difference H.sub.heat between said solid solution and said unmixed components from Eq. (31).

2. A method of claim 1 for determining the difference in specific enthalpies at a given temperature T and pressure due to heat content differences between a solid solution and unmixed components of the same composition, comprising a) providing unmixed components of a substance that is crystalline and a polymer in a known weight ratio and at a given temperature; b) providing a solid solution mixture of said substance in said polymer in the same weight ratio at said temperature; c) obtaining the specific heats of said unmixed components using DSC or other appropriate method, from the lowest temperature for which data is used T.sub.min to said temperature T or higher; d) obtaining the specific heat of said solid solution using or other appropriate method, from the lowest temperature for which data is used T.sub.min to said temperature T or higher; e) taking the difference between the specific heats of said solid solution and said unmixed components over the experimental range from T.sub.min to said temperature T or higher to construct a function C.sub.P; f) providing an approximating function c that fits said experimental C.sub.P data; g) calculating values of c for temperatures ranging from absolute zero to T.sub.min; h) determining the difference in specific enthalpy due to the heat content difference H.sub.heat between said solid solution and said unmixed components from Eq. (31).

3. A method of claim 1 for determining the difference in specific enthalpies at a given temperature T and pressure due to heat content differences between a solid solution and unmixed components of the same composition, comprising a) providing unmixed components of a substance that is amorphous and a polymer in a known weight ratio and at a given temperature; b) providing a solid solution mixture of said substance in said polymer in the same weight ratio at said temperature; c) obtaining the specific heats of said unmixed components using DSC or other appropriate method, from the lowest temperature for which data is used T.sub.min to said temperature T or higher; d) obtaining the specific heat of said solid solution using DSC or other appropriate method, from the lowest temperature for which data is used T.sub.min to said temperature T or higher; e) taking the difference between the specific heats of said solid solution and said unmixed components over the experimental range from T.sub.min to said temperature T or higher to construct a function C.sub.P; f) providing an approximating function c that fits said experimental C.sub.P data; g) calculating values of c for temperatures ranging from absolute zero to T.sub.min; h) determining the difference in specific enthalpy due to the heat content difference H.sub.heat between said solid solution and said unmixed components from Eq. (31).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a hypothetical plot of G.sub.SS per gram of polymer vs. the moles of drug per gram of polymer for solid solutions of a drug in a polymer formed from the unmixed components at a given temperature. The minimum of the plot represents .sub.1=0, and the corresponding drug concentration represents the solubility of the drug in the polymer at that temperature.

(2) FIG. 2 shows experimental DSC data from 220-310K for the drug indomethacin and the polymer PVP K-30 (polyvinylpyrrolidone). The indomethacin concentration is 2.5% by weight and the PVP-K30 concentration is 97.5% by weight (1 gram of indomethacin per 39 grams of PVP-K30). The solid line represents the specific heat vs. temperature experimental data for the solid solution, and the dashed line represents the specific heat data for the corresponding unmixed components. The dotted line represents the difference in specific heats (solid solution minus unmixed components), which is C.sub.P.

(3) FIG. 3 shows the difference in specific heats for indomethacin 2.5% by weight with PVP-K30. The solid line from 220-310 K represents the experimentally determined difference C.sub.P taken from FIG. 2, and the dashed line represents the fit using the function c given in Example 1 and the extrapolation of c to absolute zero.

(4) FIG. 4 shows a plot of G.sub.SS per gram of polymer vs. the drug weight fraction for indomethacin and PVP-K30 at 25 C., as described in Example 3. The drug weight fraction corresponding to the minimum of the plot represents the drug solubility in the polymer at 25 C.

SUMMARY OF THE INVENTION

(5) The instant invention comprises a method for predicting the solubility of a drug in a polymer, in the form of a solid solution at a given temperature. Said given temperature is not unique, and the solubility can be predicted at many temperatures using the instant invention.

(6) The instant invention provides a method for determining the difference in specific enthalpy and specific entropy between a solid solution and its corresponding unmixed components at a given temperature or temperatures, and a method to use that information to predict the solubility of a drug in a solid polymer at the temperature(s).

(7) In addition, the instant invention provides a method for determining the difference in specific enthalpy and specific entropy differences and predicting the drug-in-polymer solubility at given temperature(s) without requiring impractically long experiments, and without the need to perform experiments at elevated temperatures. In fact, the instant invention uses data obtained from experiments at temperatures below the temperature(s) at which the solubility is to be predicted.

(8) In its broadest sense, the instant invention relates to determining or predicting the solubility of any molecule in any solid matrix at many temperatures. While the example and equations treat binary systems comprised of one drug plus one polymer, the examples and equations can be extended to include systems comprised of three or more components. In addition, while one of the primary advantages of the instant invention relates to use in solid matrices that exhibit significant effects of the solid structure on specific heats, the matrix is not limited to solids, but can also be a viscous liquid or semi-solid.

(9) An important advantage of the instant invention is that, at typical storage temperatures, it provides a method for predicting the solubility that does not require the use of elevated temperatures nor impractically long experiments. While the use of elevated temperatures can allow determination of solubilities at those elevated temperatures, there is no reliable method to determine the solubility of a drug in a polymer at storage temperatures from high temperature data. In addition, performing experiments at high temperatures can accelerate chemical degradation of many drugs, thus comprising the accuracy of the experimental results.

