Analytic solutions of a simple advection-diffusion model of an oxygen transfer device
- Sean McKee^{1},
- Ewan A Dougall^{2} and
- Nigel J Mottram^{1}Email author
https://doi.org/10.1186/s13362-016-0019-3
© McKee et al. 2016
Received: 9 July 2015
Accepted: 5 April 2016
Published: 13 April 2016
Abstract
Artificial blood oxygenation is an essential aspect of cardiopulmonary bypass surgery, maintaining physiological levels of oxygen and carbon dioxide in the blood, and thus temporarily replacing the normal function of the lungs. The blood-gas exchange devices used for such procedures have a long history and have had varying degrees of success. In this paper we consider a simple model of a new approach to enhancing the diffusion of oxygen into the blood while it is contained in the artificial blood oxygenator. We show that using a transverse flow, which may be set up using mixing elements that we have previously shown experimentally to enhance blood oxygenation, will increase the oxygen levels within the blood. This simple model and associated analytic solutions can then be used to aid the optimisation of blood oxygenation devices.
Keywords
blood oxygenation advection diffusion model mathematical analysis numerical simulation1 Artificial blood oxygenation
During cardiopulmonary bypass (CPB) procedures, it is necessary to maintain physiological levels of oxygen and carbon dioxide in the blood. To do this, the patient is connected to a blood-gas exchange device (commonly known as a blood oxygenator) which replaces the respiratory function of the natural lungs. Such devices have a relatively long history, and by the end of the nineteenth century two different methods of organ perfusion had been developed: von Schröder [1] designed a bubble oxygenator in 1882 while in 1885 von Frey and Grubber [2] developed a rotating disc oxygenator. However, probably the first successful device was created by Gibbon [3] who in 1939 produced a vertical screen oxygenator, a variant of which was eventually employed in the first successful clinical cardiopulmonary bypass procedure in 1953 [4]. However, there was general agreement that direct contact between the gas and blood phases, which occurred in these devices, was not conducive to prolonged use surgically. Thus with the advent of polydimethysiloxane (silicon rubber), membrane oxygenators were constructed, initially using a parallel plate configuration, with alternating layers of blood and oxygen, separated by a semi-permeable membrane.
Oxygenator design was further modified by the utilisation of thin-walled hollow fibres (typically 250 μm internal diameter). This version of the oxygenator more closely resembled the capillary beds present in natural lungs. In this design, either blood or oxygen can flow down the hollow fibre, while the other medium flows outside. In the original hollow fibre devices blood was contained within the fibre lumen and they were termed intra-luminal flow (ILF) [5, 6]. However, it was found that this configuration required large areas of membrane material (approximately 5.0 m^{2} for adult CPB) and that the gas transfer efficiency was limited by the ability of oxygen to diffuse in blood. A further revision of the hollow fibre device therefore had blood flowing through a hollow fibre matrix with oxygen diffusing from within the fibre lumens. This is known as extra-luminal flow (ELF) and was found to be 3-4 times more efficient in gas transfer than ILF due to the advective enhancement of the transport process in the blood phase.
Whichever geometry of oxygenation device used, the process of oxygenation of blood is the same. Oxygen is pumped into the device across a semi-permeable membrane and so enters the blood by being dissolved in the blood plasma. This dissolved oxygen then becomes bound to haemoglobin in the red blood cells through a reaction that occurs at the cell membrane. The take-up of oxygen by the cells is, however, limited and there exists a maximum level of the concentration of bound oxygen. In order to model a blood oxygenation device it is necessary to consider the concentrations of both dissolved and bound oxygen and the effects of advection, diffusion and reaction.
