Content-Type: multipart/mixed; boundary="-------------0405250015170" This is a multi-part message in MIME format. ---------------0405250015170 Content-Type: text/plain; name="04-162.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-162.keywords" Marginal Zone theory;Explicit relation; Plasma Layer ---------------0405250015170 Content-Type: text/plain; name="Explicit Relation.mht" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="Explicit Relation.mht" MIME-Version: 1.0 Content-Location: file:///C:/1E4BB24E/ExplicitRelation.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" On a Explicit Relation for Plasma Layer Thickness

On a Explicit Relation for Plasma = Layer Thickness

 =

 

 

 

Dr. Kal Ren= ganathan Sharma PE

Professor, = School of Chemical and Biotechnology

Shanmugha A= rts, Science, Technology & Research Academy

SASTRA Deem= ed University

Tirumalaisa= mudram,  Thanjavur  613402

India

Phone: 91 0= 4362 264101-107

Fax: 91 043= 62 264120

Email: jyoti_kalika@yahoo.com

 =

 

 

 

Abstract=

 

The marginal zone theory is used to account for the ob= served Fahreus Linquist effect when the viscoity of blood changes with the diamete= r of the capillary. An attributable cause is the axial accumulation of cells.  The discharge rate from Hagen Pois= eulle law at steady state was derived by Haynes (1960) for the core and plasma la= yer and a total discharge rate was expressed as a function of the pressure drop along the capillary, quartic dependence on the radius of the capillary and quartic dependence on the dimensionless marginal zone thickness.  The apparent of viscosity of the b= lood is expressed as a function of the ratio of the core layer viscosity and the plasma layer viscosity.  In or= der to back out a marginal zone thickness from a given set of information, the Cha= rm and Kurland expression (1974) for the viscosity and hematocrit variation and the temperature dependence parameter of the hematocrit alpha can be used to develop two transcendental equations and two unknowns. This is the recommen= ded procedure used currently and the equations are quartic in sigma, the dimensionless ratio of marginal zone thickness with the radius of the capil= lary. In this study an alternate procedure is developed for solving for the margi= nal zone thickness explicitly. On examination of the temperature variation parameter, alpha, for a given temperature, the transcendental equation is l= inear zed by a linear regression between alpha*H and H the hematocrit. The slope = and intercept values are used in the expression of sigma =3D sqrt(alpha (c)/alp= ha(t)) and the hematocrit expression H(t) =3D H(c)*sigma^2. The resulting quadrati= c in simga^2 can be solved for and the sigma values obtained explicitly.

 

 

 

 

 

 

Introduction<= o:p>

 

The Fahraeus- Lindquist eff= ect (1) can be captured by the marginal zone theory proposed by Haynes (2).  Tube flow of blood at high shear r= ates ( > 100/sec) exhibits the dependence of diameter on the viscosity effect is the Fahreus –Lindquist effect.  When the diameter of the tube decreases below 500 microns upto 4 = 211; 6 microns,  the viscosity of t= he blood also decreases.  The mar= ginal zone theory may be used to characterize the effect from 4-6 microns to 500 microns in tune diameter.  An = expression is obtained for the apparent viscosity in terms of the plasma layer thickne= ss, tube diameter and the hematocrit. 

 

The bold flow within a tube= or vessel is divided into two regions; a central core that contains the cells = with a viscosity,  mc, and the cell free marginal or plasma layer that consists only of plasma wit= h a thickness of = d, and a viscosity equal to that of the plasma denoted by mp.  In each region the flow is conside= red to be Newtonian and at steady state.  For the core region,  t= he governing equation may be written as;

 

trz   =3D    (DP)r/2L    =3D  -mc<= /span>vzc/<= /span>r        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            (1)

The boundary conditions can= be written as;

 

BC1:  r =3D R - d,   trz   (core)  =3D  = trz   (plasma)            =             &nb= sp;            =             &nb= sp;            = (2)

BC 2: r =3D 0,   vzc/<= /span>r  =3D 0            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =       (3)

 

The first boundary conditio= ns stems from the continuity of the transfer of momentum across the interface between the core and plasma layer and the second boundary condition derives from the fact that the axial velocity would be a maximum at the center of the tube f= rom symmetry arguments.  In a simi= lar vein,  for the plasma layer the governing equation and boundary conditions can be written as;

 

trz   =3D    (DP)r/2L    =3D  -mp<= /span>vzp/<= /span>r        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;           (4)       

BC 3:  r =3D R,  vzp =3D 0        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;    (5)

BC 4:  r =3D R - d,   vzp  =3D vzc        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      (6)