(10) Unlike other methods, the instant invention allows avoiding experiments at elevated temperatures that are above the temperature(s) at which the drug-in-polymer solubility is to be predicted. In fact, the experimental data needed to predict the solubility are obtained from methods such as DSC performed at temperatures that are lower than said temperature(s) at which the solubility is to be predicted.

(11) In essence, the invention provides a method for predicting the solubility of a drug or other molecule in a solid polymer or matrix, comprising: a) providing a drug and polymer in a specific ratio, such as grams of drug per gram of polymer; b) determining the specific heat of the unmixed components in said ratio over a given range of temperatures; c) making a solid solution of said drug and polymer in the same ratio as the unmixed components; d) determining the specific heat of the solid solution over the said range of temperatures used in part (b); e) determining the difference in specific heats between the solid solution and unmixed components over said temperature range; f) calculating the difference in specific enthalpy and specific entropy due to specific heat differences between the solid solution and unmixed components at a given temperature; g) calculating the specific enthalpy of mixing and specific entropy of mixing due to forming the solid solution from the unmixed components; h) calculating the change in the specific Gibbs energy G.sub.SS resulting from forming the solid solution from the unmixed components; i) repeating steps (a) through (h) for at least two other drug-polymer ratios; j) plotting G.sub.SS per gram of polymer vs. the grams of drug per gram of polymer for each drug-polymer ratio at a given temperature; k) calculating the predicted solubility of the drug in the polymer as the grams of drug per gram of polymer at which the minimum of the plot in step (j) at said given temperature.

Glossary of Terms

(12) Matrix Polymer or other solid in which a drug is dispersed or dissolved 0 Subscript denoting polymer or matrix 1 Subscript denoting drug DSC Differential scanning calorimetry h.sub.heat Difference in specific enthalpy between a solid solution and unmixed components due to heat content differences h.sub.mix Difference in specific enthalpy between a solid solution and unmixed components due to mixing of the drug and polymer s.sub.heat Difference in specific entropy between a solid solution and unmixed components due to mixing of the drug and polymer s.sub.mix Difference in specific enthalpy between a solid solution and unmixed components due to heat content differences g.sub.SS Change in specific Gibbs energy when a solid solution is made from unmixed components at a given temperature h.sub.SS Change in specific enthalpy when a solid solution is made from unmixed components at a given temperature s.sub.SS Change in specific entropy when a solid solution is made from unmixed components at a given temperature .sub.1,SS The difference in chemical potential between the drug that is dissolved in a solid matrix and the pure, unmixed drug, as given by Eq. (1) and Eq. (4) .sub.1 Chemical potential of the drug SS Subscript denoting solid solution P Pressure T Temperature n.sub.0 Moles of polymer or matrix in a mixture n.sub.1 Moles of drug in a mixture n.sub.10 Total moles (n.sub.10=n.sub.0+n.sub.1) w.sub.0 Weight of polymer or matrix in a mixture w.sub.1 Weight of drug in a mixture w.sub.10 Total weight in a drug-polymer mixture w.sub.10=w.sub.0+w.sub.1) C.sub.P0 Specific heat of unmixed polymer or matrix C.sub.P1 Specific heat of unmixed drug C.sub.P10 Specific heat of a solid solution c Approximating function used to estimate C.sub.P at temperatures below T.sub.min C.sub.P Specific heat difference between a solid solution of a drug and polymer in a given ratio, and one gram of unmixed components in the same ratio, as given by Eq. (29) T.sub.min Lowest temperature for which experimental thermal analysis data such as DSC is reliably obtained or used .sub.0 Volume fraction of polymer in the solid solution .sub.1 Volume fraction of drug in the solid solution M.sub.0 Molecular weight of polymer or matrix (g/mole) M.sub.1 Molecular weight of drug (g/mole) f.sub.0 Weight fraction of polymer or matrix in a mixture f.sub.1 Weight fraction of drug in a mixture r Degree of polymerization R Universal gas constant=8.314 J/K/mole Drug-polymer energy interaction parameter

DISCUSSION OF THE INVENTION

(13) Mathematical Model and Methods to Obtain Numerical Values for Terms Used in the Final Equations

(14) Solid solutions of a drug in a solid polymer or other matrix are formulated to increase the dissolution rate and bioavailability of the drug relative to that of its most stable form. However, when the dissolved drug concentration is too high, these systems are physically unstable and the dissolved drug tends to precipitate during storage. In what follows, the solubility will be characterized in terms of the chemical potential of the dissolved drug compared to the undissolved (unmixed) drug, which will then be related to the change in the specific Gibbs energy resulting from forming a solid solution. A model for determining the change in specific Gibbs energy will be presented, along with experimental and theoretical methods needed to determine it as a function of the drug loading, and a discussion of how that information can be used to predict the solubility of a drug in a solid polymer.

(15) Role of the Chemical Potential

(16) Parts of this section follow the presentation of Bellantone et al. [11]. It is well known that there is a thermodynamic tendency for materials to go from higher to lower chemical potential [24], which can involve a change of form, location, or both. Here, the chemical potentials to be compared will be those of the drug that is dissolved (molecularly dispersed) in a solid matrix and the drug in its precipitated or unmixed form. In this discussion, the equilibrium case will be considered in which the precipitated drug and unmixed drug will be considered as being thermodynamically interchangeable. However, it is also possible to apply the model using thermodynamic properties of metastable precipitated drug forms (forms that are not in the lowest energy, equilibrium state), in which case the apparent solubility of the metastable drug form, rather than true or equilibrium solubility of the most stable crystalline drug form would be predicted.

