/O; ANALYTICAL A>:iJ EXPERIMEWIAL STUD^ OF BLOOD Or/GFNAIORS AND PULMONARY MASS TilA^^SFER IN LIQUlO BiiEAIHinC Ey JAMZS V;iLi.IA:-L I'ALCO A DISSERTATION PfZSENTED TO Ini: GRADUATE COUNCIL OF T:1E UNIVERSITY OF ITORIDA IN PARTIAL FULFIII^l^NT OF THE REQUIREMENTS FCK THE DEGREE OF DOCTOR Of t'HILOSOI-IiY UNIVERSITY Of FLORlDa 1971 ACKNOWLEDGEMENT The author wishes tc express his sincere appreciation Co Pi:ofessor R. D. Walker, Jr., Chairman of his Supervisory Coir.iriittee , for his interest and his alvays helpful suggestions. The author is indebted to the other rnerabers of his Supervisory Committee: to Dr. Pv. S. Eliot for his initial encouragement to undertake a study in the biological area; to Dr. J. H. Modall for extending the facilities of the Dcpartinent of Anestr.esiology and the cooperation of his staff for this research work; to Dr. T. M. Reed for his patient teaching which provided the basis for n'lch of this research work; and to Dr. A. K. V.irn-.a for generously agreeing to serve on this coirj'nittee iS a representative of the Departni.enc of Mathennatics . The author also wishes to thank the staff of the Department of Anesthesiology, particularly Dr. C. A. Hardy, for their generous assistance. Thanks go to the pu".p rocn crew at Shanus Teaching Hospital for their assistance in takii-.g dita during open-heart surgery. Thanks also go to I'essrs. J. Kalv7ay, H. Jones, T. Lambert and E. Millar for help v;itb providing equiprrient and r.aterials for this project. Finally the author expresses his grateful appreciation to Mrs. Karen Walker for her p.-^tienL v.'ork in typirg this Ji:^c-'ertation. The anther acknowledges the support of the Department of Cher.ical Engineering during this study and iihanks them for their support. a I 2. J Experimental Results — Bublle Diaireter Measurements - 2.4 Experir-ieiital i1csult=;---0xygenatci Simula'ci ca. 'i. CriSERV/iTIONS DURING OPj-:N~H^-ART SURGV.RY 3.1 Theory of Gas Transff^r Throuar: I'lood .1.2 r,.vp(?r-;iiienta] Frc ced jn-J 3.3 Experrlrren'cal Re.iults 3. A Conr. lu?i.or.s arsd Rtcnrmendaclcrs - VI XlTl . TABLE OF CONTENTS ii ACKNOi-.T.EuGEMENT V LIST OF TnBLES ■ LIST Of riGURES NOMENCLATURE ABSTRACT CHAPTERS: 1. irlE DEVELOi'MENT OF ARliiiClAL BLOOD OXYGENATORS 1 1. 1 ]-Tistorical Develcpment 1.2 The Cardiovascular-Pulmonary System 5 1. 3 The Properties of Bl eod . - 1.4 Description of Oxj'genators 1 . 5 The Lung as an Oxygenator ■ 2. SII-fJLATION OF THE BUBBLE OXYGENATOR 2 . 1 f^achematical Models 2.2 Experimental Equip.r.ent and Procedure R V3 31 36 36 50 55 71 71 7 5 SI ij 1. TABLE OF CONTENTS (Continued) 4 . THE DISC OXYGENATOR 109 4.1 Description of the Disc Model 109 4.2 Analytical Results — Computer SiiTiul.ition 117 4.3 Conclusion and Reconiriendations 131 5. GAS TRANSPORT I^^ LIQUID-FILLED LUNGS 133 5.1 Introduction 133 5.2 Theory of Diffusion 134 5.3 Theory of Imperfect Convective Mixing 140 5.4 Transient Dye Penetration in the Lung Experir.ent.-.l Proctjuure , 144 5.5 Results and Conclusions 149 APPENDICES 15 5 A. COMPUTER SIMl'LATION 156 B. 'TOME OBSERVATIONS ON MEMBRANE OXYGENATORS 191 E. 1 0ne-di:.".2n£i.cnal I.ar.inr.r Flow Model 191 3 . 2 The CSTR Model ] 93 C GAS EXCiiANGE IN AIR BREAlhING 198 D. EXPERlMlINTAL DATA 201 E 1 3LI0GP_\"UY , 212 JiXOGPJ^PHI'JAL SKETCH 215 . LIST OF T;i3LES Table £^S^ 2.4-1 Comparison of Proposed Models >7ith Experimental Results 2.4-7. Fractional Gas Holdup Voliime Vs. Function of 0^ and Blood Flow B.ates 3.2--2 Accuracy of Experimental Data Taken During Open- Heart Surgery • 53 64 2.4-3 Boundary Layer Thickness and Profile Parameters.... 70 3.2-1 Summary of Data Taken During Open-Keart Surgery.... 77 30 3.3-1 Experim.cntal Values of 0 Mass Transfer Coefficient and Other Pertinent Parameters 87 D-1 Data Taken During Bubble Measurement Experiment 202 D-2 Saline Simulation of a Blood Oxygenator D-3 Oxygenation Data from Open-Heart Surgery D-4 Transient Liquid-Breathing Experiment v.'ith Saline.. 204 206 209 D-5 Transient Liquid-Breathing Experim.ent with Fiuorocarbon (FX-80) 211 LIST OF FIGURES Figure ./ . a, ■ - ■( Z£S5 1. 2-1 The Cardiovascular System 6 1.3-1 Tlie Effect of Carbon Dioxide Partial Pressure on Oxygen Saturation in \\Tiole Blood 15 1.3-2 The Effect of Temperature on Oxygen Saturation 16 1.3-3 Effect of pH on Oxygen Saturation in \\Tiole Blood 17 J--3-4 The Effect of 0„ Saturation on Carbon Dioxide Concentration 20 1.3-5 Cascade Mechanism, for Thrombosis 21 1.3-6 Feedback Mechanism for the Grov7th of Thrombi, 22 1.4-1 Tiie Bubble Oxygenator ,. 26 l.''+-2 The Disc Oxygenator 27 2.1-1 Comparison of Concentration Profiles as a Function of Num.ber of Stages in Series 42 2,1-2 Comparison of Residence Time Distribution Functions for Varying Number of Stages in Series 43 2.2-1 Viscosity of Saline -CMC Solution As a Function of Composition , 45 2.2-2 Experim.ental Apparatus Ur-ed to Measure Bubble Diameters 46 2. 2-3 Rlood Simulation Experi.r.ent Apparatus. , . ■ 48 2.3-1 Dis tributicn of Bubble Si^es by Surface Area 51 2.3--2 Distribution of Bubble Sizes by Volume 52 2.4-1 Experimental Results of the Saline Simul^tJon Experiment , 56 2.'^-2 Gas Holdup Volume as a Function of Gas to Liquid Volume Flow Rate Ratio in the ILF Bubble Oxygenator.. 60 Gas Holdup Volume as a Function of Gas to Liquid Volume Flow Rate Ratio in the 2LF Bubble Oxygenator.. 61 V L LIST OF FIGURES (Continued) Figure Zil£^. 2.4-4 Gas Holdup VoluTie as a Function of Gas to Liqviid Volume Flow Rate Ratio in the 3LF Bubble Oxygenator... 62 2.4-3 Gas Holdup Volurae as a Function of Gas to Liquid Volume Flow Rate Ratio in the 6LF Bubble Oxygenator... 63 2.4-6 Thin Film Diffusion Model for Oxygen Absorption 67 3.2-1 Scbenatic of Surgical Operating Setup 76 3.3-1 Data Taken During Open-Heart Surgery 85 3.3-2 The F.ffect of Ter.perature on Oxygen Absorption in the ILF Bubble Oxygenator 90 3.3-3 The Effect of Temperature on Oxygen Absorption in the 2LF Bubble Oxygenator 91 3.3-4 Effect of Temperature on Oxygen Absorption in the 3LF Bubble Oxygenator 92 3.3-5 Effect of Temperature en Oxygen Absorption in the 6LF Bubble Oxygenator 93 3.3-6 The Effect of 0 to Blood Flow Rate Ratio on Arterial 0 Partial Pressure in the ILF Bubble Oxygenator 95 3.3-7 The Effect of 0 to Blood Flow Rate Ratio on Arterial 0 Partial Pressure in the 2LF Bubble Oxygenator 96 3.3-S The Fffect of 0 to Blood Flow Rate Ratio on Arter?al 0 Partial Pressure in the 3LF Bubble Oxygenator 97 3.3-9 The Effect of 0 to Blood Flow Rate on Arterial 0^ P.-irtial Pressure in tne 6LF Bubble Oxygenator 98 3.3-10 The Effect of 0 to Blood Flow Ratio on Arterial CO Partial Pressure in the ILF Bubble Oxygenator,... 99 3.3-11 The Effect of 0^ to 31oo(i Elow Ratio on the Arterial CO^ Partial Pressure in a 2LF Z.i.ibble Oxvg.sna'ior. T 100 ;i?. LIST OF FIGURES (Ccrt]nupd) Figure _ page 3.3-12 The Effect of 0 to Blood Flow Ratio on the Arterial C0„ Partial Pressure in the 3LF Babble Oxygenator... 101 3.3-13 The Effect of 0, to Blocd Flow Ratio on the Arterial C0„ Partial Pressure in the 6LF Bubble Oxygenator... 102 3.3-14 The Effect of Venous 0 Partial Pressure on Oxygen Absorption in the ILF Bubble Oxygenator 103 3.3-15 The Effect of Venous 0 Partial Pressure on Oxygen Absorption in the 2LF Bubble Oxygenator lOA 3.3-16 Effect of Venous 0 Partial Pressure on Oxygen Absorption in the 3LF Bubble Oxygenator 105 3.3-17. Effect of Venous 0 Partial Pressure on Oxygen Absorption in the 6LF Bubble Oxygenator 106 4.1-1 0^ Transfer on a Blood Filr.i Ill 4.1-2 Sche.Tiatic of Perfectly Mixed Stages in a Disc Oxygenator 116 ■'-..2-1 The Effect of Initial 0 Partial Pressure on the Boundary Layer Concentration Profile 123 4,2--2 The Effect of Ter.perature on 0 Absorption in the Disc Oxygenator 125 4.2-3 Carbon Dioxide Boundary Layer Profile ] 26 4.2-4 The Effect of Temperature on CO Desorption in the Disc Oxygenator 127 4.2-5 0,, Partial Pressure as a Function of Stage No. for a Slood Flow of 40 cc/sec 128 4.2-6 0 Partial Pressure as a Function of Stage No. for a Blood Flow of 50 cc/sec 129 4.2-/ 0^ Partial Pressure as a lurction of Stage V.o. for a Blood Flow of 75 cc/sec 130 VI XI LIST OF FIGURES (Continued) Figure Page s _ 7 - ? 5.2-1 Diffusion-Controlled Model of Liquid Breathing with Plug Flow , . . . 