# A Novel Multiscale Mathematical Model for Building Bone Substitute Materials for Children

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model and Results

#### 2.1. Modeling the Biological Components: An Inspirational Model

#### 2.2. Our Mathematical Model

- -
- collagen compartments ${x}_{1}$ and ${x}_{2}$ have been replaced by a single compartment where immature collagen crosslinks progressively moves toward the assembled collagen matrix;
- -
- mineral formation is created from precipitated crystal growing in function of collagen and aggregating in function of inhibitors and nucleators.

#### 2.3. Results

#### 2.4. Multiscale Modelling

## 3. Materials and Methods

#### 3.1. An Associated System of Ordinary Differential Equation

#### 3.2. Numerical Illustrations

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wegst, U.G.K.; Bai, H.; Saiz, E.; Tomsia, A.P.; Ritchie, R.O. Bioinspired structural materials. Nat. Mater.
**2015**, 14, 23–36. [Google Scholar] [CrossRef] [PubMed] - Sheikh, Z.; Najeeb, S.; Khurshid, Z.; Verma, V.; Rashid, H.; Glogauer, M. Biodegradable Materials for Bone Repair and Tissue Engineering Applications. Materials
**2015**, 8, 5744–5794. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dutta, S.R.; Passi, D.; Singh, P.; Bhuibhar, A. Ceramic and non-ceramic hydroxyapatite as a bone graft material: A brief review. Ir. J. Med. Sci.
**2015**, 184, 101–106. [Google Scholar] [CrossRef] [PubMed] - Fitoussi, F.; Ilharreborde, B. Is the induced-membrane technique successful for limb reconstruction after resecting large bone tumors in children? Clin. Orthop. Relat. Res.
**2015**, 473, 2067–2075. [Google Scholar] [CrossRef] [PubMed] - Li, J.; Pan, Z.; Yan, S.; Zhao, X. Single-cortex is better than double-cortex in fibula grafts for large tibia bone defect in a 2-year-old child: A case report of a successful surgery and discussion of bone graft choices. Medicine
**2017**, 96, e5965. [Google Scholar] [CrossRef] [PubMed] - Leet, A.I.; Boyce, A.M.; Ibrahim, K.A.; Wientroub, S.; Kushner, H.; Collins, M.T. Bone-Grafting in Polyostotic Fibrous Dysplasia. J. Bone Joint Surg. Am. Vol.
**2016**, 98, 211–219. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zioupos, P.; Currey, J. Changes in the Stiffness, Strength, and Toughness of Human Cortical Bone With Age. Bone
**1998**, 22, 57–66. [Google Scholar] [CrossRef] - Zimmermann, E.A.; Ritchie, R.O. Bone as a Structural Material. Adv. Healthc. Mater.
**2015**, 4, 1287–1304. [Google Scholar] [CrossRef] [PubMed] - Boskey, A.L.; Imbert, L. Bone quality changes associated with aging and disease: A review. Ann. N. Y. Acad. Sci.
**2017**, 1410, 93–106. [Google Scholar] [CrossRef] [PubMed] - Garnero, P. The Role of Collagen Organization on the Properties of Bone. Calcif. Tissue Int.
**2015**, 97, 241. [Google Scholar] [CrossRef] [PubMed] - Fyhrie, D.P.; Christiansen, B.A. Bone Material Properties and Skeletal Fragility. Calcif. Tissue Int.
**2015**, 97, 213–228. [Google Scholar] [CrossRef] [PubMed] - Burr, D. The contribution of the organic matrix to bone’s material properties. Bone
**2002**, 31, 8–11. [Google Scholar] [CrossRef] - Unal, M.; Creecy, A.; Nyman, J.S. The Role of Matrix Composition in the Mechanical Behavior of Bone. Curr. Osteoporos. Rep.
**2018**, 16, 205–215. [Google Scholar] [CrossRef] [PubMed] - Saito, M.; Marumo, K. Effects of Collagen Crosslinking on Bone Material Properties in Health and Disease. Calcif. Tissue Int.
**2015**, 97, 242–261. [Google Scholar] [CrossRef] [PubMed] - Eyre, D.R.; Koob, T.J.; Ness, K.P.V. Quantitation of hydroxypyridinium crosslinks in collagen by high-performance liquid chromatography. Anal. Biochem.
