Cell-free marginal layer model: Difference between revisions

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{{Short description|Mathematical model used in hemodynamics}}

In small [[capillary]] [[hemodynamics]], the cell-free layer is a near-wall layer of [[Blood plasma|plasma]] absent of [[red blood cell]]s since they are subject to migration to the [[capillary]] [[center]] in [[Poiseuille flow]].<ref>{{cite bookjournal|titleauthor=W. Pan, B. Caswell and G. E. Karniadakis (|year=2010), "|title=A low-dimensional model for the red blood cell",|journal= Soft Matter, DOI:|doi=10.1039/C0SM00183J|pmc=3838865|pmid=24282440|volume=6|issue=18 |page=4366|bibcode=2010SMat....6.4366P }}</ref> .'''Cell-free marginal layer model''' is a [[mathematical model]] which tries to explain [[Fåhræus–Lindqvist effect]] mathematically.

==Mathematical modeling==

===Governing equations===

Consider [[steady flow]] of [[blood]] through a [[capillary]] of [[radius]] <math>R</math>. The [[capillary]] cross section can be divided into a [[core]] region and cell-free [[plasma]] region near the wall. The governing equations for both regions can be given by the following equations<ref>{{cite book|last=Krishnan B. Chandran|first=Alit P. Yoganathan , Ajit P. Yoganathan , Stanley E. Rittgers|title=Biofluid mechanics : the human circulation|year=2007|publisher=CRC/Taylor & Francis|___location=Boca Raton|isbn=9780849373282|url=http://www.amazon.com/Biofluid-Mechanics-Circulation-Krishnan-Chandran/dp/084937328X}}</ref> :

===Governing equations===

Consider [[steady flow]] of [[blood]] through a [[capillary]] of [[radius]] <math>R</math>. The [[capillary]] cross section can be divided into a [[wikt:core|core]] region and cell-free [[Blood plasma|plasma]] region near the wall. The governing equations for both regions can be given by the following equations:<ref>{{cite book|last=Krishnan B. Chandran|first=Alit P. Yoganathan , Ajit P. Yoganathan , Stanley E. Rittgers|title=Biofluid mechanics : the human circulation|year=2007|publisher=CRC/Taylor & Francis|___location=Boca Raton|isbn=9780849373282|url=http://www.amazon.com/Biofluid978-Mechanics0-Circulation8493-Krishnan7328-Chandran/dp/084937328X2}}</ref> :

:<math> \frac{ -\Delta P}{ L }=\frac{1}{r}\frac{d}{dr}(\mu_c r \frac{du_c}{dr});</math> <math> 0 \le r\ \le R-\delta\,</math>

:<math> \frac{ -\Delta P}{ L }=\frac{1}{r}\frac{d}{dr}(\mu_p r \frac{du_p}{dr});</math> <math> R-\delta\le r\ \le R\ \,</math>

where:

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:<math>L</math> is the length of capillary

:<math> u_c </math> is [[velocity]] in core region

:<math> u_p </math> is [[velocity]] of plasma in cell-free region

:<math> \mu_{c} </math> is [[viscosity]] in core region

:<math> \mu_{p} </math> is [[viscosity]] of plasma in cell-free region

:<math>\delta</math> is the cell-free [[Blood plasma|plasma]] layer thickness

===Boundary conditions===

The [[boundary condition]]s to obtain the solution for the two [[differential equation]]s presented above are that the velocity gradient is zero in the tube center, no slip occurs at the tube wall and the [[velocity]] and the [[shear stress]] are continuous at the [[interface (chemistry)|interface]] between the two zones. These [[boundary condition]]s can be expressed mathematically as:

*<math>\left. \frac{du_c}{dr}\right|_{r= 0}=0</math>

*<math>\left.u_p\right|_{r= R}=0</math>

*<math>\left.u_p\right|_{r= R-\delta}=\left.u_c\right|_{r= R-\delta}</math>

*<math>\left.\tau_p\right|_{r= R-\delta}=\left.\tau_c\right|_{r= R-\delta}</math>

