A simple model of filtration and macromolecule transport through microvascular walls
AA SIMPLE MODEL OF FILTRATION AND MACROMOLECULETRANSPORT THROUGH MICROVASCULAR WALLS
LAURA FACCHINI, ALBERTO BELLIN, AND ELEUTERIO F. TORO
Abstract.
Multiple Sclerosis (MS) is a disorder that usually appears inadults in their thirties. It has a prevalence that ranges between 2 and 150per 100 000. Epidemiological studies of MS have provided hints on possiblecauses for the disease ranging from genetic, environmental and infectious fac-tors to other factors of vascular origin. Despite the tremendous effort spentin the last few years, none of the hypotheses formulated so far has gainedwide acceptance and the causes of the disease remain unknown. From a clin-ical point of view, a high correlation has been recently observed between MSand Chronic Cerebro-Spinal Venous Insufficiency (CCSVI) in a statisticallysignificant number of patients. In this pathological situation CCSVI may in-duce alterations of blood pressure in brain microvessels, thereby perturbingthe exchange of small hydrophilic molecules between the blood and the exter-nal cells. In the presence of large pressure alterations it cannot be excludedalso the leakage of macromolecules that otherwise would not cross the vesselwall. All these disorders may trigger immune defenses with the destruction ofmyelin as a side effect. In the present work we investigate the role of perturbedblood pressure in brain microvessels as driving force for an altered exchangeof small hydrophilic solutes and leakage of macromolecules into the interstitialfluid. With a simplified, yet realistic, model we obtain closed-form steady-state solutions for fluid flow and solute transport across the microvessel wall.Finally, we use these results (i) to interpret experimental data available in theliterature and (ii) to carry out a preliminary analysis of the disorder in theexchange processes triggered by an increase of blood pressure, thereby relatingour preliminary results to the hypothesised vascular connection to MS. INTRODUCTION
Multiple Sclerosis is an autoimmune neurodegenerative disorder of unknown ori-gin that damages the myelin, a fatty layer that envelops and protects the axons.The damage of the myelin causes neuron electrical impulses to travel slowly alongtheir axons, leading to a variety of symptoms with debilitating consequences. Therepeated damage of the myelin causes the loss of the remyelination capacity of oligo-dendrocytes and produces scar-like lesions around damaged axons. From a clinicalpoint of view, these lesions are demonstrated to be localised in the white matterand to be venocentric (i.e. these plaques are always found around venules).Recent clinical evidence suggests an association of MS with CCSVI [Zamboni,2006; Singh and Zamboni, 2009; Zamboni et al., 2009]. However, the evolution ofthe process from venous stenosis to local hypertension and leakage of hematic sub-stance, which may trigger the immune response as ultimate cause of demyelination
Mathematics Subject Classification.
Key words and phrases.
