Approximately, 98% of the oxygen carried in blood is chemically bound to hemoglobin that resides inside red blood cells (RBCs). Transferring oxygen to/from blood requires, therefore, the transport of oxygen to/from the RBCs. In the natural lungs, this transport to the RBCs is facilitated by the fact that the blood-side passages, the capillary vessels, are 5–10 μm in width and the RBCs, being of the same size, pass single file next to the exchange membranes, Figure 1A. Transport distances are on the order of 3–5 μm. The blood-side passages in contemporary artificial lungs, on the other hand, are on the order of 200–300 μm wide, Figure 1B, and efficient gas transport can be achieved only with designs that induce transverse blood mixing that bring all RBCs close to the exchange surfaces. Most current devices consist of a packed bed of hollow fibers in which blood is transversely mixed by flowing through the tortuous spaces outside the fibers, gas flows inside the hollow fibers and gas exchange occurs across the fiber walls. The present work theoretically explores the potential of various strategies for constructing multiple microchannels, 10–40 μm in width, that can be used for artificial lungs. Such small blood-side passages would reduce the need for the transverse mixing and should result in efficient, compact artificial lung devices with very low pressure drops. Another advantage of small passages is the fact that the smaller the passage, the greater the ratio of the exchange surface area to blood volume. For a circular cross-section, for instance, the surface area to volume ratio varies inversely with the diameter.
Designing an artificial lung with blood-side passages that are on the order of 10 μm in width presents two major challenges: 1) how to configure a system that has anywhere from tens of thousands to possibly hundreds of millions of small, uniform channels together with a system that effectively delivers oxygen and removes carbon dioxide to/from the RBCs flowing within the microchannels and 2) how to assure the microchannels are blood compatible and do not plug with thrombus. A system of effective microchannels would not be practical if it is not blood compatible, but neither are blood compatible microchannels practical if they cannot be assembled into a feasible and effective system. The present work attempts to address the first issue by theoretically examining the effectiveness of four different configurations of blood microchannels that can be used in artificial lungs.
Natural blood capillaries are often thought of as circular channels embedded in tissue. The capillary blood space in the lungs, however, also has been characterized “as a sheet of fluid flowing between two membranes held apart by a number of more or less equally spaced ‘posts'.”1 For artificial lungs, two basic blood-side geometries, thus, may be considered for the microchannels: circular and rectangular. A circular cross-section has the advantage of being structurally stable if, as is desirable in artificial lungs, the intralumen pressure is greater than the extralumen pressure. A circular cross-section, however, has the disadvantage of having the least amount of surface area per unit volume relative to other shapes. A rectangular cross-section, on the other hand, provides more exchange surface area per unit volume but, if the walls are compliant, requires internal or external structural support.
Any effective configuration of microchannels must meet certain hemodynamic, gas transport, and size specifications: 1) The blood-side pressure drop, ΔP, should be low. 2) Shear-induced blood trauma should be avoided by setting limits on the level and exposure time of shear stresses. 3) Both the number of required microchannels and the overall physical size of the structure holding the microchannels should be as small as possible. 4) The configured array of microchannels should be able to raise the hemoglobin saturation level from venous conditions, say 66%, to arterial conditions, say 98%.
Page et al.2,3 experimentally studied the oxygenation of both blood (hct = 30%) and hemoglobin solutions (10 gm/dl) flowing in single circular microchannels embedded in silicone rubber slabs that were surrounded by an oxygen atmosphere. They investigated a variety of flows in 10 and 25 μm circular microchannels and determined that blood oxygenation depends on the residence time, tr, in the microchannel. Their results suggest that the hemoglobin saturation of whole blood will rise from a venous saturation of about 66% to an arterial saturation of 98% in about 0.15 seconds flowing in 10 μm channels and in about 0.75 seconds flowing in 25 μm channels. The following calculations are based on these results and assume that the blood residence time is crucial.
