For narrow tubes, red blood cells concentrate in the core region, leaving an annular zone called cell-free layer. This has an impact on both the tube hematocrit level (Fåhraeus effect) and the apparent blood viscosity (Fåhraeus–Lindqvist effect). Blood flow, mass transfer across the microvessel membrane, and diffusion in the tissue affect the solute concentration profiles. The Krogh tissue cylinder concept, limiting mass transfer to a cylinder around each microvessel, and the marginal zone concept (introduced by Haynes to analyze blood flow dynamics in narrow tubes) are both used to develop a model for solute transfer from blood in microvessels to the surrounding tissues, based on fundamentals. The analysis accounts for advection and diffusion in each zone of the microvessel, solute transport in the microvessel membranes, and diffusion and reaction in the tissues. The present investigation provides an analytical solution. The approach can be extended to treat other kinetic models, while accounting for Fåhraeus and Fåhraeus–Lindqvist effects in blood microvessels. The model is validated against published results for glucose transport from blood to tissue.
Transport of a solute from blood to the surrounding tissues involves transport in the microvessel, through the microvessel wall, and transport in the tissue. Blood flow in the microvessels affects the concentration profiles. For small blood vessels below 500 μm diameter, diameter affects both the tube hematocrit (Fåhraeus effect) and the apparent blood viscosity (Fåhraeus–Lindqvist effect) with red blood cells (RBCs) concentrating in the core region [1,2,3,4,5,6,7,8]. Haynes marginal zone theory considers blood as consisting of two layers: a marginal zone (cell-free layer) surrounding a core region where RBCs concentrate . Both zones are treated as continuum fluids. The theory was used to investigate Fåhraeus and Fåhraeus–Lindqvist effects [9,10,11]. Other blood flow models include shear-induced models [12,13,14,15] using the shear-induced particle migration concept [16,17] and elastic stress-induced models [18,19,20,21,22]. Non Newtonian and viscoelastic models for blood are reviewed in Arciero et al.  and in the references therein. Mesoscale models account for the particulate aspect of blood [23,24].
The Krogh cylinder model was introduced by Krogh to solve for transport of oxygen in tissue . The assumptions used and extensions of the model are discussed in studies [23,26,27,28] and the references therein. In the case of oxygen transport, accounting for enhanced oxygen transport by myoglobin and reaction in RBCs is required [26,28]. Different geometries for RBCs were proposed . An analytical solution, assuming cylindrical RBCs to occupy the lumen of the blood vessel shows the intra-capillary high resistance to oxygen transport . Groebe  accounted for axisymmetric diffusion of oxygen through three layers assuming cylindrical RBCs with plasma filling the gap between them. An analytical solution and a discussion of the simplifying assumptions are provided in ref. . Blood is assumed to be formed by spherical RBCs, in line, equidistant, and surrounded by plasma in ref. . A frame of reference moving with the RBCs is selected . Other shapes for RBCs were selected and assumed to be surrounded by plasma: cylindrical (using a frame of reference in which RBCs are fixed)  and parachute form [33,34] (with a reference system in which RBCs are fixed in ref. , while considering RBCs motion in the frame of reference in the study ). Transport of other species like NO in relation with oxygen transport are discussed in refs. [23,27,28,30]. A model for biofluids flow in slowly and successively contracting and expanding rectangular geometry with two weakly permeable walls (allowing for flow through the walls) is solved using the homotopy method .
A model for solute transfer accounting for advection and diffusion using Krogh model for transport in tissue is presented in Fournier . For intra-capillary transport, blood velocity is assumed uniform inside the capillary and an overall transport coefficient from blood to the outer surface of the capillary wall (external side of the capillary membrane) is used in the final expressions for solute concentration in ref. .
The present model uses Krogh concept for solute transport in the tissue  (with mass transfer limited to a cylinder around each microvessel) along with Haynes marginal zone concept for blood flow  (considering the microvessel as formed of a cell-free layer and a core region where RBCs concentrate). The model for solute transport from blood to the surrounding tissue accounts for advection and diffusion in both the core region and the cell-free layer in the microvessel, solute transport in the microvessel membrane, and diffusion and reaction in the tissue (Figure 1). The approach is based on the fundamentals of fluid mechanics and mass transfer, where the velocity profiles in the two zones, core and cell-free layer, are considered to account for advection in the mass transfer equations. An analytical approach combining the different steps in solute transport is provided.
