Abstract
The implant of a femoral prosthesis is a critical process because of the relatively high temperature values reached at the bone/cement interface during the cementation of the infibulum. In fact, the cement is actually a polymer that polymerizes in situ generating heat. Moreover, the conversion of monomer into polymer is never 100%; this is dangerous because of the toxicity of the monomer. In this paper, we present a 3-D axisymmetric mathematical model capable of taking into account both the geometry of the implant and the chemical/physical properties of the cement. This model, together with its numerical simulation, thus represents a useful tool to set up the optimal conditions for the new materials developed in this orthopaedic field. The real complex geometry is assumed to be a bone/cement/metallic system having cylindrical symmetry, thus allowing the model to be reduced to two space variables. The cementation process is described by the Fourier heat equation coupled with a suitable polymerization kinetics. The numerical approximation is accomplished by semi-implicit finite differences in time and finite elements in space with numerical quadrature. The full discrete scheme amounts to solve linear positive definite symmetric systems preceded by an elementwise algebraic computation. We present various numerical simulations which confirm some critical aspects of this orthopaedic fixing technique such as thermal bone necrosis and the presence of unreacted residual monomer.
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References
Burnett, G. M., Duncan, G. L.: High conversion polymerization of vinyl system. I Methylmethacrylate. Makromolecul. Chem. 51, 154–170 (1962)
Carslaw, H. S., Jaeger, J. C.: Conduction of heat in solids. Oxford: Oxford University Press 1959
Ciarlet, P. G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978
Huiskes, R.: Some fundamental aspects of human joint replacement. Acta Orthop. Scand. Suppl. 185, 43–108 (1979)
Jefferis, C. D., Lee, A. J. C., Ling, R. S. M.: Thermal aspects of self curing PMMA. J. Bone Joint Surg. 578, 511–518 (1975)
Mazzullo, S., Paganetto, G., Simonazzi, T.: Thermal effects during cementation of an endomedullary infibulum. In: Proceedings, ECMI 1990. Teubner (to appear)
Mazzullo, S., Paolini, M., Verdi, C.: Polymer crystallization and processing: free boundary problems and their numerical approximation. Math. Engrg. Indust. 2, 219–232 (1989)
Mazzullo, S., Paolini, M., Verdi, C.: An axisymmetric analysis of thermal effects during cementation of femoral prostheses. In: Proceedings, Conference on Numerical Methods for Free Boundary Problems 1990. Basel: Birkhäuser (to appear)
Moritz, A. R., Henriques, F. C. Jr.: Studies of thermal injury II. Am. J. Pathol. 23, 695 (1947)
Paolini, M., Verdi, C.: An automatic mesh generator for planar domains. Rivista di Informatica 20, 251–267 (1990)
Pipino, F.: Il punto sulla cementazione degli impianti protesici. Firenze: OIC Medical Press 1987
Verdi, C., Visintin, A.: Numerical analysis of the multidimensional Stefan problem with supercooling and superheating. Boll. Unione Mat. It. 1-B, 795–814 (1987)
Visintin, A.: Stefan problem with phase relaxation. IMA J. Appl. Math. 34, 225–245 (1985)
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This work was partially supported by MURST (Fondi per la Ricerca Scientifica 40%) and CNR (IAN, Contract 880032601, and Progetto Finalizzato “Sistemi Informatici e Calcolo Parallelo”, Sottoprogetto “Calcolo Scientifico per Grandi Sistemi”) of Italy
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Mazzullot, S., Paolini, M. & Verdi, C. Numerical simulation of thermal bone necrosis during cementation of femoral prostheses. J. Math. Biol. 29, 475–494 (1991). https://doi.org/10.1007/BF00160473
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DOI: https://doi.org/10.1007/BF00160473