Recent advances of pharmaceutical crystallization theories

doi:10.1016/j.cjche.2019.11.008

Abstract

Abstract: As a technology of separation and purification, crystallization plays a vital role in diverse industries such as inorganic salt, pharmaceutical, and food industries, which has a huge impact on purity, crystal polymorph, crystal morphology, and particle size distribution of final products. In the past few decades, with the rapid advancement of experimental approaches and molecular simulation methods, considerable advances in the interpretation of crystallization mechanisms have been obtained, promoting the investigation and understanding of crystallization theories greatly. In this review, the advances of pharmaceutical crystallization theories in recent years from the perspectives of nucleation and crystal growth are summarized and discussed. Two thermodynamic models that are helpful in the study of the crystallization mechanisms will be introduced. In this section, the perturbed-chain statistical associating fluid theory (PC-SAFT) and a chemical-potential-gradient model will be introduced, which have been successfully applied in pharmaceutical solubility prediction, the research of dissolution mechanism as well as dissolution kinetics analysis. These two models are expected to be applied to the study of pharmaceutical crystallization process and mechanism. Furthermore, molecular simulation based on the interaction between particles can provide structural information, thermodynamics, and dynamics properties of complex systems at the molecular level, like intermolecular interaction and surface adsorption energies. Application and some shortcomings of molecular simulation, especially molecular dynamics simulation, in the field of pharmaceutical crystallization will be expounded.

Key words: Nucleation, Crystal growth, Thermodynamics, Molecular simulation

摘要： As a technology of separation and purification, crystallization plays a vital role in diverse industries such as inorganic salt, pharmaceutical, and food industries, which has a huge impact on purity, crystal polymorph, crystal morphology, and particle size distribution of final products. In the past few decades, with the rapid advancement of experimental approaches and molecular simulation methods, considerable advances in the interpretation of crystallization mechanisms have been obtained, promoting the investigation and understanding of crystallization theories greatly. In this review, the advances of pharmaceutical crystallization theories in recent years from the perspectives of nucleation and crystal growth are summarized and discussed. Two thermodynamic models that are helpful in the study of the crystallization mechanisms will be introduced. In this section, the perturbed-chain statistical associating fluid theory (PC-SAFT) and a chemical-potential-gradient model will be introduced, which have been successfully applied in pharmaceutical solubility prediction, the research of dissolution mechanism as well as dissolution kinetics analysis. These two models are expected to be applied to the study of pharmaceutical crystallization process and mechanism. Furthermore, molecular simulation based on the interaction between particles can provide structural information, thermodynamics, and dynamics properties of complex systems at the molecular level, like intermolecular interaction and surface adsorption energies. Application and some shortcomings of molecular simulation, especially molecular dynamics simulation, in the field of pharmaceutical crystallization will be expounded.

关键词: Nucleation, Crystal growth, Thermodynamics, Molecular simulation

Jingyun Weng, Yiping Huang, Dule Hao, Yuanhui Ji. Recent advances of pharmaceutical crystallization theories[J]. Chinese Journal of Chemical Engineering, 2020, 28(4): 935-948.

Jingyun Weng, Yiping Huang, Dule Hao, Yuanhui Ji. Recent advances of pharmaceutical crystallization theories[J]. 中国化学工程学报, 2020, 28(4): 935-948.

References

[1] J. Chen, B. Sarma, J.M. Evans, A.S. Myerson, Pharmaceutical crystallization, Cryst. Growth Des. 11(2011) 887-895.
[2] W.Y. Su, N. Jia, H.S. Li, H.X. Hao, C.L. Li, Polymorphism of D-mannitol:Crystal structure and the crystal growth mechanism, Chin. J. Chem. Eng. 25(2017) 358-362.
[3] Q. Chen, F. Zou, P. Yang, J. Zhou, J. Wu, W. Zhuang, H. Ying, Transformation of microstructure and phase of disodium guanosine 5'-monophosphate:thermodynamic perspectives, Chin. J. Chem. Eng. 26(2018) 2112-2120.
[4] D. Erdemir, A.Y. Lee, A.S. Myerson, Nucleation of crystals from solution:classical and two-step models, Acc. Chem. Res. 42(2009) 621-629.