Ultrafiltration; Starling’s law; Capillary wall; Nonlinear transport ofmacromolecules. a r X i v : . [ q - b i o . S C ] A ug LAURA FACCHINI, ALBERTO BELLIN, AND ELEUTERIO F. TORO and neurodegeneration typical of MS has been the subject of an intense debate withoften opposed views. This is a controversial hypothesis, yet it is consistent with thepredominantly venocentric orientation of the MS inflammatory lesions and withthe otherwise unexplained perivenular iron deposition observed in many clinicalcases [Adams, 1988]. A possible path connecting cerebrospinal venous stenosis tochronic fatigue and MS was proposed by Tucker [2011] on the basis of qualitativeconsiderations of elementary fluid mechanics.From a physiological point of view, the microvessel wall plays an importantrole in maintaining the equilibrium between intravascular and extravascular fluidcompartments. Under normal conditions the vessel walls are nearly impermeableto macromolecules, while lipophilic species and small hydrophilic substances areallowed to cross the wall and reach the surrounding tissue. Fluid flow and transportof dissolved molecules across the walls depend on the permeability and diffusivity ofthe membrane composing the wall. Therefore, alterations of the blood pressure maylead to impaired exchange processes and, in extreme cases, to leakage of hematicfluid. Several alterations of these exchange processes have been observed, mainly incompartments other than the brain, resulting in leakage of macromolecules, whichis typically attributed to reduction of osmotic pressure, or inflammatory processesthat alter the endothelial structure.In the present work we investigate the role of an altered blood pressure as thedriving force for alteration of the exchange processes and the leakage of macro-molecules. In particular, we analyze through a simplified, yet realistic, flow andtransport model, the impact of alterations in the hydrostatic blood pressure ontransport of molecules across the microvessel wall. The microvessel wall is assumedto be composed of two layers with different permeability and porosity, as assumedin previous studies on fluid flow and macromolecules transport in heteroporousmembranes. The inner layer represents the glycocalyx, a membrane composed ofextracellular polymeric material which is believed to exert an important sievingeffect on macromolecules, while the external layer represents the combined effect ofthe endothelial cells, the basal membrane and the external astrocyte feet.With this model we obtain closed-form steady-state solutions for the fluid flowand solute transport through the microvessel walls, which can be used for a prelim-inary analysis of the leakage of macromolecules due to an increase of blood pressurein CCSVI/MS patients.2.
MICROVESSEL ANATOMY
In this section we summarise anatomic features of mammalian blood vesselsuseful to describe the geometry of the computational domain used in the presentwork.The circulatory system is composed by vessels of size ranging from centimetresin the main ones to a few microns in the capillary bed. The structure of the vesselwall differs between arteries and veins and also between large vessels and capillaries.Each segment of circulation shows an optimal combination of size, wall composition,thickness and cross-sectional area that best fulfils its function. For example, arteriesare more muscular than veins because they have to bear the pumping force of theheart.Large vessels are formed of three layers: the endothelium, the middle layer com-posed by smooth muscle cells and the connective layer. On the contrary, small
SIMPLE MODEL OF FILTRATION AND MACROMOLECULE TRANSPORT 3 vessels such as capillaries, venules and arterioles are only one-cell thick, in order tooptimize the exchange of small hydrophilic molecules from the blood stream to theinterstitial volume before crossing the cell membrane.Molecules dissolved in water are driven through the vessel wall by the gradientof the net pressure p , which is given by the difference between the hydrostatic P and osmotic Π pressure: p = P − σ Π , where σ is the reflection coefficient. σ depends on the ratio between the Stokes radius of the molecule and the poreradius, or the size of the cleft between adjacent endothelial cells. When the sizeof the molecule is comparable with the pore size (or the