(17) When a solid solution is made from unmixed components, the drug is dissolved in the polymer and the resulting change in the drug chemical potential is denoted as .sub.1,SS where

(18) $\begin{array}{cc}{}_{1,\mathrm{SS}}=\left[\begin{array}{c}\mathrm{chemical}\mathrm{potential}\mathrm{of}\\ \mathrm{drug}\mathrm{in}\mathrm{solid}\mathrm{solution}\end{array}\right]-\left[\begin{array}{c}\mathrm{chemical}\mathrm{potential}\mathrm{of}\\ \mathrm{unmixed}\mathrm{or}\mathrm{precipitated}\mathrm{drug}\end{array}\right]& \left(1\right)\end{array}$
where the subscript 1 denotes the drug and SS denotes solid solution.

(19) For drug molecules that precipitate out of a solid solution, the ending form will be regions of pure precipitated drug, and .sub.1,SS represents the difference in chemical potential between the dissolved drug form and the ending precipitated form. If a solid solution is stored for a sufficient time to allow precipitation and equilibration of the precipitated form to its most stable crystalline form, it is plausible to assume that the chemical potential of the precipitated form of the drug will equal that of the original form used to make the solid solution. (However, as noted, this is not necessary for the model presented to hold, as long as the correct chemical potential of either the starting drug material or final drug material is specified.) Also, the precipitated drug may be in the form of one or many particles, but all particles would be large enough so they take on properties of unmixed drug, in the sense that the effects of the drug-polymer molecular interactions for the precipitated drug molecules can be neglected.

(20) For a given drug and polymer solution at a given temperature and pressure, precipitation of the dissolved drug out of a solid solution may or may not be thermodynamically favored, depending on the dissolved drug concentration. It is known from thermodynamics that at as more of the drug is dissolved in the polymer, the chemical potential of the dissolved drug in the polymer increases, which leads to the following: When the dissolved drug concentration in the polymer is lower than its solubility, .sub.1,SS<0 and precipitation is not favored, so the solid solution is stable. When the dissolved drug concentration in the polymer is higher than its solubility, the solid solution is supersaturated, .sub.1,SS>0 and precipitation is favored, so the solid solution is unstable. When the dissolved drug concentration in the polymer equals its solubility, the solid solution is saturated, .sub.1,SS=0 and there is no tendency to convert between dissolved and undissolved forms, so precipitation is not favored and the solid solution is stable.

(21) The above considerations show that it is important to know the solubility, since this represents the highest stable dissolved concentration of drug in a solid solution.

(22) Relationship Between Chemical Potential and the Gibbs Energy

(23) When a system is changed from an initial to final state, there is an associated change in the Gibbs energy that equals the Gibbs energy of the final state minus the Gibbs energy of the initial state. For the case of making a solid solution from unmixed components with the same starting and initial temperature, this change is denoted by G.sub.SS and is given by
G.sub.SS=H.sub.SSTS.sub.SS(2)
where H.sub.SS and S.sub.SS denote the enthalpic and entropic changes associated with the formation of the solid solution, and T is the temperature at which G.sub.SS is to be evaluated.

(24) As noted, the chemical potential of the drug plays an important theoretical role in studying solid solutions. Thermodynamically, the drug chemical potential is defined as the change in the Gibbs energy per change in amount of drug, holding all other factors constant (temperature, pressure, and the amount of polymer. This is expressed as a partial derivative of the Gibbs energy with respect to the drug content (typically in moles), given by [25]

(25) $\begin{array}{cc}{}_1={\left(\frac{G}{n_1}\right)}_{T,P,n_0}& \left(3\right)\end{array}$
where n.sub.1 and n.sub.0 denote the moles of drug and polymer, respectively. For the case of making a solid solution from pure drug, the corresponding change in drug chemical potential due to making the solid solution is related to the change in the Gibbs energy by

(26) $\begin{array}{cc}{}_{1,\mathrm{SS}}={\left(\frac{G_{\mathrm{SS}}}{n_1}\right)}_{T,P,n_0}& \left(4\right)\end{array}$

(27) It is possible to find .sub.1,SS in two mathematically equivalent ways. The first is to write equations for G.sub.SS, then take the partial derivative as given by Eq. (4). The second is to plot G.sub.SS vs. n.sub.1 (at constant of T, P and n.sub.0) and take the slope at with respect to n.sub.1. The first method works well if equations are known for given enthalpic or entropic contributions, while the graphing method is more practical of equations for any contribution to G is not known from theoretical calculations. In practice, some contributions can be calculated theoretically while others must be determined experimentally, in which case the graphing method is likely preferable. When graphing, it is often preferable to construct plots of the change in Gibbs per amount of polymer vs. the moles of drug per amount of polymer are made, or

(28) $\begin{array}{cc}\frac{G_{\mathrm{SS}}}{w_0}\mathrm{vs}.\frac{n_1}{w_0}& \left(5\right)\end{array}$

(29) Dividing both G.sub.SS and n.sub.1 by the weight of the polymer in the mixture, which is proportional to the number of moles of polymer, is equivalent to holding the amount of polymer constant as required by the partial derivative given by Eq. (4). Once the plot is constructed, the slope gives .sub.1,SS. A hypothetical example is shown in FIG. 1, which shows where a solid solution is stable (negative or zero slope, corresponding to .sub.1,SS0) and where it is unstable (positive slope, corresponding to .sub.1,SS>0) for a given temperature.