137 Diffusion-Controlled Model of Liquid Breathing with Perfect Mixing 139 5.3-1 A Model of the Lung as a Series of CSTRs 141 5.3-2 Response of the Lung to a Stpp Change in Dye Ccnc. for CSTR Limiting Case 145 5.4-1 Fxperimental Apparatus for the Liquid-Breatr.ing Experiraent ■ 146 5.5-1 Concentration Profiles After 1 Inspiration 150 5.5-2 Results of Saline Liquid-Breathing Experiment 151 5.5-3 Results of Fluorocarbcn (FX-SO) Breathing Experiment 153 32-1 CSTR Model for Turbulent Mass Transfer in Membrane Oxygenators 194 B2-2 Membrane Oxygenator Gas ExchantiC in CSTR Model 196 IX NOMENCLATURE A = interfacial surface area C = concentration C. = concentration of cor.pcnent i in a T:ixture * C. = concentration of component i in the liquid phase that is in equilibrium with component i in a second phase (C ) = n~dimensional column matrix containing the concentrations of the blood constituents at the entrance of the bubble oxygenator C^^ = the amount of component k bound to component M C. = concentration of the ith component in a liquid film ^ out C . = cou'-entration of the ith component in the liquid ph?.se between two discs in C. = concentration of component i in the inlet stream to a mixing stage D = binary diffusion coefficient [D] = n X n matrix of multicomponent diffusion coefficients D = diffusivity of oxygen in blood D - bubble diameter D. = diffusivity of component i into a mixture D = diffusi"ity of oxygen into plasma e. = concentration of dye in the alveoli of the ith stage in the lung f,, = equilibrium relatioiiship which equates the total amount of component k to the concentrations of the remaining constituents G^_, = equilibrium relationship which equates the amount of component k, bound to compc.ent M, to the rcnaining species concentr-itions in l.r'f; mixture H J. 1 K [K] k K. . N. -1 P. 1 r R (R) R. X S S volume fraction of red cells in the b]ood, i.e., heip.atocrit diffusion flux of component i binary mass transfer coefficient n X n matrix of multicomponent m.ass transfer coefficients equilibrium constant mass transfer coefficient of species i into species j mass flux of component i pressure partial pressure of component i radius radius of a bubble n-dim.ensional column matrix of reaction rates rate of reaction of the ith com.ponent fractional oxygen saturation reduced fractional saturation •" arterial 0 saturation - venous 0 saturation I saturation at P_ = 760 mm - venous 0 saturation ^2 tim.e V V temperature volume flow rate into alveoli in the itli mixing stage = volume velocity voluir.e f lov;- rate volum.e in the lung at which transport becomes diffusion- controlled volume of a stagnant film XI V = gas holdup volume n V^^„ = total lung volume TOT ° V = initial volume of the lungs X = distance parameter y = mole fraction of oxygen saturation Greek Letters a. = Henry's law constant of component i in a solvent Y - ratio of gas to liquid flow rate in the bubble oxygenator 6 = boundary layer thickness y = viscosity T = residence time G = residence time of a film at the blocd membrane interface X = effective residence time as defined in Equation 3.3-4 'i/ = effective residence time as defined in Equation 3.3-3 a: = angular velocity Xll Abstract of Dissertation Presented to the Grsduate Council of the University of Florida in Partial Fulfillment of uhe Rcauirenents for the Degree of Doctor of Philosophy AN ANALYTICAL .M-JD EXPERIMENT.^ S f UDY OF BLOOD OXYGENATORS AJ^ID PULMON/vRY MASS TRANSFER IN LIQUID BRrZATHING By JaiT.es VJillian Falco December, 19 71 Chairnian: Professor Robert D. Walker, Jr. Major Departnient : Chemical Engineering Machematical models for blood oxygenation in bubble and disc o>rygenators have been propostd. In the case of the bubble ox^'^enator, a single-s ta^e , perfectly mixed absorber model was tested and ronf irr.ied by a saline siraulation experiment which approximated oxygenator use in open-heart surgery. From data taken during sixteen open-heart operations, oxygen and carbon dioxide mass transfer coefficients vere estimated as K „ = 0.00528 --- 0^,B sec L. cni K --= n.ci31 — - CO ,B sec 'with \:'r\ezs. results, a ccmoucer program was written to simulate ihe operation of the babble oxygenator ov;;r a v/ide raiu:t ci oxygen ana blood flow rates. During the si.iiulation experiment:, it was foirnd that for each of the four sizt-s of oxygenators rested, an optimal XI 11 ratio of gas to liqu?ld flow rate was obtadned. furthermore, the oxygenation rate v:as reduced when gas flow rates exceeded 7 to 3 liters per n^inute in all four models. In the case of the disc oxygenator, the equilibrium rclacion- ships and other physical constants have been put into subroutine form for easy substitution in other systems- The problem of gas transport in liquid-filled lungs was also considered. It was proposed that oxygen ?.nd carbon diox'ide transport through the bulk of the lung by convective mixing instead of by diffusion as proposed by Kyjstra. A transient breathing experiment measuring dye penetration into saline- and f luorocarbon- filled lungs was devised and carried out. From the data obtained, it was dcterm.iiied that gas transfer th.rcugh the bulk of the luug was by convective mixing; the lung v;as approximatea by tun perfect mixing stages in series when flucrocarbon was used and twenty perfect m.ixing stages in series v;hen saline was used. XJ.V CHAPTER 1 THE DEVELOPMENT OF ARIIIICIAL BLOOD OXYGENATORS 1.1 Historical JDeve_lopnicnt_ Attempts to oxygenate blood arcificially date back to the nineteenth century. The oxygenation of blood by shaking \;ith air v:as reported by Ludvig and Schmidt (1) in 1868, vhile the first continuous process for blood oxygenation vas attenpted by Schorder (2) in 1832, who showed that blood could be oxygenated by bubbling air through it. Zeller (3) in 1903, improved the rate of oxygenation by using pure oxygen in place of air. Continued improvement in this method of oxygenation eventually led to the development of the bubble oxygenator by De '.."all and co-workers (4) in 1956. It is probably the most widely used oxyg.enator at this time cring to its relatively low cost, simplicity of .operation and complete d: sposabil iCy. Another early metliod of oxygenating blood, v.-hich eventually evolved into a successful design, was the thin film transfer unit; it was studied by Hooker (3) in 1915, and by Drinker and co-workers (6). The basic d.^sisn consisted of a glass cylinder through which oxygen was passed. A thin film of blood was distributed on the cylinder wall, the direction of blood flow induced bring countercurrent to thv^, direction of oxygen flow. This type of design eventually evolved into the screen oxygenator developed by Miller e_t_ai_^_ (7) in 1951. In the Miller and Gibbon oxygenator, a sorios of parallel screens ro"lr.c3 a ?et of concentric cyli.nlers ac the blood--f ilming sarfaca. but the essential idea of oxygen transfer into a thin blood fill- is still the .Tiain feature of the design. Bjork (3) developed an alternate thin film oxygenator in 1948. His oxygenator consisted of a series of rotating discs, exposed to a stream of oxygen, which dip into a blood reservoir. This method of oxygenation is based on oxygen transfer into a thin film which is constantly being renewed with fresh blood. These three types of oxygenators, the bubble, screen, and disc, have been tested experimentally and used clinically. These oxygenators, x.-hich might be termed first-generation oxygenators, h.-^.ve two features and disadvantages in common. Firstly, all of them require direct contact between gaseous oxygen and blood, and thus protein denaturacion becomes a problem after extended periods of operation. Secondly, these methods of oxygenation attempt to minimize resisfance to mass transfer of oxygen into the blood by minimizing diCfusional resistance in the blood phase. To date, the principal application of blood oxygenators has been as cardiac bypass units in open--heart surgery. The three oxygenators discussed above have been used for short term (up to approxin;ately three hours) by-pass of the heart ana lungs during either surgical repair or replacement of sections of the heart. The '-urrent limiti^.tion on operating ti:ne is the rate of hemolysis, or red blood cell acscruc-.ion, and the rate of protein dcnaturation. In an effort to minimize pcctein denaturation new oxygenators, \:hich might be termed second-generation oxygenators, are in development. All of these new designs e.lirninate the direct contact of gaseous oxygen and blood. It is anticipated that eliriination of the blood-gas intc-rfp.ce will reduce protein denaturation and increase the possible bypass time to the order of days rather than hours. Such a developu'ent would be of value in the treatment of heart and lung damage which cannot be corrected by surgery. Th:^rc have been a number of different schemes proposed to accomplish the oxygenaticn of blood without direct contact between gaseous oxygen and blood, two of which appear to be promising. Research on the use of membranes through v:hich gaseous oxygen can diffuse into blood has been underway for approximately the ].ast fifteen years. Koiff and Ealzer (9) attempted to oxygenate blood by flowing blood in polyethylene tubes while oxygen was passed over the tubes. Bodeli and co-workers (10) tried the reverse experim.ent of im.mersing tubes, "■-hrough wKich oxygen flovred, in blood. Others (11,12) have attei-ipted to use these lv;o methods with different membrane materials. The material that appears to be miost promising at this time is Silastic (a silicone rubber) tubing. Pierce (1'3) has also tested a