**1984**, 137, 380–388. [Google Scholar] [CrossRef] - Gineyts, E.; Borel, O.; Chapurlat, R.; Garnero, P. Quantification of immature and mature collagen crosslinks by liquid chromatography-electrospray ionization mass spectrometry in connective tissues. J. Chromatogr. B
**2010**, 878, 1449–1454. [Google Scholar] [CrossRef] [PubMed] - Berteau, J.; Gineyts, E.; Pithioux, M.; Baron, C.; Boivin, G.; Lasaygues, P.; Chabrand, P.; Follet, H. Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula. Bone
**2015**, 79, 190–195. [Google Scholar] [CrossRef] [PubMed] - Roach, H. Why does bone matrix contain non-collagenous proteins? The possible roles of osteocalcin, osteonectin, osteopontin and bone sialoprotein in bone mineralisation and resorption. Cell Biol. Int.
**1994**, 18, 617–628. [Google Scholar] [CrossRef] [PubMed] - Gamsjaeger, S.; Hofstetter, B.; Fratzl-Zelman, N.; Roschger, P.; Roschger, A.; Fratzl, P.; Brozek, W.; Masic, A.; Misof, B.; Glorieux, F.; et al. Pediatric reference Raman data for material characteristics of iliac trabecular bone. Bone
**2014**, 69, 89–97. [Google Scholar] [CrossRef] [PubMed] - Lefèvre, E.; Lasaygues, P.; Baron, C.; Payan, C.; Launay, F.; Follet, H.; Pithioux, M. Analyzing the anisotropic Hookeś law for childrenś cortical bone. J. Mech. Behav. Biomed. Mater.
**2015**, 49, 370–377. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Berteau, J.P.; Baron, C.; Pithioux, M.; Launay, F.; Chabrand, P.; Lasaygues, P. In vitro ultrasonic and mechanic characterization of the modulus of elasticity of children cortical bone. Ultrasonics
**2014**, 54, 1270–1276. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Irwin, A.; Mertz, H. Biomechanical Bases for the CRABI and Hybrid III Child Dummies. Proc. Stapp Car Crash Conf.
**1997**, 41, 1–12. [Google Scholar] - Roth, S.; Raul, J.; Willinger, R. Limitation of scaling methods in child head finite elements modelling. Int. J. Veh. Saf.
**2007**, 2, 404–421. [Google Scholar] [CrossRef] - Bala, Y.; Depalle, B.; Douillard, T.; Meille, S.; Clément, P.; Follet, H.; Chevalier, J.; Boivin, G. Respective roles of organic and mineral components of human cortical bone matrix in micromechanical behavior: An instrumented indentation study. J. Mech. Behav. Biomed. Mater.
**2011**, 4, 1473–1482. [Google Scholar] [CrossRef] [PubMed] - Komarova, S.; Safranek, L.; Gopalakrishnan, J.; Ou, M.Y.; McKee, M.; Murshed, M.; Rauch, F.; Zuhr, E. Mathematical model for bone mineralization. Front. Cell Devel. Biol.
**2015**, 3, 51. [Google Scholar] [CrossRef] [PubMed] - Ott, I.; Kienzler, R.; Schröder, R. Aging in the cortical bone: A constitutive law and its application. Arch. Appl. Mech.
**2010**, 80, 527–541. [Google Scholar] [CrossRef] - Depalle, B.; Duarte, A.G.; Fiedler, I.A.; Pujo-Menjouet, L.; Buehler, M.J.; Berteau, J.P. The different distribution of enzymatic collagen cross-links found in adult and children bone result in different mechanical behavior of collagen. Bone
**2018**, 110, 107–114. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Uzel, S.G.; Buehler, M.J. Molecular structure, mechanical behavior and failure mechanism of the C-terminal cross-link domain in type I collagen. J. Mech. Behav. Biomed. Mater.
**2011**, 4, 153–161. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Demers, J.-L.H.; Esmonde-White, F.W.; Esmonde-White, K.A.; Morris, M.D.; Pogue, B.W. ext-generation Raman tomography instrument for non-invasive in vivobone imaging. Biomed. Opt. Express.
**2015**, 6, 793–806. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Representation of bone mineralization following Komarova et al. hypothesis in [25]. Continuous lines describe activations of the proteins involved during mineralization, while dotted lines stand for regulation.

