We model RBCs as rigid oblate ellipsoids with eccentricity ratio of 0.3, mimicing the thickness to width ratio of rat RBC, as illustrated in . The density of RBC is close to the density of plasma. Accordingly, RBCs are considered to be purely buoyant in plasma flow. The depletion interaction theory applied to RBC aggregation (

Neu and Meiselman 2002) proposes that a depletion layer between two adjacent cells restricts the space available to polymer molecules (e.g. DEX) between cells by displacing solvent into the bulk phase, and this generates osmotic pressure between the depletion zone and the bulk phase. According to the theory, total interaction energy per unit surface

*W*_{T} between two infinite surfaces due to polymers such as DEX and Poly Ethylene Glycol is the sum of depletion energy

*W*_{D} and electrostatic repulsive energy

*W*_{E} (with negligible van der Waals interaction)

where

where π, Δ,

*d, δ* and

*P* are the osmotic pressure term, depletion thickness, intercellular distance, RBC glycocalyx thickness (5 nm) and penetration depth;

*σ, ε, ε*_{o} and

*k* are the surface charge density of RBC, the relative permittivity of the solvent, the permittivity of the vacuum and the Debye–Huckel length, respectively. The above formulation yields an optimum polymer concentration for interaction energy because the penetration depth,

*P*, is an inverse exponential function of the polymer concentration. The interaction energy increases, reaches the maximum and decreases as the concentration increases. Furthermore, due to the electrostatic energy, interaction energy has an optimum with respect to the intercellular distance. The interaction energy gradually increases to a maximum and decreases to zero as two surfaces approach each other. A strong repulsive force results when the distance between the surfaces is close to the sum of the thicknesses of RBC glycocalyx.

Bhattacharjee and collaborators approximated Derjaguin’s formula to find intercellular energy between two curved surfaces using the surface element integration (

Bhattacharjee et al. 1998). Based on the surface element integration, total interaction energy can be expressed as

where

**n** _{1} and

**n** _{2} are the outward unit normal vectors of the surface elements of particles 1 and 2, respectively, and

*S*_{1} is the surface of particle 1. Here,

*h* is the local distance between the two surface elements,

**k** _{1} and

**k** _{2} are the unit normal vectors of two parallel planes with the direction parallel to a line

*L*, which connects the centres of two particles. Therefore,

**k** _{1} = −

**k** _{2}. The detailed description is given in our earlier study (

Chung et al. 2007).

Based on the depletion interaction energy, from Equations

(1) to

(4), an interaction force and torque between two RBCs were derived in

Appendix A (see Equations

(A6) and

(A9)). The interaction force based on (

Neu and Meiselman 2002) was modified using the Morse potential adopted by

Liu et al. (2004). A force per unit surface by

Liu et al. (2004) is represented by

where

*h*_{o} is the intercellular distance between the two surface elements when the force becomes zero and

*A, β* are the parameters explicitly obtained from

Equation (A6) with

Equation (5) by comparing the optimum force and zero force occurred at two points at a distance. shows the force vs. intercellular distance. We adopted the force given by

Equation (5) because the constant attractive force over a range of intercellular distances shown in was regarded as non-physical; the repulsive component of the force was modified computationally to prevent RBCs overlap as explained below. Torque in

Equation (A9) is also modified using the modified force in

Equation (5).

Because of the attractive force described in

Equation (5) (see also ), two spheroids modelled as RBCs begin to approach each other as they are close enough to be in the region where the interaction energy acts. Total interaction force over the entire surface of an RBC may increase until the closest surface of the cell overlaps the other RBC. As described earlier, the minimum cellular distance (10 nm) is modelled to be the sum of the thicknesses of glycocalyx covering the entire surfaces of two cells. Accordingly, the cells cannot approach within 10 nm in the model. Energy distribution near the minimum distance shows that the electrostatic energy increases infinitely as the minimum distance is approached. Physically, when two cells approach each other, the interaction forces (attractive and repulsive) between the entire surfaces of cells reach equilibrium and the cells form an aggregate. Due to the computational limitation of the grid size, the interaction energy was scaled up to have the interaction force between cells within the computational interaction range equal to the force generated in the physiological range using an intercellular distance scale factor 50, as in our previous study (

Chung et al. 2007). After the minimum distance between the cell surfaces reaches within our minimum computational length (500 nm), if the cells overlap each other, we imposed a repulsive force which has the same magnitude with opposite direction as that of the interaction force between the cells at the time, so that the total force became zero to prevent the overlap. At this point, we assume that the two RBCs form an aggregate. Based on this aggregation criterion, we obtained distances from the apex of bifurcation and time required for two cells to form an aggregate after leaving the daughter branches.