Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 407

Once Eq. ( 24 ) is solved to determine v at a given value of t the evolution of the
radii of the tracheal lumen and the cartilage ring is calculated using

dR l
dt ¼ v ð R l ; t Þ;

dR c
dt ¼ v ð R c ; t Þ;

ð 27 Þ

with initial conditions, R l ð 0 Þ¼ R l ; 0 and R c ð 0 Þ¼ R c ; 0 ; which specify the initial
radii of tissue boundaries within the decellularised trachea.

3.4 Nondimensionalization

Nondimensionalization was performed to simplify the model by reducing the
number of parameters. The general approach used was to scale the variables so as
to make the maximum cell densities and chemical concentrations in the non-
dimensional model of the order unity, and where possible to scale the production
and proliferation rates to unity rather than rates of removal and degradation. To
this end the following set of substitutions were made:

t ¼ t = s ;

r ¼ r = L ; v ¼ sv = L ;

ð 28 Þ

R ¼ R = L ; N ¼ N = N ; C ¼ C = C

where s is a convenient time scale over which tissue growth can be measured, and
L is a characteristic length scale. In Eq. ( 28 ) and in what follows the subscript
denotes any of the subscripts for the species in the model. The following new
(dimensionless) parameters are introduced:

9
=

R ¼ R = L ; D ¼ D s = L 2 ;

p ¼ N msc = N chn ;

N mac ¼ 2I mac =ð hd mac Þ; C ilk ¼ b ilm N msc = d ilk ; C tgf ¼ b tm N mac = d tgf ;

C tnf ¼ b tnf N msc = d tnf ; C col ¼ b col = d col ; C gag ¼ b gch = d gch ;

b msc ¼ b msc N msc s ; b epi ¼ b epi N epi s ; b fib ¼ b fib N fib s ;

b chn ¼ b chn N chn s ; b tch ¼ b tch ð N chn Þ 2 s = C tgf ; b ile ¼ b ile N epi s =ð LC ilk Þ;

k cm ¼ k cm C tgf s ;

I epi ¼ I epi s =ð LN epi Þ; I msc ¼ I msc s =ð LN msc Þ;

d mac ¼ d mac s ; d chn ¼ d chn C tgf s ;

I fib ¼ I fib s =ð LN fib Þ;

;

d ilk ¼ d ilk s ; d tnf ¼ d tnf s ; d tgf ¼ d tgf s ;

d col ¼ d col N fib s ; d gch ¼ d gch N chn s ; d gag ¼ d gag s ;

K mi ¼ K mi = C ilk ; K ct ¼ K ct = C tgf ; K et ¼ K et = C tgf ;

K ti ¼ K ti = C tnf ; K mt ¼ K mt = C tgf ; K mn ¼ K mt = C tnf ;

A vc ¼ A vc C col ; A vg ¼ A vg C gag ;

T inf ¼ T inf s ;

h ¼ h = L :

ð 29 Þ

whereby the equations become in dimensionless form, dropping the carets,