[5] M.A. Sharaf, R.A. Dobbins, A comparison of measured nucleation rates with the predictions of several theories of homogeneous nucleation, J. Chem. Phys. 77(1982) 1517-1526.
[6] B.E. Wyslouzil, J. Wolk, Overview:homogeneous nucleation from the vapor phasethe experimental science, J. Chem. Phys. 145(2016) 211702.
[7] N.I. Diamantonis, G.C. Boulougouris, E. Mansoor, D.M. Tsangaris, I.G. Economou, Evaluation of cubic, SAFT, and PC-SAFT equations of state for the vapor-liquid equilibrium modeling of CO₂ mixtures with other gases, Ind. Eng. Chem. Res. 52(2013) 3933-3942.
[8] Z. Khedri, M. Almasi, A. Maleki, Thermodynamic properties of 1-hexyl-3-methylimidazolium nitrate and 1-alkanols mixtures:PC-SAFT model, J. Chem. Eng. Data 64(2019) 4465-4473.
[9] I. Abala, F.E.M.h. Alaoui, A. Sahib Eddine, F. Aguilar, N.M. Rujas, E. Montero, (ρ, VE, T) Measurements of the ternary mixture (dibutyl ether +1-heptanol + heptane) at temperatures up to 393.15 K and pressures up to 140 MPa and modeling using the Peng-Robinson and PC-SAFT equations of state, J. Chem. Eng. Data 64(2019) 3861-3873.
[10] J. Camacho, E. Diez, I. Diaz, G. Ovejero, PC-SAFT thermodynamics of EVA copolymer-Solvent systems, Fluid Phase Equilib. 449(2017) 10-17.
[11] D.-H. Yoo, W.G. Lee, B.-S. Lee, Modeling phase behavior of poly(ethylene glycol) in supercritical fluids, J. Mol. Liq. 283(2019) 332-337.
[12] Y. Ji, R. Paus, A. Prudic, C. Lübbert, G. Sadowski, A novel approach for analyzing the dissolution mechanism of solid dispersions, Pharm. Res. 32(2015) 2559-2578.
[13] R. Shahriari, M.R. Dehghani, Prediction of thermodynamic properties of aqueous electrolyte solutions using equation of state, AIChE J. 63(2017) 5083-5097.
[14] B. Seo, T. Kim, S. Kim, J.H. Ryu, J. Ryu, J. Yoon, W.B. Lee, Y.-W. Lee, Interfacial structure analysis for the morphology prediction of adipic acid crystals from aqueous solution, Cryst. Growth Des. 17(2017) 1088-1095.
[15] M. Salvalaglio, T. Vetter, M. Mazzotti, M. Parrinello, Controlling and predicting crystal shapes:the case of urea, Angew. Chem. Int. Ed. Engl. 52(2013) 13369-13372.
[16] V.I. Kalikmanov, Classical nucleation theory, Nucleation Theory, Springer 2013, pp. 17-41.
[17] G.C. Sosso, J. Chen, S.J. Cox, M. Fitzner, P. Pedevilla, A. Zen, A. Michaelides, Crystal nucleation in liquids:Open questions and future challenges in molecular dynamics simulations, Chem. Rev. 116(2016) 7078-7116.
[18] E. Sanz, C. Vega, J. Espinosa, R. Caballero-Bernal, J. Abascal, C. Valeriani, Homogeneous ice nucleation at moderate supercooling from molecular simulation, J. Am. Chem. Soc. 135(2013) 15008-15017.
[19] J. Merikanto, E. Zapadinsky, A. Lauri, H. Vehkamäki, Origin of the failure of classical nucleation theory:Incorrect description of the smallest clusters, Phys. Rev. Lett. 98(2007) 145702.
[20] J. Julin, I. Napari, J. Merikanto, H. Vehkamäki, A thermodynamically consistent determination of surface tension of small Lennard-Jones clusters from simulation and theory, J. Chem. Phys. 133(2010)(044704).
[21] S.T. Yau, P.G. Vekilov, Direct observation of nucleus structure and nucleation pathways in apoferritin crystallization, J. Am. Chem. Soc. 123(2001) 1080-1089.
[22] B. O'Malley, I. Snook, Crystal nucleation in the hard sphere system, Phys. Rev. Lett. 90(2003)(085702).