membrane oxygenator which has blood flow channels embedded in spaced layers of membranes through wb. '.ch oxygen is passed. The second method of blood oxygenation which would eliminate the direct contact of gaseous oxygen with blood involves the use of "Iriert f luorooarbons as an "txchange riediumi. Basically, the process works as follow?: i flucrocarbon (or other inert, v;ater-insoluble liquid) is oxygenated by bubbling oxygen througli it, then the oxygenated f Iviorccarbon is brought iito contact with venous blood. Oxygen is transferred frcni the saturated fluorocarbon into the blood and carbon dioxide is transferred from the blood into the fluorocarbon. Sinc3 fluorocarbon is insoluble in blood, the blood-f luorocarbon mixture can be separated and the fluorocarbon can be recycled for decarbonation and reoxygenation. Research in this area is quite recent and the literature on this r.ethod of oxyg>2nation is sparse. Nose end co-workers (14) have designed a thin film oxygenator using fluorocarbon (FX-80) as a transfer medium. Dundas (15) has perfcrr.ed siiailar experiments with FX-80 as well as DC-;200 silicone oil. Results so far are promising, but this method of oxygenation will require a great deal of further research before a working fluorocarbon oxygenator can be developed. Although tlie bubble, disc and screen oxygenators h.ave bee.n in use for over a c.-.cade, no mathematical models have been developed which describe their operation adequately. furthermore, only in the case of the screen oxygenator (16) has there been an attempt to describe mathematically the rate-lir.itiug process in oxygen transfer. Significantly, the initial work done thus far with fluorocarbon oxygenators does not involve matheiT.uticai models. It was th-a original goal of this research to develop and test mathematical models for the disc and bubble oxygenator.s in the hope that t'nese models would provide a basis for similar modelling of fluorocarbon oxygenators. Fur iherrr.ore, such matheTiatical models should pro/e useful in developing in vitro expevimento to determine the efff.cts of anesthetics aad other drugs on bleed oxygenation, and should prc'ide a clinical tool for open-heart sur.'jetv. TLe development cf models tor membrane oxygenators, in contrast to other oxygenators, has been the subject of a number of research studies. Bradley (1/) has done a thorough study of gas exchange through silastic tubes through which blood is pumped. LLghtfoot (IS), and Iveisman and Mockros (19) have also constructed models for the design of membrane oxygenators. 1 . 2 The Cardiovascular-Pulmonary System Since blood oxygenators are designed to Lake over the functiors normally perform.ed by the heart and lungs, the evaluation of sucli artificial oxygenators requires a thorough undei'standing of the hiur^an cardiovascular-pulmonary system. This system consists of the heart, lungs and a network of veins, arteries and capillary beds. The function cf the kidneys is also important to consider as some of the problem.s that arise in artificial blood oxygenation are directly attributable to these organs. A schematic diagram, of the cardiovascular system is shown in Tigure 1.7. .1. The heart serves as a pump to circulate bi'ood tb.rough the netw'ork of veins and artsries to the various points in the body /.'here oxygen, carbon dioxide, and other blood constituent:? are exchanged. It is a four- chambered vessel: Venous bluod flov;s into the right atriuin and tlitnce into the right ventricle wlilch acts as a positive displacement pum.p to force the blood through the pulmonary system, ard into the left atrium. Arterial blood flows into the left ventricle which i-.i turn pumps blood through the arterle.=; and veins which compose the circulatory system. Vein co^ C^ Lungs Heart ■Jaste Products Ar t e ry ■--<-■ Capillary Bed 1 Waste Material Figure 1.2-1. The Cardio ovascu-ar jysten. The arteries are a network of flov? channels V7hich transport blcod to the pul.monary capillary bed and to systemic capilla:y beds which are distributed throughout the body tissues. Gas exchange takes place in these capillary beds. In the case of pulmonary capillaries, oxygen is transferred into the blood while carbon dioxide is transferred out, and in the case of the systemic capillaries, carbon dioxide is transferred into the blood while oxygen is transferred into the tissue. Upon exiting from the systemic capillaries the blood is transported back to the heart through a network of veins. The function of the kidneys is to remove waste material from the blood stream. About 25% of the total blood flow passes through the renal arteries into these two bean-shaped organs. Once in the kidney, blood is distributed to approximately 1 million transfer units called nephrons. A nephron consists of an entrance called Eo^.jTTian's capsule and a series of transfer units in which four processes occur. The first unit, the glomerulus, is an ultrafilter which separates erythroctes, lipids and plasma proteins from the rem.aining plasm.a constituents. The second unit consists of the proximal tubule, Henlc's loop, and distal tubule. Tliis section is basically a long tubule folded and looped in sections in which some plasma constituents and water are reabsorbed and other constituents are secreted into the tr.bule. The final unit is the collecting duct in which, as its name implies, waste products and water are collected to be eventually excreted froi.i the body. The details of the plienomena wnich occur in the kidneys are quite coiifplex and numerous, and a coniprehensive discussion cf them is teyoad the scope of this work. There are a number of text and papers •Jhich treat the kidneys, among which is a recent and concise sur..mary by Pitts (20). The lung, of course, also forms an important part of the cardiovascular-pulmonary system supplying oxygen to the blood and facilitating carbon dioxide removal from the same. Since it is an oxygenator in its own right, we have chosen to discuss it in cor.,parison with artificial oxygenators in Section 1.5 rather than in connection wit]i the cardiovascular -pulm.onary system. 1.3 The Properties o f Blood Blocd has been the subject of much research, generally concentrating on either biochemiical interactions or rheology. In the following paragraphs we have drawn heavily from, texts by Forruscn (21), and ^'h: tmore (22) to collect a pertinent summary. Whole blood is essentially a suspension of red blood cells in pla:-ma. Ct'r.er formed elem.ents in the plasma include white blood cells and platelets. The plasma is an aqueous .solution containing about 7% proteins, C.9% inorganic salt, and 2.1% organic substances other than proteins. /•s 'Stated in Section 1.2, bleed is distributed throughout the body through a netv7ork of veins and aiceries. In addition to supplying all body tissues with oxygen snd removing carbon dioxide and waste materials, blood also carries nutrients to the tissues, and it also serves as a heat transfer me.dium to control temperature within the narrov; range necc^ssary for norzial functioning of the body. Further- more, blood regulates the fluid balance throughout the body and provides a defense mechanism against diseases. The red blood cells, or eyrthrocy tes, V7hich carry most of the oxygen in the blood stream are biconcave discoids. The dimensions of the human eyrthrocy tes quoted by Lehman (23), and Britton (24) are as follows : diiimeter = 7.8 microns thickness = 1.8A - 2.06 microns volune = 88 cubimicrons . The red cell is quite flexible and thus easily distorted. It is th.is property that permits the calls to pass through capillaries th?L are smaller in diameter than themselves. It is interesting to note chat the eyrthrocytes pass through the capillaries in slip floxv. Although a number of studies on microcirculation hc^ve been reported by Copley (25), Lew (26), U'ells (27,28), and Goldsmith (29), a mathematical m.odel based upon slip flow has not been proposed or tested at this time. Since good microcirculation is necessary for adeqi.iate oxygenation of cell tissue and since microcirculation is affected by hemolysis, that is, cell breakage, during cardiac bypass, it appears that an understanding of the slip flow mechanism in capillaries would provide ^?aluable insight into the development of better blood uxygcnators. An erythrocyte consists of a membrane enclosing fluid without a nucleus. Th-: ri.embrane is formed of a bimolecu.lar layer of lipids and Lhe cell fluid contains approximately 33% hemoglobin. Hemoglobin is 10 v.ha constituent which is responsible for the large oxygen-carrying capacity of blood, vide infra. I>Tilte blood cells, or leucoytes, provide a defense "leclianisin against disease. They are classified into three groups according to size; ranging from 7 to 22 'nicrous. Froin the smallest to the largest the three types of leucocytes are l>rr.phocytes , granulocytes, and monocytes. The total concentration of thece cells in normal blood is negligible compared to red cell concentration, the ratio of eyrthocyte to l=acocyLe cells being jpi;roximately lOGO to 1. The white cell is more rigid than the red cell, but it has a gelatiuous membrane which easily deforms to adjust to local conditions. Platelets are disc-shaped cell remnants much sm.alier in size than ct'ner formed elem.ents and having a diameter between 0.5 dnd 3 microns as reported by Bell (30), Merrill (31), and Britten (24). Platelets play an important role in the blood coagulation process, which v;e shall discuss shortly. Plasma, the fluid in which all of these form.od elem.ents are suspended, is both a molecular and ionic solution. The i;;ajor ions which are dissolved in the solution are sodium, potassium., calciuin, magnesium:, chlorine, and bicarbonate. The principal m.olecular proteins in the solution are fibrinogen, a, 8, and ■(- globulins, and albumin. Fibrinogen, v/riich polymerizes to fibrin during coagulation, is one of the largest of the protein molecules. The globulins, whose specific functions are pcI understood, are extremely im.portant as carrievTS of lipids and other water soluble substance. Albumin, the plasma prct^ir in highest c^'ucentrntion , is important in maintaining the balance of vater metabolism. 