Four cases are considered below: open circular channels imbedded in gas-permeable sheets, open rectangular channels, wide rectangular channels with support posts, and screen-filled rectangular channels. The latter three are bounded top and bottom with silicone rubber membranes. For open circular channels, it is assumed that venous blood will be arterialized with a residence time of 0.20 seconds in 12 μm circular channels and 0.75 seconds in 25 μm channels. Lacking any direct data for oxygenation in rectangular channels, these same required residence times of 0.20 and 0.75 seconds are assumed for blood flow in open rectangular channels that are 12 and 25 μm high, respectively. Similar residence times can be expected in the rectangular channels with support posts. Some blood-side mixing can be expected in screen-filled channels and shortened required times can be expected. The height of screen-filled microchannels, however, is somewhat dictated by the availability of appropriate screens and a 40 μm height is considered.
An example of the first strategy, constructing open circular channels in gas-permeable sheets material, is illustrated in Figure 2. The gases would transfer to/from the blood by diffusion from gas channels running along the edges of the sheets in a direction transverse to the blood channel axes. A variety of techniques may be used to construct the multiple microchannels. One method for creating such microchannels consists of imbedding a bundle of microfibers in a block of gas-permeable material, slicing the block into thin sheets, chemically removing the microfibers and leaving the circular microchannels in the sheet material. Two possible template materials are optical glass fibers and polylactic acid yarn fibers. Our experiments using this technique have shown that it is possible to have a 12 μm microchannel density, ρc, of about 250,000/cm2 (about 28% of the area) and a 25 μm microchannel density of about 60,000/cm2 (about 29% of the area).
The acceptable width, W, of the sheet strips, Figure 2, would be determined by the ability to transversely transport oxygen through the support material to the blood flowing in the microchannels relative to the oxygen required by that blood. The ability to transport oxygen will depend on the effective permeability, 𝒫 or mass transfer resistance, ℜ = 1/ 195 of the support material, the cross-sectional area of the strip, the half width of the strip, and the partial pressure difference, δ(pO2), from the gas channel to the center of the strip. The rate at which oxygen can be transversely transported per centimeter of the strip is thus calculated: 2𝒫Lδ(pO2)/(W/2). The required rate at which the oxygen must be delivered depends on the blood volume flow rate per channel, Qc, the number of channels in a strip, ρcW, and the mmoles of O2/(ml of blood), co2, required to raise the hemoglobin saturation level from, say, 66%–98%, i.e., QcρcWco2. Two types of support material matrices could be considered: a solid material or a material with gas-filled, connected open pores. Silicone rubber is the logical choice for a solid material; its permeability to oxygen and carbon dioxide is at least an order of magnitude greater than that of virtually all other possible choices and it is reasonably blood compatible. A gas-filled porous matrix could have a structure material that is essentially impermeable to the gases or one that is permeable, such as silicone rubber. For a solid support material with embedded microchannels, the effective permeability is calculated as a weighted average of the permeability of the support material and the permeability of plasma in the channels. For an open-pore matrix, the effective permeability is calculated as a weighted average of the permeability in the open pores, assuming the support material is impermeable, and in the plasma in the microchannels. The permeability (diffusivity × solubility) values to oxygen used in the calculations are 2.60 × 10−10, 2.78 × 10−11 and 9.43 × 10−6 mmoles/(sec cm2 (mm Hg/cm)) in silicone rubber,4 plasma,5,6 and oxygen,7 respectively.
An example of the second strategy, constructing open rectangular channels with thin, gas-permeable membrane walls, is illustrated in Figure 3. The gases would transfer to/from the blood by diffusion from transverse-running gas channels on the other side of the membrane walls. Such microchannels can be developed using photolithographic methods. A positive photoresist can be employed to prepare a patterned silicon wafer that is used as a master for an imprinted silicone membrane.8 The microchannel height, H, and the thickness of patterned membrane, Tpm, depend on the speed and time during spin coating of the photoresist and polydimethylsiloxane (PDMS) solution, respectively. The height can be adjusted in the range of 10–40 μm and the thickness in the range of 50–300 μm. Unsupported straight open blood-side channels would necessitate fairly stiff walls and relatively small channel widths.
