Sains Malaysiana 42(5)(2013): 661–666
Simulations of Hirschsprung’s Disease Using Fractional
Differential Equations
(Simulasi Penyakit Hirschsprung Menggunakan Persamaan Pembezaan
Pecahan)
F.A. Abdullah*
School of Mathematical Sciences, Universiti Sains Malaysia
11800 USM, Pulau Pinang, Malaysia
Diserahkan: 20 Julai 2011/Diterima: 20 Oktober 2012
ABSTRACT
In this paper, we examined a model of cell invasion focusing on
the wavefront of the neural crest (NC) cells in the case of
Hirschsprung’s disease (HSCR). Hirschsprung’s disease (HSCR)
is a congenital defect of intestinal ganglion cells and causes patients to have
disorders in peristalsis. This simulation model was performed using the fractional
differential equations (FDEs) based upon two basic cell
functions. Here, we simulated the mathematical model in a one-dimensional
setting, based on the fractional trapezoidal numerical scheme and the results
showed an interesting outcome for the mobility of the cellular processes under
crowded environments.
Keywords: Fractional differential equation; Hirschsprung disease;
simulation
ABSTRAK
Dalam penyelidikan ini, kami mengkaji model berkaitan penyerangan
sel dan fokus kajian adalah pada gelombang penyerangan sel neural dalam
penyakit Hirschsprung (HSCR). Penyakit Hirschsprung (HSCR)
adalah penyakit yang berkaitan dengan kecacatan semasa lahir atau sebelum lahir
dan berpunca daripada sel ganglion sehingga menyebabkan proses periltalsis
menjadi tidak normal. Model simulasi adalah berdasarkan
persamaan pembezaan pecahan (FDES) ke atas dua sel asas. Kajian ini mensimulasikan model matematik dalam satu dimensi
berpandukan kepada kaedah berangka trapezoid pecahan. Hasil
keputusan daripada simulasi ini menunjukkan wujud hasil yang menarik daripada
pergerakan sel dalam keadaan bersesak.
Kata kunci: Penyakit Hirschsprung; persamaan
pembezaan pecahan; simulasi
RUJUKAN
Abdullah, F.A. 2009. Numerical
methods for fractional differential equations and their applications to system
biology. PhD Thesis, University of Queensland, Brisbane, Australia
(unpublished).
Diethelm, K., Ford, N.J. & Freed, A.D. 2002. A Predictor-corrector approach for the numerical solution of
fractional differential equations. Nonlinear Dynam. 29: 3-22.
Diethelm, K., Ford, N.J. & Freed, A.D. 2004.
Detailed error analysis for a fractional Adams method. Numerical Algorithms 36:
31-52.
Diethelm, K., Ford, N.J., Freed, A.D. & Luchko, Y. 2005. Algorithms for the fractional
calculus: A selection of numerical methods. Comput. Methods Appl. Mech.
Engrg. 194: 743-773.
Ferreti, P., Copp, A.
Tickle, C. & Moore, G. 2006. Embryos, Genes and Birth Defects. 2nd ed. UK: John Wiley and Sons.
Ford, N.J. & Simpson, C. 2001. Numerical approaches to the
solution of some fractional differential equations. Proceedings of HERCMA 100-107
Landman, K.A., Simpson, M.J. & Newgreen, D.F.
2007.Mathematical and experimental insights into the development of enteric
nervous system and Hirschsprung’s disease.Develop. Growth Differ. 49:
277-286.
Landman, K.A., Simpson, M.J., Slater, J.L. & Newgreen, D.F.
2005. Diffusive and chemotactic cellular migration: Smooth and discontinous
travelling wave solutions. SIAM J. Appl. Math. 65(4): 1420-1442.
Oldham, K.B. & Spanier, J. 1974. The
Fractional Calculus, 111, Mathematics in Science and Engineering. New York: Academic Press.
Passarge, E. 2002. Dissecting Hirschsprung disease. Nat. Genet. 31: 11-12.
Simpson, M.J., Landman, K.A., Hughes, B.D. &
Newgreen, D.F. 2006. Looking inside an invasion
wave of cells using continuum models: Proliferation is the key. J. Theor.
Biol. 243: 343-360.
Simpson, M.J., Zhang, D.C., Mariani, M.,
Landman, K.A. & Newgreen, D.F. 2007. Cell
proliferation drives neural crest cell invasion of the intestine. Dev. Biol. 302: 553-568.
Skaba, R. 2007. Historic milestones of
Hirschsprung’s disease (commemorating the 90th anniversary of Professor Harald
Hirschsprung’s death). J. Pediatr. Surg. 42: 249-251.
Yntema, C.L. & Hammond, W.S. 1954. The
origin of intrinsic autonomic ganglia of trunk viscera in the chick embryo. J. Comp. Neurol. 101: 515-541.
Yuste, S.B., Acedo, L. & Lindenberg, K. 2004. Reaction front in an A+B→C reaction- subdiffusion process. Phys. Rev. E 69: 036126.
Yuste, S.B. & Lindenberg, K. 2001. Subdiffusion limited A+A reactions. Phys. Rev. Lett. 87: 118301.
Yuste, S.B. & Lindenberg, K. 2002. Subdiffusion-limited
reactions. Chem. Phys. 284: 169-180.
*Pengarang untuk surat-menyurat; email: farahaini@usm.my