Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 262

the contribution of the DPD particle self-diffusion to the diffusive transport of
the nutrient substrate is significant. Consequently, the real nutrient substrate
diffusion coecient in a DPD model must be determined for given λ ij and
T by calibrating the DPD model using the known analytical solutions for
one-dimensional diffusion equation.
The biofilm density, c b,i , is found from equation (7.13):
dc b,i / dt = c b,i {

c s,i /( k 2 + c s,i ) − k 3 }

(7.19)

Equation (7.19) should be solved for biofilm DPD particles only where c b,i > 0.
Once the biomass density, c b , exceeds the maximum biomass density (1.0 for
the normalized biomass density), the excess biomass is transferred to the near-
est fluid DPD particle in the cut-off range with c b = 0, and this fluid DPD
particle is spontaneously changed to a biofilm DPD particle. In the rare event
of an absence of fluid DPD particles in the cut-off region of particle i (this
may happen if particle i lies deep inside biofilm domain), the excess biomass is
assumed to be lost. In practice, this “unphysical” procedure is highly unlikely
to occur because biomass growth is concentrated mainly on the interface, and
decay dominates inside the biomass domain. This biomass spreading mecha-
nism is similar to the discrete rules used to redistribute the biomass in cellular
automata models (Picioreanu et al. 1998), where a search for a “free-space”
element among the nearest-neighbor elements is performed.
Lennard-Jones or bi-harmonic potentials have been used in MD studies
of crack propagation in solids (Abraham et al. 1997; Buehler et al. 2003).
Similarly, a simple harmonic potential with an equilibrium distance, r e , that is
slightly smaller than the cut-off distance, r 0 , is used to model the interactions
between biofilm DPD particles, and this results in a soft solid-like biofilm
structure. The harmonic potential energy and force between biofilm particles
i and j are given by

S bb
2 r e ( r ij −

r e ) 2 e ij = − (7.20)

−

f ji = S bb (1

− r ij /r e ) r ij

f ij = (7.21)

When the distance, r ij , exceeds the cut-off distance, r 0 , the force between
particles i and j falls to zero because the “bond” between these particles is
ruptured.
The biofilm-biofilm particle interaction strength, S bb , and the equilibrium
distance, r e , control the mechanical properties of the biofilm, such as the elas-
tic modulus and the maximum strain at failure. In the model, the interaction
between a liquid DPD particle and a biofilm DPD particle is assumed to be
equal to the purely repulsive liquid-liquid interaction or it can be assumed to
have a harmonic functional form with an equilibrium distance of r e = r 0 , and
the strength, S lb , can be tuned for various liquid-biofilm interaction strengths
to satisfy the no-slip boundary condition at the liquid-biofilm interface. Here
S lb and S ll are the strength coecients used in equation (7.15) to compute the
conservative forces between liquid-biofilm particles and liquid-liquid particles.