A simple model of the liver microcirculation

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A simple model of the liver microcirculation

  1. 1. A SIMPLE MODEL OF THE LIVER MICROCIRCULATION1. INTRODUCTION:The liver is the largest internal organ, and the body’s second largest, after the skin. Ithas four lobes and is surrounded by a capsule of fibrous connective tissue calledGlisson’s capsule. Glisson’s capsule in turn is covered by the visceral peritoneum (tunicaserosa), except where the liver adheres directly to the abdominal wall or other organs.The parenchyma of the liver is divided into lobules, which are incompletely partitionedby septa from Glisson’s capsule. Figure-1: Liver Figure-2: Classic lobule of liver Page 5 of 43
  2. 2. Figure-3: Classic lobule of liverThe blood vessels supplying the liver (portal vein and hepatic artery) enter at the hilum(or porta hepatis), from which the common bile duct (carrying bile secreted by theliver) and lymphatic vessels also leave. Within the liver sinusoids, the oxygen-poor butnutrient rich blood from the portal vein mixes with the oxygenated blood from thehepatic artery. From the sinusoids, the blood enters a system of veins which convergeto form the hepatic veins. The hepatic veins follow a course independent of the portalvessels and enter the inferior vena cava.The classic liver lobule is considered to be the functional unit of the organ. Each is aroughly hexagonally shaped, three dimensional unit, demarcated by connective tissueand constructed of the parenchymal cells of the liver, hepatocytes, in large numbers.On the picture to the right we can see the diagrammatic representation of a classicliver lobule, showing the organization of branches of the portal vein and hepatic artery,along with bile duct branches at the periphery of the lobule (in portal triads), and thecentral vein in the center of the lobule.As mentioned above, the liver receives a dual blood supply. The hepatic portal veinprovides about 60% of the incoming blood, the rest comes from the hepatic artery. The Page 6 of 43
  3. 3. blood from the portal vein has already supplied the small intestine, pancreas andspleen, and is largely deoxygenated. It contains nutrients and noxious materialsabsorbed in the intestine, blood cells and their breakdown products from the spleen,and endocrine secretions from the pancreas. The blood from the hepatic artery suppliesthe liver with oxygen. Because the blood from the two sources intermingles as itperfuses the liver cells, these cells must carry out their many activities under lowoxygen conditions that most cells could not tolerate.In the process of transplanting a new liver in a patient, it is possible that the liver issmaller than the correct size and it takes some time for the new liver to grow andmatch the correct size in the new body. During the growth process, hypertensions inhepatic portal vein can occur due to the micro-vascular changes and the high rate ofperfusion. Allowing some extra blood to bypass, a shunt may be inserted between thethe portal veins and hepatic veins. Knowledge about the resistance of the flow throughthe liver, the relation between the pressure drop across the liver and the blood flow, ishighly important for a proper shunt design.In this simple model, we present the pressure distribution and velocity profle across thehepatic lobules, which are the basic morph functional units of the liver. The lobule hasapproximately a hexagonal shape and the diameter of each lobule is about 1 mm (seeFigure 2 and 3). We small focus on one lobule and calculate the pressure distributionand velocity profle across one lobule.2. MATHEMATICAL MODEL OF THE fzOW IN THE LIVER LOBULE:We consider the liver as a two dimensional porous media and therefore we apply theassumptions of porous media in our modelling.2.1 Flow in porous medium:We model the liver as a porous medium and consider it as a solid that has manyinterconnected pores. Due to the complexity of the geometry, solution of the Navier-Stocks equation in each individual pore, even in the linear form is complicated. In thiscase one good approach is to use the average velocity across the liver. Using theaverage velocity is valid as long as: S TA  S IPWmere,S TA  Special scales of my whole area andS IP  Special scales of individual pores Page 7 of 43
  4. 4. Also we consider the medium to be homogenous (characteristics of the medium is thesame everywhere) and isotropic (no direction preference). In addition, we consider theporosity ɸ of the porous media as: V  P VTWmere,V P =Volume of pores,VT = Total volume of tme areaWe consider the blood to be a Newtonian fluid and therefore the flow in the porousmedia is governed by Darcy equation: k q   p ( 1) Where  q: volume flux per unit area of the medium. q is called the apparent velocity and is not the actual velocity. The actual average velocity within the pores is- u  q q  L T  k: permeability of the medium. Measures how strong are the solid against the flow. The larger the permeability constant, the lower is the resistance of the medium against the flow. k depends on porosity (ɸ) and the geometry of the pores (tortousity and the degree of interconnection). It is important to mention that k does not depend on the theology of the fluid. k   L2  μ: dynamical viscosity  p: fluid pressureFor an incompressible fluid owing through an incompressible solid, the coninuitityequation is .q  0Taking the divergence of the Darcy equation we get 2.p  0Which means that the velocity field of an incompressible fluid is harmonic.2.2 Modeling assumptions:We use a simple model as in Figure 4 with the following assumptions:  The lobules are treated as identical regular hexagonal prisms arranged in a lattice.  The diameter of the portal tracts and centrilobular veins are much smaller than the axial length of the lobules. Page 8 of 43
