A set of measures of red blood cell (RBC) aggregates are developed and applied to examine the aggregate structure under plane shear and channel flows. Some of these measures are based on averages over the set of red blood cells which are in contact with each other at a given time. Other measures are developed by first fitting an ellipse to the planar projection of the aggregate, and then examining the area and aspect ratio of the fit ellipse as well as the orientations of constituent RBCs with respect to the fit ellipse axes. The aggregate structural measures are illustrated using a new mesoscale computational model for blood cell transport, collision and adhesion. The sensitivity of this model to change in adhesive surface energy density and shear rate on the aggregate structure is examined. It is found that the mesoscale model predictions exhibit reasonable agreement with experimental and theoretical data for blood flow in plane shear and channel flows. The new structural measures are used to examine the differences between predictions of two- and three-dimensional computations of the aggregate formation, showing that two-dimensional computations retain some of the important aspects of three-dimensional computations.
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The authors acknowledge financial support to J.K.W. Chesnutt from the Presidential Fellowship and IIHR–Hydroscience and Engineering at the University of Iowa and from the School of Engineering at the University of Vermont. Funding for J.S. Marshall was provided by the U.S. Department of Transportation (grant number DTOS59-06-G-00048) and by Vermont EPSCoR (grant number EPS 0701410).
Author information Authors and AffiliationsDepartment of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA, 52242, USA
J. K. W. Chesnutt
School of Engineering, The University of Vermont, Burlington, VT, 05405, USA
J. K. W. Chesnutt & J. S. Marshall
Correspondence to J. S. Marshall.
Additional informationAssociate Editor James B. Bassingthwaighte oversaw the review of this article.
Appendix: Fitting an Equivalent Aggregate Ellipse to Extrema Points Appendix: Fitting an Equivalent Aggregate Ellipse to Extrema PointsIn the following, we describe the method used for fitting the equivalent aggregate ellipse to a set of extrema points defined in the main text. This method follows some of the ideas proposed by Halir and Flusser22 for fitting an ellipse to a set of points, with the difference that the center of the equivalent aggregate ellipse is constrained to coincide with the centroid of the extrema points.
An ellipse can be expressed as a quadratic form in vector notation as
$$ P\left( \xi \right) = \xi \cdot {\mathbf{q}} = 0 , $$
(A-1)
where \( \xi \equiv \left[ {\begin{array}{*{20}c} {x^{2} } & {xy} & {y^{2} } & 1 \\ \end{array} } \right] \), \( \mathbf{q} \equiv [\mathbf{q}_{1}^{T} |q_{{33}} ]^{T} \), and \( {\mathbf{q}}_{1} \equiv \left[ {\begin{array}{*{20}c} {q_{11} } & {q_{12} } & {q_{22} } \\ \end{array} } \right]^{T} \). To fit an ellipse to a set of N points, the coefficients q of the ellipse are found such that the sum of squared distances of the points to the ellipse is minimized subject to the constraint
$$ 4q_{11} q_{22} - \left( {2q_{12} } \right)^{2} = 1 . $$
(A-2)
This constraint is derived by scaling of the coefficients q, noting that multiplication of q by a non-zero constant yields an equivalent quadratic. The ellipse-fitting problem can be stated mathematically as finding
$$ \min \left\| {{\mathbf{Dq}}} \right\|^{2} $$
(A-3)
subject to the constraint
$$ {\mathbf{q}}^{T} {\mathbf{Cq}} = 1 , $$
(A-4)
where D is an N × 4 matrix given by
