A Mathematical Model for the Blood Plasma Flow Around Two Aggregated Low-Density Lipoproteins

Abstract

The rheological behaviour of low-density lipoprotein (LDL) particles within the blood plasma and their role in atherogenesis, as well as their ability to aggregate under certain circumstances, is the subject of many clinical tests and theoretical studies aiming at the prevention of atherosclerosis. In the present study we develop a mathematical model that describes the flow of the blood plasma around two aggregated LDLs. We consider the flow as a creeping steady incompressible axisymmetric one, while the two aggregated LDLs are described by an inverted oblate spheroid. The mathematical methods of Kelvin inversion and the semi-separation of variables are employed and analytical expressions for the stream function are derived. These expressions are expected to be useful for further model developing and screening as well as the theoretical justification and validation of laboratory results.

Appendix

From [21] we have that

$$ \frac{1}{r}=\frac{1}{\overline{c}}{\displaystyle \sum_{n=0}^{\infty}\frac{{\left(-1\right)}^n\left(4n+1\right)(2n)!}{2^{2n}{\left(n!\right)}^2}\;i}\;{Q}_{2n}\left(i\lambda \right){P}_{2n}\left(\zeta \right). $$

(11.25)

Employing the relations connecting the Legendre functions with the Gegenbauer functions [22], using recurrence relations and after performing calculations, we arrive at

$$ \frac{\left(1-{\zeta}^2\right)\left({\lambda}^2+1\right)}{\sqrt{\lambda^2-{\zeta}^2+1}}=\left[-2{G}_1\left(i\lambda \right)-\frac{18}{5}{H}_2\left(i\lambda \right)-\frac{12}{5}{H}_4\left(i\lambda \right)\right]\kern0.24em i{G}_2\left(\zeta \right)+{\displaystyle \sum_{n=2}^{\infty }{w}_{n-1}{e}_{n-1}{d}_{n-1}{H}_{2n-2}\left(i\lambda \right)\;i\;{G}_{2n}\left(\zeta \right)}+{\displaystyle \sum_{n=2}^{\infty}\left(-{w}_{n-1}{e}_{n-1}^2-{w}_n{d}_n^2\right){H}_{2n}\left(i\lambda \right)\;i\;{G}_{2n}\left(\zeta \right)}+{\displaystyle \sum_{n=2}^{\infty }{w}_n{d}_n{e}_n{H}_{2n+2}\left(i\lambda \right)\;i\;{G}_{2n}\left(\zeta \right)}, $$

(11.26)

where

$$ {d}_n=\frac{2n\left(2n-1\right)}{4n+1},\kern1em {e}_n=\frac{2\left(2n+1\right)\left(n+1\right)}{4n+1},\kern1em {w}_n=\frac{{\left(-1\right)}^n\left(4n+1\right)(2n)!}{2^{2n}{\left(n!\right)}^2}. $$

(11.27)

We know from [17] that the solution of (11.18) is

$$ \psi \left(\tau, \zeta \right)={g}_0\left( i\lambda \right){G}_0\left(\zeta \right)+{g}_1\left( i\lambda \right){G}_1\left(\zeta \right)+{\displaystyle \sum_{n=2}^{\infty}\left[{g}_n\left( i\lambda \right){G}_n\left(\zeta \right)+{h}_n\left( i\lambda \right){H}_n\left(\zeta \right)\right]}, $$

(11.28)

with

$$ {g}_2\left(\tau \right)={A}_2{G}_2\left(\tau \right)+{B}_2{H}_2\left(\tau \right)+{C}_2{G}_0\left(\tau \right)+{D}_2{G}_1\left(\tau \right)+{E}_2{G}_4\left(\tau \right)+{F}_2{H}_4\left(\tau \right), $$

(11.29)

and

$$ {g}_{2n}\left(\tau \right)={A}_{2n}{G}_{2n}\left(\tau \right)+{B}_{2n}{H}_{2n}\left(\tau \right)+{C}_{2n}{G}_{2n-2}\left(\tau \right)+{D}_{2n}{H}_{2n-2}\left(\tau \right)+{E}_{2n}{G}_{2n+2}\left(\tau \right)+{F}_{2n}{H}_{2n+2}\left(\tau \right),\kern1em n\ge 2. $$

(11.30)

Using orthogonality arguments in (11.28) from (11.26) we can calculate coefficients B _2n , D _2n , F _2n , while we have to calculate A _2n , C _2n , E _2n and we have only two equations to apply. Therefore we need one more condition. We use the fact that the oblate spheroid becomes a sphere when the semifocal distance tends to zero, so the solution in the oblate coordinates becomes the solution in the spherical coordinates. This allows us to find the solution of which is given in (11.24) with

$$ {g}_2\left( i\lambda \right)={A}_2{G}_2\left( i\lambda \right)+i\frac{9b\;\overline{c}\;U}{5}{H}_2\left( i\lambda \right)+ ib\;\overline{c}\;U{G}_1\left( i\lambda \right)+{E}_2{G}_4\left( i\lambda \right)+i\frac{6b\;\overline{c}\;U}{5}{H}_4\left( i\lambda \right), $$

(11.31)

$$ {g}_{2n}\left(i\lambda \right)={A}_{2n}{G}_{2n}\left(i\lambda \right)+i\frac{b\;\overline{c}\;U}{2}\left({w}_{n-1}{e}_{n-1}^2+{w}_n{d}_n^2\right){H}_{2n}\left(i\lambda \right)-i\frac{b\;\overline{c}\;U}{2}{w}_{n-1}{e}_{n-1}{d}_{n-1}{H}_{2n-2}\left(i\lambda \right)+{E}_{2n}{G}_{2n+2}\left(i\lambda \right)-i\frac{b\;\overline{c}\;U}{2}{w}_n{d}_n{e}_n{H}_{2n+2}\left(i\lambda \right), $$