aperture), the vessel wallbehaves as a perfect membrane and σ → . On the other hand, when the moleculesare much smaller than the pore size, the membrane effect vanishes and σ → .In the latter case, transport across the vessel wall is controlled by the gradient ofthe hydrostatic pressure. For a given pore (or cleft) size, the role of the osmoticpressure increases with the size of molecule. In the present work we consider twolayers, one represented by the glycocalyx and the other by the cleft. The sievingeffect of glycocalyx on macromolecules is represented by a σ value that approaches1, while in the stratum representing the endothelial cells σ is typically smaller, toreflect the larger aperture of the tight junctions connecting the two sides of the cleftat the border between adjacent cells [Levick, 2010].3. CONCEPTUAL MODEL
Let us approximate the microvessel geometry as a rigid circular cylinder, infin-itely long in the z -direction, i.e. in the direction of the blood stream. We assume thevessel wall composed by one or more permeable layers of a given thickness. Physi-cal properties, such as permeability and molecular diffusion are assumed constantwithin a layer, but may vary across the layers. The porosity is assumed the samein all layers. Molecules of a given Stokes radius are dissolved into the blood plasmaat a concentration that does not modify its density and viscosity. Furthermore, tosimplify the analysis we assume that the pressure gradient is small in the longitudi-nal direction, such that blood flow through the vessel lumen can be decoupled fromthe filtration through its wall. In general, the osmotic pressure changes with thesolute concentration c . For small concentrations, the following linear relationshipis often considered: Π = φRT c , where φ is a parameter that depends on the Stokesradius of the molecule, R is the gas constant and T is the absolute temperature.Consequently, the flow and transport equations are coupled through the concen-tration c that feeds back through Π to the flow. This leads Levick and Michel[2010] to conclude that microvessels cannot absorb fluid from the interstitial space,as is often argued. However, this feedback is important mainly when hydrostaticpressure is abruptly reduced, as in the Landis experiment [Landis, 1932], whereashere we are interested in the increase of hydrostatic pressure. We therefore neglectthis feedback and solve the flow and transport equations separately.Under the above assumptions, mass balance of the solvent and the solute leadsto the following governing equations for the pressure p = p ( x, y, z, t ) (1) ∂p∂t = kρgµS s ∇ p, and for the concentration c = c ( x, y, z, t ) (2) ∂c∂t + q n · ∇ c = ∇ · ( D · ∇ c ) . LAURA FACCHINI, ALBERTO BELLIN, AND ELEUTERIO F. TORO where k is the wall permeability, ρ is the blood density, g is the acceleration dueto gravity, µ is the blood dynamic viscosity, S s is the specific storage of the porousmaterial, n is the porosity of the material and D is the diffusion tensor.The specific water (solvent) discharge q = q ( x, y, z, t ) is proportional to the netpressure gradient through the Starling equation [Levick, 2010](3) q = − Kρg ∇ p, where(4) K = kρgµ . Finally, the mass flux of the solute f m = f m ( x, y, z, t ) is given by(5) f m = (1 − σ ) q c − n D · ∇ c. The above equations written in cylindrical coordinates ( r, θ, z ) and assuming radialsymmetry take the following form ∂p∂t = kρgµS s r ∂∂r (cid:18) r ∂p∂r (cid:19) , (6) q = − Kρg ∂p∂r , (7) ∂c∂t + qn ∂c∂r = (cid:18) dr + ∂d∂r (cid:19) ∂c∂r + d ∂ c∂r , (8) f m = (1 − σ ) qc − nd ∂c∂r . (9)Since the initial and boundary conditions are independent from the coordinates z and θ , we only consider the radial component d of the diffusion tensor D .Furthermore, we assume that d is given by the sum of the molecular diffusion d m and the hydrodynamic dispersion d h = Aq , where A is the dispersivity and q isthe radial component of the specific discharge q .In the next section we consider the steady-state solution of the above flow andtransport equations. 4. ANALYSIS
The steady-state equations for the solvent and for the solute in cylindrical coor-dinates assume the following form ∂∂r (cid:18) r ∂p∂r (cid:19) , (10) qn ∂c∂r = (cid:18) dr + ∂d∂r (cid:19) ∂c∂r + d ∂ c∂r . (11)4.1. Steady-state solutions for a single-layer vessel wall.
