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Showing content from https://link.springer.com/article/10.1007/s10439-010-0184-2 below:

Numerical Investigation of the Effects of Channel Geometry on Platelet Activation and Blood Damage

The presence of TAT in the samples was measured using an Enzygnost TAT microassay (Dade Behring, Deerfield, IL, USA). Absorbance was determined using a SpectraMax Plus 384 microplate reader (Molecular Devices, Sunnyvale, CA, USA) at a wavelength of 492 nm. Samples exceeding the highest standard were diluted and re-assayed.

PF4 was measured using a Zymutest PF4 microassay (Hyphen Biomed, Neuville sur Oise, France). Absorbance was determined using a SpectraMax Plus 384 (Molecular Devices) at a wavelength of 450 nm. The standard curve ranged from 0.5 to 10 g/L. Samples with higher concentrations were diluted with sample diluent prior to the start of the assay.

Hemolysis in the samples was determined using a microplate assay. The plasma samples were mixed with Drabkins reagent in a 1:9 ratio of plasma to Drabkins reagent, and 200 mL of the mixture was added to each of the wells. The resulting color intensities of the samples were measured with a microtiter absorbance plate reader (SpectraMax Plus 384, Molecular Devices) at a wavelength of 540 nm. A standard curve was constructed with known amounts of whole blood and Drabkins reagent, and from the standard curve the concentration of hemoglobin in each of the samples was determined. The standards were prepared by measuring the RBC count and hemoglobin concentration of the whole blood using a cell counter (Coulter MAX). This whole blood was then mixed with different amounts of Drabkins reagent to obtain a range of hemoglobin concentrations from 0 to 5.48 mg/mL.

The EBF is the fluid–solid interaction force used to impose the no-slip boundary condition at the platelet surface. Let Π_s and Π_f represent the continuum solid and fluid domains, Γ the fluid–solid boundary, and Γ_s and Γ_f be subsets for the solid and fluid boundary nodes. F ^fsi(x, t) and g(x, t) represent the force density (force per unit volume) acting, respectively, on the solid and the fluid at the boundary nodes x on Γ at time t. Therefore, by Newton’s third law, F ^fsi(x, t) = −g(x, t) for x \(\in \) Γ. The motion of the fluid is governed by the Navier–Stokes and continuity equations with the inclusion of the EBF, which can be written as

$$ \left. {\begin{array}{*{20}l} {\rho \left( {{\frac{\partial {\user2 u}}{\partial t}} + {\user2 u} \cdot \nabla {\user2 u}} \right) = - \nabla P + \mu \nabla^{2} {\user2 u} + {\user2 g}\left( {{\user2 x},t} \right)} \\ \nabla \cdot {\user2 u} = 0 \\ \end{array} } \right\}, $$

(4)

where  \(\user2{x}\in\Pi_{\rm f}\). In the first equation, g(x, t) = 0 when  \(\user2{x} \; \notin \; \Gamma.\) In the discretized formulation, the EBF g is evaluated on the fluid boundary node as shown in Eq. (7).

To calculate the fluid–solid interaction force, we first calculate the fluid velocity \( {\user2 U}_{\text{f}} \left( {{\user2 x}_{j}^{i} ,t} \right) \) at solid boundary node \( {\user2 x}_{j}^{i} \) at time t,

$$ {\user2 U}_{\text{f}} \left( {{\user2 x}_{j}^{i} ,t} \right) = \int\limits_{{\Pi_{\text{f}} }} {{\user2 u}\left( {{\user2 x}^{\text{e}} ,t} \right)} D\left( {{\user2 x}^{\text{e}} - {\user2 x}_{j}^{i} } \right)d{\user2 x}^{\text{e}} ,\quad {\user2 x}_{j}^{i} \in \Gamma_{\text{s}} , $$

(5)

where \( {\user2 x}_{j}^{i} \) is the position vector for the jth solid node on the ith platelet, x ^e represents the position vector for the fluid nodes, and \( D\left( {{\user2 x}^{\text{e}} - {\user2 x}_{j}^{i} } \right) \) is the Dirac delta function.21