[23] D.W. Oxtoby, Crystal nucleation in simple and complex fluids, Philos. Trans. R. Soc. London, Ser. A 361(2003) 419-428.
[24] P.G. Vekilov, Two-step mechanism for the nucleation of crystals from solution, J. Cryst. Growth 275(2005) 65-76.
[25] D. Gebauer, M. Kellermeier, J.D. Gale, L. Bergström, H. Cölfen, Pre-nucleation clusters as solute precursors in crystallisation, Chem. Soc. Rev. 43(2014) 2348-2371.
[26] P.R. ten Wolde, D. Frenkel, Enhancement of protein crystal nucleation by critical density fluctuations, Science 277(1997) 1975-1978.
[27] O. Galkin, K. Chen, R.L. Nagel, R.E. Hirsch, P.G. Vekilov, Liquid-liquid separation in solutions of normal and sickle cell hemoglobin, Proc. Natl. Acad. Sci. U. S. A. 99(2002) 8479-8483.
[28] B.A. Garetz, J. Matic, A.S. Myerson, Polarization switching of crystal structure in the nonphotochemical light-induced nucleation of supersaturated aqueous glycine solutions, Phys. Rev. Lett. 89(2002) 175501.
[29] S. Chattopadhyay, D. Erdemir, J.M. Evans, J. Ilavsky, H. Amenitsch, C.U. Segre, A.S. Myerson, SAXS study of the nucleation of glycine crystals from a supersaturated solution, Cryst. Growth Des. 5(2005) 523-527.
[30] Y. Peng, F. Wang, Z. Wang, A.M. Alsayed, Z. Zhang, A.G. Yodh, Y. Han, Two-step nucleation mechanism in solid-solid phase transitions, Nat. Mater. 14(2015) 101.
[31] N.D. Loh, S. Sen, M. Bosman, S.F. Tan, J. Zhong, C.A. Nijhuis, P. Král, P. Matsudaira, U. Mirsaidov, Multistep nucleation of nanocrystals in aqueous solution, Nat. Chem. 9(2017) 77.
[32] C. Guo, J. Wang, J. Li, Z. Wang, S. Tang, Kinetic pathways and mechanisms of twostep nucleation in crystallization, J. Phys. Chem. Lett. 7(2016) 5008-5014.
[33] P.G. Vekilov, Nucleation, Cryst. Growth Des. 10(2010) 5007-5019.
[34] W. Pan, A.B. Kolomeisky, P.G. Vekilov, Nucleation of ordered solid phases of proteins via a disordered high-density state:Phenomenological approach, J. Chem. Phys. 122(2005) 174905.
[35] P.G. Vekilov, The two-step mechanism of nucleation of crystals in solution, Nanoscale 2(2010) 2346-2357.
[36] D. Gebauer, A. Völkel, H. Cölfen, Stable prenucleation calcium carbonate clusters, Science 322(2008) 1819-1822.
[37] J.P. Jolivet, E. Tronc, C. Chaneac, Iron oxides:From molecular clusters to solid. A nice example of chemical versatility, Compt. Rendus Geosci. 338(2006) 488-497.
[38] K. Onuma, A. Ito, Cluster growth model for hydroxyapatite, Chem. Mater. 10(1998) 3346-3351.
[39] H.J. Schöpe, G. Bryant, W. van Megen, Two-step crystallization kinetics in colloidal hard-sphere systems, Phys. Rev. Lett. 96(2006)(175701).
[40] M.H. Nielsen, S. Aloni, J.J. De Yoreo, In situ TEM imaging of CaCO₃ nucleation reveals coexistence of direct and indirect pathways, Science 345(2014) 1158-1162.
[41] R. Li, X. Zhang, H. Dong, Q. Li, Z. Shuai, W. Hu, Gibbs-Curie-Wulff theorem in organic materials:a case study on the relationship between surface energy and crystal growth, Adv. Mater. 28(2016) 1697-1702.
[42] V.A. Cimmelli, A. Romano, G. Starita, The Gibbs principle for the equilibrium of crystals, Int. J. Eng. Sci. 28(1990) 677-688.