11 Having described the constituents of the blood we are nov7 • prepared to venture into a discussion of how these various coir.ponents in Che blood coT.bine with oxygen and carbon dioxide, transport these two gases to the appropriate locations in the body, and th^n release them to the body tissue and lungs, respectively. Of n;ajor importance are Che reactions hemoglobin undergo but we will also comment on thrcrabcsis, protein denaturation, and heaolysis which are three serious problems which may occur during or shortly after cardiac bypass. Heracglcbin is a large protein molecule with a molecular weight of 67,000 containing approximately 10,000 atoms (32) and an effective diameter of 50 to 64 X (33). It is a tetram.er, each polymeric chain containing an iron atom combined with a heme group connected to a polypeptide chain. The heme group is an iron porphyrin complex which reversibly binds oxygen. It is important to note that heme iron bound to oxygen remains in the ferrous state (i.e., oxidation of iron by oxygen does net take place), and consequently the oxygen molecule maintains its identity. Both oxygen and hemoglobin r.re paramagnetic, but oxyhemoglobin is diamagnetic, 5 ndicating a covalent bond between iron and oxygen. This bond is, in f act^ very weak, and the reaction can be shifted by a slight charge in pH. Consider a dissociable hydrogen ion attached to a hemiOglobin molecule. H-Kb"*" + 0^ t ^^^2 *" '""-^ (1-3-1) If the pK of the hemoglobin environm.ent decreases, i.e., the hydrogen ion concentration Increases, the reaction is shifted Co the left with a release of oxygen. If the pH increases, the reaction will shift to 12 the right and oxygen will be taken up. In the body the pH decreases when carbon dioxide is released into the blood stream in the foLin of CO^ + H^O t H^CO^ t H^ 4- HCO^ (1.3-2) Thus, oxygen release to body tissue is facilitated by carbon dioxide absorption into the blood. In the lungs, where carbon dioxide is released, the pH increases and oxygen binding to h,eir.c:^lob in is facilitated again by the CO transfer. The kinetics of hemoglobin-oxygen reactions have been the subject of a number of research studies and several models ha\e been proposed for the -.nechanisra of reaction. Among the earliest is the mechanism proposed by Hill (34) Hb -I- nO^ t Hb(0^)n (1.3-3) for which it can be easily sho\.'n that the mole fraction of hemoglobin saturated is y=-^~- (1.3-4) 1 + KP" wliere K is the equilibrium constant and F is the partial pressure of oxyi?en in the inixtura. Since each hemoglobin m.olecule combines with 4 oxygen ::iolscuj.es , the value of n should be equal to 4, but experimental values range between 1,4 and 2.9 and they depend on the ionic strength of the solution. Although this fact undermines the theoretical basis for Hill's equation. Equation 1.3-4 is still used because cf ius simplicity. K, of ^ourse, is a function of both ionic strength and pH. 13 Adair (35), In 1925, proposed a four-parameter model for hemoglobin oxygenation involving the following series of steps: Hb, + 0^ t Hb,0„ 4 2 4 2 Hb^O, + 0^ t Hb^O^ (1.3-5) 'V4 " °2 ^ ^*4°6 It can be shovn after a fair amount of algebra that the fractional saturation is k^p + 2k^k.^p^ + 3k^k^k^p^ + ak^k^k^k^p 4(1 + k^P + K^K^P^ + K^K^K^P^ + ^^^2^3^^? ) (1.3-&) whpre K throuoh K, are the equilibrium constants for the reactions '1 ^4 shown in Equation L.3-5. Note that Equation 1.3--6 does not take into account pH or ionic concentration and thus the equilibrium constants must be functions of these two variables in addition to temperature. In 1935, Pauling (17) developed a model v;hich did take into .occount i"iH effects and assumes a heme-heme interaf.tion . His resulting equilibrium relationship - 2 2 2-3 3 4-4 4 KP + (2a + 1)K P + 3a KP + a K P ,. o^.n y ;^^-2 2^ri "4-4 4 y^-^ 'J 1 + 4K? + (4a + 2)K-P + 4a K' P + a K P ■•'here PTf,na is the decrease in free energy due to the intcra.^tion ot two groups HbO,,. Tf R.T>nb is the difference in the change free energies of hydrogen ion dissociation from ox>he:Mgl obin and "rom hemoglobin, the pH dependency of K is given by r, _ .,, (1 + bA/[H^])- (1 + A/[H^])^ (1.3-8) v/here A is the acid ionization constant. A modification of Pauling's model was proposed by Margaria (36) in 1963. His final result is y = 1 + KP ] ^ , ^ icp (1.3-9) + m - 1 where m is constant found experimentally to be equal to 125. It should be noted that Margaria' s equation is a one-parameter model and tiiat K is a function of pH as well as ionic concentration again. There are a number of other possible models which have been suminarized by Adcodato de Sou^a (37). We have chosen to use an updated version of Adair's eq\:ation developed by Kelman (38) to approximate the saturation curve. V.\^. modified Adair equation incluiiing temperature, 'p}\ and carbon cii'vxide concei;tration corrections has been written as a subroucine for convenient 'vomputer solution by KeLi-an. It is thus particularly st'ited for this v:ork. The actual equations used along v;ith the subroutine are listed in Appendix A. Typical saturation curves as functions of oxygen partial pressure, carbon dioxide partial pressure, temperature, and p.4 are shown in Figures 1-3-1, 1.3-:^ and 1.3-3- It should be noted that as dH Increases, the saturation curve 15 1.000 0.800 •ij 0.600 u n) >3 o./^oo o o CI) 1-1 0.200 0.0 i^-O mm pH - 7.4 Tempe.i-ature - 37°C- L 0.0 20.0 AO.O 60.0 SO.O JL lO'J.O 0 Partial Pressure (mm) Figure 1.3-1. The Effect of Carbon Oioxide Partial Pressure on Oxygen Saturation in Vn-iolo Blood. 16 1.000 0.800 CO ^ 0.600 ij to u •u cS a o •H O O.AOO 0.200 0.0 P;:' = 40.0 mm 0.0 20. 0 ^0.0 60.0 J l_ „L 80.0 100.0 0 Partial Pressure Figure 1.3--; The Effect of Teraperature on Oxygen Saturation. 17 w a o u u 3 4J CO CO C o ■rt a 1 . 000 0.800 0.500 G.AOC 0.0 0-0 20.0 /.O.O 60.0 SCO C^ Partial Pressure (mm) 100.0 Figure 1.3-3. Effect of pH on Oxygen Satruratr'on in Uliole Blood. 18 shj.fts tov.ards lower partial pressures of oxygen. This effect, known as the Bohr effect, plays a vital role in facilitating the exchange and transport of oxygen in the body as stated earlier in this section. Carbon dioxide also interacts with blood in a number of ways. The niajority of carbon dioxide is carried in both the plasma and red cells in the fom of bicarbonate io!is. The reaction of CO with water to form carbonic acid, and subsequently bicarbonate ions, takes place rrainly with.in the red cell, where the reaction is catalyzed by carbonic anhvdross. The hydrogen ion released vjhen H„CO„ dissociates reacts with the nitrogen of the imidazole group of the hemoglobin molecule. This r3action buffers the blood and regulates the pK v/ithin a narrow range for large changes in CO concentration. CO also reacts directly with the amine groups of hemoglobin as v;c Ll as proteins in general. These reactions can be suTumarizsd as fellows: CO^ -r Kb-NH^ Z Hb-NH-COOH 2 2 Hb-:\H-COOH i Hb-NH-COO" + h"^ carbonic anhydrose CO.^ 4 li^O - H CO, (1.3-10) I i. 2 3 H CO i H' + !iCO~ h"*" + (IlbO )~ 'L Hb -I- 0 Bradley (1/) suggested that CO concentrations car: be represented by a two -parameter model C - 0.373 - 0.07A85S-i- 0.00456 P„^ (1.3-11) *-0„ 19 where C 53 the total concentration of carbon dioxide, S is the fractional saturation of hemoglobin by oxygen, and P is the partial pressure of carbon dioxide. Since the total CO- concentration is not a function of total hemoglobin concentration and does not equal zero at zero CO partial pressure. Equation 1.3-11 must be viewed as semienipirical; it is valid for CO partial pressures ranging from approximately 30 to 60 mm.. We note in passing that there is a variation in the CO saturation curve with respect to hemoglobin saturation, similar to the Bohr effect for oxygen saturatiun. This is shown in Figure 1.3-4 and is known as tha Kaldane effect. In addition to blood-gas chemistry, ve are interested in a series of reactions tri3gered by trauma which induce blood clotting and, more generally, thrombosis. Thrombosis is alv.