Abstract
A variety of natural or synthetic calcium phosphate (CaP)-based scaffolds are currently produced for dental and orthopaedic applications. These scaffolds have been shown to stimulate bone formation due to their biocompatibility, osteoconductivity and osteoinductivity. The release of the \(\hbox {Ca}^{2+}\) ions from these scaffolds is of great interest in light of the aforementioned properties. It can depend on a number of biophysicochemical phenomena such as dissolution, diffusion and degradation, which in turn depend on specific scaffold characteristics such as composition and morphology. Achieving an optimal release profile can be challenging when relying on traditional experimental work alone. Mathematical modelling can complement experimentation. In this study, the in vitro dissolution behaviour of four CaP-based scaffold types was investigated experimentally. Subsequently, a mechanistic finite element method model based on biophysicochemical phenomena and specific scaffold characteristics was developed to predict the experimentally observed behaviour. Before the model could be used for local \(\hbox {Ca}^{2+}\) ions release predictions, certain parameters such as dissolution constant (\(k_{\mathrm{dc}}\)) and degradation constant (\(k_\mathrm{sc}\)) for each type of scaffold were determined by calibrating the model to the in vitro dissolution data. The resulting model showed to yield release characteristics in satisfactory agreement with those observed experimentally. This suggests that the mathematical model can be used to investigate the local \(\hbox {Ca}^{2+}\) ions release from CaP-based scaffolds.
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Adachi T, Osako Y, Tanaka M, Hojo M, Hollister SJ (2006) Framework for optimal design of porous scaffold microstructure by computational simulation of bone regeneration. Biomaterials 27:3964–3972
Arifin DY, Lee LY, Wang C-H (2006) Mathematical modeling and simulation of drug release from microspheres: implications to drug delivery systems. Adv Drug Deliv Rev 58:1274–1325
Bianchi M et al (2014) Substrate geometry directs the in vitro mineralization of calcium phosphate ceramics. Acta Biomater 10:661–669
Bléry P et al (2014) Evaluation of new bone formation in irradiated areas using association of mesenchymal stem cells and total fresh bone marrow mixed with calcium phosphate scaffold. J Mater Sci Mater Med 25:2711–2720
Bohner M, Baumgart F (2004) Theoretical model to determine the effects of geometrical factors on the resorption of calcium phosphate bone substitutes. Biomaterials 25:3569–3582
Brazel CS, Peppas NA (2000) Modeling of drug release from swellable polymers. Eur J Pharm Biopharm 49:47–58
Byrne DP, Lacroix D, Planell JA, Kelly DJ, Prendergast PJ (2007) Simulation of tissue differentiation in a scaffold as a function of porosity. Young’s modulus and dissolution rate: application of mechanobiological models in tissue engineering. Biomaterials 28:5544–5554
Carlier A, Chai YC, Moesen M, Theys T, Schrooten J, Van Oosterwyck H, Geris L (2011) Designing optimal calcium phosphate scaffold-cell combinations using an integrative model-based approach. Acta Biomater 7:3573–3585
Chai YC et al (2012a) Current views on calcium phosphate osteogenicity and the translation into effective bone regeneration strategies. Acta Biomater 8:3876–3887
Chai YC et al (2012b) Ectopic bone formation by 3D porous calcium phosphate-Ti6Al4V hybrids produced by perfusion electrodeposition. Biomaterials 33:4044–4058
Chai YC et al (2012c) Mechanisms of ectopic bone formation by human osteoprogenitor cells on CaP biomaterial carriers. Biomaterials 33:3127–3142
Charles-Harris M, Koch MA, Navarro M, Lacroix D, Engel E, Planell JA (2008) A PLA/calcium phosphate degradable composite material for bone tissue engineering: an in vitro study. J Mater Sci Mater Med 19:1503–1513