[43] E. Schonherr, K. Matsumoto, K. Murakami, The morphology of vapor grown C-60 crystals as an ideal example of the Gibbs-Wulff's law, in:S. Saito, T. Ando, Y.Iwasa, K. Kikuchi, M. Kobayashi, Y. Saito (Eds.),Nanonetwork Materials:Fullerenes, Nanotubes and Related Systems 2001, pp. 405-408.
[44] J. Prywer, Explanation of some peculiarities of crystal morphology deduced from the BFDH law, J. Cryst. Growth 270(2004) 699-710.
[45] W.J. Li, E.W. Shi, W.Z. Zhong, Z.W. Yin, Growth mechanism and growth habit of oxide crystals, J. Cryst. Growth 203(1999) 186-196.
[46] T.D. Turner, L.E. Hatcher, C.C. Wilson, K.J. Roberts, Habit modification of the active pharmaceutical ingredient lovastatin through a predictive solvent selection approach, J. Pharm. Sci. 108(2019) 1779-1787.
[47] J.D.H. Donnay, D. Harker, A new law of crystal morphology extending the law of Bravais, Am. Mineral. 22(1937) 446-467.
[48] X.B. Zhou, J.R. Zhu, J.M. Gu, X.R. Hu, A new polymorph of the gastrokinetic drug cisapride monohydrate, Acta Crystallogr, Sect. C:Struct. Chem. 74(2018) 690-695.
[49] P. Hartman, H.K. Chan, Application of the periodic bond chain (PBC) theory and attachment energy consideration to derive the crystal morphology of hexamethylmelamine, Pharm. Res. 10(1993) 1052-1058.
[50] M. Zhang, Z. Liang, F. Wu, J.-F. Chen, C. Xue, H. Zhao, Crystal engineering of ibuprofen compounds:From molecule to crystal structure to morphology prediction by computational simulation and experimental study, J. Cryst. Growth 467(2017) 47-53.
[51] F. Shen, P. Lv, C. Sun, R. Zhang, S. Pang, The crystal structure and morphology of 2, 4, 6, 8, 10, 12-hexanitro-2, 4, 6, 8, 10, 12-hexaazaisowurtzitane (CL-20) p-Xylene solvate:A joint experimental and simulation study, Molecules 19(2014) 18574-18589.
[52] Q. Yi, J. Chen, Y. Le, J. Wang, C. Xue, H. Zhao, Crystal structure and habit of dirithromycin acetone solvate:A combined experimental and simulative study, J. Cryst. Growth 372(2013) 193-198.
[53] K.J. Roberts, R. Docherty, R. Tamura, Engineering Crystallography:From Molecule to Crystal to Functional Form, Springer, 2017.
[54] J.W. Mullin, 6-Crystal growth, in:J.W. Mullin (Ed.), Crystallization, Fourth edition, Butterworth-Heinemann, Oxford 2001, pp. 216-288.
[55] F. Frank, The influence of dislocations on crystal growth, Discuss. Faraday Soc. 5(1949) 48-54.
[56] E. Budevski, G. Staikov, W. Lorenz, Electrocrystallization:Nucleation and growth phenomena, Electrochim. Acta 45(2000) 2559-2574.
[57] M. Uwaha, Introduction to the BCF theory, Prog. Cryst. Growth Charact. Mater. 62(2016) 58-68.
[58] K. Tsukamoto, In-situ observation of crystal growth and the mechanism, Prog. Cryst. Growth Charact. Mater. 62(2016) 111-125.
[59] L. Jia, M. Svärd, Å.C. Rasmuson, Crystal growth of salicylic acid in organic solvents, Cryst. Growth Des. 17(2017) 2964-2974.
[60] K.A. Jackson, The interface kinetics of crystal growth processes, Interface Sci. 10(2002) 159-169.
[61] J.D. Weeks, G.H. Gilmer, Dynamics of crystal growth, Adv. Chem. Phys. (1979) 157-228.
[62] S.T. Chui, J.D. Weeks, Phase transition in the two-dimensional Coulomb gas, and the interfacial roughening transition, Phys. Rev. B 14(1976) 4978-4982.
[63] P. Bennema, H. Meekes, S.X.M. Boerrigter, H.M. Cuppen, M.A. Deij, J. van Eupen, P. Verwer, E. Vlieg, Crystal growth and morphology:new developments in an integrated Hartman-perdokconnected netroughening transition theory, supported by computer simulations, Cryst. Growth Des. 4(2004) 905-913.