The ability to transfer oxygen depends on the membrane thicknesses, Tm and Tpm, mass transfer resistance of the membranes, ℜm, the gas-side oxygen partial pressure, the plasma mass transfer resistance, ℜp, and the blood channel length, width and height. Assuming that membrane thicknesses ranges from 25 to 400 μm, and that the oxygen must transfer through the membrane and through plasma to the center of the blood channel, the potential rate of oxygen transfer is δ(pO2)(WL)/(ℜmTm + ℜp(H/2)) + δ(pO2)(WL)/( ℜmTpm + ℜp(H/2)). The required rate of oxygen transfer per channel depends on the rate of blood flow in each microchannel, the hemoglobin concentration, and the entering venous blood saturation, i.e., the required oxygen uptake per ml of blood: Qcco2.
An example of the third strategy, constructing broad open rectangular channels with support posts, is illustrated in Figure 4. The internal support posts provide the structure needed to assure channel stability and channel height uniformity. The channels with such a design can be of unlimited width. As with the straight open rectangular channels, broad open channels with posts can be constructed using the photolithographic methods.8 The technique could be used, for instance, to construct a system that essentially reproduces the idealized model of the natural lung proposed by Fung and Sobin.9 This model idealizes the endothelial membranes as two parallel flat plates and the observed, somewhat irregular, posts as uniform, circular cylinders arranged in a doubly periodic pattern, Figure 4A. Lee10 calculated a friction factor, f, for this model as a function of dimensionless parameters ε, H/a, and ω1/ω2, in which ε = void fraction, H = distance between the parallel plates, a = the post diameter, ω1 = distance between the centers of the posts in the ×-direction, and ω2 = distance between the centers of the posts in the y-direction. The void fraction ε = 1 − πa2/(2ω1ω2) and the pressure gradient dp/dx = 12fμV/H2.
The ability to transfer oxygen depends on the membrane thicknesses, Tm and Tpm, mass transfer resistance of the membranes, ℜm, the gas-side oxygen partial pressure, the plasma mass transfer resistance, ℜp, and the blood channel length, width and height. Assuming that the oxygen must transfer through the membrane and through plasma to the center of the blood channel, the potential rate of oxygen transfer is δ(pO2)(WLε)/(ℜmTm + ℜp(H/2)) + δ(pO2)(WLε)/(ℜmTpm + ℜp (H/2)). The required rate of oxygen transfer per channel depends on the rate of blood flow in each microchannel, the hemoglobin concentration, and the entering venous blood saturation, i.e., the required oxygen uptake per milliliter of blood: Qcco2.
An example of the fourth strategy, constructing screen-filled wide rectangular microchannels, providing structural support, channel uniformity, and some transverse blood mixing, is illustrated in Figure 5. Such screen-filled channels also could be of unlimited width; short length, very wide channels are possible. An accurate estimate of the pressure drop is difficult in this case, but a rough estimate can be made using the Ergun equation for flows in a packed bed, assuming the packed-bed particles are the wires of the screen, i.e., cylinders of diameter d (See Appendix). Because the Reynolds number for these flows is very low, less than one, only the first term of the Ergun equation is necessary. The ability to transfer oxygen would be enhanced by the blood-side mixing induced by the screens, and the required residence time should be somewhat less than that in the corresponding open channel. An estimate for the required residence times is based on experimental results reported.11 The authors used a 40 μm channel filled with a screen that resulted in an effective channel void fraction, ε, of 0.80. The screen wires have a diameter, d, of 0.0018 cm. With oxygen transport from only one side, one of their results shows a saturation increase from 0.68 to 0.94 with a residence time of 0.52 seconds. Because the required saturation increase should be 0.66–0.98, a residence time of, say, 0.80 seconds might be an estimate for this one-sided transport. For two-sided transport, as depicted in Figure 5, a residence time of 0.40 seconds can be assumed.
The ability to transfer oxygen depends on the membrane thicknesses, Tm, mass transfer resistance of the membranes, ℜm, the gas-side oxygen partial pressure, the plasma mass transfer resistance, ℜp, and the blood channel length, width, and height. Assuming that the oxygen must transfer through the membrane and through plasma to the center of the blood channel, the potential rate of oxygen transfer is δ(pO2)(2WLε)/(ℜmTm + ℜp(H/2)). The required rate of oxygen transfer per channel depends on the rate of blood flow in each microchannel, the hemoglobin concentration, and the entering venous blood saturation, i.e., the required oxygen uptake per milliliter of blood: Qcco2.