Danoux CB et al (2015) Elucidating the individual effects of calcium and phosphate ions on hMSCs by using composite materials. Acta Biomater 17:1–15
Dash S, Murthy PN, Nath L, Chowdhury P (2010) Kinetic modeling on drug release from controlled drug delivery systems. Acta Pol Pharm 67:217–223
Fredenberg S, Wahlgren M, Reslow M, Axelsson A (2011) The mechanisms of drug release in poly (lactic-co-glycolic acid)-based drug delivery systems—a review. Int J Pharm 415:34–52
Frenning G (2003) Theoretical investigation of drug release from planar matrix systems: effects of a finite dissolution rate. J Control Release 92:331–339
Frenning G (2004) Theoretical analysis of the release of slowly dissolving drugs from spherical matrix systems. J Control Release 95:109–117
Frenning G, Strømme M (2003) Drug release modeled by dissolution, diffusion, and immobilization. Int J Pharm 250:137–145
Frenning G, Tunón Å, Alderborn G (2003) Modelling of drug release from coated granular pellets. J Control Release 92:113–123
Frenning G, Brohede U, Strømme M (2005) Finite element analysis of the release of slowly dissolving drugs from cylindrical matrix systems. J Control Release 107:320–329
Gao P, Fagerness PE (1995) Diffusion in HPMC gels. I. Determination of drug and water diffusivity by pulsed-field-gradient spin-echo NMR. Pharm Res 12:955–964
Guyot Y, Papantoniou I, Chai YC, Van Bael S, Schrooten J, Geris L (2014) A computational model for cell/ECM growth on 3D surfaces using the level set method: a bone tissue engineering case study. Biomech Model Mechanobiol 13:1361–1371
Hecht F (2012) New development in freefem++. J Numer Math 20:251–266
Hong M-H, Kim S-M, Om J-Y, Kwon N, Lee Y-K (2014) Seeding cells on calcium phosphate scaffolds using hydrogel enhanced osteoblast proliferation and differentiation. Ann Biomed Eng 42:1424–1435
Horbett T, Waldburger J, Ratner B, Hoffman A (1988) Cell adhesion to a series of hydrophili-hydrophobic copolymers studies with a spinning disc apparatus. J Biomed Mater Res 22:383–404
Karadzic I, Vucic V, Jokanovic V, Debeljak-Martacic J, Markovic D, Petrovic S, Glibetic M (2015) Effects of novel hydroxyapatite-based 3D biomaterials on proliferation and osteoblastic differentiation of mesenchymal stem cells. J Biomed Mater Res Part A 103:350–357
Kaunisto E, Tajarobi F, Abrahmsen-Alami S, Larsson A, Nilsson B, Axelsson A (2013) Mechanistic modelling of drug release from a polymer matrix using magnetic resonance microimaging. Eur J Pharm Sci 48:698–708
Lanao RPF, Leeuwenburgh SC, Wolke JG, Jansen JA (2011) Bone response to fast-degrading, injectable calcium phosphate cements containing PLGA microparticles. Biomaterials 32:8839–8847
Lao LL, Peppas NA, Boey FYC, Venkatraman SS (2011) Modeling of drug release from bulk-degrading polymers. Int J Pharm 418:28–41
Lobo SE, Glickman R, da Silva WN, Arinzeh TL, Kerkis I (2015) Response of stem cells from different origins to biphasic calcium phosphate bioceramics. Cell Tissue Res 361: 477–495
Masaro L, Zhu X (1999) Physical models of diffusion for polymer solutions, gels and solids. Prog Polym Sci 24:731–775
Mazón P, García-Bernal D, Meseguer-Olmo L, Cragnolini F, Piedad N (2015) Human mesenchymal stem cell viability, proliferation and differentiation potential in response to ceramic chemistry and surface roughness. Ceram Int 41:6631–6644
Mezahi F, Oudadesse H, Harabi A, Lucas-Girot A, Le Gal Y, Chaair H, Cathelineau G (2009) Dissolution kinetic and structural behaviour of natural hydroxyapatite vs. thermal treatment. J Therm Anal Calorim 95:21–29
Nava MM, Raimondi MT, Pietrabissa R (2013) A multiphysics 3D model of tissue growth under interstitial perfusion in a tissue-engineering bioreactor. Biomech Model Mechanobiol 12:1169–1179
Polakovič M, Görner T, Gref R, Dellacherie E (1999) Lidocaine loaded biodegradable nanospheres: II. Modelling of drug release. J Control Release 60:169–177