We now considera geometrical situation as depicted in Figure 1(a), which shows a cylinder whoseinner surface of the endothelial cells is represented by radius r and whose outerwall is determined by radius r .In this case, we obtain two generic solutions p ( r ) = α + β ln r, r ∈ [ r , r ] (12) c ( r ) = δ + γh ( r ) , r ∈ [ r , r ] (13) SIMPLE MODEL OF FILTRATION AND MACROMOLECULE TRANSPORT 5 (a) (b)
Figure 1. (a) The domain is an infinitely long hollow cylindercomposed by one layer only, whose inner and outer radii are r and r . (b) Cross section depicting boundary conditions, where p c = P c − σ Π c and c c refer to the net blood pressure and soluteblood concentration, respectively and p i = P i − σ Π i and c i indicatethe blood pressure and solute concentration in the interstitial fluid.each of them depending on two parameters which can be computed by imposingthe boundary conditions, where h ( r ) is an auxiliary function depending on thepermeability and on the boundary conditions(14) h ( r ) = − µnkβ [ µd m r − kβA ] − kβnµd m , if k ( p c − p i ) (cid:54) = 0ln ( µd m r ) d m , if k ( p c − p i ) = 0 for r ∈ [ r , r ] , with(15) β = p c − p i ln r − ln r . We suppose that the boundary conditions are independent from the coordinates z and θ . So we set constant pressures and concentrations at the boundary, asdepicted in Figure 1(b), p ( r ) = p c , (16) p ( r ) = p i , (17) c ( r ) = c c , (18) c ( r ) = c i , (19)where p c = P c − σ Π c refers to the net blood pressure, c c to the solute bloodconcentration, p i = P i − σ Π i indicates the blood pressure in the interstitial fluidand c i the interstitial solute concentration. So we obtain closed-form steady-state LAURA FACCHINI, ALBERTO BELLIN, AND ELEUTERIO F. TORO solutions for r ∈ [ r , r ] , given by p ( r ) = p c ln( r /r ) + p i ln( r/r )ln( r /r ) , (20) q ( r ) = − kµr p c − p i ln r − ln r , (21) c ( r ) = c c [ h ( r ) − h ( r )] + c i [ h ( r ) − h ( r )] h ( r ) − h ( r ) . (22)The mass flux depends on the values of the permeability and on the boundaryconditions f m ( r ) = kβµr (cid:20) σc ( r ) − c i h ( r ) − c c h ( r ) h ( r ) − h ( r ) (cid:21) , if k ( c c − c i ) (cid:54) = 0 − nr c c − c i h ( r ) − h ( r ) , if k ( c c − c i ) = 0 (23)for r ∈ [ r , r ] , with(24) β = p c − p i ln r − ln r . Steady-state solutions for a vessel wall composed by two layers.
Wenow consider a more complex case in which the vessel wall is composed by two layers ( r , r ) and ( r , r ) , as depicted in Figure 2(a), with different values of permeability,diffusivity and reflection coefficient.The general case of m layers can be treated similarly.(a) (b) Figure 2. (a) The domain is an infinitely long hollow cylindercomposed by two layers ( r , r ) and ( r , r ) . (b) Cross section de-picting boundary conditions, where p c = P c − σ Π c and c c refer tothe net blood pressure and solute blood concentration, respectivelyand p i = P i − σ Π i and c i indicate the blood pressure and soluteconcentration in the interstitial fluid.The generic solution for the net pressure assumes the following form(25) (cid:26) p ( r ) = α + β ln r, r ∈ [ r , r ] p ( r ) = α + β ln r, r ∈ [ r , r ] while the solute concentration is given by(26) (cid:26) c ( r ) = δ + γ h ( r ) , r ∈ [ r , r ] c ( r ) = δ + γ h ( r ) , r ∈ [ r , r ] SIMPLE MODEL OF FILTRATION AND MACROMOLECULE TRANSPORT 7 where, similarly to the previous case, h j ( r ) is a function that depends on the ge-ometry and the permeability of the layer h j ( r ) = − µnk eq B (cid:2) µd m j r − k eq BA j (cid:3) − k