(30) Other plots can be derived from the one described by Eq. (5), in which G.sub.SS or a quantity proportional to it decreases as some function of the drug concentration to a minimum and then increases, and in which the minimum of the plot can be used to identify the drug solubility. For instance, since n.sub.1=w.sub.1/M.sub.1, where w.sub.1 denotes the weight of the drug in the formulation and M.sub.1 denotes the molecular weight of the drug, the plot given by Eq. (5) can be replaced by

(31) $\begin{array}{cc}\frac{G_{\mathrm{SS}}}{w_0}\mathrm{vs}.\frac{w_1}{w_0}& \left(6\right)\end{array}$
The minimum of this plot corresponds to the drug-in-polymer solubility as grams of drug per gram of polymer. This can be related to the weight fractions of the polymer and drug in the formulation, which are denoted by f.sub.0 and f.sub.1 and are given as

(32) $\begin{array}{cc}{f}_0=\frac{w_0}{w_{10}}{f}_1=\frac{w_1}{w_{10}}{w}_{10}=w_0+w_1& \left(7\right)\end{array}$
Since w.sub.1/w.sub.0=f.sub.1/f.sub.0 the solubility in terms of the drug fraction is also obtained from the minimum of a plot of

(33) $\begin{array}{cc}\frac{G_{\mathrm{SS}}}{w_0}\mathrm{vs}.f_1& \left(8\right)\end{array}$

(34) It is convenient to express the thermodynamic quantities in terms of the specific Gibbs, specific enthalpy and specific entropy, which are the Gibbs energy per gram, the enthalpy per gram, and the entropy per gram of sample, respectively. These will be denoted by lowercase g, h and s, respectively. Denoting the change in specific Gibbs energy due to forming a solid solution (S.sub.SS per gram) as g.sub.SS, and noting that

(35) $\frac{G_{\mathrm{SS}}}{w_0}=\frac{G_{\mathrm{SS}}/w_{10}}{w_0/w_{10}}=\frac{g_{\mathrm{SS}}}{f_0},$
the plot of Eq. (8) can be replaced by an alternative plot of

(36) $\begin{array}{cc}\frac{g_{\mathrm{SS}}}{f_0}\mathrm{vs}.f_1& \left(9\right)\end{array}$
The minimum of a plot Eq. (9) will also identify the solubility of a drug in a solid polymer.

(37) It should be noted that the solubility of a drug in a polymer depends on the temperature, and typically increases as the temperature at which the solubility is evaluated increases. Thus, the general form of the plot shown in FIG. 1 or described by Eq. (6)-(9) would still apply in the sense that the plot would show a negative slow at low drug fractions, a minimum at some drug fraction, and a positive slope at higher drug fractions. However, the exact values of the slopes and drug concentrations corresponding to the minima would be different for different temperatures. In general, it would be expected that raising the temperature at which the solubility data is evaluated would result in the minimum of the curve shifting to the right (corresponding to higher concentrations). This consideration is important, because it can be of interest to evaluate the drug-in-polymer solubility at more than one temperature.

(38) Determining g.sub.SS

(39) In forming a solid solution, the initial state consists of pure precipitated or undissolved drug plus empty polymer at a given temperature, and the final state consists of a solid solution of drug dissolved in the polymer at the same temperature. It is assumed that all of the drug present goes into the solid solution, and the approach is to evaluate g.sub.SS for all solutions and determine the solubility as the solution at which g.sub.SS is minimized. (If excess drug that does not dissolve is included, it would increase the specific Gibbs energy of the initial and final states by the same amount, so neither the slopes of plots such as Eq. (5) nor the predicted solubility would not be affected.)

(40) The Gibbs, enthalpy and entropy are thermodynamic state functions, so the change in their values depends only on the difference between the initial and final states, and not the particular process that brought about the change [25]. Also, if temperatures of the starting and final states are the same and denoted by T, it is been shown that [11]

(41) 0$\begin{array}{cc}\left[\begin{array}{c}\mathrm{total}\mathrm{Gibbs}\\ \mathrm{energy}\mathrm{change}\end{array}\right]=\left[\begin{array}{c}\mathrm{total}\mathrm{enthalpy}\\ \mathrm{change}\end{array}\right]-T\left[\begin{array}{c}\mathrm{total}\mathrm{entropy}\\ \mathrm{change}\end{array}\right]& \left(10\right)\end{array}$
which is analogous to Eq. (2). Since the specific change in Gibbs energy, specific change in enthalpy and specific change in entropy are also state functions, Eq. (10) can be written using the analogous specific Gibbs energy, specific enthalpy and specific entropy.

(42) Eq. (10) is applicable here because the solubility is determined by comparing the unmixed and solid solution states at the same temperature. For the formation of a solid solution from pure unmixed drug and polymer at a given temperature and pressure, the changes in the specific enthalpy and specific entropy arise from the following contributions. The specific enthalpy of mixing h.sub.mix, which reflects differences in the potential energy content per gram of the mixture due to differences in the intermolecular interaction energies. The specific entropy of mixing s.sub.mix, which reflects the change in entropy per gram of mixture due to mixing of the drug in the polymer. The difference in the specific enthalpy between the solid solution and unmixed components due to differences in the heat content, which reflects differences in the kinetic energy of the molecules in the solid solution and unmixed form. This is denoted as s.sub.heat. The difference in specific entropy due to the differences in heat content between the solid solution and unmixed components, which is denoted by s.sub.heat.