[64] A.J. Malkin, G. Kuznetsov Yu, T.A. Land, J.J. DeYoreo, A. McPherson, Mechanisms of growth for protein and virus crystals, Nat. Struct. Biol. 2(1995) 956-959.
[65] F. Rosenberger, Protein crystallization, J. Cryst. Growth 166(1996) 40-54.
[66] G. Li, D. Wang, Z. Huang, Study on interface-phase of crystal growth, J. Synth. Cryst. 30(2001) 171-177.
[67] X. Yu, J. You, Y. Wang, Z. Cheng, B. Yu, S. Zhang, D. Sun, G. Jiang, Microprobe of structure of crystal/liquid interface boundary layers, Sci. China, Ser. E:Technol. Sci. 44(2001) 265-273.
[68] L. Chen, L. Song, G. Lan, J. Wang, Solubility and metastable zone width measurement of 3,4-bis(3-nitrofurazan-4-yl)furoxan (DNTF) in ethanol + water, Chin. J. Chem. Eng. 25(2017) 646-651.
[69] L. Zhou, Z. Wang, M. Zhang, M. Guo, S. Xu, Q. Yin, Determination of metastable zone and induction time of analgin for cooling crystallization, Chin. J. Chem. Eng. 25(2017) 313-318.
[70] A. Prudic, Y. Ji, G. Sadowski, Thermodynamic phase behavior of API/polymer solid dispersions, Mol. Pharm. 11(2014) 2294-2304.
[71] J. Gross, G. Sadowski, Perturbed-chain SAFT:An equation of state based on a perturbation theory for chain molecules, Ind. Eng. Chem. Res. 40(2001) 1244-1260.
[72] J. Gross, G. Sadowski, Application of the perturbed-chain SAFT equation of state to associating systems, Ind. Eng. Chem. Res. 41(2002) 5510-5515.
[73] A. Prudic, T. Kleetz, M. Korf, Y. Ji, G. Sadowski, Influence of copolymer composition on the phase behavior of solid dispersions, Mol. Pharm. 11(2014) 4189-4198.
[74] A. Prudic, Y. Ji, C. Luebbert, G. Sadowski, Influence of humidity on the phase behavior of API/polymer formulations, Eur. J. Pharm. Biopharm. 94(2015) 352-362.
[75] A. Prudic, A.-K. Lesniak, Y. Ji, G. Sadowski, Thermodynamic phase behaviour of indomethacin/PLGA formulations, Eur. J. Pharm. Biopharm. 93(2015) 88-94.
[76] R. Paus, E. Hart, Y. Ji, G. Sadowski, Solubility and caloric properties of cinnarizine, J. Chem. Eng. Data 60(2015) 2256-2261.
[77] R. Paus, Y. Ji, L. Vahle, G. Sadowski, Predicting the solubility advantage of amorphous pharmaceuticals:a novel thermodynamic approach, Mol. Pharm. 12(2015) 2823-2833.
[78] A.A. Noyes, W.R. Whitney, The rate of solution of solid substances in their own solutions, J. Am. Chem. Soc. 19(1897) 930-934.
[79] A.T. Lu, M.E. Frisella, K.C. Johnson, Dissolution modeling:factors affecting the dissolution rates of polydisperse powders, Pharm. Res. 10(1993) 1308-1314.
[80] G.K. Vudathala, J.A. Rogers, Dissolution of fludrocortisone from phospholipid coprecipitates, J. Pharm. Sci. 81(1992) 282-286.
[81] T. Higuchi, Mechanism of sustained-action medication. Theoretical analysis of rate of release of solid drugs dispersed in solid matrices, J. Pharm. Sci. 52(1963) 1145-1149.
[82] T. Higuchi, Rate of release of medicaments from ointment bases containing drugs in suspension, J. Pharm. Sci. 50(1961) 874-875.
[83] L.P. De Almeida, S. Simões, P. Brito, A. Portugal, M. Figueiredo, Modeling dissolution of sparingly soluble multisized powders, J. Pharm. Sci. 86(1997) 726-732.