eq Bnµd m j , if k k ( p c − p i ) (cid:54) = 0ln (cid:0) µd m j r (cid:1) d m j , if k k ( p c − p i ) = 0 (27)for r ∈ [ r j , r j +1 ] , recalling that d m j is the molecular diffusion and A j is the disper-sivity of the j -th layer. B and k eq are now defined as B = p c − p i , (28) k eq = k k k (ln r − ln r ) + k (ln r − ln r ) . (29)The constants appearing in the above solutions are obtained by imposing suitableboundary conditions for both the net pressure and solute concentration, as depictedin Figure 2(b) p ( r ) = p c , (30) p ( r ) = p i , (31) c ( r ) = c c , (32) c ( r ) = c i , (33)where p c = P c − σ Π c refers to the net blood pressure, c c to the solute bloodconcentration, p i = P i − σ Π i indicates the blood pressure in the interstitial fluidand c i the interstitial solute concentration. These boundary conditions shouldbe supplemented by the conditions resulting from imposing the continuity of thespecific discharge and the solute flux at the interface between the two layers at r = r q ( r ) = q ( r ) , (34) f m, ( r ) = f m, ( r ) , (35)and that both pressure and solute concentration are continuous at r = r p ( r ) = p ( r ) , (36) c ( r ) = c ( r ) . (37)With all these conditions the pressure within the first and second layer are givenby(38) p ( r ) = k [ p i ln( r /r ) − p c ln( r /r )] + p c k ln( r /r ) k ln( r /r ) + k ln( r /r ) and(39) p ( r ) = k [ p i ln( r /r ) − p c ln( r /r )] − p i k ln( r /r ) k ln( r /r ) + k ln( r /r ) respectively. LAURA FACCHINI, ALBERTO BELLIN, AND ELEUTERIO F. TORO
The resulting expression for the specific discharge is the same in the two regions,indeed q ( r ) = − k j µ · ∂p j ∂r ( r ) = − k k ( p c − p i ) µ [ k ln( r /r ) + k ln( r /r )] 1 r , (40)for r ∈ [ r , r ] , where j ∈ { , } indicates the layer we are considering.Similarly, under steady-state conditions, the solute concentration assumes thefollowing expression(41) c ( r ) = c ( r ) = S + T h ( r ) V , r ∈ [ r , r ] c ( r ) = S + T h ( r ) V , r ∈ [ r , r ] where the parameters S , T , S , T , V depend on the value of k k ( p c − p i ) .Indeed, these parameters are defined as(42) S = c c (1 + σ − σ ) h ( r ) h ( r ) − c i h ( r ) h ( r )++ c c ( σ − σ ) h ( r ) h ( r ) , if k k ( p c − p i ) (cid:54) = 0 − c i h ( r ) − c c [ h ( r ) − h ( r ) + h ( r )] , if k k ( p c − p i ) = 0 (43) T = [ c i − c c (1 − σ + σ )] h ( r ) + c c ( σ − σ ) h ( r ) , if k k ( p c − p i ) (cid:54) = 0 c c − c i , if k k ( p c − p i ) = 0 (44) S = c c h ( r ) h ( r ) + c i ( σ − σ ) h ( r ) h ( r )+ − c i (1 − σ + σ ) h ( r ) h ( r ) , if k k ( p c − p i ) (cid:54) = 0 c i [ h ( r ) − h ( r ) + h ( r )] − c c h ( r ) , if k k ( p c − p i ) = 0 (45) T = [ c i (1 + σ − σ ) − c c ] h ( r ) + c i ( σ − σ ) h ( r ) , if k k ( p c − p i ) (cid:54) = 0 c c − c i , if k k ( p c − p i ) = 0 (46) V = ( σ − σ )[ h ( r ) h ( r ) − h ( r ) h ( r )]+ − (1 − σ + σ ) h ( r ) h ( r )++(1 + σ − σ ) h ( r ) h ( r ) , if k k ( p c − p i ) (cid:54) = 0 h ( r ) − h ( r ) + h ( r ) − h ( r ) , if k k ( p c − p i ) = 0 . The resulting solute flux is the following f m,j ( r ) = k eq Bµr (cid:20) σ j c j ( r ) − S j V (cid:21) , if k k ( p c − p i ) (cid:54) = 0 − nT j V r , if k k ( p c − p i ) = 0 (47) SIMPLE MODEL OF FILTRATION AND MACROMOLECULE TRANSPORT 9 for r ∈ [ r j , r j +1 ] , where j ∈ { , } indicates the layer we are considering and σ j represents the reflection coefficient that may be different in the two layers.Similar expressions may be obtained for three and more layers.4.3. The travel time through the vessel wall.