$$ {\mathbf{D}} = \left[ {{\mathbf{D}}_{1} |{\mathbf{D}}_{2} } \right], \quad {\mathbf{D}}_{1} = \left[ {\begin{array}{*{20}c} {x_{1}^{2} } & {x_{1} y_{1} } & {y_{1}^{2} } \\ \vdots & \vdots & \vdots \\ {x_{i}^{2} } & {x_{i} y_{i} } & {y_{i}^{2} } \\ \vdots & \vdots & \vdots \\ {x_{N}^{2} } & {x_{N} y_{N} } & {y_{N}^{2} } \\ \end{array} } \right], \quad {\mathbf{D}}_{2} = \left[ {\begin{array}{*{20}c} 1 \\ \vdots \\ 1 \\ \vdots \\ 1 \\ \end{array} } \right]_{N \times 1} , $$
(A-5)
and C is a 4 × 4 constraint matrix given by
$$ {\mathbf{C}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ {} & {{\mathbf{C}}_{1} } & {} \\ {} & {} & {} \\ \end{array} } & {\left| {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \right.} \\ \end{array} } \\ {\overline{{\begin{array}{*{20}c} 0 & 0 & 0 & {\left| 0 \right.} \\ \end{array} }} } \\ \end{array} } \right], \quad {\mathbf{C}}_{1} = \left[ {\begin{array}{*{20}c} 0 & 0 & 2 \\ 0 & { - 1} & 0 \\ 2 & 0 & 0 \\ \end{array} } \right]. $$
(A-6)
To solve the minimization problem, a quadratically constrained least-squares minimization is employed with the Lagrange multiplier λ to write
$$ {\mathbf{Sq}} = \lambda {\mathbf{Cq}}, $$
(A-7)
where S is a 4 × 4 matrix given by
$$ {\mathbf{S}} \equiv {\mathbf{D}}^{T} {\mathbf{D}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathbf{S}}_{1} } & | & {{\mathbf{S}}_{2} } \\ \end{array} } \\ {\overline{{\begin{array}{*{20}c} {{\mathbf{S}}_{2}^{T} } & | & {{{S}}_{3} } \\ \end{array} }} } \\ \end{array} } \right], $$
(A-8)
and S 1 = D T1 D 1, S 2 = D T1 D 2, and S 3 = D T2 D 2 = N. The system of equations (A-4) and (A-8) can be solved using generalized eigenvectors.
Noting that
$$ \left\| {{\mathbf{Dq}}} \right\|^{2} = {\mathbf{q}}^{T} {\mathbf{D}}^{T} {\mathbf{Dq}} = {\mathbf{q}}^{T} {\mathbf{Sq}} = \lambda {\mathbf{q}}^{T} {\mathbf{Cq}} = \lambda , $$
(A-9)
the eigenvector q k corresponding to the minimal non-negative eigenvalue λ k is the solution to the minimization problem. However, in the case that the projections of the particle extrema points come near to exactly falling on an ellipse, the constraint matrix C will become singular and the matrix S will become ill-conditioned, being singular if all N scatter points lie on an ellipse. This numerical instability can be addressed by reformulating the problem using partitioned vectors and partitioned matrices. Specifically, (A-7) can be rewritten as the system
$$ {\mathbf{S}}_{1} {\mathbf{q}}_{1} + {\mathbf{S}}_{2} q_{33} = \lambda {\mathbf{C}}_{1} {\mathbf{q}}_{1} , \quad {\mathbf{S}}_{2}^{T} {\mathbf{q}}_{1} + S_{3} q_{33} = 0. $$
(A-10)
Solving (A-10)2 for q 33 and substituting into (A-10)1 gives
$$ \left( {{\mathbf{S}}_{1} - {\mathbf{S}}_{2} S_{3}^{ - 1} {\mathbf{S}}_{2}^{T} } \right){\mathbf{q}}_{1} = \lambda {\mathbf{C}}_{1} {\mathbf{q}}_{1} . $$
(A-11)
Since C 1 is non-singular by definition, we can multiply (A-11) by the inverse of C 1 to obtain
$$ {\bar{\mathbf{S}}\mathbf{q}}_{1} = \lambda {\mathbf{q}}_{1} , $$
(A-12)
where \( {\bar{\mathbf{S}}} \equiv {\mathbf{C}}_{1}^{ - 1} \left( {{\mathbf{S}}_{1} - {\mathbf{S}}_{2} S_{3}^{ - 1} {\mathbf{S}}_{2}^{T} } \right) \) is a 3 × 3 matrix. Similarly, the constraint (A-4) can be written in terms of these partitioned matrices as
$$ {\mathbf{q}}_{1}^{T} {\mathbf{C}}_{1} {\mathbf{q}}_{1} = 1. $$
(A-13)
We solve the reduced system (A-12)–(A-13) for the minimum non-negative eigenvalue λ k and the corresponding eigenvector q 1, which can then be used in (A-1) to obtain the equivalent aggregate ellipse for the given set of extrema points.
About this article Cite this articleChesnutt, J.K.W., Marshall, J.S. Structural Analysis of Red Blood Cell Aggregates Under Shear Flow. Ann Biomed Eng 38, 714–728 (2010). https://doi.org/10.1007/s10439-009-9871-2
Received: 01 April 2009
Accepted: 07 December 2009
Published: 19 December 2009
Issue Date: March 2010
DOI: https://doi.org/10.1007/s10439-009-9871-2
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