2 Governing equations
Haynes theory considers blood as consisting of two phases: an annular cell-free layer and a core layer containing RBCs. The velocity profile is discussed in Fournier , Chebbi , and the references therein. Only boundary conditions and related results of the present work are presented here. A schematic diagram of a microvessel and its Krogh cylinder is shown in Figure 1. The cylindrical coordinates r and z are used.
For fully developed flow, the velocity profile is independent of z. The following boundary conditions express continuity of the velocity and shear stress at the two zones interface r = r i, in addition to finite shear stress at the axis of the microvessel and no-slip boundary condition at the inner surface of the microvessel wall
Applying the momentum balance for the two zones yields
where v denotes velocity, μ represents viscosity, r is the radial coordinate, R is the microvessel radius, p gr is the absolute value of the pressure gradient, and subscripts a, c, and i denote the annular region, the core region, and the two-zone interfacial area, respectively.
The velocity profiles in the core region and the cell-free layer (annular region) are given by
The average velocity v * is given by
where Q is the blood volumetric flow rate.
The core hematocrit H C is assumed uniform. The conservation of hematocrit yields the following relations [3,11]:
where H T is the tube hematocrit, H D is the discharge hematocrit, and .
The apparent viscosity μ app is given by the Hagen–Poiseuille equation and is related to the core zone viscosity by:
The core zone viscosity is assumed to be a function of H C (up to 0.6) according to ref. :
and T is blood temperature in Kelvin.
The mass transfer model accounts for advection-diffusion in the microvessel side, mass transfer across the microvessel membrane, and diffusion and zero-order reaction in the surrounding tissues demarcated by the Krogh cylinder. For small concentrations and nearly constant temperature, the diffusion coefficients are constant, and the use of conservation of mass of solute yields the following partial differential equations assuming steady state (zero accumulation term):
where z is the axial cylindrical coordinate, c denotes the solute concentration, D represents the diffusion coefficient of the solute, and subscripts c and a refer to the core and annular zones, respectively. The effective diffusion coefficient is used for the core region as it consists of plasma and RBCs. In the case of oxygen transport for instance, a reaction term involving the solute consumption needs to be accounted for in the solute mass balance. The present model does not include the case where reaction involving the solute occurs in the blood side.
The following boundary conditions express continuity of concentration and fluxes at the interface between the core and annular zones, in addition to finite concentration gradient at the centerline:
The effective diffusion coefficient D c can be obtained using Maxwell equation [3,36]:
where D RBC is the solute diffusion coefficient in RBCs. Mass transfer through the microvessel membrane is mainly due to diffusion, with negligible transport by filtration . Mass transfer in the membrane is considered in the present model as purely diffusional, and the boundary condition at the microvessel wall can be written at the inner side of the membrane as :
at the outer side of the microvessel membrane, where subscript T refers to the tissue, and P M and t m denote the solute permeability and membrane thickness, respectively.
At the limit of the Krogh tissue cylinder, r = r T, we have :
which restricts mass transfer to the Krogh tissue cylinder surrounding the specific microvessel. This condition prevents mass transfer between the different Krogh cylinders.
In case the critical length L cr, at which the solute concentration is zero, is larger than the microvessel length, then at the Krogh radius r T, Eq. (22) is still valid, but the reaction term in the model needs to be restricted to a smaller part of the Krogh cylinder where the solute is non zero.
In the tissue, we assume zero-order reaction occurring at a molar rate of R 0 per unit volume. Assuming constant D T for the same arguments listed before for D a and D c, and using the constant density approximation, transport of solute can be easily shown to be governed by (Appendix B.11 ):
Solving for c T using boundary condition 22 gives :
where is the r-component of the solute concentration as defined in Eq. (25), in consistency with Krogh–Erlang solution [25,27].
In order to determine the concentration profile, an expression for is required.
Assuming sufficiently large Péclet number and large microvessel length, we neglect axial diffusion in the microvessel side and in its Krogh tissue cylinder. Excluding a small region near z = 0, we seek a solution of the form
where , , and depend on r and are independent of z, c 0 is the average solute concentration in blood at z = 0, and b a, b c, and b T are constants. A similar form of the solution with an r-component and a linear term proportional to z was used to treat the case of laminar fluid flow in a tube subject to a constant heat flux  (Chapter 10). The present case is different and accounts for two zones inside the microvessel, mass transfer across the membrane, and mass transfer and reaction in the surrounding Krogh cylinder. The results show that the solution is valid except very near the tube inlet  (Chapter 12).
The use of boundary conditions 16 and 20 yields , with subscripts c, a, and T dropped, and b c, b a, and b T simply denoted as b.