(11.32)

$$ {A}_2=-i\;b\;\overline{c}\;U\frac{\left|\begin{array}{cc}\hfill \frac{9}{5}{H}_2\left(i{\lambda}_0\right)+{G}_1\left(i{\lambda}_0\right)+\frac{6}{5}{H}_4\left(i{\lambda}_0\right)\hfill & \hfill {G}_4\left(i{\lambda}_0\right)\hfill \\ {}\hfill \frac{9}{5}{H}_2^{\prime}\left(i{\lambda}_0\right)+{G}_1^{\prime}\left(i{\lambda}_0\right)+\frac{6}{5}{H}_4^{\prime}\left(i{\lambda}_0\right)\hfill & \hfill {G}_4^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}{\left|\begin{array}{cc}\hfill {G}_2\left(i{\lambda}_0\right)\hfill & \hfill {G}_4\left(i{\lambda}_0\right)\hfill \\ {}\hfill {G}_2^{\prime}\left(i{\lambda}_0\right)\hfill & \hfill {G}_4^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}, $$

(11.33)

$$ {E}_2=-i\;b\;\overline{c}\;U\frac{\left|\begin{array}{cc}\hfill {G}_2\left(i{\lambda}_0\right)\hfill & \hfill \frac{9}{5}{H}_2\left(i{\lambda}_0\right)+{G}_1\left(i{\lambda}_0\right)+\frac{6}{5}{H}_4\left(i{\lambda}_0\right)\hfill \\ {}\hfill {G}_2^{\prime}\left(i{\lambda}_0\right)\hfill & \hfill \frac{9}{5}{H}_2^{\prime}\left(i{\lambda}_0\right)+{G}_1^{\prime}\left(i{\lambda}_0\right)+\frac{6}{5}{H}_4^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}{\left|\begin{array}{cc}\hfill {G}_2\left(i{\lambda}_0\right)\hfill & \hfill {G}_4\left(i{\lambda}_0\right)\hfill \\ {}\hfill {G}_2^{\prime}\left(i{\lambda}_0\right)\hfill & \hfill {G}_4^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}, $$

(11.34)

$$ {A}_{2n}=-i\frac{b\;\overline{c}\;U}{2}\frac{\left|\begin{array}{c}\hfill \left({w}_{n-1}{e}_{n-1}^2+{w}_n{d}_n^2\right){H}_{2n}\left(i{\lambda}_0\right)-{w}_{n-1}{e}_{n-1}{d}_{n-1}{H}_{2n-2}\left(i{\lambda}_0\right)-{w}_n{d}_n{e}_n{H}_{2n+2}\left(i{\lambda}_0\right)\kern1em {G}_{2n+2}\left(i{\lambda}_0\right)\hfill \\ {}\hfill \left({w}_{n-1}{e}_{n-1}^2+{w}_n{d}_n^2\right){H}_{2n}^{\prime}\left(i{\lambda}_0\right)-{w}_{n-1}{e}_{n-1}{d}_{n-1}{H}_{2n-2}^{\prime}\left(i{\lambda}_0\right)-{w}_n{d}_n{e}_n{H}_{2n+2}^{\prime}\left(i{\lambda}_0\right)\kern1em {G}_{2n+2}^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}{\left|\begin{array}{cc}\hfill {G}_{2n}\left(i{\lambda}_0\right)\hfill & \hfill {G}_{2n+2}\left(i{\lambda}_0\right)\hfill \\ {}\hfill {G}_{2n}^{\prime}\left(i{\lambda}_0\right)\hfill & \hfill {G}_{2n+2}^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}, $$

(11.35)

and

$$ {E}_{2n}=-i\frac{b\;\overline{c}\;U}{2}\frac{\left|\begin{array}{c}\hfill {G}_{2n}\left(i{\lambda}_0\right)\kern1em \left({w}_{n-1}{e}_{n-1}^2+{w}_n{d}_n^2\right){H}_{2n}\left(i{\lambda}_0\right)-{w}_{n-1}{e}_{n-1}{d}_{n-1}{H}_{2n-2}\left(i{\lambda}_0\right)-{w}_n{d}_n{e}_n{H}_{2n+2}\left(i{\lambda}_0\right)\hfill \\ {}\hfill {G}_{2n}^{\prime}\left(i{\lambda}_0\right)\kern1em \left({w}_{n-1}{e}_{n-1}^2+{w}_n{d}_n^2\right){H}_{2n}^{\prime}\left(i{\lambda}_0\right)-{w}_{n-1}{e}_{n-1}{d}_{n-1}{H}_{2n-2}^{\prime}\left(i{\lambda}_0\right)-{w}_n{d}_n{e}_n{H}_{2n+2}^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}{\left|\begin{array}{cc}\hfill {G}_{2n}\left(i{\lambda}_0\right)\hfill & \hfill {G}_{2n+2}\left(i{\lambda}_0\right)\hfill \\ {}\hfill {G}_{2n}^{\prime}\left(i{\lambda}_0\right)\hfill & \hfill {G}_{2n+2}^{\prime}\left(i{\lambda}_0\right)\hfill \end{array}\right|}. $$

(11.36)