-ays triggered by either chemically or physically induced trauma. The first step in the process is aggregation of platelets; the next step involves the polNT.ierizaCicn of fibrinogen to fibrin, vjhich forms a matrix for the thrombus. During the physiological changes, a series of chemical reactions occurs fcr which a cascade m.echcinism has been proposed (39,40). The reaction is an activation of an enzyme called Kagem.an factor. This enzyme acts as a catalyst to activate another enz^mie, etc., finally forming tnrom.bin which, catalyzes the polym.erlzation of tibrincgen to fibrin (Figure 1.3-j). The formation as growth of thrombi is enhanced by a feedback mechanism as shown in Figure 1.3-6. Since adenosine diphosphate and thrcm.bin cause aggregation, their formation during the casctide reaction proviiJes a feedback mechanism for further T'rc'.'t'i and f or-w-.c-; on . 20 o o I— 1 u o o H 0.700 0.600 r- 0.500 O.AQO o ■H u s ■i) o c: o rsj 3 0.300 0.200 20.0 30.0 40.0 50.0 60.0 70.0 CO Partial Pressure Figure 1.3-4. The Effect of 0 Saturation on Carbon Dioxide Conc'ntr.-ition. 21 x + c 0) o fi •rl Vl ,Q •rl T3 •H P. •H hJ „ CN + nJ 0) U w oi CO + c #1 c: a •H > •H u 42 s <^ — >5- o —•k. O X ji^ _>. u u r- x: H u o -^ -H •H Pi* P^ WO •H .n ^ I— I ca M o X in ^^ ;:: v-i v-l o U -U •H d M X H CO -H o .n fi o s •H o 0) a Id to o u in I X CD J-- S O CO CO n •H '■^ ctl ^-^ U o + + -r a u *r^ u n S + o r-* !l^ (— q £••« •H 0) o 0) d M J-i !-i A ,-1 TJ) '^ PL, < •< ^-1 to 'fl o u M r-i ^--1 U-i o 4-1 cu m 60 Q) rH a CO "J cd « iJ x: VJ nj L3 P4 0) CO 0) Pi i-l CI Q) r— ( U lU O 4J XJ Pj O M nJ Ph U< a I o .JD 'J 01 a o -a •a c 23 The problem of eliminatxiig blood clots is currently surmounteJ by the use of anticoagulants such as ACD (acid, Citrate, Dextrose), and heparin. These anticoagulants do not completely eliminate the problem as poor oxygen distribution, indicative of embolism and clot formation, is scmetim.es observed both during and after surgery. Research to provide a better understanding of clotting and thrombosis mechanisms appears to be a prerequisite for improved clinical techniques in this area. 1.4 Description of Oxygenators The title "oxygenator" for artificial heart-lungs is a misnomer, since both CO and 0 are exchanged in these units; gas exchangers would appear to be a more descriptive term. In all of the direct contact exchangers, three processes occur: 1. Oxvgen is transported to the blood-gas interface and carbon dioxide is transported away from this interface by convection; 2. Oxyger and CO are Lranspotted through the bloud by diffusion and ,;onvcc;ive mixing; 3. Chemical reactions involving CO and oxygen take place vjithin r'le red blood cell. Ta the design of bJ.ocd oxygenators, it is desirable to oxygenate as i.uich blood a« possible in as short a time as possible, and consequently, vario-.'s resist'^nces to masr. transfer in both gas and liquid phases should be minini/.ed. The gas-phase resistance is essentially eliminated xu all cn're^it direct contact oxygenators b> si'pplying a 24 very high gas to liquid volume flow rate (in fact ipuch more oxygen is supplied than required for complete saturation of the blood). This leaves the liquid-phase resistance to be dealt with. In mathematical form, the rate of mass transfer of a gas through a liquid can be written as (-'^1) rate of accumulation - net flux of component i by diffusion of component 1 + net flux of component i by convection + rate of formation of component i by reaction or (1.4-1) 3C. 7~ = -V.J.- 7.(c.v)-!- R. (1.4-2) ot ~--l- 1- 2. Using a mul ticomponent generalization of Pick's law and assuming inco^iipressibiiiLy , we obtain --^ = [D]V^(C) - vV(C) + (R) (1.4-3) o t ~ ~ Since difiusicn is, in general, a ranch slower process tlian convection, the major 'itsistance to mass transfer occurs in regions which are stagnant. The minimization of these diffusion layers is the major design ccusidtration in all oxygenators. In this stagnant boundary layer, Equation 1.4-3 reduces to ~- - lC]v2(C) -;- (R) (1.4-4) 25 In the bubble oxygenator (Figure 1.4-1), venous blood and ; oxygen are pumped cocurrently into the bottom of the oxygenation ch?.mber. Oxygen enters through a sparger and apparently bubbles through the chamber in plug flov7. Oxygen diffuses into the blood fro:n the gas bubbles, and CO^ diffuses into the bubbles from the blood. There is a stagnant layer of blood which surrounds each gas bubble through which both gases must diffuse. After passing through the oxygenation chamber, the arterial blood flows through a stainless steel mesh which defoams the blood and then through a collecting reservoir. The major advantages of this type of oxygenator are as Follows; 1. the bubble oxygenator is inexpensive and completely disposable; 2. the entire system requires a small blood priming volume: 3. the cocurrent flow of oxygen and blood minimizes the pressure drop across the system; 4. the equipment is easy to operate; 5. the large num.ber of bubbles provides a large blood-gas ■ i£iter facial area for gas exchange. The major disadvantage of bubble oxygenators if that the turbulent motion of blood in the oxygenation causes hemolysis and thus limits the Linie bypass can be sustained. The disc oxygenator, as shown in Figure 1.4-2, consists or a series of discs mounted in a horizontal cylinder. Venous blood is pumped into one end of the cylinder, the flow rates being regulated at both ends by ♦"wo pumps to maintain the blood level at a depth of Vsnous Blood t Degasing Steel Kire Arterial Blood — 0, Figure 1.4-1. The Bubble Oxygenator. 27 Csl T3 O O •-A pq + T-'jzizz:: ITZZIZH pT-C u o u nj C (U O a 0) 0) H CM I o u 3 •H 0) > -I O 28 one-third tha dianeter of the cylinders. As the blood flovs through the chamber J a portion of it is picked up on the rotating discs as thin filiTis. ?3 the film is carried around by the rotating disc, oxygen is absorbed from the surrounding atmosphere, and carbon dioxide is released. It can be shown that blood flow betveen each of the discs is turbulent \;hen the equipment is operated at the conditions normal for surgery. The diffusion layer which limits gas transfer, in this case, is the thin blood film on the surface of the discs. This surface is rene\7ed once e\'ery revolution by blood in which the discs are partially submerged. The main advantage of the disc oxygenator is that turbulence is restricted to the spaces between rotating discs. Ihis minimizes hemolysis due to mechanical breakage of blood cells via turbulence. The m.ajor disadvaiitages of this oxygenator are as follows: 1. equipment and required resterilization procedures are expensive ; 2. a large blood priming volume is required. The screen oxygenator has evolved from a seu of concentric cylinders to an arrangement of parallel screens. i^lood is pumped to a fixture at Lho top of the screens where it is distributed. It then flows down both sides of each screen contacting oxygen. The thin film provides efficient gas transfer, particularly if the blood flow is tuibalerit. The major advantage of the screen oxygenator is that small scale turbulence minimizes heuiolvsis. The n;ajor disadvantages are 29 large boldup volumes and. intermittent channel ling blood flow which causes variations in oxygen and carbon dioxide transfer. These three direct contact oxygenators have two limitations which restrict their operating time as we have stated previouL=ly. Hemolysis, or release of hemoglobin from the red blood cell iuuo the plas-.aa, occurs in cardiac bypass using any of che three oxygenators now in clinical use. There are two ways in which hemoglobin release can occur. If red blood cells are placed in distilled water, they swell, loosing their discoid shape, and become spherical. The cell membrane expands until it ruptures, releasing hemoglobin into the surrounding distilled water. The driving force for cell expansion is osmotic pressure, which is caused by the impermeability of the cell membrane to various electrolytes and proteins. The other cause of hemolysis is mechanical breakage. This type of hemolysis is generally due to turbulent flow and mechanical punping. It appears that since tui/bulence in puraps is severe, this particular problem will be overcome largely by better pump designs. The problem of protein denaturation is also common to all clinUally u-^d direct contact oxygenators. Protein denaturation is the alteration of the molecular structure of the protein molecule which leads to chanc.r.s in the properties of the molecules. The most likely sxpiaraticn for the denaturation caused by erposure of blood to direct contact -ith oxygen ±v the influ.mce of Interfacial forces on the procoia mcleculcs an'; the subsequent reaction of these molecules with ox-.