(43) The above contributions can be added, so the change due to making a solid solution in the specific enthalpy and specific entropy are given by h.sub.SS=h.sub.mix+h.sub.heat and s.sub.SS=s.sub.mix+s.sub.heat, respectively. From Eq. (10) in terms of specific quantities, the resulting specific Gibbs energy change g.sub.SS is given by
g.sub.SS=h.sub.SSTs.sub.SS(11)

(44) The specific enthalpy and specific entropy of mixing can be calculated using known equations and models. For drugs that are uniformly mixed in the polymer, equations for the enthalpy and entropy of mixing are commonly calculated using the Flory-Huggins model [6-8]. These can be converted into the specific enthalpy and specific entropy by dividing by the weight of the mixture w.sub.10, and are given by

(45) $\begin{array}{cc}{h}_{\mathrm{mix}}=n_1{}_0\mathrm{RT}/w_{10}& \left(12\right)\\ s_{\mathrm{mix}}=-\frac{R}{w_{10}}\left[n_1\ln {}_1+n_0\ln {}_0\right]& \left(13\right)\end{array}$
where n.sub.0 and n.sub.1 denote the moles of polymer and drug in the solid solution, respectively, and .sub.0 and .sub.1 denote the corresponding volume fractions of polymer and drug, respectively. The parameter denotes the drug-polymer energy interaction parameter (analogous to the Flory-Huggins parameter), and R is the universal gas constant.

(46) To evaluate the above equations, values must be known for the moles of drug and polymer n.sub.1 and n.sub.2, the ratio of the polymer molar volume to the drug molar volume, denoted by r. These can be calculated from the weights of drug and polymer w.sub.1 and w.sub.0 in the solid solution, and their respective molecular weights M.sub.1 and M.sub.0, using known equations [8]. The moles of drug and polymer were calculated as
n.sub.1=w.sub.1/M.sub.1 n.sub.0=w.sub.0/M.sub.0(14)
The drug and polymer volume fractions were calculated as

(47) $\begin{array}{cc}{}_1=\frac{n_1}{n_1+{\mathrm{rn}}_0}{}_0=\frac{{\mathrm{rn}}_2}{n_1+{\mathrm{rn}}_0}=1-{}_1& \left(15\right)\end{array}$
where r is the ratio of the molar volume of the polymer to the molar volume of the drug and is commonly approximated as the degree of polymerization, or

(48) $\begin{array}{cc}r\mathrm{degree}\mathrm{of}\mathrm{polymerization}=\frac{\mathrm{MW}\mathrm{polymer}}{\mathrm{MW}\mathrm{monomer}}& \left(16\right)\end{array}$

(49) The volume fractions are not the same as weight fractions, but are be related as

(50) $\begin{array}{cc}{}_1=\frac{\frac{f_1}{M_1}}{\left(\frac{f_1}{M_1}+\frac{{\mathrm{rf}}_0}{M_0}\right)}{}_0=\frac{\frac{{\mathrm{rf}}_0}{M_0}}{\left(\frac{f_1}{M_1}+\frac{{\mathrm{rf}}_0}{M_0}\right)}& \left(17\right)\end{array}$

(51) The interaction parameter can be obtained from solubility parameters calculated for each drug and polymer combination using the method of van Krevelen [7] as

(52) $\begin{array}{cc}=\frac{{v_1\left({}_1-{}_0\right)}^2}{\mathrm{RT}}+0.34& \left(18\right)\end{array}$
where .sub.1 and .sub.0 denote the solubility parameters for the drug and polymer, respectively, and .sub.1 denotes the molar volume of the drug. The factor of 0.34 was not included in the original Flory-Huggins publications, but has been introduced to improve correlations between calculated and experimental results when the predicted value of is less than 0.3 or so [8].

(53) Alternative methods of determining have been proposed by using the melting point method [25] and solubility determined using the endpoint method [22]. These have the potential advantage of being experimentally determined, but also suffer from the disadvantage of being determined at elevated temperatures, so the values to be used at room temperature need to be estimated. However, the of method chosen to determine does not affect the invention itself.

(54) The specific enthalpy and specific entropy changes due to differences in the heat content of solid solutions vs. unmixed components, h.sub.heat and S.sub.heat, can be determined from experimental data using known thermodynamic equations as follows. It is known that the changes in the enthalpy at constant pressure resulting from a temperature change are related to the heat capacity, which is defined as the change in heat content per degree change in temperature. When done at a constant pressure, which corresponds to open pan DSC conditions, this is referred to as the constant pressure heat capacity. (For brevity, unless otherwise noted, the terms heat capacity and specific heat will refer constant pressure conditions.) The constant pressure heat capacity per gram of material is referred to as the specific heat and is denoted by C.sub.P=dh/dT, where h is the specific enthalpy.