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Integrating governing equations with respect to ''r'' and applying the above discussed boundary conditions will result in:

:<math> u_pu_c=\frac{ \Delta P R^2}{ 4\mu_cmu_p L }[1-(\frac{ R-\delta}{R})^2-\frac{\mu_p}{\mu_c}(\frac{r}{R})^2+\frac{\mu_p}{\mu_c}(\frac{ R-\delta}{R})^2]</math>

:<math> u_p=\frac{ \Delta P R^2}{ 4\mu_p L }[1-(\frac{r}{R})^2]</math>

===Volumetric flow rate for cell-free and core regions===

<math>Q_p = \int\limits_{R-\delta}^{R} 2\pi*u_prdr = \frac{\pi \Delta P}{8\mu_p L}(R^2-(R-\delta)^2)^2</math>

<math>Q_c = \int\limits_{0}^{R-\delta} 2\pi*u_crdr=\frac{\pi\Delta P*(R-\delta)^2}{8L}[\frac{(R-\delta)^2}{\mu_c}+\frac{2(R^2-(R-\delta)^2)}{8\mu_p}]</math>

Total [[volumetric flow rate]] is the algebraic sum of the flow rates in core and plasma region. The expression for the total [[volumetric flow rate]] can be written as:

Total [[volumetric flow rate]] is the algebraic sum of the flow rates in core and plasma region. The expression for the total [[volumetric flow rate]] can be written as:

:<math> Q=\frac{ \pi \Delta P R^4}{ 8\mu_p L }[1-(1-\frac{\delta}{R})^4(1-\frac{\mu_p}{\mu_c})]</math>

:<math> Q=Q_c+Q_p=\frac{ \pi \Delta P R^4}{ 8\mu_p L }[1-(1-\frac{\delta}{R})^4(1-\frac{\mu_p}{\mu_c})]</math>

Comparison with the [[viscosity]] which applies in the [[Poiseuille flow]] yields effective [[viscosity]], <math> \mu_{e} </math> as:

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:<math> \mu_{e}=\frac{\mu_p}{[1-(1-\frac{\delta}{R})^4(1-\frac{\mu_p}{\mu_c})]} </math>

It can be realized when the radius of the [[blood vessel]] is verymuch larger than the thickness of the cell-free [[Blood plasma|plasma]] layer, the effective [[viscosity]] is equal to bulk [[blood viscosity]] <math> \mu_c </math> at high shear rates ([[Newtonian fluid]]).

'''Relation between hematocrit and apparent/effective viscosity'''

[[Conservation of mass|Conservation of Mass]] Requires:

<math>QH_D=Q_cH_c</math>

<math>\frac{H_T}{H_C}=\sigma^2</math>

<math>H_T</math> = Average Red Blood Cell (RBC) volume fraction in small capillary

<math>H_D</math>= Average RBC volume fraction in the core layer

<math>\frac{H_T}{H_D}=\frac{Q}{Q_c}\sigma^2</math>, <math>\sigma = \frac{R-\delta}{R}</math>

<math>u_e = \frac{\pi\Delta PR^4}{8Q}</math>

<math>\frac{u_p}{u_e}=1+\sigma^4[\frac{u_a}{u_c}-1]</math>

Blood viscosity as a fraction of [[hematocrit]]:

<math>\frac{u_e}{u}=1-\alpha H</math>

==See also==

*[[Fåhræus–Lindqvist effect]]

*[[Blood viscosityHemorheology]]

*[[hemodynamicsHemodynamics]]

==References==

{{Reflist}}

* {{cite journal | last1 = Chebbi | first1 = R | year = 2015 | title = Dynamics of blood flow: modeling of the Fahraeus-Lindqvist effect | journal = Journal of Biological Physics | volume = 41| issue = 3 | pages = 313–26| doi = 10.1007/s10867-015-9376-1 | pmid = 25702195 | pmc = 4456490 }}

[[Category:Fluid dynamics]]