Ribeiro AC, Barros MC, Teles AS, Valente AJ, Lobo VM, Sobral AJ, Esteso M (2008) Diffusion coefficients and electrical conductivities for calcium chloride aqueous solutions at 298.15 K and 310.15 K. Electrochimica Acta 54:192–196
Roberts SJ, Geris L, Kerckhofs G, Desmet E, Schrooten J, Luyten FP (2011) The combined bone forming capacity of human periosteal derived cells and calcium phosphates. Biomaterials 32:4393–4405
Shannon RT (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr Sect A Cryst Phys Diffr Theor Gen Crystallogr 32:751–767
Shih Y-RV et al (2014) Calcium phosphate-bearing matrices induce osteogenic differentiation of stem cells through adenosine signaling. Proc Natl Acad Sci 111:990–995
Siepmann J, Göpferich A (2001) Mathematical modeling of bioerodible, polymeric drug delivery systems. Adv Drug Deliv Rev 48:229–247
Siepmann J, Siepmann F (2008) Mathematical modeling of drug delivery. Int J Pharm 364:328–343
Siepmann J, Peppas NA (2011) Higuchi equation: derivation, applications, use and misuse. Int J Pharm 418:6–12
Siepmann J, Siepmann F (2013) Mathematical modeling of drug dissolution. Int J Pharm 453:12–24
Siepmann J, Kranz H, Bodmeier R, Peppas N (1999) HPMC-matrices for controlled drug delivery: a new model combining diffusion, swelling, and dissolution mechanisms and predicting the release kinetics. Pharm Res 16:1748–1756
Snorradóttir BS, Jónsdóttir F, Sigurdsson ST, Thorsteinsson F, Másson M (2013) Numerical modelling and experimental investigation of drug release from layered silicone matrix systems. Eur J Pharm Sci 49:671–678
Sonnaert M, Luyten FP, Schrooten J, Papantoniou I (2015) Bioreactor-based online recovery of human progenitor cells with uncompromised regenerative potential: a bone tissue engineering perspective. PloS ONE 10:e0136875
Sun X, Kang Y, Bao J, Zhang Y, Yang Y, Zhou X (2013) Modeling vascularized bone regeneration within a porous biodegradable CaP scaffold loaded with growth factors. Biomaterials 34:4971–4981
Wu L, Ding J (2005) Effects of porosity and pore size on in vitro degradation of three-dimensional porous poly (D, L-lactide-co-glycolide) scaffolds for tissue engineering. J Biomed Mater Res Part A 75:767–777
Young M, Carroad P, Bell R (1980) Estimation of diffusion coefficients of proteins. Biotechnol Bioeng 22:047–955
Acknowledgments
Céline Smekens is gratefully acknowledged for her work as a master student on measuring the \(\hbox {Ca}^{2+}\) ions release using micro ion electrode. Bachelor students Antoine Delacroix and Simon Bruneau (ESEO, Angers, France) are gratefully acknowledged for their work on development of Morphing CiTy (ULg, Liege, Belgium). Varun Manhas and Yann Guyot are funded by Belgian National Fund for Scientific Research (FNRS) Grant FRFC 2.4564.12. Greet Kerckhofs and Yoke Chin Chai are financed by the postdoctoral Grant of the Research Foundation—Flanders (FWO/12R4315N and 1.5.172.13N-Interdisc.—http://www.fwo.be/en/). The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 279100. The microCT images have been generated on the X-ray computed tomography facilities of the Department of Materials Engineering of the KU Leuven, financed by the Hercules Foundation (Project AKUL 09/001: Micro- and nanoCT for the hierarchical analysis of materials). This work is part of Prometheus, the KU Leuven R&D Division of Skeletal Tissue Engineering (http://www.kuleuven.be/prometheus).
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Manhas, V., Guyot, Y., Kerckhofs, G. et al. Computational modelling of local calcium ions release from calcium phosphate-based scaffolds. Biomech Model Mechanobiol 16, 425–438 (2017). https://doi.org/10.1007/s10237-016-0827-9
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DOI: https://doi.org/10.1007/s10237-016-0827-9