Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel

In a recent paper by the authors, a well-known governing nonlinear PDE used to model oxygen transport was formulated in a generalized coordinate system where the Laplacian was expressed in metric tensor form. A reduction of the PDE to a simpler problem, subject to specific integrability conditions, was shown, and in the present work, a novel approximate analytical solution is obtained in terms of the degenerate Weierstrass P function using a compatibility relation through the factorization of the reduced almost linear ode and subject to similar boundary conditions for a microfluidic channel used in recent work by the authors. A specific form of the initial equation which was reduced has been used by Nair and coworkers describing the intraluminal problem of oxygen transport in large capillaries or arterioles and more recent work by the corresponding author describing the release of adenosine triphosphate (ATP) in micro-channels. In the present problem, a channel with a central core, rich in red blood cells, and with a thin plasma region near the boundary wall, free of RBCs is considered.


Introduction
Various biophysical phenomena are modeled using nonlinear differential equations. Such is the case of a model used by Nair et al. [1][2][3] to describe oxygen transport in large capillaries [1][2][3]. This model incorporates two regions of blood flow. One is a core region of the blood with RBCs present, and in this core, oxygen dissociates into blood oxyhemoglobin. The velocity of blood in the core region is a function of the plasma velocity and rate of oxygen dissociating. The second region is a thin strip of flowing plasma with no RBCs at the wall of the micro-fluidic channel. The appropriate Robin condition and no-flux conditions are incorporated at and around a permeable membrane with oxygen transport through the membrane. Since there are two distinct regions of flow of liquid, RBCs with plasma and plasma alone, it is necessary to match the rate of change of partial pressure of oxygen at the common boundary of the liquid in each of the two regions. This kind of model has been used previously by Moschandreou et al. [4] studying the influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport. In that study a numerical approach was used to solve the governing equations. In the present work we seek an analytical solution for a different highly nonlinear problem in a micro-fluidic channel. Flows that are fully developed at inlet were studied by Ng [5] who studied oscillatory dispersion in a tube with chemical species undergoing linear reversible and irreversible reactions at the tube wall. Relative importance to the present work is that fully developed flow occurs at inlet of channel with boundary conditions specified on the wall. No inlet mass transport is specified at inlet of channel similar to [5] but unlike [6], for example. Using Painlevé analysis [7], certain nonlinear second-order ordinary differential equations (ODE) can be factorized and solved. We consider the model of Nair et al. [1][2][3], where the nonlinear PDE reduced in [8], to an "almost linear" second-order ode is considered. It is well-known that the Weierstrass P function in its series form is problematic to use in computational work due to very slow convergence of numerical methods. It is the aim of the present work to show that a degenerate form of the special function can be used as in [9] in the reduction of the nonlinear PDE in [1][2][3]. A thorough and recent review of oxygen control with microfluidics has been carried out in [10] and all of its references within. In this work we see how the microscale can be leveraged for oxygen control of RBCs.

General tensorial mass transport
Regardless of the kinematics of a surface (dynamic or stationary), all surfaces, S ¼ ∂Ω, enclosing a solid volume, Ω, obey the following intuitive conservation relation for an enclosed observable mass, m o of some arbitrary substance: where j o is the flux of the observable out or into the surface and Σ represents the net increase or decrease in the observable's mass.
The relation states intuitively that any change of the observable's mass within the solid, plus all observable mass entering or leaving the boundary, should represent the net change in the mass of the object.
Any mass transport can be derived from the above relation converted into the differential form. We first recognize that the observable's mass can be represented through the observable's density: In addition, we can make the same statement about the net equilibrium constant. Suppose there is a local equilibrium density, σ, such that We then can obtain a full integral form of the conservation relation: In general, it can be shown that for a general surface that is moving, the time derivative of a volume integral defined over the dynamic volume enclosed by the surface can be summarized as [11] d dt whereC is defined as the normal projection of the surface's perpendicular speed, also named, the surface velocity. This is encapsulated (using Einstein summation convention) by the tensorial equation: where N ¼ Z i N i is the unit normal to the surface and Z i is the contravariant basis for the coordinate space. Thus, we can simplify our conservation relation: ð We can also simplify the vector surface element, dS ¼ NdS, and convert the flux term into a tensorial formation, by recognizing Finally, we will use the definition of the surface velocity and combine the two surface integral terms into one: ð We can use Gauss' divergence theorem on the surface integral term and unite all the terms under one volume integral: Using the localization theorem, the integrand must be zero inside the volume integral, and we obtain the differential form of the conservation relation: We can simplify the equation, further by considering a particular form of the flux. In this, we consider advective flux (flux due to bulk movement of an observable's mass) and diffusive flux (flux due to a concentration gradient). Using advective formulas and Fick's first law, we obtain the flux to be where v o ¼ v i o Z i is the velocity of the observable within the volume. Substituting these into the differential conservation relation, we obtain We simplify the equation, expanding the covariant derivative to obtain the final form of the conservation relation: This form can also be put into an invariant tensorial form by utilizing the invariant time derivative operator _ ∇ À Á from the calculus of moving surfaces [11]: This equation can also be put into a vector form:

Application to oxygen transport
We consider a biological application of the observable's mass transport equation to microfluidic arterial oxygen transport. In this case, o ¼ O 2 . For this, we are required to make a few assumptions: (A1) We first tentatively assume that the arteriole's surface is stationary and not. This would necessarily imply V i ⊥ Z i ¼ 0: (A2) We also consider steady-state solutions, by assuming that the density is not dependent on time. This means that ∂ ∂t ρ O 2 ¼ 0:. (A3) In addition, we restrict our studies to microfluidic environments which are in equilibrium. This would imply that the net local density change is zero, or σ ¼ 0: (A4) We also assume that the diffusion constant is a constant.
Using the above relations, we reduce the conservation relation to Both of the operators can be expanded using the Voss-Weyl formula [11] and restated in terms of partial derivatives with respect to the spatial coordinates, Z i , and the spatial metric tensor, Z ij : We assume for a moment that the coordinate system chosen is some general axial coordinate system consisting of two arbitrary coordinates, Z 1 , Z 2 À Á , and a third coordinate corresponding to the standard z coordinate found in cylindrical and Euclidean coordinate systems.
This forms a three-dimensional coordinate system of Z 1 , Z 2 , z À Á : We first assume that the velocity of the observable is only along the z coordinate. This means that the term on the left is greatly simplified: We then assume that the velocity of the observable does not depend on the zcoordinate. This means that the particle moving along a streamline parallel to the length of the tube will not accelerate. This will produce the equation: From here, if we assume the coordinates Z 1 , Z 2 À Á ¼ x, y ð Þ, then we can simplify greatly to obtain the final form: In addition, we assume that ρ O 2 does not depend on x. This produces the final equation of In addition, for all the above equations, since by Boyle's law, at a constant temperature, all instances of oxygen density can be equivalently replaced by oxygen pressure.

Governing equation for oxygen transport
As can be extrapolated from above, the general form of the nonlinear PDE for consideration defining oxygen transport in core region with Poiseuille hemodynamic flow is given by Eq. (13). The boundary conditions are shown in Section 10.1 of the Appendix, the velocity profile is shown in Section 10.2, and Figure 1 shows the geometry of the problem: There is a core region of blood flow with RBCs and plasma and a cell-free region with only plasma flowing. In the plasma region near the wall, the governing equation is as in Eq. (13), without the second term in the square brackets. In general the geometry of the problem can be either a tube or a channel, and the Laplacian is generalized in [8]. In the present work, we confine the problem to a channel flow. The blood plasma velocity is v p , and v RBC is the velocity of RBCs together with plasma in the cell-rich region. The velocity of the RBCs is lower due to the slip between plasma and RBCs. The distribution of RBCs is such that the hematocrit is higher at the center of the channel and lower near the wall. The term dSO 2 dPO 2 is the slope of the oxyhemoglobin dissociation curve and is a highly nonlinear function of oxygen tension PO 2 [1][2][3]. The dissociation curve is approximated by the Hill equation [4,6], where an empirical constant N is used and a constant P 50 appears which is the oxygen tension that yields 50% oxygen saturation. Hb T ½ is the total heme concentration, D p is the given oxygen diffusion coefficient in plasma and has units of μm 2 =s and K RBC , K p are the solubilities of O 2 in the RBCs and plasma, respectively, and have units of M=mmHg. The values of the respective parameters are taken from [8].
We thus have a PDE of the form: Choosing a separation form of PO 2 whereZ ¼Z 1 ,Z 2 indicates a semi-general coordinate system; we assume the following form of the solution: LetZ ij be the metric tensor of the coordinate system composed ofZ 1 ,Z 2 . As derived in [8], we express the Laplacian of an arbitrary function, ψZ À Á , in terms of the metric tensor in curvilinear coordinates, i.e., Applying this definition to ∇ 2 Σ À1 PL 1 þ L 2 ð Þsimilar to [8], we show that Rearranging in terms of dΣ À1 dΣ , we obtain Isolating Σ terms and allowing a zero separation constant, we obtain where L 1 6 ¼ 0 and P must obey the metric conditioñ Also, Σ À1 must obey the conditions It has been shown in [8] that for L 2 z ð Þ ¼ 0, Eq. (13) can be reduced to where  2 is a separation constant, v pZ À Á ¼ c À d r k k 2 is Poiseuille flow, and where A 1 , A 2 are arbitrary constants.
In the present work, we consider the general case where the forms are chosen for L 1 and L 2 : where C, C 1 , and m i are constants. C 1 is a free constant, C is to be determined using boundary conditions, and m i is a fixed known constant. The reason for this choice of functions L 1 z ð Þ and L 2 z ð Þ will be made apparent in Sections 4 and 5. The constant m i is chosen as in Figure 1, and  1 is a free constant.