[84] A. Hixson, J. Crowell, Dependence of reaction velocity upon surface and agitation, Ind. Eng. Chem. 23(1931) 923-931.
[85] R.W. Korsmeyer, R. Gurny, E. Doelker, P. Buri, N.A. Peppas, Mechanisms of solute release from porous hydrophilic polymers, Int. J. Pharm. 15(1983) 25-35.
[86] S. Dash, P.N. Murthy, L. Nath, P. Chowdhury, Kinetic modeling on drug release from controlled drug delivery systems, Acta Pol. Pharm. 67(2010) 217-223.
[87] M. Dejmek, C.A. Ward, A statistical rate theory study of interface concentration during crystal growth or dissolution, J. Chem. Phys. 108(1998) 8698-8704.
[88] R. Paus, Y. Ji, F. Braak, G. Sadowski, Dissolution of crystalline pharmaceuticals:Experimental investigation and thermodynamic modeling, Ind. Eng. Chem. Res. 54(2015) 731-742.
[89] R. Paus, E. Hart, Y. Ji, A novel approach for predicting the dissolution profiles of pharmaceutical tablets, Eur. J. Pharm. Biopharm. 96(2015) 53-64.
[90] Y. Ji, A.K. Lesniak, A. Prudic, R. Paus, G. Sadowski, Drug release kinetics and mechanism from PLGA formulations, AICHE J. 62(2016) 4055-4065.
[91] Y. Ji, M. Lemberg, A. Prudic, R. Paus, G. Sadowski, Modeling and analysis of dissolution of paracetamol/Eudragit^® formulations, Chem. Eng. Res. Des. 121(2017) 22-31.
[92] R. Paus, Y. Ji, Modeling and predicting the influence of variable factors on dissolution of crystalline pharmaceuticals, Chem. Eng. Sci. 145(2016) 10-20.
[93] R. Paus, A. Prudic, Y. Ji, Influence of excipients on solubility and dissolution of pharmaceuticals, Int. J. Pharm. 485(2015) 277-287.
[94] T. Kawasaki, H. Tanaka, Formation of a crystal nucleus from liquid, Proc. Natl. Acad. Sci. U. S. A. 107(2010) 14036-14041.
[95] T. Schilling, S. Dorosz, H.J. Schöpe, G. Opletal, Crystallization in suspensions of hard spheres:a Monte Carlo and molecular dynamics simulation study, J. Phys.:Condens. Matter 23(2011)(194120).
[96] M. Matsumoto, S. Saito, I. Ohmine, Molecular dynamics simulation of the ice nucleation and growth process leading to water freezing, Nature 416(2002) 409-413.
[97] L. Vrbka, P. Jungwirth, Homogeneous freezing of water starts in the subsurface, J. Phys. Chem. B 110(2006) 18126-18129.
[98] M. Mucha, P. Jungwirth, Salt crystallization from an evaporating aqueous solution by molecular dynamics simulations, J. Phys. Chem. B 107(2003) 8271-8274.
[99] T. Li, D. Donadio, G. Russo, G. Galli, Homogeneous ice nucleation from supercooled water, Phys. Chem. Chem. Phys. 13(2011) 19807-19813.
[100] W. Zhu, F.S. Romanski, X. Meng, S. Mitra, M.S. Tomassone, Atomistic simulation study of surfactant and polymer interactions on the surface of a fenofibrate crystal, Eur. J. Pharm. Sci. 42(2011) 452-461.
[101] M. Salvalaglio, C. Perego, F. Giberti, M. Mazzotti, M. Parrinello, Molecular-dynamics simulations of urea nucleation from aqueous solution, Proc. Natl. Acad. Sci. U. S. A. 112(2015) E6-14.
[102] Y. Yang, D. Han, S. Du, S. Wu, J. Gong, Crystal morphology optimization of thiamine hydrochloride in solvent system:Experimental and molecular dynamics simulation studies, J. Cryst. Growth 481(2018) 48-55.
[103] C.J. Schram, S.P. Beaudoin, L.S. Taylor, Polymer inhibition of crystal growth by surface poisoning, Cryst. Growth Des. 16(2016) 2094-2103.