Assuming that the platelet will move with the velocity at the previous time step, the fluid–solid interaction force \( {\user2 F}^{\text{fsi}} \left( {{\user2 x}_{j}^{i} ,t} \right) \) at the platelet boundary node is given by

$$ {\user2 F}^{\text{fsi}} \left( {{\user2 x}_{j}^{i} ,t} \right) = \rho_{\text{f}} {{\left( {{\user2 U}_{\text{f}} \left( {{\user2 x}_{j}^{i} ,t} \right) - {\user2 U}_{\text{p}} \left( {\user2{x}_{j}^{i} ,t - \Updelta t^{\text{LBM}} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\user2{U}_{\text{f}} \left( {\user2{x}_{j}^{i} ,t} \right) - {\user2 U}_{\text{f}} \left( {x_{j}^{i} ,t - \Updelta t^{\text{LBM}} } \right)} \right)} {\Updelta t^{\text{LBM}} }}} \right. \kern-\nulldelimiterspace} {\Updelta t^{\text{LBM}} }},\quad {\user2 x}_{j}^{i} \in \Gamma_{\text{s}} , $$

(6)

where \( \rho_{\text{f}} \) is the density of the fluid and \( \Updelta t^{\text{LBM}} = 1 \) is the LBM time step. The resulting force acting on the fluid is given by

$$ {\user2 g}\left( {\user2{x}^{\text{e}} ,t} \right) = - \int\limits_{\Gamma } {{\user2 F}^{\text{fsi}} \left( {{\user2 x}_{j}^{i} ,t} \right)} D\left( {{\user2 x}^{\text{e}} - {\user2 x}_{j}^{i} } \right)d{\user2 x}_{j}^{i} ,\quad {\user2 x}^{\text{e}} \in \Gamma_{\text{f}} , $$

(7)

where g will be used as an EBF term in the lattice-Boltzmann equation as discussed in the following section.

Once the total force and torque are obtained for each platelet, the velocity and angular velocity of the suspending platelets can be computed by the numerical solution of the Newtonian dynamics equations.

The state of the fluid at node x ^e at time t is described by the distribution function f _k (x ^e, t), which is calculated through the lattice-Boltzmann equation2,14,17

$$ f_{k} \left( {{\user2 x}^{\text{e}} + {\user2 e}_{k} ,t + \Updelta t^{\text{LBM}} } \right) = f_{k} \left( {{\user2 x}^{e} ,t} \right) + {\frac{1}{\lambda }}\left[ {f_{k}^{\text{eq}} \left( {{\user2 x}^{\text{e}} ,t} \right) - f_{k} \left( {{\user2 x}^{\text{e}} ,t} \right)} \right]. $$

(8)

Here, \( f_{k}^{\text{eq}} \left( {{\user2 x}^{\text{e}} ,t} \right) \) is the equilibrium distribution function at x ^e at time t, λ the single relaxation time constant, and e _k the discrete velocity vector. The fluid density \( \rho_{\text{f}} \) and the macroscopic fluid velocity \( {\user2 u}\left( {{\user2 x}^{\text{e}} ,t} \right) \) are obtained from the first two moments of the distribution function, given by

$$ \rho_{\text{f}} \left( {{\user2 x}^{\text{e}} ,t} \right) = \sum\limits_{k} {f_{k} \left( {\user2{x}^{\text{e}} ,t} \right)\quad {\text{and}}\quad \rho_{\text{f}} \left( {{\user2 x}^{\text{e}} ,t} \right){\user2 u}\left( {\user2{x}^{\text{e}} ,t} \right) = \sum\limits_{k} {f_{k} \left( {{\user2 x}^{\text{e}} ,t} \right) \cdot {\user2 e}_{k} } .} $$

(9)

The most common lattice model for 3D simulations is the D3Q19 model, which uses nineteen discrete velocity directions. The equilibrium distribution function is defined as

$$ f_{k}^{\text{eq}} = w_{k} \rho \left[ {1 + 3{\user2 e}_{k} \cdot {\user2 u} + \frac{9}{2}\left( {\user2{e}_{k} \cdot \user2{u}} \right)^{2} - \frac{3}{2}\left| {\user2 u} \right|^{2} } \right], $$

(10)

where \( w_{0} = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3},\quad w_{1 - 6} = {1 \mathord{\left/ {\vphantom {1 {18}}} \right. \kern-\nulldelimiterspace} {18}} \) (non-diagonal directions), and \( w_{7 - 18} = {1 \mathord{\left/ {\vphantom {1 {36}}} \right. \kern-\nulldelimiterspace} {36}} \) (diagonal directions) in the 3D D3Q19 model. For the present model, the speed of sound is \( c_{\text{s}} = \sqrt {{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \) and the kinematic viscosity is \( \nu = {{\left( {2\lambda - 1} \right)} \mathord{\left/ {\vphantom {{\left( {2\lambda - 1} \right)} 6}} \right. \kern-\nulldelimiterspace} 6}. \)