The potential and tradeoffs of the four designs for microchannel artificial lungs are determined by some relatively simple calculations. The assumed specifications are that the 1) device should be capable of oxygenating a total blood flow rate, Qb, of 4 L/min, 2) blood has a hematocrit of 30%, 3) pressure drop should be 10 mm Hg or less, and 4) shear-induced trauma should be avoided. The range of channel lengths, L, considered in all cases is from 0.25 to 5.00 mm. A blood with a 30% hematocrit would be expected to have an apparent viscosity, μ, of about 2.5 cP in large channels. In the small channels considered in these calculations, however, the viscosity would be reduced because of Fahraeus-Lindquist effects12 and an apparent viscosity of 2 cP is assumed. An array of shear stresses and exposure times will exist in most flows, but an estimate of the potential for blood trauma may be calculated by using an average exposure time and an average shear stress. One summary of research on shear-induced hemolysis and platelet activation13 suggests that trauma can be minimized if τ2 − 900/t − 225 <0, in which τ = shear stress in dynes/cm2 and t = exposure time in seconds. The potential for blood trauma, thus, is estimated by calculating a trauma factor T = τ2 − 900/t − 225, which should be less than zero to be acceptable. The average shear stress used in the calculations is the root-mean-square value and the average exposure time is the average transit time. The root-mean-square shear stress, related to the rate of mechanical energy dissipation,13 can be calculated from the viscosity, pressure drop, flow rate and blood volume: [μ(ΔP)Q/V]1/2, in which μ = viscosity, ΔP = pressure drop, Q = flow rate, and V = fluid volume.
The hemodynamics of flow in each of the four cases is calculated as follows: 1) Assume a microchannel diameter or height and the range of microchannel lengths, L. 2) Assume the required residence time, tr, to fully oxygenate venous blood. 3) Calculate the average blood velocity, V = L/tr, and blood flow rate in each channel, Qc. 4) Calculate the pressure drop, ΔP, root-mean-square shear stress, τ, and the trauma factor, T. 5) Calculate the required number of microchannels, n = Qb/Qc, and the total physical size. The equations for the latter calculations are given in Table 1.
Circular Microchannels Imbedded in Gas-permeable Sheets
Figure 6 shows theoretical hemodynamic tradeoffs for circular microchannels imbedded in strips. The channel length and specified residence time determine the velocity and flow rate in the channels, which, in turn, determine the pressure drop, estimated shear stress and the required number of channels. Figure 6A shows the pressure drop as a function of channel length, and, as expected, the pressure drop with 25 μm channels is significantly less than that with 12 μm channels. If, for instance, 10 mm Hg is a specified maximum allowable pressure drop, the 12 μm channels would have to be 0.8 mm or less in length, whereas the 25 μm channels could be as long as 3.1 mm. Figure 6B shows that blood trauma would be avoided in the 12 μm channels if the channel length is 1.45 mm or less, whereas 25 μm channels could be 5 mm or longer.
Figure 7 shows that the number of required microchannels significantly decreases as the length of the microchannel increases from 0.25 to 2.0 mm and is only slightly greater for the 12 μm channels than for the 25 μm microchannels for any specific channel length. If, as indicted above, pressure loss restrictions limit the microchannel length to 0.8 mm or less for the 12 μm microchannels whereas the length could be as long as 3.1 mm for the 25 μm microchannels, the number of required microchannels would be much larger for the former than for the latter: about 140,000,000 vs. 35,000,000. The required total size is just 57 ml for 12 μm microchannels and 207 ml for 25 μm microchannels. The blood prime volume is 13 and 50 ml for the 12 and 25 μm channels, respectively.