-gM i^l). The-, protein molocule is a surface accive agent owing t-r^ ,-v,, .-,,t -h-^r T^r'-s of -*ts molecular chain are hydrophobic and other 30 parts are hydrophilic . In solution, the hydrophobic sections tend! to align themselves in the interior of a molecular coil while the hydro- philic s-'^ctions tend to lie exposed to the water. At an incerface, the protein nolccule tends to unfold or unravel so that the hydrophobic sections orient themselves toward the gas phase, and the hydrophilic sections orient themselves toward the liquid phase. This orientation exposes protein bonds to attack by the gaseous oxygen and thus alters the protein structiire. The membrane and fluorocarbon oxygenators are supposed to minimize this problem by eliminating the blood-gas interface. Although there are a number of feasible designs for both types of oxygenators, none have been put iiito clinical use to our knowledge. The experimental designs all conform to the thin film model. In the case of the membrane oxygenator, it appears that the most successful models have emp]oyed small diameter tubes or flcv; channels surrounded by flov.'ing oxygen. The tube diameter or cross-sectional area of the flow channel must be minimized since near the meinbrane v/all gas transport js diffusion-controlled in a stagnant layer of blood. Also t of tmpcrtance is resistance to gas transport i?i "rhe membrane. in Appendix B^ we have presented more detailed comments on this problem 'Bradley (17) has present?d a dxccussion on the effect of silicone membrane thickness and has shoxvm that resistance to CO transport through the membrane becomes the rate-limiting 3tep as wall thickness is ir.creased. The opposite is true in direct contact oxygenators in which oxygen transport is the limiting rate process. 31 along V7ith a siir.ple example to illuGtrate resistance to transfer in series. The Major drawback to r.ieir.brane oxygeaators is that the transport of gases throiigh the membrane and stagnant blood boundary layer is dif f usion-ccntrolled : hence a large priming volume is required to gener^ite the surface area needed for adequate oxygenation and decartona- ticn. Attempts to increase efficiency by reducing the diameter of the tubes must be balanced against increased pumping and resultant hemolysis. Further increases in efficiency of gas transport by reducing membrane thickness are also limited by structural requirements. The fiuorocarbon oxygenator has promise as a long term artificial blood-gas exchanger. As stated earlier, current experimental designs are limited to the thin film types. Other, more efficient, methods of contacting oxygen- saturated fiuorocarbon and venous blood are available, av.d these should be tested. The principal drawback of these possible methods, including the thin film process, is that there is some indication that a blood fiuorocarbon emulsion forms which is difficult to break. Since more than trace amounts of fiuorocarbon in the blood can cause embolisms (15), auy such emulsion m.ust be scrupulously removed. 1.5 The Lang as an Oxygenator The lung, of course, is an oxygenator supplying oxygen to the blood and rer.oving carbon dioxide fromi the same. The respiratory systen'. congenation of blood at atmospheric pressure. The description of gas transport through the lung by diffusion alone appears to be quite unsatisfactory and we shall show that diffusion cannot account for the aaiounts of oxygen and carbon dioxide transferred in liquid-filled lungs. Furthermore we shall develop an A similar diffusion model lias been proposed for gas-filled lungs by LaForce (51) J and supporting data have been quoted by Kylstra (AC). The ass'jmpticn of dif f usicn-ccntrclled transport in gas -filled lungs is further from reality than in the case of liquid-filled lungs and argur.ents for a more realistic model are presented in Appendix C, 35 alterr.ata model, based upon imperfect mixing theor\ , v.'liicli we believe will describe the functioning of the lung more accurately than models proposed previously. CHA-Pj'ER 2 SDOJLATION OF THE BUBBLE OXYGENATOR 2.1 Mathematical Models From our original observations of the bubble cxygenatcr during open-heart surgery, we concluded that both blood and gas were in turbulent flov; in the oxygenation chamber. We vrere also able to ascertain that oxygen gas bubbles passed through the chamber in essentially plug flow. The blood flow patterns, however, could not be determined precisely; thus a saline simulation experiment was devised to investigate liquid flow patterns in the oxygenation chamber. It was decided to sinulate blood with a normal saline solution to which was added a small amount of carboxy -methyl-cellulose (C.M.C.) to increase the viscosity of saline to that of whole blood. Since oxygen dci's not react or physically bind with saline, it was felt tliat the flov/ characteristics of blood could be obtained by measuring the race of absorption of oxygen into saline, i.e., fluid mechanical effects would effectively be separated from chemical kinetic effects. In such an absorption process, occurring in a turbulent flow channel, there are two limiting cases which are of physical signif ic>ince. The first case is plug tlo;; of a liquid through the column. In such a column, li^iuid and oxygen, entering the bottom and flowing cocurrently , would pass through the oxygenating chamber in a slug, and any mixing T As ve discovered in the simulation, this is not always the case when the o::ygenator it- operated incorrectly. 36 37 which occurred in the liquid phase v.'ould be local. In such a situation, .?. mass balance acrcss a slug of infinitesimal volun.e V wo-ald predict the. rate of nass transfer as where C = concentration of the oxygen in the saline solution "•'2 c" = concentration of oxygen at the gas-liquid interface K -- r.ass transfer coefficient A = 0 bubble-saline interfacial area V - Yolune of the slug. If ve further assume that the liquid density is constant and irhat the range of absorption is small compared to the flow rate of gas, Equation 2.1-1 becomes dt V ^ 02 0^^ Upon integration of this equation, we obtain where the initial condition has bean used. Equation ?.l-3 can be written in reduced form as 38 X — , exp v:here KA V C - C O2 O2 (2.1-4) Now, t is the residence tire of a slug of liquid, i.e., the time that a slug or element of fluid remains in the oxygenation column, and since these elements are in plug flow, the residence time of any element of fluid is equal to the average residence tim.e of the liquid, r, or T = V where V is the volum.e flow rate of saline. Thus the final form of Equation 2.1-3 is exp - KA V (2.1-5) The other limiting case is a single perfectly mixed stage. In this r.odel, an element of fluid entering the bottom of the oxygenator is immediately distributed throughout the oxygenation char.ber. Thus, all of the llo'.'id in the co!'v:i.'n is at the exit composition C, . Of ^2 course, the instantaneous mixing of entering liquid is a hypothetical case which cannot be physically roali;;ed; if, hov;ever, the time required for distribution of liquid is small compared to the average residence time, the above as-ua.ption predicts accurately the physical behavior of the system. A mass balance across the oxygenator in this case yields 39 '■\ - ^2^ = -^-^^S, - ^0^) (2.1-6) or follovir/g andlogous steps to those taken in tha plug flew case 1 + AK V (2.1-7) We had criginally expected that inixing in the oxygenator would lie between these two extremes and that the best mathematical model for the system w.iuld be n perfectly mixed stages in series for which it can be shown Tiy extension of Equation 2.1-6 that 1 + AK nV 1 + >^r Kt (2.1-8) Although tha results of our experiments indicated that 1 perfectly mixed stage described the system accurately, we have included a comparison of our results with n = 2 for illustrative purposes. There is an alternate derivation of Equation 2.1-7 which parallels Kramer 2nd Westerterp's (52) analysis of a 1st order chemical reactiori carried out in a continuously stirred tank reactor (CSTR) . In th.is development, it is noted the rate of change in concentration of an element of rluid which remains in the oxygenator for a given l(--n:7,th of time, r, is gi^'on by Equation 2.1-2 and the concentration of this element is given by fKA 1 X = exp --- tj (2.1-9) Nov, the elements which constitute the liojid in the oxygenation column 40 remain ia the column for diffarenc periods of time. The probability of an element remaining in the celunin for a given time, r, i? given by the residence time distribution function which, for a CSTR, can be calculated as folloivs: Since an element of fluid entering the chamber mixes perfectly with the bulk