Substituting for velocities into Eqs. (14) and (15) while using Eqs. (6), (7), and (25) gives:
Boundary conditions (16)–(18) and (20) yield:
Integrating Eqs. (26) and (27) while using Eq. (29) gives:
where K is an integration constant.
In order to find b, a solute mass balance is made considering a control volume including the Krogh cylinder, the membrane, and blood limited to the interval z – z + Δz:
is independent of z. Solving for b yields :
Considering the Krogh cylinder as our selected control volume gives:
Rearranging the above equation yields
Using Eq. (36) shows that the solute concentration profile, given by Eq. (24), is consistent with boundary condition (21).
Using the following definitions:
The dimensionless variables are defined as
The dimensionless profiles are obtained using Eqs. (6), (7), (31), and (32).
Converting Eqs. (8) and (9) to a dimensionless form gives:
Using the relationship between concentrations across the microvessel membrane, Eq. (30), yields in a dimensionless form:
Applying Eqs. (43) and (30) at the inner side of the microvessel wall, we get:
Differentiating Eq. (41) yields:
Applying Eq. (45) at and using Eq. (44) gives:
Assuming an average concentration c 0 at z = 0 and applying a mass balance yields:
Using Eq. (34) and substituting for b in Eq. (47) gives:
Splitting the integral term in Eq. (48) into two terms to account for the existence of two zones yields in dimensionless form:
Substituting for the velocity and concentration profiles in Eqs. (39)–(41), we get:
where J c and J a are given by:
Applying Eq. (48) at the interface between the core and annular zones yields:
where Λ is given by:
Combining Eqs. (49), (52), (53), and (56) yields:
Using Eq. (44) yields the following expression for the centerline value :
The mass transfer coefficient can be determined using:
The average solute concentration in blood c av is given by:
in which v and c represent v a and c a in the annular zone, and v c and c c in the core zone, respectively. Then, using Eqs. (25) and (48) yields:
Substituting Eqs. (62) and (25) into Eq. (61) gives:
Then, combining Eqs. (63) and (36) provides:
4 Computational procedure and results
Solving the problem requires a number of steps. The dimensionless profiles and require the values of σ, the core region hematocrit, H C, and the ratio D a/D c. The steps are as follows: (i) for specific values of HT and σ, calculate the core zone hematocrit H C, the dimensionless core zone viscosity μ c/μ a, the discharge hematocrit H D, and the dimensionless blood viscosity μ app/μ a using Eqs. (10)–(13), (ii) determine and using Eq. (42), (iii) obtain , , and Λ using Eqs. (39), (46), and (56), respectively, and (iv) find B 1, B 2, B 3, , and using Eqs. (54) and (55). Finally, Eqs. (58), (56), and (59) can be used to provide the values of , , and at the inner side of the wall, the interface between the two zones inside the microvessel, and the centerline, respectively. The use of Eq. (44) and the dimensionless value of D a/(P M R) provides the value of . For H D and microvessel diameter (2R) given, finding H T and σ requires trial and error calculations using the blood viscosity correlation in Pries et al. .
The dimensionless profiles for and are given by Eqs. (40) and (41) as functions of r/R, and in a dimensional form by Eqs. (31), (32), and (24) for , , and as functions of r. The dimensionless variables are defined by Eqs. (37) and (38). The concentrations can be obtained using Eq. (25) where b, providing the rate of change of concentration in the axial direction, is given by Eq. (34). As mentioned previously, the expressions are not valid in the near vicinity of z equal to zero. The range for D c/D a can be obtained using Maxwell equation for two limiting cases:
and an intermediate case: D c/D a = 1. The following table provides an application of the abovementioned procedure as an example.
Table 1 shows the impact of diffusion in RBCs on the concentrations inside the microvessel, with obviously less deviations at high diffusion coefficient D c as anticipated, as higher diffusion coefficient yields less resistance to mass transfer. The concentrations decrease with increase in r at constant z. This allows solute transport from blood to tissue. Some negative values for are expected since the profile is expected to satisfy Eq. (49). The negative values are deviations and do not represent concentrations as seen from Eq. (25). Using the analytically determined profiles and performing numerical calculations confirm that Eq. (49) is satisfied as seen from Table 1.
|Constant||Limiting low D c value||Intermediate case: D c = D a||Limiting high D c value|
|μ c/μ a||1.947||1.947||1.947|
|D c/D a||0.4771||1.000||3.192|
|μ app/μ a||1.469||1.469||1.469|
|I c||9.132 × 10−3||4.591 × 10−3||1.747 × 10−3|
|I a||−9.132 × 10−3||−4.591 × 10−3||−1.747 × 10−3|
As an example, the model is applied to glucose transport from blood flowing in a capillary to the surrounding Krogh tissue cylinder. For the purpose of comparison, the data and values from ref.  are used and summarized in Table 2.