(55) The specific heat, specific enthalpy and specific entropy are all functions of the temperature. The specific heat can be determined using thermal analysis methods such as DSC, and changes in the specific enthalpy and specific entropy due to adding or removing heat to change the temperature from T.sub.1 to T.sub.2 can be calculated from the specific heat as [10]

(56) $\begin{array}{cc}h={}_{T_1}^{T_2}{C}_{P}dT& \left(19\right)\\ s={}_{T_1}^{T_2}\frac{C_P}{T}dT& \left(20\right)\end{array}$
From thermodynamics, it is known that at absolute zero (0 K, or 0 Kelvin), there is no heat content and the entropy due to heat content also equals zero. Thus, the specific enthalpy and specific entropy content arising from adding heat to raise the temperature from absolute zero to a temperature T are given by

(57) $\begin{array}{cc}{h}_{\mathrm{heat}}={}_0^T{C}_{p}d{T}^{}& \left(21\right)\\ s_{\mathrm{heat}}={}_0^T\frac{C_p}{T^{}}d{T}^{}& \left(22\right)\end{array}$
For a mixture of w.sub.0 grams of a polymer with specific heat C.sub.P0 plus w.sub.1 grams of a drug with specific heat C.sub.P1, the total specific enthalpy and specific entropy per gram of the unmixed components due to the heat content at temperature T are given by

(58) $\begin{array}{cc}{h}_{\mathrm{heat}}={}_0^T\frac{\left(w_0{C}_{P0}+w_1{C}_{P1}\right)}{w_{10}}d{T}^{}={}_0^T\left(f_0{C}_{P0}+f_1{C}_{P1}\right)d{T}^{}\left(\mathrm{unmi}\mathrm{xed}\mathrm{components}\right)& \left(23\right)\\ s_{\mathrm{heat}}={}_0^T\frac{\left(f_0{C}_{P0}+f_1{C}_{P1}\right)}{T^{}}d{T}^{}\left(\mathrm{unmixed}\mathrm{components}\right)& \left(24\right)\end{array}$
where f.sub.0 and f.sub.1 denote the weight fractions of the polymer and drug in the mixture, respectively, and are given by Eq. (7). The specific enthalpy and specific entropy for the corresponding solid solution at the same temperature T are given by

(59) $\begin{array}{cc}\begin{array}{cc}{h}_{\mathrm{heat}}={}_0^T{C}_{p_{10}}d{T}^{}& \left(\mathrm{solid}\mathrm{solution}\right)\end{array}& \left(25\right)\\ \begin{array}{cc}{s}_{\mathrm{heat}}={}_0^T\frac{C_{p10}}{T^{}}d{T}^{}& \left(\mathrm{solid}\mathrm{solution}\right)\end{array}& \left(26\right)\end{array}$
where C.sub.P10 denotes the specific heat of the solid solution. Thus, the differences between the specific enthalpy and specific entropy per gram (solid solution minus the unmixed components) are given by

(60) 0$\begin{array}{cc}{h}_{\mathrm{heat}}={}_0^T\left(C_{p_{10}}-f_1{C}_{P1}-f_0{C}_{P0}\right)d{T}^{}& \left(27\right)\\ s_{\mathrm{heat}}={}_0^T\frac{\left(C_{P_{10}}-f_1{C}_{P1}-f_0{C}_{P0}\right)}{T^{}}d{T}^{}& \left(28\right)\end{array}$

(61) In practice, there is a lower limit to the temperature at which thermal analysis instruments such as DSC can provide accurate experimental data. As a result, it is not possible to experimentally obtain the necessary specific heats over all temperatures down to absolute zero, but instead they only be evaluated to a minimum temperature, denoted by T.sub.min. The experimental data can be used to construct a temperature-dependent difference in specific heats, denoted as C.sub.P, which is defined as the difference between the specific heat of the solid solution minus the specific heat of the unmixed components in the same ratio, over the temperature range of T.sub.min to any higher temperature T for which experimental specific heat data is available. This is given by
C.sub.P=C.sub.P10f.sub.1C.sub.P1f.sub.2C.sub.P0 (from T.sub.min to T)(29)

(62) Because of the thermal analysis instrument limitations, C.sub.P provides no data between absolute zero and T.sub.min, so it is necessary to estimate the values of specific heat differences at temperatures below T.sub.min. One way to do this is to extrapolate the plot of the experimentally obtained C.sub.P vs. the temperature from T.sub.min to absolute zero. This can be done as follows. Obtain the experimental values for C.sub.P from T.sub.min up to at least temperature Tat which the solubility of the drug in the polymer is to be evaluated. Construct an approximating function c that is a function of the temperature, which will be used to estimate C.sub.P at temperatures lower than T.sub.min. Fit the function c to the experimental values of C.sub.P from T.sub.min to T using any appropriate method, such as linear or nonlinear regression. Using the fitted equation for c, calculate values of c over temperatures from absolute zero to T.sub.min. These values will be used as substitutes for experimental specific heat difference for temperatures below T.sub.min.