liquid, its current position is independent of its previous history. Consequently, the probability of it remaining in the column longer than a specified time x -f At, is the product of the probability of the elem.ent remaining longer than time T and the probability of the element remaining longer than At. If F(t) is the volume fraction in the outlet stream, having a residence time less than i, then this probability is given by 1 - F(t + At) = [1 - F(t)1[1 - F(At)] (2.1-10) Now, since element position is independent of past history, all elem.ents have an equal chance of leaving the column in the time period Ax, namely, F(At) = I At = —- (2.1-U) Substituting this equation into Equation 2.1-10 gives ^+yF(T) =i (2.1-12) P.ecognizing that at t = 0, no fluid element has left the oxygenatoc , i.e., F(0) = 0 (2.1-13) Equation 2.1-12 becomes, upon integration, 41 F(t) = 1 - exp[-T/t] Fur uhermore, since the change in the bulk concentration between the entrance and exit of the oxygenation chaniber is simply the volume average of the various eler^.ents, the bulk concentration is . f exp KA V T = 0 dFd) or exp KA V 1 = 0 — exp[-T/t]dT (2.1-14) Upon integration of Equation 2.1-12, one obtains the result , , KA (2.1-15) which is identical to hquation 2.1-7. This second derivation reveals rr.ore about the physical phenomenon of ideal mixing than the first development since it not only predicts bulk exit concentration but also the residence time distribucion function. A graphical comparison in Figures 2.1-1 -^nd 2.1-2, sTicws the difjJerences between the tnrae cases under condition. It can be s;;en easily from Figure 2.1-1 that, for a given residence time, the greater the number of mi:ring st.^.ge?, the smaller the value of x, i.e., oxygenation is more efficiently accoruplished by a larger number of mixing stages. The second salient feature that should be noted is that as n approaches "infinity the resulting x cuirve approaches the plug flow curve. A2 1.0 c o -H U f f "\ ■» 9 * ' * • 9 ^ ¥ iCTD \ Light Source R'?servoir Pump Figure 2.2-2. Experimental Apparatus Used to Maasure Bubble Diameters. 47 a 12-15. cer cap^.city saline reservoir, and a multiple finger variable drive pump. Accessories included calibrated gas and liquid flow meters, a pressure-reducing valve, and a thermocouple. Photographs of rising bubbles were taken using a Unitron Series N Metallograph with Polaroid camer-3 attachment and auxiliary light source. The magnification was set at 5X. As it was found that the saline solution corroded metallic surfaces, Tygon tubing connected by glass and plastic joints was used exclusively. The experimental procedure used to measure bubble diameters was as follows: 1. Saline was pum.ped through the oxygenator at a flow rate of 1.4 f. /min. 2. Air flow through the oxygenator was regulated at 5.9 l/min. 3. Three sets of photographs were taken of rising bubbles at 10, 1/, and 30 cm above the sparger entrance. The camera was focused as closely as possible on the center of the oxygenation chamber to minimize the distortion of the rounded surface. Measurements were taken at 23'C (room temperature) ac.d 1 atmosphere pressure. The experimental apparatus used to determine the mixi-ig model and to measure O^-saline mass transfer coefficient is shown in Figure 2.2-3. It cous5sted of two Miniprim.e oxygenators in series, three pumps, high pressure oxygen and nitrogen sources, plus all the accessory equipment used in the- bubble merisurement experimenr. with the exception of the m- cror^copc acd camera. The first oxygenator was used 05 u ti u fO U ■ a (U s •H U a) w c o ■H u CO rH 3 s H LO 13 O o rH M t— , I CM 60 •rH 49 to saturate the saline solution with oxygen, and the second was used as an oxygen stripper. A bypass v;as installed betveen the saline reservoir ana the first oxygenator to vary liquid flow through the system. Tv70 solenoid valves were also installed betv7een the high pressure oxygen source and the sparger entrance, and between the saline reservoir and the oxygenator entrance. These solenoids were used to stop simultaneously the flow of oxygen and saline for the purpose of measuring holdup volumes. Sampling ports x;ere installed at the entrance and exit of the first oxygenator in order to measure the change in oxygen content across the oxygenation chamber. A drain v;as also provided at the bottom of the oxygenation chamber to facilitate the measurement of holdup volumes. A galvanic cell oxygen analyzer was used to measure oxygen concentration of liquid samples. As the name implies the analyzer is a galvanic cell with a lead anode and a silver cathode. An aqueous KOK solution is used as an electrolyte and together with anode and cathode it is enclosed by a polyethylene membrane which is permeable to oxygen. rhe experimental procedure for testing each of the four oxygenators (,]-, 2-, 3- and 6-liter capacity units) was as follows: 1. Oxygen flow ratr^ was adjusted to 5 ". /mln. 2. Saline flow rate was adjusted to a predetermined value. 3. Oxygen flow rate was adjusted to predetermined value. 4. Nitrogen flow rate was set at a value not less than 7.0 ;. /min. 50 5. After waiting 10 minates to allow the svstera to coi^e to steady state, a 50 ml sa:nple was drawn from the entrance to the oxygenator and analyzed for oxygen concentration. 6. 50 ml saraples were then taken and analyzed until two successive oxygen readings were recorded which varied less than 0.6% of the full scale. 7. 50 ml samples were then taken from, the oxygenator exit and analyzed until two successive oxygen readings were recorded whicli varied less than 0.6Z of the full scale. 8. The temperature of the saline in the oxygenator was recorded im.mediately after each 50 ml sample was drawn. The oxygen analyzer "u^as calibrated at the beginning of each day in saline solution saturated with air. The analyzer was also recalibrated at the end of each day for a period of two to three days after the probe m.embrane had been changed and electrodes cleaned. 2.3 Experimental Results — Bubble Diameter Mea surements The results of the bubble measurem.ent experiment are shown in Figures 2.3-1 and 2.3-2. Actual data are given in Appendix D. From the photograp'iis taken, it was determined that the r'sing bubbles were not peri'ect spheres but tended to be eli.psoidal in shape. Consequently, the formulas used to calculate the surface area and volume of each bubble v;ere, respectively, ^ „ ,,2 ab . ~1 b = 2-,T |b + -■- sm t: V - -J rab 51 10 8 - 03 "^ 5 :3 pa c u J2 -'4 ._ 3 2 - 0,0 10.0 20.0 30.0 Surface Area (nm j Figure 2.3--1. Distribution of Bubble Sizes by Surface Area. 52 'J J3 pq 10.0 ?.0 . 0 Volume (;nrp ) Figure 2.3-2, Distribution of Bubble Sizes by Volume . 53 where a = najor raaius b ■= minor radius e = eccentricity As can be seen, both distribution curves for volume and surface area are skew syirmietric. Graphical integration of the distribution curve 3 for volume gives an a^^erage bubble volume of 13.69 mm and an apparent average spherical diam.eter of 2.97 ram. Graphical integration of the surface area distribution function gives an average bubble 2 surface area of 26.9 mm" and an apparent average spherical diameter of 2.93 mm. Since the apparent spherical diameters calculated from the average volume and average surface area were virtually identical, it was assumed that the effective spherical diameter of the bubbles was the average of these two values, i.e., 2.95 mm, in all following calculations involving blood and saline. Analysis of photographs of bubbles taken at various heights in the oxygenation chamber indicated little change in bubble size throughout the column. There appeared to be a slight increase in the avecage diamieter of approximately 10% from the bottom to the I'op of the column, but data points were too few, particularly at the top of the column, to definitely confirm this trend. Furthermore^ a calculation of maximum hydrostatic pressure drop across the oxygenation columns of all f our M iniprime oxygenator models predict" a maximum, gas volume change of 7%. During norir.al operation of the oxygenators even this small change will not be obtained since a fraction of the oxygenation chamber volume is occupied by gas thus reducing the hydrostatic head. Finally, it is the total surface of cnc bubbles in the oxygenator that is im.portant; 5A including bubble variation as a function of position in the calculation of the average diameter, as has been done, should give a valid estiTiate of the surface area for mass transfer coefficient estimation. A more serious source of error could arise from the assumption that the bubble diameter is independent of change in gas and liquid flow rate. The most concrete evidence to support this assumption is that the term KA vras found to be directly proportional to the gas hold- up volume in the experiments performed to measure the mass transfer coefficient. If the average bubble diam.eter varied, this would not have been the case. An attempt was made to correlate bubble diameter data with the single-bubble regime model summarized by Perry (53) which predicts 6Da 1/2 (2.3-1) where D = bubble diam.eter D = orifice diameter a = gas-liquid interfacial surface tension p^- - liquid density. The average bubble diam.eter size, using Equation 2.3-1, was calculated to be 7.54 mm which is about twice as large as the estimated value. The experimentally obtained average diameter was also compared with the empirical correlation 0.5 0. 