Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 320

• W ð 0 Þ¼ 0;
• The increments, W ð t k þ 1 Þ W ð t k Þ and W ð t k Þ W ð t k 1 Þ are independent for any
0 t k 1 \t k \t k þ 1 ;
• For 0 s t ; the increment W ð t Þ W ð s Þ has the Gaussian distribution with
mean 0 and variance t s ; i.e. W ð t Þ W ð s Þ N ð 0 ; t s Þ:

Further, W ð t Þ is 'stochastically continuous' (lim t ! s P ðj W ð t Þ W ð s Þj [ Þ¼ 0),
where P stands for the probability. The formal analytic solution,

X ð t Þ¼ X 0 þ rW ð t Þ; Y ð t Þ¼ Y 0 þ rW ð t Þ; Z ð t Þ¼ Z 0 þ rW ð t Þ;

ð 2 Þ

for t [ 0 ;

can be given, however the differential form is more useful in this study from a
practical point of view. The equations are classically numerically solved using the
Euler-Maruyama Method, given by

X i þ 1 ¼ X i þ rDW i þ 1 ; X 0 ¼ð X 0 ; Y 0 ; Z 0 Þ:

ð 3 Þ

Here each component of DW is a normally distributed stochastic parameter with
zero mean and variance Dt ; denoted by N ð 0 ; Dt Þ; and it can be proved that [ 24 ]
each component m satisfies DW i þ 1 ¼ W i þ 1 W i N ð 0 ; 1 Þ

p

; in other words, a
Gaussian distribution with zero mean and a variance of Dt : We show a run of the
solution of the stochastic differential equations with one bacterium initially located
at ð 0 ; 0 ; 0 Þ with mobility r ¼ 2 : 6833 10 5 m/

Dt

p : This value was chosen from
[ 25 ] and corresponds to the classical bacillum in Fig. 1 . Figure 1 shows the tra-
jectory of the bacterium over time in three dimensions. Since Fig. 1 only gives one
specific run, the trajectory itself is a stochastic parameter and hence for many
purposes the probability density function is of more importance. To this extent,
since dW ð t Þ N ð 0 ; 1 Þ

p

and W ð t Þ N ð 0 ; 1 Þ t

p

; the probability density for the
position of the bacterium at time t for each coordinate direction satisfies

dt

exp ð ð m m 0 Þ 2
2r 2 t

1
2pr 2 t

p

f m ð t ; m Þ¼ Þ; m 2ð X ; Y ; Z Þ: ð 4 Þ

Since the Brownian motion in each coordinate direction is an independent
stochastic event, the multi-variate probability density is given by

ð 2pr 2 t Þ 2 exp ð ð x X 0 Þ 2

1

f ð x ; y ; z Þ¼ Þ; ð 5 Þ

2r 2 t

3

which solves the initial value problem in R

ot r 2

of

2 Df ¼ 0 ;

f ð 0 ; ð x ; y ; z ÞÞ ¼ d ð x X 0 Þ: ð 6 Þ

Here d ð x Þ represents the Dirac Delta Distribution in three dimensions, with
characteristics