An important quantity in theexchange process is the time a single solute molecule takes to cross the vessel wall.We call this time the travel time τ , in analogy with transport in porous media.For the single layer case, τ may be approximated by neglecting the diffusivecomponent of the mass flux τ = (cid:90) r r n (1 − σ ) q ( r ) d r = − nµβk (1 − σ ) r − r . (48) 5. PRELIMINARY RESULTS
The structure of the vessels is very specialized in relation to their functionalityand this specialization results in different permeability and reflection coefficientsof the vessel wall. Table 1 shows typical values of the geometrical properties ofmicrovessels together with the hydraulic conductivity to serum albumin and thereflection coefficient. Although the permeability of the venules is expected to belarger than the permeability of the arterioles, in the absence of specific data, and forillustration purposes in the subsequent exercise we assumed the same permeabilityfor both microvessels.Parameter [unit] Value Reference K [kg sec − (cm H O) − ] 2.49 · − [Michel and Curry, 1999] σ n r A [ µ m] 15 [Silverthorn, 2010] r V [ µ m] 10 [Silverthorn, 2010] ∆ x A [ µ m] 6 [Silverthorn, 2010] ∆ x V [ µ m] 1 [Silverthorn, 2010] Table 1.
Typical values of the parameters used in the computa-tion. K is the hydraulic conductivity for serum albumin, σ is thereflection coefficient for serum albumin, n is the porosity, r is themean radius of the vessel and ∆ x is the vessel thickness. A refersto the arteriolar end of the capillary bed, while V to the venousend.In addition, venules and arterioles are subjected to different internal hydrostaticpressures and external osmotic pressures. Table 2 shows the typical mean pressuresin different microvessels.The difference of the net pressure p between the internal (subscript c ) and theexternal side (subscript i ) of the microvessels, i.e.(49) ∆ p = p c − p i = ( P c − σ Π c ) − ( P i − σ Π i ) , provides a first rough quantification of the expected flux through the vessel wallper unit area, i.e. the specific discharge. In Table 2, we observe that ∆ p is positive Location P c P i σ Π c σ Π i ∆ p [cm H O] [cm H O] [cm H O] [cm H O] [cm H O]arteriolar endof capillary 47.62 -2.72 38.10 0.14 12.38venular endof capillary 20.41 -2.72 38.10 4.08 -10.88
Table 2.
Mean pressures in human body, taken from [Boron andBoulpaep, 2005]. P represents the hydrostatic pressure, while Π is the osmotic pressure and σ is the reflection coefficient. Thesubscript c refers to the pressure measured inside the vessel, whilethe subscript i is measured just outside the vessel. ∆ p is definedas the difference of the net pressure p between the internal and theexternal side of the microvessels, i.e. ∆ p = p c − p i = ( P c − σ Π c ) − ( P i − σ Π i ) .for arterioles (12.38 cm H O) and negative for venules (-10.88 cm H O). This leadsto a tendency for absorpion at the venular end of the capillary bed, which maybe contrasted by the parallel increase of the osmotic pressure within the cleftsjust downstream the glycocalyx, the membrane coating the internal surface of theendothelial cells [Levick, 2010]. As mentioned before, in the present work we neglectthis feedback mechanism.We start by considering the microvessel wall composed by a single layer. Figure 3shows the specific discharge q crossing the vessel wall as a function of the hydrostaticpressure P c for both arterioles and venules.In the case in which the vessel wall is composed by only one layer, we can studythe behavior of the discharge per unit length and of the travel time of a molecule,assuming that the external pressures P i and Π i and the internal osmotic pressure Π c are constant. The internal hydrostatic pressure P c is the residual pressure,controlled by the cardiac pressure, so we can represent our quantities with respectto it.For typical values of venular pressure (see the black bullet on the thick straightline in Figure 3), the specific discharge is negative, meaning that venules absorbfluid and the dissolved molecules from the interstitial volume. On the other hand,arteriolar pressure is positive letting the oxygen and the nutrients nourish the sur-rounding tissues.In Figure 4, the travel time τ of a target molecule (in this case, serum albumin)is depicted with respect to the internal hydrostatic pressure P c for arteriolar (thincurve) and venular end (thick curve) of the capillary bed.For typical values of internal pressure (see the black bullets in Figure 4), τ ispositive for arterioles and negative for venules, reflecting the opposite directionof the flow in the two cases. An increase of the hydrostatic pressure leads to areduction of τ for the arterioles. In the case of venules, the same increase leads toa larger travel time | τ | . Both occurrences may induce a significant alteration of theexchange mechanisms between the interstitial fluid and the cells. If the hydrostaticpressure increases above a given threshold (about cm H O, in the present case)
SIMPLE MODEL OF FILTRATION AND MACROMOLECULE TRANSPORT 11
Figure 3.