The fluid–solid interaction force g from Eq. (7) becomes an additional term to the collision function and is included in the lattice-Boltzmann equation as

$$ f_{k} \left( {\user2{x}^{\text{e}} + \user2{e}_{k} ,t + \Updelta t^{\text{LBM}} } \right) = f_{k} \left( {{\user2 x}^{\text{e}} ,t} \right) + {\frac{1}{\lambda }}\left[ {f_{k}^{\text{eq}} \left( {{\user2 x}^{\text{e}} ,t} \right) - f_{k} \left( {\user2{x}^{\text{e}} ,t} \right)} \right] + \frac{3}{2}w_{k}\;{\user2 g} \cdot \user2{e}_{k}.$$

(11)

The following are the validation cases of suspension flow for the LBM-EBF method. Additional validation cases can be found in the methodology paper of Wu and Aidun.26

The motion of a solid ellipsoid in a simple shear flow is analyzed in this section. The boundary of this particle is given by

$$ {\frac{{x^{2} }}{{a^{2} }}} + {\frac{{y^{2} }}{{b^{2} }}} + {\frac{{z^{2} }}{{c^{2} }}} = 1 $$

(12)

When one of the principal axes of the ellipsoid is kept parallel to the vorticity vector, as shown in Fig. 12, the rotation angle, ϕ, and the angular rate of rotation, ϕ, are given by Jeffery15

$$ \phi = \tan^{ - 1} \left( {\frac{b}{a}\tan {\frac{abGt}{{a^{2} + b^{2} }}}} \right) $$

(13)

$$ \dot{\phi } = {\frac{G}{{a^{2} + b^{2} }}}(b^{2} \cos^{2} \phi + a^{2} \sin^{2} \phi ) $$

(14)

where G is the shear rate and t is time. In our simulation, the computational domain is 120 × 120 × 60 lattice nodes. The Reynolds number is Re = Gd ²/ν, where d = 2a. For different aspect ratios b/a, the computational results agree very well with Jeffery’s analytical solution, as shown in Fig. 13. This demonstrates that the no-slip boundary condition on the ellipsoid surface is well satisfied.

A solid ellipsoid immersed in simple shear flow, known as Jeffery’s orbit

G = 1/6000, a = 12, ν = 1.5, Re = 0.064. Case(1): b = c = 9, the solid line is Jeffery’s solution15 and the crosses (×) are the simulation results. Case(2): b = c = 3, the dash line is Jeffery’s solution and the open squares are the simulation results. The LBM-EBF simulation results agree very well with Jeffery’s solution

It is well known from past experiments that the RBC shape changes into a parachute shape in capillary pressure-driven flow, as shown in Fig. 14, retain their shape through the capillary tube, and recover to its original shape in the post-capillary region. This unique deformation of the RBC is necessary in nature for high fluidity in microvessels and for high efficiency of oxygen diffusion to tissue by increasing the surface area and interaction with the endothelial cells.

Axisymmetrically deformed RBC in a ‘parachute configuration’ in capillary pressure-driven flow

Several investigators have used this phenomenon to measure the RBC’s deformability. In the recent experimental setup of Tsukada et al. 23 they use a set of transparent crystal microchannels and a high-speed video camera to capture high-resolution pictures and achieve quantitative data. Dilute suspensions of RBCs passed through a glass capillary tube with a diameter of 9.3 μm were imaged and analyzed. The velocity and the deformation index DI _P of a RBC depends on the pressure gradient in the channel. In this experiment,23 DI _P is given by

$$ {\text{DI}}_{P} = \frac{c}{d} $$

(15)

where d is the diameter of the deformed RBC in the parachute configuration and c is the length of the RBC along the axial direction as shown in Fig. 14. The simulation results are compared with the experimental results.23 The Capillary number Ca _P in Fig. 15 is defined as

$$ Ca_{P} = {\frac{{\mu U_{x} }}{S}} $$

(16)

where μ is the viscosity of the suspending fluid, U _x the RBC velocity, and S the membrane shear modulus.

Deformation index DI _P vs. capillary number Ca _P . The solid squares are the experimental data from Tsukada et al. 23 and the open squares are the LBM-EBF simulation results. The numerical and experimental results agree very well up to Ca _P  ≈ 0.35

The RBC deformation index DI _P is shown in Fig. 15 as a function of Capillary number Ca _P . The simulations agree well with experiments up to Ca _P  ≈ 0.35 where we see a deviation between the results.

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