The boundary condition 3 co= mes from the zero velocity condition at the wall and the 4th boundary condition is that the velocity need be the same at the interface of the two phases.  Equation (1) and (4) = may be integrated and the core and plasma flow rates given by the following;

 

Qp   =3D  p(DP)/8mpL ( R2  - (R - d)2 )2     =             &nb= sp;            =             &nb= sp;            =             &nb= sp;       (7)

Qc  =3D  = p(DP) R2/8mpL ( (R - = d)2   -  ( 1 - /s2) (R - d)4/ R2  - mp/mc (R - = d)4/ 2R2)

           &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =         (8)

 

The total discharge rate of= the blood would equal the sum of the flow rates in the core and plasma regions = and is given by;

 

Q  =3D    p(DP) R4/8mpL ( 1 -  ( 1 - d/R)4 ( 1 - = mp/mc) )       =             &nb= sp;            =             &nb= sp;       (9)

 

Equation (8) can be used to= fit the apparent viscosity data and obtain values of the plasma layer thickness and= the core hematocrit as a function of the tube diameter.   A relation between the core hematocrit, Hc and the feed hematocrit, HF and the thickness of the plasma layer is needed.&n= bsp; An equation is needed to describe the dependence of the blood viscos= ity on the hematocrit since the value of the Hc  will be larger than HF b= ecause of the axial accumulation on the RBCs.   This relative increase in th= e core hematocrit will make the equation in the core have a higher viscosity than = the blood in the feed.  The follow= ing equation by Charm and Kurland (3) may be used to exp= ress the dependence of the viscosity of blood at high shear rates on the hematoc= rit and temperature;

 

m   =3D  mp ( 1/( 1- = aH)   )            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;        (10)

or aH  =3D  1 - = mp/m        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;        (11)

where

a  =3D  0.070 exp( 2.49 H + 1107/T exp(-1.69H))            =             &nb= sp;            =          (12)

where the temperature is  in Kelvin. Theses equations are va= lid to a hematocrit of 0.6 with a stated accuracy of 10 %.   If a subscript occurs on the viscosity then the corresponding values of H and a in the above equations will carry the same subscript.  If s  =3D  1 - = d/R then the solution for the plasma layer thickness are implicit and require solution of two equations and two unknowns:

 

mapp/ mF  =3D  ( 1 - = aFHF) /( 1 - = s4 = aCHC )       =             &nb= sp;            =             &nb= sp;            =             &nb= sp; (13)

HC/HF=   =3D  1  +    (   1 - s2)2/s2 ( 2 (   1 - s2)  +   s2mp/mc)        &= nbsp;           &nbs= p;            &= nbsp;          (14)

 

An explicit expression for = the plasma layer thickness is desirable and a method is explored in this study.=

 

Methodology

 

Equation (12) is examined u= sing a spreadsheet.  It can be observ= ed that  aH  is linear with H at a given temper= ature for the range of hematocrit for which the Charm and Kurland expression is valid.  Thus for= the core and tube ;

 

ac Hc   =3D  m Hc  + C            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp; (15)

aTHT   =3D  m HT + C        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;     (16)

 

The slope and intercept can= be obtained by a least squares regression between the aH  and H as given by equation (12).   From a balance of cells in t= he two phases in the tube it can be seen that;

 

HT  =3D  = s2 Hc     = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;        (17)

Dividing equation (14) by (= 15),

 

ac Hc  / aTHT   =3D  m Hc  + C    / m HT + C          &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      (18)

 

From equation (11) it can b= e seen that;

 

 aTHT  =3D   1 -  mp/ mapp        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p; (19)

From equation (9)   with minor rearrangement it c= an be seen that;

aCHC  s 4  =3D  1 -  mp/ = mapp        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;        (20)

Equating (19) and (20) and combining with equation (17),

s  =3D   ÖaT/aC        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =   (21)

Plugging equation (21) and = (17) into equation (18);

s 4  =3D  (m HT + C)/ (m HT/ s2+ C)        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;        (22)

with,   s2 =3D p

the quadratic can be solved= ;

C p2  +=   (m HT – 1)p  + C =3D 0

 

Thus an explicit expression= for the plasma layer thickness in terms of the tube hematocrit is developed.  The tube hematocrit can be read fr= om the linear regression developed between the aH  and H at a given temperature once = the apparent viscosity of the tube is known.

 

References

]<= /b>

  1.   1999,  Basic Transport Phenomena in Biomedical Engineering, Taylor & Francis,  Philadelphia, PA, USA.

 

  1.   Physical B= asis of the Dependence of Blood Viscosity on Tube Radius,  Am. J. Physiol.,  198, 1193.

 

  1.   G. S. Kurland,  1974,  Blood Flow and Microcirculation,  John Wiley, New York, NY, USA.

 

 

 

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