Figure 8A shows the ratio of the calculated required rate of oxygen transfer to the rate of delivery as a function of strip width for the case with a solid silicone rubber support material. Assuming that the plasma takes up 28% of the area, the area-based effective weighted average permeability is 1.94 × 10−10 mmoles/(sec cm2 (mm Hg/cm)). Widths ranging from 0.5 to 10 mm are considered. The narrower the width, the greater the rate of oxygen delivery and the lesser the rate required. Even with a width of only 0.5 mm with 25 μm channels, however, the possible rate of delivery is four times less than the rate required to oxygenate the blood. For the 12 μm channels, the ratio of the required rate to the supplied rate is about 14 for the 0.5 mm width. Widths <0.5 mm do not seem practical. Figure 8B, on the other hand, shows the same ratio for an example of an open-pore support matrix. The assumption in this case is that the support matrix material is impermeable to oxygen but that 25% of the matrix is open, gas-filled pores that are interconnected. The weighted average permeability, assuming 25% open-pore matrix and a 28% channel area, is about 2.11 × 10−6 mmoles/(sec cm2 (mm Hg/cm)). The rate of oxygen delivery would be adequate for strip widths of 14 mm or less with 12 μm microchannels and 13 mm or less with 25 μm microchannels.
Open Rectangular Microchannels with Gas-permeable Walls
Figure 9 shows theoretical hemodynamic tradeoffs for straight, parallel rectangular microchannels that are 12 or 25 μm high and 0.3 mm wide. Figure 9A shows that the pressure drop in the 12 μm channels would be <10 mm Hg if the channel length is 1.25 mm or less. The pressure drop in 25 μm microchannals, on the other hand, is <10 mm Hg for all channel lengths considered. Figure 9B shows that the trauma factor in 12 μm channels is negative if the channel length is 2.3 mm or less, whereas it is always negative for all lengths considered in the 25 μm channels.
Figure 10 shows that the number of required microchannels sharply decreases as the channel length increases from 0.25 to 1.3 mm. The required number of 12 μm microchannels would be about 2,900,000 if the channel length were its maximum, as required by pressure drop considerations, of 1.25 mm, whereas the required number of 25 μm microchannels would be about 1,340,000 if the channel lengths were 5.0 mm. The required total physical size is 300 ml for 12 μm microchannels that are 1.25 mm long and 583 ml for 25 μm microchannels that are 5.0 mm long. The blood prime volume is 13 and 50 ml for the 12 and 25 μm channels, respectively.
The ability to deliver oxygen to the flowing blood will depend, among other things, on the thickness of the silicone rubber membranes. Membrane thicknesses ranging from 25 to 400 μm were considered. Figure 11 shows that the rate of delivery of oxygen would exceed the rate at which oxygen is required if silicone rubber membranes of 230 μm thickness or less were used with the 12 μm high microchannels, but membranes could be as thick as 400 μm with the 25 μm microchannels.
Broad Open Rectangular Spaces with Support Posts
If, instead of completely open rectangular channels, a patterned membrane with periodic posts, such as that shown in Figure 4, is used, the channels have internal support and the width can be extended without limitations. The posts provide channel stability and uniformity and possibly induce some transverse blood mixing that would reduce the blood-side resistance to mass transfer. Because the mixing more likely is longitudinal rather than transverse, however, no transverse mixing was assumed in this study.
Figure 12 shows theoretical hemodynamic tradeoffs for broad open rectangular spaces with support posts that are 12 or 25 μm high and 15 mm wide. Figure 12A shows that the pressure drop in the 12 μm channels would be <10 mm Hg if the channel length is 1.0 mm or less. The pressure drop in 25 μm microchannals, on the other hand, is <10 mm Hg if the channel length is 3.9 mm or less. Figure 12B shows that the trauma factor in 12 μm channels is negative if the channel length is 1.8 mm or less, whereas in 25 μm channels it is negative for all the channel lengths considered.
Figure 13 shows that the required number of microchannels decreases sharply as the channel length increases from 0.25 to 1.5 mm. The required number of 12 μm microchannels would be about 87,000 if the channel length were its maximum, as required by pressure drop considerations, of 1.0 mm, whereas the required number of 25 μm microchannels would be about 40,000 if the channel length were 3.9 mm. The required total physical size is 252 ml for 12 μm microchannels that are 1.0 mm long and 453 ml for 25 μm microchannels that are 3.9 mm long. The blood prime volume is 13 and 50 ml for the 12 and 25 μm channels, respectively.
