# Mathematical modeling of calcium phosphate precipitation in biologically relevant systems: scoping review

- 111 Downloads

## Abstract

Biologically guided precipitation of calcium phosphates is important for the formation of calcified human tissues, such as bone and teeth, and is of practical significance in numerous industrial and agricultural processes, such as wastewater treatment and dairy ultrafiltration. Mineral precipitation is physicochemically complex and becomes even more complex in the presence of biological materials. The theoretical foundation of phase transition in general has been developed and is vital for many applications, such as metallurgy and weather prediction. The goal of this scoping review was to identify and evaluate established mathematical approaches developed to describe the formation of calcium precipitates in biological systems. A scoping review was conducted using MathSciNet, Scopus, and Web of Science databases to retrieve eligible mathematical modeling papers on calcium precipitates in biological systems. From the 2096 studies screened, 115 studies were included. The major biological systems of interest were tissues of the human body (49/115), water research (38/115), and agricultural and earth sciences applications (17/115). The majority of studies described precipitation of calcium phosphate (79/115), followed by calcium carbonate (22/115). Mathematical modeling of calcium precipitation was dominated by classical nucleation (64/115) and kinetic (38/115) theories. Only a minority of studies explicitly modeled chemical reactions in the aqueous phase (33/115). Biological components were explicitly described in 45/115 studies and included as physicochemical limitations in 70/115 studies. The majority of the studies (91/115) attempted to quantitatively compare the model predictions to the experimental data, with 59/115 reporting good to reasonable fit. This scoping review suggests that broad theories, such as classical nucleation and kinetic theories, may be adapted for modeling calcium precipitation in biologically relevant systems; however, detailed mathematical descriptions of biological, chemical, and physicochemical aspects of calcium precipitation are required.

## Keywords

Crystallization Nucleation Growth Modeling Precipitation## Notes

### Acknowledgements

The authors of this paper would like to acknowledge Ms. April Colosimo, McGill University, for her help in building the search strategy used to retrieve the final library for the scoping review.

### Funding

This study was funded by Natural Sciences and Engineering Research Council of Canada (Grant Number 288253).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