  5. 5.  The axial length of the lobules is long compared to the length of an edge of the hexagon.  End effects are neglected and the flow is treated as two-dimensional. Figure-4: Simple model for liver lobule2.3 Problem Formulation:Our goal is to find the pressure distribution and velocity profile at every point in theliver lobule. We assume that:  The lobule is a two dimensional porous media  Portal tracts are modeled as point sources  Centrilobular veins are modeled as point sinkUsing the polar coordinates, the apparent velocity q  q r , q  of a two dimensionalsource/sink point centered in the origin satisfies the followings:  If we denote the total flux through any circle centered in origin as Q and the distance from the source/sink point as r, then 0  rq d  2rq 2 r r Q Q q 2r  Due to symmetry q  0mere for sources we have Q > 0 and for sinks Q < 0. Using Darcy law [equation (1)],we can obtain the pressure distribution as Q p log r  c 2k Page 9 of 43
  6. 6. Using the above scheme, the mathematical problem can be formulated asp.n  0 along the hexagon sidesWhere Q  and Q  denote the intensity of the sources and sinks, respectively and weassume Q   0 and Q   0 . Moreover, in the above expressions  is the Dirac functionand n denotes the unit vector normal to the hexagon side.2.4 Pressure distribution:To enforce p.n  0 , a good technique is to add infinite sources and sinks to our model.In a lattice of hexagons there are twice as many sources (angles of the hexagons) thansinks (centers of the hexagons). In order for the mass to be conserved the intensity ofa sink has to be twice that of a source.The solution for the pressure in a single point (x,y) within the hexagon can then beexpressed as:  2N Q   Q    2N  p( x, y )  lim    N    log  x  x   2   y  y i  2    log  x  x   2   y  y  2  2k   j 1 2k   i j j  i 1 Where N  denotes the number of sinks (and N   2 N  ), xi , y i  denotes the positionof the i th source and xi , y i  the position of the j th sink.2.4.1 Dimenii nzess variabzes: x x*  L q q*  U p p*  UL k Page 10 of 43
  7. 7. Where L is the length of the side of the lobule, U is the characteristic velocity of theblood, and the pressure scaling is derived from the Darcy equation. Also, we assumethat the flux per unit length Q  coming out of portal tracts can be written as: Q   UdWhere d is the diameter of a portal tract.2.5 Pressure distribution in the dimensionless form reads:  2N d    log x   y   log x   y  2N dp ( x , y )  lim    * 2 * 2    2 2 * * *  * x * y  *  x* *  y * N      j 1 2 L   i i j j  i 1 2 L q *  * p *Now, using the following dimensional values extracted from the lecture notes, we cancompute the velocity profile and pressure distribution. L  500m d  50m U  4  10 3 ms 12.6 Simulation of the pressure distribution and velocity profie:2.6.1 Description of the algorithm: 1. We take a rectangular domain with area 2 x n 2 y n  where x n and y n denote the number of points in the positive x and y axis, respectively. 2. We generate the sources and sinks using 4 vectors as follows: v1  1,0 1 3  v2   ,  2 2  3 3  w1   ,  2 2   3 3 w2   ,   2 2 The sinks and sources are the linear combination of these vectors. Page 11 of 43
  8. 8. 3. Since the above source/sink generation wont end up in a symmetric domain, we choose a circular symmetric domain to calculate the pressure distribution formulated in equation (2), for example a good symmetric domain could be: 2 xn  y n  2 2 x n if x n  y n 3 2 x n  y n  y n if x n  y n 2 2 3 4. Choosing the dimensional values as L  500m , d  50m and U  4  10 3 ms 1 , we will calculate the pressure distribution and velocity profile using the dimensionless equations (2) and (3) in one hexagonal lobule. 5. Finally, we sketch the velocity profile and pressure distribution in one hexagonal lobule. Figure-5: Point generator vectorsFEW PAGES ARE MISSING DUE TO @COPY RIGHT OF THIS ARTICLE Page 12 of 43
  9. 9. 4. RESULTS:For the simulation part, we consider x n  15, y n  15 , and the grid size to be h  0.05 . Figure-5: Tme Source/sink distribution in liver lobules Figure-6: Pressure distribution in one mexagonal lobule Page 16 of 43
  10. 10. Figure-7: Pressure distribution in one mexagonal lobuleFigure 6 and 7 show the pressure distribution within a hexagonal liver lobule. It can beimmediately noticed that the pressure in the portal tracts (blood entering points locatedat each of the vertices of the hexagonal lobule) and centrilobular veins (blood exitingpoints located in the center of the cross-section) are larger than the other areas. As aconclusion, the deformation of tissues which is the result of high pressure in the livermay occur in these entering/exiting areas. Figure-8: Velocity distribution in the liver lobule Page 17 of 43
  11. 11. Figure 8 illustrates the velocity distribution in the liver lobule. It can be observed thatp.n  0 is satisfied in all the sections of the hexagon, where n is the outward normalto each side of hexagon.5. ConclusionsAlthough our model is simple, it gives useful information about the pressure distributionand velocity pofile in a cross-section of the liver lobule. Note that considering the bloodarcheology, one can give a slightly more accurate distribution, especially consideringthe shear-thinning behavior of blood. This modification is useful in the implementationof a small liver and the shunt design. Page 18 of 43

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