Effect of micro-cracks on the angiogenesis and osteophyte development during degenerative joint disease
Osteoarthritis, one of the most common types of arthritis, is characterized by the development of osteophytes. The main cause of joint degeneration is mechanical loading, but there are also several other factors that influence the development of osteophytes. In order to formulate a mathematical model of bone spurs’ development we have selected the most important factors, such as angiogenesis, micro-damage of the tissue structure and cell signaling. The proposed system of integro-differential equations describes the degenerative changes in the joint. Numerical calculations were implemented into the COMSOL Multiphysics software and the obtained results thus reflect relationships between certain parameters and variables. Additionally, the results correspond with those obtained from medical observations.
Keywordsosteoarthritis, angiogenesis, mathematical modelling, mechanical loading, micro-damage, osteophytes,
References E. Bednarczyk, T. Lekszycki. A novel mathematical model for growth of capillaries and nutrient supply with application to prediction of osteophyte onset. Z. Angew. Math. Phys., 67(4), 2016.
 E. Bednarczyk, T. Lekszycki. Osteophyte development during osteoarthritis (OA) – consideration of angiogenesis, mechanical loading and tissue microstructure. Engineering Transactions, 64(4): 533–540, 2016.
 C.S. Bonnet, D.A. Walsh. Osteoarthritis, angiogenesis and inflammation. Rheumatology, 44(1): 7–16, 2005.
 D.B. Burr, E.L. Radin. Microfractures and microcracks in subchondral bone: are they relevant to osteoarthrosis? Rheumatic Disease Clinics of North America, 29(4): 675–685, 2003.
 T.R. Coughlin, O.D. Kennedy. The role of subchondral bone damage in post-traumatic osteoarthritis. Ann. N.Y. Acad. Sci., 1383(1): 58–66, 2016.
 R.E. Frans`es, D.F. McWilliams, P.I. Mapp, D.A. Walsh. Osteochondral angiogenesis and increased protease inhibitor expression in OA. Osteoarthritis and Cartilage, 18(4): 563–571, 2010.
 I. Giorgio, U. Andreaus, F. dell’Isola, T. Lekszycki. Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mechanics Letters, 13: 141–147, 2017.
 I. Giorgio, U. Andreaus, A. Madeo. The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids. Continuum Mech. Thermodyn., 28(1–2): 21–40, 2016.
 S. Hashimoto, L. Creighton-Achermann, K. Takahashi, D. Amiel, R.D. Coutts, M. Lotz. Development and regulation of osteophyte formation during experimental osteoarthritis. Osteoarthritis and Cartilage, 10(3): 180–187, 2002.
 T. Hayami, M. Pickarski, G.A. Wesolowski, J. Mclane, A. Bone, J. Destefano, G.A. Rodan, L.T. Duong. The role of subchondral bone remodeling in osteoarthritis: The role of subchondral bone remodeling in osteoarthritis: Reduction of cartilage degeneration and prevention of osteophyte formation by alendronate in the rat anterior cruciate ligament transection model. Arthritis and Rheumatism, 50(4): 1193–1206, 2004.
 Y. Henrotin, L. Pesesse, C. Sanchez. Subchondral bone and osteoarthritis: biological and cellular aspects. Osteoporos Int., Suppl. 8: S847–851, 2012.
 M.A. Karsdal, D.J. Leeming, E.B. Dam, K. Henriksen, P. Alexandersen, P. Pastoureau, R.D. Alltman, C. Christiansen. Should subchondral bone turnover be targeted when treating osteoarthritis? Osteoarthritis and Cartilage, 16(6): 638–646, 2008.
 F.C. Ko, C. Dragomir, D.A. Plumb, S.R. Goldring, T.M. Wright, M.B. Goldring, M.C.H. van der Meulen. In vivo cyclic compression causes cartilage degeneration and subchondral bone changes in mouse tibiae. Arthritis and Rheumatism, 65(6): 1569–1573, 2013.
 F. Lekszycki, T. dell’Isola. A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM, 92(6): 426–444, 2012.
 G. Li, J. Yin, J. Gao, T.S. Cheng, N.J. Pavlos, C. Zhang, M.H. Zheng. Subchondral bone in osteoarthritis: insight into risk factors and microstructural changes. Arthritis Research and Therapy, 15(6): 223, 2013.
 J. Monod. The growth of bacterial cultures. Annual Review of Microbiology, 3: 371–394, 1949.
 D. Pfander, D. K¨ortje, R. Zimmermann, G. Weseloh, T. Kirsch, M. Gesslein, T. Cramer, B. Swoboda. Vascular endothelial growth factor in articular cartilage of healthy and osteoarthritic human knee joints. Ann Rheum Dis., 60(11): 1070–1073, Nov 2001.
 S. Suri, S.E. Gill, S.M. de Camin, D. Wilson, D.F. McWilliams, D.A. Walsh. Neurovascular invasion at the osteochondral junction and in osteophytes in osteoarthritis. Ann Rheum Dis., 66(11): 1423–1428, 2007.
 P.-F. Verhulst. Notice sur la loi que la population suit dans son accroissement. Correspondance math´ematique et physique, 10: 113–121, 1838. (available in https://books.google.pt/books?id=8GsEAAAAYAAJ&printsec=frontcover&hl=pt-PT&source=gbs ge summary r&cad=0#v=onepage&q&f=true).
 X.L. Yuan, H.Y. Meng, Y.C.Wang, J. Peng, Q.Y. Guo, A.Y.Wang, S.B. Lu. Bone-cartilage interface crosstalk in osteoarthritis: potential pathways and future therapeutic strategies. Osteoarthritis and Cartilage, 22(8): 1077–89, 2014.
 L. Zhang, H. Zheng, Y. Jiang, Y. Tu, P. Jiang, A. Yang. Mechanical and biologic link between cartilage and subchondral bone in osteoarthritis. Arthritis Care and Research, 64(7): 960–967, 2012.