## References

- Abdelkebir K (2012) Biomimetic layer-by-layer templates for calcium phosphate biomineralization. Acta Biomater 8(9):3419–3428Google Scholar
- Barat R, Montoya T, Seco A, Ferrer J (2011) Modelling biological and chemically induced precipitation of calcium phosphate in enhanced biological phosphorus removal systems. Water Res 45(12):3744–3752Google Scholar
- Berthoud R (1912) Thorie de la formation des faces dun cristal. J Chim Phys 10:624–635Google Scholar
- Blair HC, Larrouture QC, Li YN, Lin H, Beer-Stoltz D, Liu L et al (2017) Osteoblast differentiation and bone matrix formation in vivo and in vitro. Tissue Eng Part B Rev 23(3):268-+Google Scholar
- Boistelle R, Lopez-Valero I (1990) Growth units and nucleation: the case of calcium phosphates. J Cryst Growth 102(3):609–617Google Scholar
- Brar T, France P, Smirniotis PG (2001) Heterogeneous versus homogeneous nucleation and growth of zeolite. J Phys Chem B 105(23):5383–5390Google Scholar
- Chu YA, Moran B, Olson GB, Reid ACE (2000) A model for nonclassical nucleation of solid-solid structural phase transformations. Metall Mater Trans 31(5):1321–1332Google Scholar
- Christian JW (1975) The theory of transformations in metals and alloys, 2nd edn. Pergamon, OxfordGoogle Scholar
- Davies C (1962) Ion associations, 1st edn. Butterworths, LondonGoogle Scholar
- De Yoreo JJ (2003) Principles of crystal nucleation and growth. Rev Mineral Geochem 54(1):57–93Google Scholar
- Debye P, Hckel E (1923) Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte erscheinungen. Phys Z 24:185–206Google Scholar
- Dorozhkin SV, Epple M (2002) Biological and medical significance of calcium phosphates. Angew Chem Int Ed 41(17):3130Google Scholar
- Drake F, Pierce GW, Dow MT (1930) Measurement of the dielectric constant and index of refraction of water and aqueous solutions of KCl at high frequencies. Phys Rev 35:613Google Scholar
- Gibbs JW (1878) On the equilibrium of heterogeneous substances. Am J Sci 96:441–458zbMATHGoogle Scholar
- Hanrahan G (2010) Modelling of pollutants in complex environmental systems, 2nd edn. ILM Publications, HertfordshireGoogle Scholar
- Helgeson HC (1969) Thermodynamics of hydrothermal systems at elevated temperatures and pressures. Am J Sci 267(7):729–804Google Scholar
- Helt JE (1969) Effects of supersaturation and temperature on nucleation and crystal growth in a MSMPR crystallizer. RTD 6213Google Scholar
- Holt C (2004) An equilibrium thermodynamic model of the sequestration of calcium phosphate by casein micelles and its application to the calculation of the partition of salts in milk. Eur Biophys J 33(5):421–434Google Scholar
- Isopescu R, Mateescu C, Mihai M, Dabija G (2010) The effects of organic additives on induction time and characteristics of precipitated calcium carbonate. Chem Eng Res Des 88(11):1450–1454Google Scholar
- Kalikmanov VI (2013) Nucleation theory, 1st edn. Springer, NetherlandsGoogle Scholar
- Karthika S, Radhakrishnan TK, Kalaichelvi P (2016) A review of classical and nonclassical nucleation theories. Cryst Growth Des 16(11):6663–6681Google Scholar
- Komarova S, Safranek L, Gopalakrishnan J, Ou Mj, Mckee M, Murshed M et al (2015) Mathematical model for bone mineralization. Front Cell Dev Biol 3(51):6663–6681Google Scholar
- Liu XY (1999) A new kinetic model for three-dimensional heterogeneous nucleation. J Chem Phys 111(4):1628–1635Google Scholar
- Liu XY (2001a) Influence of nucleation nature on Ca mineral/substrate structural synergy and implications for biomineralization in microgravity. J Chem Phys 115(21):9970–9974Google Scholar
- Liu XY (2001b) Interfacial process of nucleation and molecular nucleation templator. Appl Phys Lett 79(1):39–41Google Scholar
- Liu Y, Wu W, Sethuraman G, Nancollas GH (1997) Intergrowth of calcium phosphates: an interfacial energy approach. J Cryst Growth 174(1):386–392Google Scholar
- Lu X, Leng Y (2005) Theoretical analysis of calcium phosphate precipitation in simulated body fluid. Biomaterials 26(10):1097–1108Google Scholar
- Loewenthal GE, Marais GR (1989) Mixed weak acid/base systems part 1 mixture characterisation. Water SA Manuscr 15(1):496Google Scholar
- Martin B (1994) Mathematical model for the mineralization of bone. J Orthop Res 12(3):375–383Google Scholar
- McKee MD (1994) Osteopontin deposition in remodeling bone: an osteoblast mediated event. J Bone Miner Res 11(6):873–874Google Scholar
- Mohan C (2006) Buffers—a guide for the preparation and use of buffers in biological systems, 3rd edn. BMD Biosciences, San DiegoGoogle Scholar
- Mullin JW (2001) Crystallization, 4th edn. Butterworth, OxfordGoogle Scholar
- Murshed M, McKee MD (2010) Molecular determinants of extracellular matrix mineralization in bone and blood vessels. Curr Opin Nephrol Hypertens 19(4):359–365Google Scholar
- Musvoto EV, Wentzel MC, Loewenthal RE, Ekama GA (2000) Integrated chemical physical processes modellingI. Development of a kinetic-based model for mixed weak acid/base systems. Water Res 34(6):1857–1867Google Scholar
- Nancollas GH, Koutsoukos PG (1980) Calcium phosphate nucleation and growth in solution. Prog Cryst Growth Charact 3(1):77–102Google Scholar
- Nielsen AE (1984) Electrolyte crystal growth mechanisms. J Cryst Growth 67(2):289–310Google Scholar
- Noyes AA, Whitney WR (1897) The rate of solution of solid substances in their own solutions. J Am Chem Soc 19(12):930–934Google Scholar
- Oliveira C, Ferreira A, Rocha F (2007) Dicalcium phosphate dihydrate precipitation:characterization and crystal growth. Chem Eng Res Des 85(12):1655–1661Google Scholar
- Ostwald W (1900) ber die vermeintliche Isomerie des roten und gelben Quecksilbersoxyds und die Oberflchenspannung fester Krper. Z Phys Chem 34:495–503Google Scholar
- Ouzzani MH, Fedorowicz Z, Elmagarmid A (2016) Rayyan a web and mobile app for systematic reviews. Syst Rev 5(1):210Google Scholar
- Pitzer KS (1973) Thermodynamics of electrolytes. I. Theoretical basis and general equations. J Phys Chem 77(2):268–277Google Scholar
- Poduri R, Chen LQ (1996) Non-classical nucleation theory of ordered intermetallic precipitatesapplication to the AlLi alloy. Acta Mater 44(10):4253–4259Google Scholar
- Rice G, Barber A, OConnor A, Stevens G, Kentish S (2010) A theoretical and experimental analysis of calcium speciation and precipitation in dairy ultrafiltration permeate. Int Dairy J 20(10):694–706Google Scholar
- Shnel O, Mullin JW (1988) Interpretation of crystallization induction periods. J Colloid Interface Sci 123(1):43–50Google Scholar
- Solomon T (2001) The definition and unit of ionic strength. J Chem Educ 78(12):1691Google Scholar
- Szilgyi B, Muntean N, Barabs R, Ponta O, Lakatos BG (2015) Reaction precipitation of amorphous calcium phosphate: population balance modelling and kinetics. Chem Eng Res Des 93(1):278–286Google Scholar
- Volmer MWA (1926) Keimbildung in bersttigten Gebilden. Z Phys Chem 119(1):277–301Google Scholar
- Wang L, Nancollas GH (2008) Calcium orthophosphates: crystallization and dissolution. Chem Rev 108(11):4628–4669Google Scholar
- Wang K, Leng Y, Lu X, Ren F, Ge X, Ding Y (2012a) Theoretical analysis of protein effects on calcium phosphate precipitation in simulated body fluid. Cryst Eng Comm 14(18):5870Google Scholar
- Wang LJ, Ruiz-Agudo E, Putnis CV, Menneken M, Putnis A (2012b) Kinetics of calcium phosphate nucleation and growth on calcite: implications for predicting the fate of dissolved phosphate species in alkaline soilstheoretical analysis of protein effects on calcium phosphate precipitation in simulated body fluid. Environ Sci Technol 46(2):834–842Google Scholar
- Wong ATC, Czernuszka JT (1993) Transformation behavior of calcium–phosphate.1. Theory and modeling. Colloids Surf A 78(1):245–253Google Scholar
- Wyman J (1930) Measurements of the dielectric constants of conducting media. Phys Rev 35(6):623–634MathSciNetGoogle Scholar
- Xie Y, Liu X, Chu PK, Ding C (2006) Nucleation and growth of calcium–phosphate on Ca-implanted titanium surface. Surf Sci 600(3):651–656Google Scholar
- Yamamoto H, Sakae T (1987) Brushite in fibrous dysplasia of the jaw bone. Acta Pathol Jpn 37(10):1699–1705Google Scholar