General form of nonlinear equation
The following form of the nonlinear nonhomogeneous PDE, Eq. (25), is considered: where and G ξ, z; ϵ 2 ð Þinvolve a small parameter by choice of L 1 and L 2 and can be made arbitrarily small, whereas Þcan be made large due to choice of b 0 z ð Þ. In light of work in [7], upon substitution of F 1 and F 2 for λ as defined in Eq. (33): and where Z is defined such that where F À1 3 ¼ 1=F 3 is defined by Eq. (35), ϕ ξ, z ð Þ is defined by Eq. (34), and W Z ð Þ is defined by Eq. (32). It follows that substitution of Eq. (30) for Y into Eq. (26) will cancel part of the coefficient of the Y term leaving the first term on the right side of Eq. (30). (The second term on the right side of Eq. (30) is the simplification of the last term on the right-hand side of Eq. (25) except the F À1 3 term). Next, ε 2 terms will cancel in Eq. (26) leaving a form of the equation which is homogeneous similar to that in [7] (where we have assumed that sup z b 0 z ð Þ is large for z far down the channel), i.e.: It can be verified that as  1 ! 0 and ϵ ! 0, the solution obtained from Eq. (26) is a linear function in ξ and independent of z, and one possible solution is a decreasing function on the height of the channel, which is intuitive as the PO 2 should drop at the wall of the channel (see Figures 1 and 2). In an approximating sense, though, the other terms will contribute in Eq. (26) as shown below.
The functions λ ξ, z ð Þ,ϕ ξ, z ð Þ and function b 0 z ð Þ chosen to be large in magnitude in the supremum sense for all z values are selected so that the transformed equation, Eq.(26), through Eqs. (29)-(31) is written as one of the Painlevé classifications of the second-order differential equations [7].
There are two independent canonical forms for this equation [7], one of which is According to Estevez et al. [7], the functions must be of the form and The functions F 1 , F 2 and F 3 should satisfy the compatibility relation [7]: The previous equation, Eq.(35), after substitution of Eqs. (27)-(29) reduces considerably to the following for F 3 : The two forms of the solution for F3 (one being the coefficient of P simplified in Eq. (25) and the other Eq. (36)) are equated to each other and compared: For z large m i =2 is dropped from the total expression. The ξ 4=5 term is scaled by multiplying by a factor of 4 to obtain ξ approximately (see Figure 3).
Since z can be large downstream, even for some small  1 , then the term 12c À 7ξ drops out. This results in equality of the two forms of F3 presented. A final substitution for z with a large number multiplied with itself maintains the equality (i.e., z ! αz, for α large and positive). The solution is valid for z downstream, and we exclude the interval 0, m i ½ in Figure 1 with no inlet PO2 value specified as a boundary condition at z ¼ 0 as this would give erroneous results in the upstream region.
The final form of the general solution to Eq. (25), downstream (defined on the interval m i , m f Â Ã in Figure 1) using Eq. (30) is where C and Q are constants and ϵ ¼ À 1 ð Þ 1=n , small, for n = 5, 4, 3, 2. As an approximation, if we divide Eq. (26) by z, using Eq. (30), for large z downstream in the channel, it can be seen that the following equation emerges: term will oscillate to zero as z approaches minus infinity.
A best line of fit can be made by scaling the term 2 CM ð Þ 9=5 . Multiplying this by approximately 0.01040 results in a best approximation as shown in Figure 4. This allows us to cancel almost all z dependence in the equation except oscillating term and get a homogeneous "almost" z-independent equation as in [7]. Hence, the PDE of Eq. (26) can be reduced to an ode in ξ of similar form but homogeneous as z approaches minus infinity.