|D||Diffusion coefficient in water||0.91 × 10−5 cm2/s||p. 221|
|D a||Diffusion coefficient in plasma||0.63 D||Eq. (5.60)|
|2R||Capillary diameter||10 μm||Table 5.1|
|v*||Average blood velocity||0.05 cm/s||Table 5.1|
|t m||Wall thickness||0.5 μm||Table 5.1|
|L||Length||1,000 μm||Table 5.1|
|R 0||Rate of glucose consumption||0.01 μmol/cm3 s||p. 221|
|P M||Membrane permeability||5.76 × 10−5 cm/s||p. 221|
|c 0||Average concentration at z = 0||5 μmol/cm3||p. 221|
|a||Glucose molecular radius||0.36 nm||p. 221|
|D T/D||Ratio of D T to D||≈0.9||Figure 5.12|
|r T||Krogh cylinder radius (assumed in )||40 μm||p. 229|
The dimensional concentration profiles are obtained from Eq. (25) as:
At the critical value z cr given by:
As noticed from Figure 2, all glucose concentrations drop as z increases due to solute transport from blood to tissue, followed by diffusion and chemical reaction in the Krogh tissue surrounding the capillary. The results for the average glucose concentration in blood, c av (z), and the glucose concentration at the outer surface of the capillary wall, c T(R + t m, z), are both very consistent with Fournier’s results. The results for the concentrations at the Krogh tissue radius, c T(r T, z), are very close to those at the outer surface of the capillary wall as seen from Figure 2, and from Figure 5.18 in ref. . The profiles for the deviations of concentration in blood from the average concentration, c(r,z)−c av(z), show a decrease in concentration as r gets closer to the capillary inner wall as glucose is transported from blood to tissue, with less deviations from the average concentration of glucose, at higher D c/D a (Figure 3). The drop in glucose concentration in the Krogh cylinder from its value at the outer surface of the wall, c T(r, z) − c T (R + t m, z), is plotted in Figure 4. Considering the previous results in Figures 2–4 and Table 3, we conclude that the significant change in the r-direction occurs in the membrane, as noted by Fournier .
|Constant||Limiting low D c value||Intermediate case: D c = D a||Limiting high D c value|
|b, μmol/cm3 μm||1.256 × 10−3||1.256 × 10−3||1.256 × 10−3|
|, μmol/cm3||2.738 × 10−2||2.738 × 10−2||2.738 × 10−2|
|c T(R + t m,z)−c av(z), μmol/cm3||−2.7356||−2.7309||−2.7280|
|k m, cm/s||1.516 × 10−2||2.769 × 10−2||5.750 × 10−2|
|z cr, μm||1,791||1,795||1,797|
The mass transfer coefficient results are reported in Table 3, showing higher values at higher D c/D a. Nevertheless, the significantly lower permeability P M plays the predominant role in determining the overall mass transfer coefficient as also mentioned by Fournier . The critical values z cr are close in the three D c/D a cases (about 1.8 mm), and larger than the capillary length in Table 2.
The concentration profiles are determined based on solute conservation of mass, mass transport equations, and relevant boundary conditions. The steady state model accounts for convection-diffusion in the microvessel, transport through the microvessel membrane, and diffusion and zero-order reaction in the Krogh tissue cylinder. Velocity gradients in the microvessel are accounted for in mass transfer calculations using the marginal zone theory results in the cell-free layer and the core region where RBCs concentrate. The solution for mass transfer requires a detailed procedure starting with blood flow calculations, and ending with the determination of the concentrations at the core zone-cell-free layer interface, and both sides of the microvessel membrane. The use of the concentration values at the boundary points yields different profiles in the microvessel and in the Krogh cylinder using the relevant equations discussed above. The solution excludes a small region near z equal to zero. Large Péclet number and microvessel length are required for neglecting the axial diffusional terms in the microvessel and tissue sides. The estimation of diffusion coefficients and permeability is not discussed here and can be found in Fournier . The approach used here can be extended to treat different kinetic models, while accounting for Fåhraeus and Fåhraeus–Lindqvist effects in blood microvessels. The use of Maxwell equation to determine the effective diffusivity of the solute in the core region of the microvessel is not restricted to low volume fractions of the dispersed phase (RBCs in our case) as noted in ref. . The model is validated against the results of Fournier  for the average glucose concentration in blood, and the glucose concentrations at the outer surface of the capillary wall and at the Krogh cylinder radius.
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