**Figure 2.**Representation of bone mineralization following the Komarova et al. hypothesis in [25] with modifications: the collagen is not considered to be in two compartments but in a continuous maturity evolution (immature $a\in [0,\alpha )$, to mature $(\alpha ,+\infty )$). Mineralization is supposed to precipitate (size 0) to crystals (size $x>0$). Aggregation is represented by the purple “balls” as a very schematic representation of crystal forming. Continuous lines describe activations of the involved during mineralization, while dotted lines stand for the regulation.

**Figure 3.**Evolution of $I\left(t\right)$ (

**Left**), $N\left(t\right)$ (

**Middle**), and ${w}_{p}\left(t\right)$ (

**Right**) with $I\left(0\right)=0.55$, $N\left(0\right)=0.7$, and ${w}_{p}\left(0\right)=1$.

**Figure 4.**(

**Left**) Population of minerals $m(t,x)$. (

**Right**) Evolution of collagen $c(t,a)$. Parameters are $\alpha =8$, $\eta =0.2$.

**Figure 5.**Evolution of minerals of all sizes $M\left(t\right)={\int}_{0}^{+\infty}m(t,x)dx$ (

**Left**) and the total mineral mass $P\left(t\right)={\int}_{0}^{+\infty}xm(t,x)dx$ (

**Right**).

**Figure 7.**Parameters are $E=10$, $\alpha =8$, and $\eta =0.2$. The values of the parameter t are as follows: (

**a**) $t=0.1$; (

**b**) $t=1$; (

**c**) $t=1.5$; (

**d**) $t=2$; (

**e**) $t=2.2$.

**Figure 8.**The graph of functions $s\mapsto k(N,I,s)$ (

**Left**) and $s\mapsto V\left(s\right)$ (

**Right**). The graph of function $s\mapsto k(s,N(1),I(1\left)\right)$ was plotted at time $t=1$, that is $N\left(1\right)=0.2010$ and $I\left(1\right)=0.5426$.

**Figure 9.**Three states of mineralization of the collagen fiber that we modeled using the interactions between collagen crosslink maturity, inhibitors, nucleators and cAp precipitation. Step 1 corresponds to non-mineralization where we found only divalent CXL in the collagen fibril, no cAp minerals, and no OPN–OC complex. Step 2 is the intermediate mineralization where we found mainly divalent CXL in the collagen fibril, cAp minerals, and an OPN–OC complex between the minerals. Step 3 is the late stage of mineralization where we found large aggregated crystals and no OPN–OC complex.

t | $\mathit{F}\left(\mathit{t}\right)$ | $\mathit{M}\left(\mathit{t}\right)$ | Elastic Energy + Plastic Energy | ||
---|---|---|---|---|---|

$0.1$ | $0.7960$ | $0.5598$ | $5.0901$ | ||

1 | $0.7964$ | $0.5576$ | $5.2620$ | ||

$1.5$ | $0.7974$ | $0.5569$ | $5.7648$ | ||

2 | $0.7993$ | $0.5565$ | $6.9175$ | ||

$2.2$ | $0.8004$ | $0.5563$ | $7.6916$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chekroun, A.; Pujo-Menjouet, L.; Berteau, J.-P.
A Novel Multiscale Mathematical Model for Building Bone Substitute Materials for Children. *Materials* **2018**, *11*, 1045.
https://doi.org/10.3390/ma11061045

**AMA Style**

Chekroun A, Pujo-Menjouet L, Berteau J-P.
A Novel Multiscale Mathematical Model for Building Bone Substitute Materials for Children. *Materials*. 2018; 11(6):1045.
https://doi.org/10.3390/ma11061045

**Chicago/Turabian Style**

Chekroun, Abdennasser, Laurent Pujo-Menjouet, and Jean-Philippe Berteau.
2018. "A Novel Multiscale Mathematical Model for Building Bone Substitute Materials for Children" *Materials* 11, no. 6: 1045.
https://doi.org/10.3390/ma11061045