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A ballon model analysis with Cu-blood medicated nanoparticles as drug agent through overlapped curved stenotic artery having compliant walls

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Abstract

In this speculative study, main focus is to examine the Cu-blood medicated application in a curved artery with overlapping stenosis. This analysis investigate the combined impact of variable and constant Cu-blood transportation with shape factor. The walls of the stenotic artery are considered to be compliant in nature. Flow of blood in a curved stenotic artery having balloon is analyzed mathematically by taking its behavior as viscous fluid. The mild stenosis approximation is used for the dimensionless terms of velocity, temperature and stress on wall of curved stenotic artery. The copper nanoparticles are used as drug agent. At the end, the comparison of curvature and non-curavture artery shows that the curved artery minimized the stress in the presence of copper as drug agent. Moreover, the use of platelets nanoparticles is more appropriate to reduce hemodynamics effects of curved catheterized artery in comparison to cylinders and bricked shape nanoparticles. Therefore, the use of Cu-blood as drug agent finds valuable application in bio-inspired field.

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan

    Muzammil Ayub, Iqra Shahzadi & S. Nadeem

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  1. Muzammil Ayub
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  2. Iqra Shahzadi
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  3. S. Nadeem
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Correspondence to Iqra Shahzadi.

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Appendix

Appendix

$$ t0 = q/64 $$
$$ \begin{aligned} t1 & = 1/(64({ \ln }h - { \ln }R)2)(q{ \ln }h2(3R4 - 4R2s0 - 4h2s0{ \ln }R) + h2{ \ln }R(16 - { \ln }R( - 16 + q(h2 - 4R2 + 4s0) \\ & \quad + 4qs0{ \ln }R)) + { \ln }h( - 16R2 + { \ln }R(h4q - 3qR4 - 4h2(4 + q(R2 - s0)) + 4qR2s0 + 8h2qs0{ \ln }R))) \\ \end{aligned}, $$
$$ \begin{aligned} t2 & = (16( - h2 + R2) + ( - h4q + 4h2(4 + q(R2 - s0)) + q( - 3R4 + 4R2s0) \\ & \quad + 4h2qs0({ \ln }h - { \ln }R))({ \ln }h - { \ln }R))/(64({ \ln }h - { \ln }R)2) \\ \end{aligned}, $$
$$ t3 = (4(1 + { \ln }R) + q({ \ln }h - { \ln }R)( - R2 + s0 + s0{ \ln }R))/(16({ \ln }h - { \ln }R)), $$
$$ t4 = - ((qs0)/16) + 1/(4( - { \ln }h + { \ln }R)), $$
$$ t5 = ((h - R)(36 + q(2(h2 + hR + R2) - 9s0)( - { \ln }h + { \ln }R)))/(36({ \ln }h - { \ln }R)2), $$
$$ t6 = q/18, $$
$$ t7 = - ((qs0)/4) + 1/( - { \ln }h + { \ln }R), $$
$$ \begin{aligned} t8 & = (2qR( - 2R2 + 9s0){ \ln }h + h{ \ln }R( - 36 + q( - 2h2 + 9s0){ \ln }R) + { \ln }h(36R + q(2(h3 + R3) \\ & \quad - 9(h + R)s0){ \ln }R))/(36({ \ln }h - { \ln }R)2) \\ \end{aligned}, $$
$$ s0 = ((h2 - R2)/({ \ln }h - { \ln }R)), $$
$$ s1 = ( - 216h2{ \ln }h + 216R2{ \ln }h + 216h2{ \ln }R - 216R2{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)), $$
$$ s2 = ( - 72h4R2{ \ln }h + 72h2R4{ \ln }h + 72h4R2{ \ln }R - 72h2R4{ \ln }R)/(1152(h2 - R2)({ \ln }h{ \ln }R)), $$
$$ s3 = (144h4 - 288h2R2 + 144R4)/(1152(h2 - R2)({ \ln }h - { \ln }R)), $$
$$ s4 = (72h4{ \ln }h + 144h2R2{ \ln }h - 216R4{ \ln }h - 216h4{ \ln }R + 144h2R2{ \ln }R + 72R4{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)),$$
$$ s5 = ( - 576h2t2{ \ln }h + 576R2t2{ \ln }h + 576h2t2{ \ln }R - 576R2t2{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)), $$