(63) There is no theoretical model or equation that dictates the mathematical form of c, and more than one functional form can be chosen. Possible forms for c include linear forms, polynomials, logarithmic forms, exponential forms, power series, etc. It is also possible to use different forms in different temperature ranges. (For example, a polynomial above T.sub.min, and a linear form with an intercept of zero below that temperature.) However, thermodynamic considerations require that the value of c approach zero as the temperature approaches absolute zero. This is because the heat capacities of materials is zero at absolute zero, so the differences must also be zero at absolute zero. Thus, a preferred form would allow a fit of c to C.sub.P and an extrapolation for which c=0 at absolute zero. One preferred form would be a sum of a constant plus exponentials, such as
c=a.sub.0+a.sub.1exp(a.sub.2T)+a.sub.3exp(a.sub.4T)+a.sub.5exp(a.sub.6T)+a.sub.7exp(a.sub.8T)(30)
where a.sub.0, a.sub.1 . . . a.sub.8 are temperature independent parameters with values that can be determined or fit by nonlinear regression or other methods so Eq. (30) will be a good approximation to the experimental C.sub.P vs. temperature data over the experimental temperature range. In such fits, the constraint that a.sub.0+a.sub.1+a.sub.3+a.sub.5+a.sub.7=0 satisfies the condition that c=0 at absolute zero.

(64) The function c will in general be different for each combination of drug and polymer. In practice, c is defined by its chosen form and the fitted values of the adjustable parameters subject to the constraint that c equal zero at absolute zero. Thus, even the same drug and polymer, there will in general be a distinct c for each weight ratio, even if the form of c does not change, due to different values of a.sub.0, a.sub.1 . . . a.sub.8 for each ratio.

(65) Alternative forms of c can also be used. For instance, the heat capacities of the solid solution and unmixed components can be separately obtained and fit with distinct approximating functions, which could be extrapolated separately to zero and the difference between them taken as a final step to obtain a function c. However, in addition to requiring that the individual extrapolated heat capacities vanish at absolute zero, an addition constraint must be imposed requiring that the slopes with respect to the temperature of both c and the experimental C.sub.P at T.sub.min be the same. Thus, fitting the experimental C.sub.P directly is likely more convenient and preferred. However, the choice procedure to construct c does not change the invention.

(66) Using the approximating function c and the experimentally obtained C.sub.P data, the final equation for the difference in enthalpy and entropy between the solid solution and unmixed components due to heat differences is given as

(67) $\begin{array}{cc}{h}_{\mathrm{heat}}={}_0^{T_{\min }}cd{T}^{}+{}_{T_{\min }}^T\left(C_{p_{10}}-f_1{C}_{P1}-f_0{C}_{P0}\right)d{T}^{}& \left(31\right)\\ {}_{\mathrm{heat}}={}_0^{T_{\min }}\frac{c}{T^{}}d{T}^{}+{}_{T_{\min }}^T\left(\frac{f_{10}{C}_{P10}-f_1{C}_{P1}-f_0{C}_{P0}}{T^{}}\right)d{T}^{}& \left(32\right)\end{array}$
and, from Eq. (11), the specific change in Gibbs energy is given by

(68) $\begin{array}{cc}{g}_{\mathrm{SS}}=\frac{n_1{}_0\mathrm{RT}}{w_{10}}+\frac{\mathrm{RT}}{w_{10}}\left(n_1\ln {}_1+n_0\ln {}_0\right)+{}_0^{T_{\min }}cd{T}^{}+{}_{T_{\min }}^T\left(C_{p_{10}}-f_1{C}_{P1}-f_0{C}_{P0}\right)d{T}^{}-T{}_0^{T_{\min }}\frac{c}{T^{}}d{T}^{}-T{}_{T_{\min }}^T\left(\frac{C_{P10}-f_1{C}_{P1}-f_0{C}_{P0}}{T^{}}\right)d{T}^{}& \left(33\right)\end{array}$

(69) The above model and equations consider binary systems in which one drug forms a solid solution with one polymer. However, the invention is not limited to binary systems, and the equations can be extended to include ternary systems, etc. Cases can include the solubility of a drug in a polymer mixture, or the solubility of a drug in a mixture of a polymer and another dissolved molecule, which could be a drug or an excipient. In those cases, the solubility of the drug in the polymer would still be determined from the minimum of a plot of g.sub.SS f.sub.0 vs. f.sub.1, as prescribed by Eq. (9), with the equations used to calculate g.sub.SS being appropriately modified.

(70) Many of the equations above are thermodynamic in nature, and thus most correctly apply to final states that are most thermodynamically stable (e.g., final and lowest energy states). For instance, the equations assume precipitation of drug molecules dissolved in a polymer result in the chemical potential of the precipitated drug being the same as the original undissolved crystalline form. However, as noted, the model can also be applied to metastable forms of the drug and/or polymer.

Example 1. Determining c, hheat and sheat for Indomethacin 2.5% by Weight and PVP-K30 at 20 C. and 30 C.

(71) The drug indomethacin and the polymer PVP-K30 were taken in a ratio of 2.5% indomethacin and 97.5% PVP-K30 by weight (e.g., 1 part indomethacin and 39 parts PVP-K30 by weight). DSC was performed on the pure indomethacin and pure PVP-K30, and the specific heats for the indomethacin C.sub.P1 and PVP-K30 C.sub.P0 were obtained as a function of the temperature over the range of 220K to 310K. The DSC instrument provided reliable data as low as 220K, which was taken as T.sub.min. Solid solutions were formed by mixing the drug and polymer in a mortar and pestle, allowing them to stand at 403 K (130 C.) for 5 days, then allowing them to cool to room temperature. DSC was also performed to determine the specific heat of the solid solution C.sub.P10 from 220K to 310K. All DSC experiments were performed using a pinhole in the lid, which approximated constant pressure conditions, so the specific heats were considered to be obtained at constant pressure. All heating and cooling rates were 5 C. per minute.