Boundary and matching conditions at wall and core-plasma interface
We employ the Robin boundary condition Eq. (58) in Appendix and derivative matching conditions at the interface of plasma and RBC core regions, respectively, at z ¼ m i and z ¼ m f , shown in Figure 1, to determine constants Q and C as well as two additional constants for linear solution in plasma layer. The solution to the linear part of Eq. (13) defined in the plasma layer, i.e., is where CylinderU and CylinderV are parabolic cylinder functions and CP 1 and CP 2 are constants to be determined.  1 is a separation constant for Eq. (42). The following boundary matching condition is utilized at the interface of the plasma layer (1 micron in height) and RBC core region (49 microns in height): at z ¼ m i and z ¼ m f , respectively (see Figure 1). Four equations in four unknowns were solved in Maple 18, where two constants are from each of the two solutions in plasma and core regions, respectively. The total of eight constants was determined and is shown in Table 1 for low and high hematocrit. The value of D p is obtained from Table 1 in [6] and c ¼ 1250 and d ¼ 0:5. For high hematocrit as shown in the Appendix, the solution incorporates different values of c and d. Also we let  1 ¼ À0:911 Â 10 À8 in the core region and  1 ¼ À0:211 Â 10 À3 the in plasma layer. Here there are two different separation constants for two different regions.
The fourth equation used was that the change in flux at the wall at the edge of the plasma layer far downstream z < < m f À Á is zero. In addition the no-flux condition shown in Figure 1 (i.e., impermeable membrane) at r ¼ 0 was satisfied exactly for all z for the core region solution. This is the fifth boundary condition. There is no inlet PO 2 specified at z ¼ 0. The oxygen tension in core is PO 2 core ð Þ¼ 33:07440049 þ 3:63804058210 À7 P which is in the form of Eq. (22) and gives a PO 2 of approximately 150 mmHg at r ¼ 0, z ¼ m i ¼ À7000 microns.

Weierstrass elliptic function
The Weierstrass P function is defined as As in [9], we consider the following expression: and derive a function of η which behaves like the Weierstrass P function at η ¼ 0 .
The development of η in the neighborhood of u ¼ 0 is Observe that the function is zero of the second order at all the points u ¼ 2μω μ ¼ 0, AE 1, AE 2, AE 3, :: ð ).
As in [9] the Weierstrass P function is shown to have the following degenerate form which is used in Section 5:

Results and discussion
The present work shows that near the inlet of the permeable membrane (see Figure 1), there is a significant drop at the wall of the channel in PO 2 as compared to downstream values as shown in Figures 2 and 5. It is worthy to note that the structure of the solution obtained in terms of degenerate Weierstrass P function forms a near constant solution across the height of a micro-fluidic channel downstream at end of the membrane region. This is seen in both Figures 2 and 5, and in Figure 5, the contours flatten out as the flow of blood proceeds far downstream. The release of ATP has been shown to be caused by a change in oxygen saturation [6]. It is concluded that there is a significant decrease in oxygen tension to the right Contour plot for channel from z 0 ¼ À7000 to z ¼ À7900 microns. Vertical axis is r variation across the channel. Horizontal axis is z variation. of the permeable membrane downstream. It is in this region that there is a significant concentration of ATP released. It is important to mention that a rapid decrease in oxygen saturation is one means to produce ATP; it is also accomplished by means of applying shear stress on the RBC as shown in [12].

Conclusion
A well-known governing nonlinear PDE used to model oxygen transport was formulated in a recent paper in a generalized coordinate system where the Laplacian is expressed in metric tensor form. A reduction of the PDE to simpler problem subject to specific integrability conditions was shown there. A reduced almost linear ode was derived, and in the present paper, a solution has been obtained using a well-known factorization method for the second-order ode where a compatibility equation has been used in equating it to a specific form of the original differential equation. Approximate oxygen tension profiles have been determined downstream in a micro-channel in the vicinity of a permeable membrane with an oxygen supply on the other side of the membrane. Although it is expected that ATP will be released as blood flows past the permeable membrane downstream, it has been shown mathematically that this is the case and increases in hematocrit produce more ATP. Future work remains to apply tensor equations for a moving arterial surface as generalized at the start of the present work.

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.