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Antonio Fasano Dipartimento di Matematica U. Dini, Univ . Firenze IASI – CNR Roma

Antonio Fasano Dipartimento di Matematica U. Dini, Univ . Firenze IASI – CNR Roma. A new model for blood flow in capillaries. FASANO, A. FARINA, J. MIZERSKI. A new model for blood flow in fenestrated capillaries with application to ultrafiltration in kidney glomeruli , submitted.

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Antonio Fasano Dipartimento di Matematica U. Dini, Univ . Firenze IASI – CNR Roma

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  1. Antonio Fasano Dipartimento di Matematica U. Dini, Univ. Firenze IASI – CNR Roma A newmodelforblood flow in capillaries

  2. FASANO, A. FARINA, J. MIZERSKI. • A new model for blood flow in fenestrated capillaries with application to ultrafiltration in kidney glomeruli, submitted

  3. Bloodcomposition • 7% of the human body weight, • average density of approximately 1060 kg/m3 • average adult blood volume 5 liters • plasma 54.3% • RBC’s (erythrocytes) 45% • WBC’s (leukocytes) 0.7% • platelets (thrombocytes) negligible volume fraction

  4. RBC’s properties Volume VRBC  90 μm3 Diameter dRBC 78 m Veryflexible

  5. Quoting from G. Mchedlishvili, Basic factor determining the hemorheological disorders in the microcirculation, Clinical Hemorheologyand Microcirculation 30 (2004) 179-180. the available fluid mechanical laws cannot be applied for a better understanding of the microcirculation in the living capillaries, and the hemorheologyin the microcirculation requires another approach than regularities of the fluid mechanics

  6. General trend: Adaptingrheologicalparametersto the vessel size

  7. Ourmodelignoresfluiddynamics and considers just Newton’s law

  8. TranslatingsequenceofRBCs and plasma elements Translatingelement Drivingforce Pressuregradient drag The drag is due to the highlysheared plasma film

  9. TranslatingsequenceofRBCs and plasma Translatingelement Drivingforce a/R  1/3 when the hematocritis 0.45 Pressuregradient drag The drag is due to the highlysheared plasma film

  10. If the capillaryisfenestratedthe plasma loss causes a progressive decreaseof the elementlength

  11. The renalglomerulusis a bundle ofcapillarieshosted in the Bowman’s capsule fenestratedcapillaries

  12. Plasma cross flow iscausedby TMP (transmembranepressure): TMP = hydraulicpressuredifference minus oncoticpressure (bloodcolloidosmoticpressure)

  13. R Vel= R2a + VRBC Element volume  Hematocrit

  14. The motionof a single element (Newton’s law) Pressuredrop density Frictioncoefficient Variableowingto plasma loss

  15. A guessof the frictioncoefficient Take = 0.45 = 1 mm/s = 40 mm Hg/mm = in the steady state equation g/s. = 8.8

  16. in a 1.5mm capillary there are about 200 of such elements  Itmakessenseto pass to a continuousmodel Divide by

  17. ConservationofRBCs Tobecoupledwith a lawforplasma outflow … Momentumbalance

  18. Balanceequationfor plasma externalpressure constant Starling’s law C = albuminconcentration RTC = “oncotic” pressure(Van’t Hofflaw)

  19. Dimensionlessvariables (L  1.5mm) convectiontime 1.5 sec

  20. Momentumbalance  inertiaisnegligible 

  21. Combining the RBC’s and plasma balanceequations or

  22. Introducing the “filtrationtime” weget the dimensionless system comparesoncotic and hydraulicpressures

  23. The case of glomeruli can beestimated, knowingthat the relative changeof during the convectiontimeis  1/3  We are interested in the (quasi) steady state

  24. Eliminating …

  25. A secondorder ODE Cauchy data:

  26. Experimentingwithvariousfrictioncoefficients and Os

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