The ability to deliver oxygen to the flowing blood will depend, among other things, on the thickness of the silicone rubber membranes. Membrane thicknesses ranging from 25 to 400 μm were considered. The results would be the same as those for the open rectangular channels without posts and are shown in Figure 11. The rate of delivery of oxygen would exceed the rate at which oxygen is required if silicone rubber membranes of 235 μm thickness or less were used with the 12 μm high microchannels, but membranes could be as thick as 400 μm with the 25 μm microchannels.
Screen-filled Rectangular Microchannels with Gas-permeable Walls
Figure 14 shows theoretical hemodynamic tradeoffs for an example of screen-filled rectangular microchannels. The case considered assumes the channels are 40 μm high and 15 mm wide. With the screen support, the channel width can be of arbitrary size, and with the induced transverse mixing, the channel length can be shortened, both favorable characteristics. The screen, with a void fraction, ε, equal to 0.80, is assumed to have the dimensions of that used by Kung et al.11Figure 14A shows that the pressure drop would be <10 mm Hg if the channel length is 4.6 mm or less. Figure 14B shows that the trauma factor is negative for all channel lengths considered.
Figure 15 again shows that the number of required microchannels decreases as the channel length increases. The required number microchannels would be about 12,120, assuming the pressure drop maximum channel length of 4.6 mm. The required total physical size is 266 ml for 40 μm microchannels that are 4.6 mm long. The blood prime volume is 27.0 ml.
These calculations are based on the characteristics of the screen used by Kung et al.11 If a wire screen of higher opening ratio, i.e., higher void fraction, is available, the pressure drop and trauma factor would be less. For example, if the void fraction is increased from 0.80 (the screen used by Kung et al.11) to 0.86 (by using screen with smaller wire diameter, i.e., 15 μm instead of 18 μm), the pressure drop would be <10 mm Hg for the lengths considered here and the trauma factor also would be decreased, Figure 14. The number of required channels would be 8,600 and total size required would be 247 ml for 6.0 mm long channels. The blood prime volume is also 27 ml for the 40 μm channels with the looser screen.
Figure 16 shows that the rate of delivery of oxygen would exceed the rate at which oxygen is required if silicone rubber membranes of 86 and 120 μm thickness or less were used with screen void fractions of 0.80 and 0.86, respectively.
Tables 2–5 summarize the results of sample calculations, based on the assumption that the maximum allowable pressure drop is 10 mm Hg, for the four cases considered. Setting the maximum allowable pressure drop to, say, 5 mm Hg would produce a different set of altered results, most of which can be determined from the provided figures. Table 2 indicates that a connected open-pore support matrix material is essential for circular microchannels imbedded in gas-permeable sheets. Table 3 is based on the assumption that the straight, open rectangular channels are 0.3 mm wide. Tables 4 and 5 are based on the assumption that the rectangular channels are 15 mm wide. The width of the straight, open rectangular channels is limited by the physical stability of the channel. The width of the rectangular channels with support posts or the screen-filled rectangular channels, on the other hand, is completely open to the designer and the effect of having wider channels than those assumed would result in a reduced number of required channels.
The purpose of this study is to address the question of the feasibility of developing artificial lungs with blood-side channels that are on the order of 10–40 μm in height. Such designs lend themselves to devices that have efficient gas transfer, low blood prime, and low pressure drops. The calculations are based on the observations2,3 that erythrocyte oxygenation in microchannels depends on transit time in the channel. In the examples considered, the assumptions are that the blood flow rate is 4 L/min, the hemoglobin concentration is 10 gm/dl, the design is limited by the requirements for oxygen transfer and the hemoglobin saturation should increase from 66% to 98%. The different schemes considered are compared by specifying that the pressure drop should be no more than 10 mm Hg and that shear-induced blood trauma should be controlled. The designs in all cases considered are limited by the pressure drop limit and not blood trauma restrictions. The value of 10 mm Hg is a reasonable, but completely arbitrarily chosen. Other pressure drop limits, both lower and higher, could be considered with, perhaps, altered conclusions. Limiting the pressure drop to a lower value would have the effect of shortening the channels, but increasing the required number. Allowing a larger pressure drop would have the opposite effect on channel length and number, but also could force the design to be limited by shear-induced trauma considerations. The physical size, on the other hand, is independent of channel length. The channel dimensions considered also are somewhat arbitrary, but are reasonable sizes that can be implemented and illustrate the approximate scale of the issues.