There have been numerous early studies conducted into mathematically and numerically modelling this type of gas transfer (see e.g. [7–16]). These modelling studies have tended to predict greater efficiency with the use of ELF. More recent work includes that of Hewitt et al. [17] who considered an intravenous membrane oxygenator with a pulsating balloon catheter, Suitek and Federspiel [18] who focuses on CO_{2} removal, Taskin et al. [16] who considered microscale modelling of blood flow and oxygen transfer and Potkay [19] who produced a limited, but nonetheless closed form solution, thus avoiding the normally extensive numerical computation. More detailed models of gas exchange in the pulmonary capillaries have been explored by Whiteley et al. [20] (see also Whiteley et al. [21]) and Vadapalli et al. [22]. Their models, although closely related, are not directly applicable to artificial lungs (i.e. oxygenators). Furthermore, their models are restricted to a single blood cell, or, in the case of Vadapalli et al., three equidistant blood cells. Other related work on oxygen transfer in blood includes that of Caputo et al. [23] and Formaggia et al. [24] and the references therein although this work largely concentrates on rectilinear flows and considers the nonlinear aspects of the oxygen binding-unbinding reaction kinetics. There is also a vast literature on the analytic and numerical modelling of the flow of blood in physiological situations, for instance with arterial branching and compliant arterial walls (see [25] and references therein). However, in our present situation we are considering an artificial oxygenator with a single capillary and non-compliant walls and therefore significant simplification of the mathematical model is possible.
The blood was heated to 37° and the oxygen levels were raised to physiological venous levels and measured. As indicated in Figure 2, a syringe pump (60 ml JMS SP-100) was used to drive the blood through the membrane. Blood samples were taken at the device entrance and exits via three-way taps. These samples were used to measure the oxygen partial pressure levels (using a Siemens RAPIDLab 248 blood/gas analyser) and the haematocrit and oxygen saturation of the blood (using a Hawksley Micro Haematocrit Centrifuge and a Radiometer OSM-2 Haemoximeter) before and after the blood had passed through the oxygenator. From these measuements the increase in blood oxygen fractional saturation (S, defined below in Section 2) could be calculated. The oxygenator was ventilated with 100% oxygen, at a flow rate of 30 ml min^{−1}. Further experimental details are included in [26].
As testing was performed on single capilliaries, blood flow rates remained at low levels. The blood flow rates were varied between 1 ml min^{−1} and 4 ml min^{−1} in 1 ml min^{−1} increments. Preliminary experiments were conducted with water instead of blood and conclusively showed that the presence of the mixer enhanced the levels of dissolved oxygen. A three dimensional numerical study, included the mixing elements, using COMSOL Multiphysics [27] was also undertaken and agreed very well with the experimental results [26]. It was found that, with no mixing element present, the oxygen transfer from the membrane towards the centre of the lumen was due to diffusion only. However, with a mixing element inserted, the introduction of a transverse flow velocity, from the region of high oxygen concentration at the membrane to low oxygen concentration away from the wall, enhanced the diffusion process and led to greater oxygen levels, of up to 100%, by the time the water had passed through the device. The scaling of this effect was tested by introducing different lengths of mixing elements and, rather than being a mixing phenomena which increased the effective diffusion constant, it was found that the increased oxygenation was due to the advection of oxygen saturated blood from the membrane wall to the centre of the capillary [26].
In this paper, inspired by these results, we consider a much more general model of transverse flow enhanced oxygenation, for a planar rather than a cylindrical device. The model presented below is therefore a significant simplification of the experiment described above but contains the essential features: the dominant flow of a liquid in one direction; with a semi-permeable membrane introducing oxygen through diffusion in a direction perpendicular to the flow; together with a transverse flow from the membrane into the main flow region. Through the use of Laplace transforms, analytic and asymptotic solutions are obtained to the resulting advection-diffusion problem. Numerical solutions are obtained for a more complete model, and compared to the analytic and asymptotic solutions. Good agreement between the numerical and analytical solutions suggests the simplifying assumptions are justifiable and leads to the possibility of prediction and optimisation for one type of blood oxygenation device.