[104] S. Baghel, H. Cathcart, N.J. O'Reilly, Polymeric amorphous solid dispersions:a review of amorphization, crystallization, stabilization, solid-state characterization, and aqueous solubilization of biopharmaceutical classification system class Ⅱ drugs, J. Pharm. Sci. 105(2016) 2527-2544.
[105] B. Wang, D. Wang, S. Zhao, X. Huang, J. Zhang, Y. Lü, X. Liu, G. Lü, X. Ma, Evaluate the ability of PVP to inhibit crystallization of amorphous solid dispersions by density functional theory and experimental verify, Eur. J. Pharm. Sci. 96(2017) 45-52.
[106] S.K. Poornachary, V.D. Chia, Y. Yani, G. Han, P.S. Chow, R.B. Tan, Anisotropic crystal growth inhibition by polymeric additives:Impact on modulation of naproxen crystal shape and size, Cryst. Growth Des. 17(2017) 4844-4854.
[107] T. Mandal, W. Huang, J.M. Mecca, A. Getchell, W.W. Porter, R.G. Larson, A framework for multi-scale simulation of crystal growth in the presence of polymers, Soft Matter 13(2017) 1904-1913.
[108] M.R. Walsh, C.A. Koh, E.D. Sloan, A.K. Sum, D.T. Wu, Microsecond simulations of spontaneous methane hydrate nucleation and growth, Science 326(2009) 1095-1098.
[109] J. Anwar, D. Zahn, Uncovering molecular processes in crystal nucleation and growth by using molecular simulation, Angew. Chem. Int. Ed. 50(2011) 1996-2013.
[110] W. Huang, T. Mandal, R.G. Larson, Computational modeling of hydroxypropylmethylcellulose acetate succinate (hpmcas) and phenytoin interactions:A systematic coarse-graining approach, Mol. Pharm. 14(2017) 733-745.
[111] T. Schilling, H.J. Schope, M. Oettel, G. Opletal, I. Snook, Precursor-mediated crystallization process in suspensions of hard spheres, Phys. Rev. Lett. 105(2010) (025701).
[112] R.C. Bernardi, M.C. Melo, K. Schulten, Enhanced sampling techniques in molecular dynamics simulations of biological systems, Biochim. Biophys. Acta 1850(2015) 872-877.
[113] T. Mandal, R.L. Marson, R.G. Larson, Coarse-grained modeling of crystal growth and polymorphism of a model pharmaceutical molecule, Soft Matter 12(2016) 8246-8255.
[114] M. King, S. Pasler, C. Peter, Coarse-grained simulation of CaCO₃ aggregation and crystallization made possible by nonbonded three-body interactions, J. Phys. Chem. C 123(2019) 3152-3160.
[115] M. De Vivo, M. Masetti, G. Bottegoni, A. Cavalli, Role of molecular dynamics and related methods in drug discovery, J. Med. Chem. 59(2016) 4035-4061.
[116] G.M. Torrie, J.P. Valleau, Nonphysical sampling distributions in Monte Carlo free-energy estimation:Umbrella sampling, J. Comput. Phys. 23(1977) 187-199.
[117] A. Barducci, M. Bonomi, M. Parrinello, Metadynamics, Wiley Interdiscip, Rev.:Comput. Mol. Sci. 1(2011) 826-843.
[118] A. Laio, M. Parrinello, Escaping free-energy minima, Proc. Natl. Acad. Sci. U. S. A. 99(2002) 12562-12566.
[119] J. Juraszek, G. Saladino, T. Van Erp, F. Gervasio, Efficient numerical reconstruction of protein folding kinetics with partial path sampling and pathlike variables, Phys. Rev. Lett. 110(2013)(108106).
[120] T.S. Van Erp, P.G. Bolhuis, Elaborating transition interface sampling methods, J. Comput. Phys. 205(2005) 157-181.
[121] R.J. Allen, C. Valeriani, P.R. ten Wolde, Forward flux sampling for rare event simulations, J. Phys.:Condens. Matter 21(2009)(463102).
[122] C. Dellago, P.G. Bolhuis, P.L. Geissler, Transition path sampling, Adv. Chem. Phys. 123(2002) 1-78.
[123] L. Filion, M. Hermes, R. Ni, M. Dijkstra, Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling:A comparison of simulation techniques, J. Chem. Phys. 133(2010) 244115.