The width of the three rectangular channel configurations is arbitrary and may have to be determined, perhaps, by other practical design considerations such as the ability to assure uniform channel height over wide expanses or overall size and shape constraints. The required number of channels would be inversely proportional to the selected channel width if the wall width between channels is small compared to the channel widths themselves. For stability, the open, straight rectangular channels would have to be fairly narrow and a 0.3 mm width was chosen with wall widths of 0.2 mm. The wall widths are 67% of the channel width in that case. The rectangular channels with support posts and the screen-filled channels, on the other hand, could be quite wide and widths of 15 mm were arbitrarily chosen. The wall widths in these cases are small compared to the channel width and as a first approximation could be neglected. The 15 mm wide screen-filled design required 12,120 channels with the ε = 0.80 screen. If the channel width were only 0.3 mm, for instance, the required number of channels would be about 605,750 for these screen-filled channels, compared to the required 1,340,000 for the 25 μm straight open channels. If, on the other hand, the width of the screen-filled channels were 50 mm, the required number of microchannels would be only 3,640.
Table 6 is a comparison of the eight configurations considered in this study. The smaller microchannel results in smaller overall physical size in every case considered. For the circular channels imbedded in a gas-permeable material, devices with 12 μm channels are only 27% as big as those with 25 μm channels. For the straight open rectangular channels and the open rectangular channels with support posts, devices with the 12 μm channels are <60% as big as those with 25 μm channels. Devices with 12 μm circular microchannels imbedded in gas-permeable material clearly would be the smallest of all configurations considered, requiring <57 ml of space and a blood prime of <14 ml. These physical sizes are only for the exchange sections, however, and do not include any space required for inlet and outlet manifolds. The 12 μm circular channel device would require 140 million channels that are only 0.8 mm long. Other attractive options are the 12 μm rectangular channels with support posts and the 40 μm screen-filled rectangular channels, which require a much smaller number of channels. Their physical sizes are in the 250–270 ml range with blood primes in the range of 13–27 ml. The former require 87,000 channels that are 1 mm long and the latter require only 12,000 channels that are 4.6 mm long. The general design tradeoff is that smaller microchannels have the advantage of resulting in smaller physical size but the disadvantage of requiring more channels with the potential problems of assuring channel uniformity. The overall physical sizes of the gas-exchange sections of these microchannel devices are much smaller than that of the natural lungs. They are not that much smaller than contemporary cross-flow artificial lungs, however, which are in the 350–450 ml range. These cross-flow devices, on the other hand, have blood primes in the 200–250 ml range, compared to the 13–27 ml range of the microchannel devices. The major advantages of these microchannel artificial lungs, thus, would be very small blood primes and very low pressure drops. A major question is whether or not the large number of required parallel, uniform channels can be developed reliably and economically.
This research was supported by the National Institutes of Health Grant No. R01 HL 071928. The authors also thank Dr. Mayfair C. Kung for her helpful discussions during the development of this work.
Theoretical Pressure Drop in Packed Beds (Porous Media)
The standard Ergun Equation for the pressure drop in a packed bed is:
in which V = the superficial velocity, Q/A, and Q = flow rate and A = gross cross-sectional area (not the interstitial area); μ = fluid viscosity; L = length of packed bed; ε = void fraction; ρ = fluid density; and Dp = “effective” particle diameter. The effective diameter, Dp, is defined as 6/av, in which av = surface area of a particle divided by its volume.
If the “particles” are cylinders with a diameter = d and length = l:
av = (πdl)/(πd2l/4) = 4/d and Dp = 6/(4/d) = (3/2)d
and the Ergun equation would be:
The Reynolds Number, Re, for these flows is:
If the Reynolds Number is less than about 5, which is true in this case, the second term in the Ergun equation can be neglected.
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