2 Mathematical modelling
Equations of this type have been used to model a vast array of real-world problems where the concentration of a dissolved substance evolves in time because of diffusion, advection by the flow of the background liquid, and the point-wise increase or the decrease of the concentration level from a source/sink term. Of particular note in the present context, are those applications where the introduction of the substance occurs at a semi-permeable boundary. In soil science, for instance, the rate at which a chemical constituent moves through soil is determined by several transport mechanisms: advection, diffusion, dispersion, adsorption and zero-order or first-order production and decay. Van Genuchten and Alves [28] have provided a substantial compendium of analytic solutions to a wide range of problems in soil science. A more recent paper by Yuan and Lu [29] deals with an analytic solution to vertical flow in unsaturated, rooted soils with variable surface fluxes; the article contains an extensive list of references. Another relevant area is the transport of particulate matter (pollution) through the atmosphere with deposition at ground level: McKay et al. [30], for instance, developed an analytic solution through Laplace transforms and contour integration. More recently, interesting analytic solutions have been obtained by Kumar et al. [31] and Jaiswal and Kumar [32] for the one-dimensional advection-diffusion equation with a variable diffusion coefficient and a variable velocity in both a finite and infinite domain. In a biological setting, an application also arises from drug diffusion through tissue. In the case of drug-eluting stents the pressure difference between the lumen and the surrounding tissue means that advection also plays a role. McGinty et al. [33] produced an analytic solution; mathematically this was interesting as the inverse Laplace transform contained three branch points. Pontrelli and de Monte [34], on the other hand, considered the problem of drug diffusion through multiple layers and obtained analytic solutions using a separation of variables approach.
This short list of similar systems in which a concentration influx occurs at the boundary (i.e. from the soil surface or drug-eluting stent) leads to a type of mixed boundary condition that is similar to the situation we consider in this paper. Here we find that this set of equations is in fact analytically solvable through the application of Laplace transforms and can be simplified in some limiting, but physically relevant, cases to provide information which may be used to inform device construction and optimisation.
In almost all the previous studies of such a system mentioned above [7–16, 36] the problem is solved numerically, and some assume the flow in the direction normal to the main pressure-driven flow is small, i.e. \(\bar{u} = 0\). We shall not make this assumption. Rather, we shall make the very reasonable assumption that bound oxygen is primarily transported by advection, rather than diffusion, that is, \(D_{b} = 0\).
To simplify the analysis of this equation we will non-dimensionalise appropriate variables and parameters. We will use the lengthscale L introduced in equation (15) (and which will later be specified as a typical device width) to rescale both \(x=L\tilde{x}\) and \(z=L\tilde {z}\). The partial pressure could be scaled with a number of quantities but we choose to scale it with \(P_{\max} = c_{\max}/\alpha_{\mathrm{O}_{2}}\), the effective maximum partial pressure of bound oxygen, so that \(c = P_{\mathrm{O}_{2}}/P_{\max}\).
Through these scalings the following non-dimensional parameters appear: the constant ratio of typical x-component to z-component flow speeds, \(\nu= \bar{u}/\bar{w}\); a ratio of diffusive and advective terms, \(\delta= D_{d}/L\bar{w}\); and the rescaled partial pressures, \(c_{0.5} = P_{0.5}/P_{\max}\), \(c^{*} = P_{g}/P_{\max}\), \(c_{0} = P_{i}/P_{\max}\).
Values of dimensional and non-dimensional parameters
Dimensional Parameters | Symbol | Value | Dimensions |
---|---|---|---|
Geometry | |||
Lengthscale of device | L | 2 × 10^{−3} | m |
Oxygen/Haemoglobin | |||
Diffusion coefficient | \(D_{d}\) | 1.5 × 10^{−9} | m^{2}s^{−1} |
Solubility of \(\mathrm{O}_{2}\) in blood | \(\alpha_{\mathrm{O}_{2}}\) | 9.98 × 10^{−6} | mol m^{−3}Pa^{−1} |
Oxygen binding capacity | \(\beta_{\mathrm{O}_{2}}\) | 5.98 × 10^{−2} | mol kg^{−1} |
Haemoglobin concentration | \(c_{Hb}\) | 120 | kg m^{−3} |
Maximum bound concentration | \(c_{\max}\) | 7.176 | mol m^{−3} |
Max. part. press. of bound conc. | \(P_{\max}\) | 7.19 × 10^{5} | Pa |
Part. press. of half saturation | \(P_{0.5}\) | 3.2 × 10^{3} | Pa |
Index constant | n | 2.8 | |
Boundaries | |||
Average longitudinal velocity | w̄ | 4 × 10^{−5} | m s^{−1} |
Average transverse velocity | ū | 1 × 10^{−6} | m s^{−1} |
Inlet \(\mathrm{O}_{2}\) partial pressure | \(P_{i}\) | 5 × 10^{3} | Pa |
Membrane \(\mathrm{O}_{2}\) partial pressure | \(P_{g}\) | 20 × 10^{3} | Pa |
Non-dimensional parameters | Symbol | Value |
---|---|---|
Oxygen/Haemoglobin | ||
Ratio of diffusion/advection | δ | 1.875 × 10^{−2} |
Boundaries | ||
Ratio of velocities | ν | 0.025 |
Scaled inlet \(\mathrm{O}_{2}\) partial pressure | \(c_{0}\) | 0.70 × 10^{−2} |
Scaled membrane \(\mathrm{O}_{2}\) part. press. | \(c^{*}\) | 2.78 × 10^{−2} |
Scaled part. press. of half sat. | \(c_{0.5}\) | 0.45 × 10^{−2} |
Wall Sherwood number | Sh | 50 |
Inlet correction parameter | ϵ | 1 × 10^{−3} |
2.1 A simplified model
We shall make two further assumptions so that we can employ the analytic expressions of the next section. These assumptions are justified below and also allow us to obtain useful qualitative (and quantitative) results which, possibly surprisingly, compare favourably with the numerical results of the model for the oxygenator device.