(72) At each temperature for which experimental data was obtained, the specific heat of the unmixed components was calculated as

(73) $\left[\begin{array}{c}\mathrm{Specifi}c\mathrm{heat}\mathrm{of}\\ \mathrm{unmixed}\mathrm{components}\end{array}\right]=0.025C_{P1}+0.975C_{P0}$
and the function C.sub.P was constructed from the experimental data at each temperature as

(74) $C_P=C_{P10}-\left[\begin{array}{c}\mathrm{Specifi}c\mathrm{heat}\mathrm{of}\\ \mathrm{unmixed}\mathrm{components}\end{array}\right]$

(75) FIG. 2 shows the specific heat vs. temperature data for the unmixed components, the corresponding solid solution, and the difference of the data C.sub.P, over the temperature range of 220K to 310K. The form used for c was the same as that given by Eq. (30), and was fitted to the data for C.sub.P over temperature range of 220K to 310K by nonlinear regression, subject to the constraint that a.sub.0+a.sub.1+a.sub.3+a.sub.5+a.sub.7=0 (which sets c=0 at absolute zero). The constrained fitting of c to C.sub.P was done using the Solver function in Excel, and the following values were obtained for the parameters: a.sub.0=0.017447, a.sub.1=0.147513, a.sub.2=0.005839, a.sub.3=0, a.sub.5=0.557832, a.sub.6=0.0, a.sub.7=0.427767, and a.sub.8=0006406. (The value for a.sub.4 was not determined because the coefficient multiplying the exponential a.sub.3 was zero. Also, a.sub.6 was set to zero because its fitted value was less than 110.sup.10.) These values were inserted as the values for the a.sub.n in Eq. (30), so the actual function c for these data was given by
c=0.017447+0.147513exp(0.005839T)0.557832+0.427767exp(0.006406T)(34)
Values of c were calculated over temperature from zero to 220K. The fitted plot and calculated values of c are shown in FIG. 3.

(76) Using the values c calculated from Eq. (34) for temperatures from 0 to 220K, and the experimental values of C.sub.P, values of h.sub.heat and s.sub.heat were calculated from Eq. (31) and (32), respectively, for various temperatures. For this combination of indomethacin and PVP-K30, the calculated specific enthalpy values at 10 C. and 25 C. (263 K and 298 K) were h.sub.heat=9.66 J/g and 18.347 J/g, respectively. The calculated specific entropy values at 10 C. and 25 C. were s.sub.heat=0.0479 J/K/g and 0.0787 J/K/g, respectively.

Example 2. Determining the Solubility of Indomethacin in PVP-K30 at 25 C.

(77) The drug indomethacin and the polymer PVP-K30 were taken in various ratios, corresponding to weight percentages of indomethacin in the mixture of 2.5%, 10%, 15%, and 20%. DSC was performed on the pure indomethacin and pure PVP-K30, and the specific heat for each was obtained as a function of the temperature over the range of 220K to 310K. The DSC instrument provided reliable data as low as 220K, which was taken as T.sub.min. Solid solutions were formed by mixing the drug and polymer in a mortar and pestle, allowing them to stand at 130 C. (403 K) for 5 days, then allowing them to cool to room temperature. DSC was also performed to determine the specific heat of the solid solutions from 220K to 300K. All DSC was performed using a pinhole in the lid and was thus considered open pan, so the specific heats were considered to be obtained at constant pressure. All heating and cooling rates were 5 C. per minute.

(78) For each drug-polymer mixture, C.sub.P and c were obtained using the procedure and functional form for c described in Example 1. (For each drug-polymer ratio, separate sets of a.sub.0, a.sub.1 . . . a.sub.8 were obtained by the fitting.) The values of h.sub.heat and s.sub.heat were obtained at 25 C. as described in Example 1 for each drug-polymer ratio. For instance, for indomethacin 2.5% and PVP-K30, h.sub.heat=18.347 J/g and s.sub.heat=0.0787 J/K/g.

(79) The entire procedure was repeated for each indomethacin-PVP combinations (2.5% m 10%, 15% and 20% indomethacin by weight), including obtaining the adjustable parameters a.sub.0, a.sub.1 . . . a.sub.8 for each indomethacin/PVP-K30 combination. The same molecular weights (357.8 g/mole for indomethacin 40,000 g/mole for PVP-K30), value of r (360, based on a monomer molecular weight of 111 g/mole) and interaction parameter (0.747 at 25 C. [11]) were used for all combinations. The values of g.sub.SS at 25 C. were calculated for each drug-polymer ratio from Eq. (33), and subsequently used to calculate values of g.sub.SS/f.sub.0.

(80) The plot of the resulting g.sub.SS/f.sub.0 vs. the indomethacin weight fraction f.sub.1, corresponding Eq. (9), is shown in FIG. 4. Although additional weight percentages would be used in practice to construct the plot, the plot shows that there is a minimum value of g.sub.SS/f.sub.0 occurring at the indomethacin weight fraction is approximately 0.15 (approximately 1 gram of indomethacin per 5.67 grams of PVP-K30, or 0.176 gram indomethacin per gram of PVP-K30). Thus, the solubility is taken as 0.176 g indomethacin per gram of PVP-K30 at 25 C. and the maximum stable weight fraction of an indomethacin in PVP-K30 solid solution is 15% by weight.

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