$$ \begin{aligned} s6 & = ( - 108h4 - 27h6q + 27h4qR2 + 108R4 + 27h2qR4 - 27qR6 + 144h4{ \ln }h + 24h6q{ \ln }h \\ & \quad - 24qR6{ \ln }h + 960h6t0{ \ln }h - 960R6t0{ \ln }h + 864h4t3{ \ln }h - 864R4t3{ \ln }h \\ & \quad + 72h4t4{ \ln }h - 72R4t4{ \ln }h - 512h3t5{ \ln }h + 512R3t5{ \ln }h + 48h6t6{ \ln }h - 48R6t6{ \ln }h \\ & \quad + 144h4t7{ \ln }h - 144R4t7{ \ln }h + 384h3t8{ \ln }h - 384R3t8{ \ln }h + 576h2t2{ \ln }h \\ & \quad + 864h4t4{ \ln }h2 + 384h3t5{ \ln }h2 - 144h4{ \ln }R - 24h6q{ \ln }R + 24qR6{ \ln }R \\ & \quad - 960h6t0{ \ln }R + 960R6t0{ \ln }R - 864h4t3{ \ln }R + 864R4t3{ \ln }R - 72h4t4{ \ln }R \\ & \quad + 72R4t4{ \ln }R + 512h3t5{ \ln }R - 512R3t5{ \ln }R - 48h6t6{ \ln }R + 48R6t6{ \ln }R \\ & \quad - 144h4t7{ \ln }R + 144R4t7{ \ln }R - 384h3t8{ \ln }R + 384R3t8{ \ln }R - 576h2t2{ \ln }h{ \ln }R \\ & \quad - 576R2t2{ \ln }h{ \ln }R - 864h4t4{ \ln }h{ \ln }R - 864R4t4{ \ln }h{ \ln }R - 384h3t5{ \ln }h{ \ln }R \\ & \quad - 384R3t5{ \ln }h{ \ln }R + 576R2t2{ \ln }R + 864R4t4{ \ln }R \\ & \quad + 384R3t5{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)) \\ \end{aligned}, $$
$$ s7 = ( - 384h2t5{ \ln }h + 384R2t5{ \ln }h + 384h2t5{ \ln }R - 384R2t5{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)), $$
$$ \begin{aligned} s8 & = (512h2t5{ \ln }h - 512R2t5{ \ln }h - 384h2t8{ \ln }h + 384R2t8{ \ln }h - 512h2t5{ \ln }R \\ & \quad + 512R2t5{ \ln }R + 384h2t8{ \ln }R - 384R2t8{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)) \\ \end{aligned}, $$
$$ \begin{aligned} s9 & = ( - 144h2 - 36h4q + 144R2 + 72h2qR2 - 36qR4 - 864h2t4{ \ln }h + 864R2t4{ \ln }h + 864h2t4{ \ln }R \\ & \quad - 864R2t4{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)); \\ \end{aligned}, $$
$$ \begin{aligned} s10 & = (108h2 + 27h4q - 108R2 - 54h2qR2 + 27qR4 - 36h2qR2{ \ln }h + 36qR4{ \ln }h \\ & \quad - 864h2t3{ \ln }h + 864R2t3{ \ln }h - 72h2t4{ \ln }h + 72R2t4{ \ln }h - 144h2t7{ \ln }h + 144R2t7{ \ln }h + 144h2{ \ln }R \\ & \quad + 36h4q{ \ln }R - 144R2{ \ln }R - 36h2qR2{ \ln }R + 864h2t3{ \ln }R - 864R2t3{ \ln }R \\ & \quad + 72h2t4{ \ln }R - 72R2t4{ \ln }R + 144h2t7{ \ln }R - 144R2t7{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)); \\ \end{aligned}, $$
$$ \begin{aligned} s11 & = (12h2q{ \ln }h - 12qR2{ \ln }h - 960h2t0{ \ln }h + 960R2t0{ \ln }h - 48h2t6{ \ln }h + 48R2t6{ \ln }h \\ & \quad - 12h2q{ \ln }R + 12qR2{ \ln }R + 960h2t0{ \ln }R - 960R2t0{ \ln }R + 48h2t6{ \ln }R \\ & \quad - 48R2t6{ \ln }R)/(1152(h2 - R2)({ \ln }h - { \ln }R)) \\ \end{aligned}, $$
$$ \begin{aligned} s12 & = (108h4R2 + 27h6qR2 - 108h2R4 - 54h4qR4 + 27h2qR6 - 144h4R2{ \ln }h \\ & \quad - 24h6qR2{ \ln }h + 24h2qR6{ \ln }h - 960h6R2t0{ \ln }h + 960h2{\text{R}}6t0{ \ln }h \\ & \quad - 864h4R2t3{ \ln }h + 864h2R4t3{ \ln }h - 72h4R2t4{ \ln }h + 72h2R4t4{ \ln }h \\ & \quad + 512h3R2t5{ \ln }h - 512h2R3t5{ \ln }h - 48h6R2t6{ \ln }h + 48h2R6t6{ \ln }h \\ & \quad - 144h4R2t7{ \ln }h + 144h2R4t7{ \ln }h - 384h3R2t8{ \ln }h + 384h2R3t8{ \ln }h \\ & \quad - 576h2R2t2{ \ln }h - 864h4R2t4{ \ln }h - 384h3R2t5{ \ln }h + 144h4R2{ \ln }R \\ & \quad + 24h6qR2{ \ln }R - 24h2qR6{ \ln }R + 960h6R2t0{ \ln }R - 960h2R6t0{ \ln }R \\ & \quad + 864h4R2t3{ \ln }R - 864h2R4t3{ \ln }R + 72h4R2t4{ \ln }R - 72h2R4t4{ \ln }R \\ & \quad - 512h3R2t5{ \ln }R + 512h2R3t5{ \ln }R + 48h6R2t6{ \ln }R - 48h2R6t6{ \ln }R \\ & \quad + 144h4R2t7{ \ln }R - 144h2R4t7{ \ln }R + 384h3R2t8{ \ln }R - 384h2R3t8{ \ln }R \\ & \quad + 1152h2R2t2{ \ln }h{ \ln }R + 864h4R2t4{ \ln }h{ \ln }R + 864h2R4t4{ \ln }h{ \ln }R + 384h3R2t5{ \ln }h{ \ln }R \\ & \quad + 384h2R3t5{ \ln }h{ \ln }R - 576h2R2t2{ \ln } - 864h2R4t4{ \ln }R2 - 384h2R3t5{ \ln }R2)/(1152(h2 - R2)({ \ln }h - { \ln }R)) \\ \end{aligned}. $$

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Ayub, M., Shahzadi, I. & Nadeem, S. A ballon model analysis with Cu-blood medicated nanoparticles as drug agent through overlapped curved stenotic artery having compliant walls. Microsyst Technol 25, 2949–2962 (2019). https://doi.org/10.1007/s00542-018-4263-x

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