We first assume that diffusion in the z-direction, parallel to the membrane, is negligible in comparison with advection in that direction. This is a common assumption in such devices due to the speed at which blood must be flowed through the device in order to mimic the flow in a human body.
3 Analysis of the advection-diffusion problem
In this section we shall employ Laplace transforms to obtain an analytic solution to the model specified by Eqs. (22-25); in addition some asymptotic forms are provided.
Lemma 1
Lemma 2
The first of these Lemmas is standard and may be found in, for example, [39] while the second is given by McGinty et al. [40].
3.1 Large Sherwood number
We see that the simplified model produces a solution that is very close (\(<0.0025\%\) error for the region we consider) to the more complete model. We can therefore conclude that the simplifying assumptions, that diffusion in the z-direction is neglected and that in the x-direction is constant, are reasonable for our set of parameter values. Even in the large Sherwood approximation the error is small if we are sufficiently far away from the point \(x=0\), \(z=0\), i.e. less than 10% error for \(x>0.15\), \(z>0.5\).
There are two further limits that we will explore: considering the concentration profile at the membrane, where \(x=0\), and close to the inlet at \(z=0\).
3.2 Near to the membrane: \(x=0\)
3.3 Near to the inlet: small z asymptotic expansion
3.4 Large x and z asymptotics
4 Device optimisation
An oxygenation device must re-oxygenate the blood entering the inlet by the time it leaves at the other end of the device. The longer the device the greater the amount of oxygen that can enter the blood. However, more compact devices are preferred, particularly if they are to be implanted, or portable. We would therefore like to reduce the length of the device while maintaining the oxygenation levels. We will concentrate on the oxygenation level at a fixed value \(x=1\), i.e. we consider a device of a nominal width of L.
If we assume that the device must increase the blood oxygenation level at all points of the outlet to at least 90% of the saturation level of blood then we must achieve \(S=0.9\) at \(x=1\). We denote the distance through the device at which \(S=0.9\) is achieved as \(z_{c}\). Figure 9(b) shows the dependence of \(z_{c}\) on the transverse flow ratio. We see that even moderate values of transverse flow will significantly reduce the necessary length of the oxygenation device.
5 Conclusions
In this article we have provided a brief history of the artificial lung, or oxygen transfer device and described a mathematical model consisting of two coupled advection-diffusion-reaction equations. A number of simplifications and assumptions were made reducing the problem to a single stationary advection-diffusion equation which we then showed could be solved analytically using Laplace transforms and convolution. Some special cases were considered: large Sherwood number; solution close to the membrane (\(x = 0\)); solution near to the inlet (small z); and large x and z asymptotics. These results were then compared with a numerical solution of the original model and good agreement was obtained, justifying the assumptions and simplifications made. Thus predictions and some degree of device optimisation have been proposed as useful applications of the theoretical model.
Declarations
Acknowledgements
This work was partially funded by the UK Engineering and Physical Sciences Research Council Doctoral Training Centre in Medical Devices at the University of Strathclyde. The supervision and guidance of the late Dr John Gaylor, from the Department of Bioengineering at Strathclyde, was invaluable during this project.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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