Discharge per unit length depending on the internalhydrostatic pressure, in the arteriolar case (thin straight line) andin the venular case (thick straight line). The dots represent thetypical values of internal blood pressure in both cases.
Figure 4.
Travel time of a molecule of serum albumin, in arteri-oles (thin curve) and in venules (thick curve). The black bulletsrepresent the typical values of internal blood pressure in both cases.the flux is inverted across the venule wall and the travel time becomes positive,thereby leading to leakage of hematic fluid from the venules into the interstitialvolume. Close to this threshold τ is large, but it reduces rapidly as the hydrostaticpressure increases further. This may provide a plausible explanation for streaks ofblood observed in the histology of MS brain plaques [Singh and Zamboni, 2009].Finally, we observe that our simple model is in agreement with the early exper-iments conducted by Landis [Landis, 1932] in frog mesenteric capillaries. CONCLUSIONS
We have presented a simplified analytical model of steady-state flow and trans-port of a target molecule through the wall of microvessels. The advantage of thismodel is that it allows us to easily explore the explicit influence of the many pa-rameters controlling the process and thereby avoiding, for the time being, the useof numerical methods. With this model we have performed a preliminary analysisof the flux across arterioles and venules by using parameters taken from existingstudies on mesenteric capillaries. In both cases we computed the time a targetmolecule (with a given reflection coefficient) spends crossing the wall, which mayprovide an indication of the alteration of exchange mechanisms due to modificationof the hydrostatic pressure at the arteriolar and venular ends.An increase of the hydrostatic pressure above the value observed in normal condi-tions leads to an increase of the flux crossing the wall of arterioles and a correspond-ing reduction of the travel time. In this condition more hydrophilic molecules arereleased in the interstitial fluid surrounding the vessel, thereby potentially reducingdownstream the availability of such substances needed for the cell metabolism. Onthe venular side a threshold hydrostatic pressure separates two different ways offunctioning. For hydrostatic pressures below such a threshold the flux is negativeand the venule absorbs fluid from the interstitial space, while above this thresholdthe venule leaks hematic fluid to the interstitial space. An increase of the hydro-static pressure has then a different impact according to the reference hydrostaticpressure. For low reference pressure (i.e. below the threshold) an increase of thehydrostatic pressure leads to a reduction of the absorption and a parallel increase ofthe travel time. However, if the reference pressure is larger than this threshold thevenule behaves similarly to the arteriole and leaks hematic fluid to the interstitialspace with a travel time that reduces rapidly with the increase of the hydrostaticpressure. The impact of these alterations on the cell metabolism may be significantand potentially may be responsible of the suffering status of oligodendrocytes ofpatients affected by MS and CCSVI.
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Department of Mathematics, University of Trento (Italy), via Sommarive 14,38123 Trento (TN)
E-mail address : [email protected] Department of Civil, Environmental and Mechanical Engineering, University ofTrento (Italy), via Mesiano 77, 38123 Trento (TN)
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