33 D„ = 0.13 D N„ B Re (2.3-2) which V7as developed by Leibson and co-workers (54). Equation 2.3--2 55 predicts the average bubble diair.eter to be O.JiO r.m or an order of magnitude too small. The range of Reynolds numbers for v/hich Equation 2.3-2 is valid covers flow rates above the single bubble range to Reynolds nuinbers bolow 2000, and this region is known as the transition region. There is no clear division between the single bubble region and the transition region, but the low gas Reynolds numbers at which the oxygenator is operated, N^ = 30 to N„ = 80 Re Re probably lie in the region in v/hich surface tension effects are important. In such a region, the variation of bubble diameter v.'ith respect to K' , and thus with flow rate, would be a secondary effect, Re no effect at all according to f^quation 2.3-1. 2.4 Experimental Results — Oxygenator Simulation Representative results of the oxygenator simulation experiments are shown in Figure 2.4-1 and a complete data listing is given in Appendix D. The variables plotted are % oxygenation, or 1-x , versus a reduced residence tim.e V .^--c (2.4-1) The variable V , the holdup volume of oxvgen. was chosen since it g was assumed that average bubble diameter was independent of flow rates. Thus V is related to the inter facial surface area A by the propcrt-Lona 1 -; !-v 56 r^ O o \o W ca o 3 6 •H c L-l •H r-l o CO in .^-x 0) (U rC 6 w •H -J- H U-; • O o Q) U CO c u Q) l-l Id 3 •H CO 03 CU ^ 1 •> S S •H -H og OJ (U • a P- o .-H 1 r-i ■ • CM o o fi. o (U.OT^BjnTBS "[B'lOjqOP.Tif - X) X 57 A - V S D, A least squares fit of the data to a 1 CSTR, 2 CSTR, and plug flov/ model v.'as performed. The linear method outlined by Mickley, Sherwood, and Reed (53) was used by rewriting Equations 2.1-5, 2.1-7, and 2.1-8 as V ^ = X = ^- M^ V D .§. . -6- !t Y = 1 X 1/2 = 1 + B V V D, Y = y,n X = - (2.4-2) (2.4-3) (2.4-4) Ihe results of these operations are summarized in Table 2.4-1. As can be seen, the standard deviations for the 2 CSTR and plug flow models are almost twice as large as for the 1 CSTR model. Fuithermore, although an F-test indicates nonrandom errors in all three analyses, a qualitative inspection of the data suggests that the nonrandom error is greater for the 2 CSTR and plug flow models than fcr the 1 l:STR model. It v.'as concluded, therefore, that the oxygenator could be apprcyimated by a 1 CSTR model for all four sizes of oxygenators . There r-.ppears to be soras discrepancy between the model and the data for smallest size bag, the 1-liter capacity oxygenator. The data suggest that the o.-.ygenation of saline was less than would be obtained if the system were perfectly mixed. Since both the sparger 58 TABLE 2.4-1 COMPARISON OF PROPOSED MODELS WITH EXPERIMENTAL RESULTS ___Model _ 1 CSTR 2 CSTR Plug Flow K cm sec 2.29 X 10 -2 4.53 X 10 51 X 10 -3 6K , -1. -— (sec ) B 4.65 X 10 9.22 X 10 1.73 X 10 -1 Standard Deviation 4.7% (5.4%") 8.5% 8.5% Data from the 1-liter capacity oxygenator v:ere deleted froir. final estimation of K. The value in brackets indicates the standard deviation v/ith these data omitted. 59 and rhanber for the 1- and 2-liter bags are the same geometrically ;and, in fact, identical dimensionally , v;e cannot attribute this phenomenon to sca-le up factors, I.e., change in mass transfer coefficient or bubble diameter. It '■.''as noted in later experiments, that any tilting of the oxygenation chamber caused channelling floxj ii certain portions of the oxygenator V7hile in other regions, stagnation and back, mixing occurred. Accompanying this, type of unstable flow was a marked reduction in oxygenation of saline. The data from the holdup volum.c measurem.ents v;ere correlated as a function of gas flow rate divided by liquid flow rate for each size oxygenator. These reduced data for each case were then fitted to 10th degree polynomials which were subsequently v.-ri tten in the form of a computer subroutine to be used in blood data analysis. These results are shown graphically in Figures 2.A-2, 2.4-3, 2.4-4, and 2.4-5. The final values of the m.ass transfer coefficients, obtained by the method of Gauss elimination (56) applied to a least squares fit, are tabulated in Table 2.4-2. The experim.ental data are also listed in Appendix D. The relatively constant value of gas holdup volumies at high gas flow races is due to fact that as the volume flow rate increases the velocity of the bubbles increases, and thus, the increase in tlie number of bubbles generated per unit time Ls offset by the speed at which the bubbles move. It '..'as also noted that at very high flow rates, bubbles coa]esct';d into large pockets of gas VThich rose rapidly through the oxygenation cha'.uber. This effect could also reduce gas holdup volume, and, in addition, ireduce the surface area available for mass transfer. 60 o e o > O 33 CO CO O .f Gas Flow Rate (liter/min) [ Liquid Flow Rdte (liter /rnin) F leu re 2,4-2, Gas Holdup Volume as a Functicjn of Gas to Liquid Volume Flow Rate Ratio in the ILF Bubble Oxygenator. 61 150 125 - u o > a. •Xi o ta ■n O 100 Gas Flow Rate (liter/min) j ^' Liquid Flow Rate (liter/min) J Fi'^ure 2. '4-3, Ca^ .Holdup Volume as a Function of Gas to Liquid VoluiT.e Flow Rate Ratio in the 2LF Bubble Oxygenator. 62 180 150 o 120 o c: 3 O > •5 r-l O CO U 90 - 60 30 0.5 1.0 1.5 2.0 ;.5 Gas Flow Rate (liter/inin) ___ Liq'jid Flow Rar.e (liter/min) Figure 2.4-4, Can Holdup Volume as a "unction of Gas to Liquid Volume Flow Rate Ratio iii tba 31, F Bubble Oxygenator. 63 300 250 - o 0) e D —I O > a- t-l O a Fi,"ure 2.4-5. f Gas Flow Pgte (liter/inln) ^[ Liquid Flow Fate (liter/niin) Gas Holdup Volume as a Function cf Gas to Liqind Volume Flow Rate Ratio in the 6I.F Bubble Oxygenator. 64 H < Pi 13 O n o o pa O O O 2 o + l-l CM H - <3- ■^ un • a JSI C^ + M • >-J OJ ro « > ^ <: •vT H a >: O :_j + -J o CM > 'P-i u '^ Q + hJ O ?- ■X CS II e 3 .-H O > a. O o •H U O o u c M o CM rH (N CM t O o r2^ 1 o rH ^ rH .-H X X X X O o CA — 1 ^r o^ -d- r^ ^3- o in Csl o CO o K) ^o 1 O rH rH rH rH X X X X C?i -cr in vJD en rH en in 00 O CO r^ rH r-i r^ rH r-- o CO C^l • • • • ,H CM cr\ 00 •n 1 CO 1 1 1 o o o o rH rH rH rH X X X X r^ a> i:r> CM in o O O rH ^ O u CO o B 3 rH O > 'JH o o •H II >- 65 u c 4-1 c o u I -^ CM ►J o rH X X X X, u o a. m i'^ LH ^^ r-H r^ CJ m rsi (O 1-H vO CO o r^i CO • • • ■ CN rH ^' a. 1 O 66 It was found that this phenoinenon. occurred at a gas flow rate of 7.0 to 3.0 £/riin depending on the size of oxygenator used. The starting point of our analysis of this absorption process was Equation 2.1-6. This equation itself is based upon the more priniitive model of diffusion through a thin filn or boundary layer. To derive Equation 2.1-6, it is assumed that the process is diffusion-controlled, i.e., the rate is controlled by a diffusion resistance in a thin layer close to the gas-liquid interface; and, furthermore, that the concentration profile across this boundary layer is linear. A schematic representation of this r.odel is shown in Figure 2.4-6. A niass balance across the diffusion layer gives dC, V "0, +AJ R+6 or d(C, - c" ) dt V O3 3r R+6 (2.4-5) Suppose that the bulk concentration changes slowly in time in comparison with the rate of change in concentration profile within th.e boundary layer, i:hen, for any small interval of time, C car be assumed as 2 independent of tinie and the diffusion of oxygen through the layt-.r can be described by dt 1 _ 3_ 1 Si 2 Ic. 3r (2.4-6) 67 0^ Transfer Figure 2.4-6. Thin Film Diffusion Model for Oxygen Absorption. 68 v.'ith the boundary corditions C(0,r) - C 0 a -■ r < a + 6 C(t,a) = C (2.4-7) C(t,6) = Cq The solution to this set of equations has been given by Crank (57) as „* [(a + 5)C. - aC ](r - a) C = lk_ .^. 0_ ____ r ro 2 « (a+5)(C --C )cos ni - a(C -C ) ^'^ n=l an nTr(r - a) 2 2 2 (2.4-8) X e For values of 6 approaching zero all of the terns in the right-hand size of Equation 2.4-£ except the first two approach zero. Differentiating these remaining terms gives (C - c^l 5r -a [(a + 6)C^ - ac ] C + r__ - _a 2' (2.4-9) Moreover, if 5 i3 much smaller than r, r ^ a ^ L '^ 1. ^ and Equation ?.4-9 can be approximated by 69 " C - C lC_I_C_i .. _0 (2.4-10) 9r 6 It 3hould thus be noted that tlie assumption of a small boundary layer thickness, such that 6 << r and 6^ < D „ti (2.4-11) 0 yields the linear concentration profile which we required. Substituting Equation 2.4-10 into Equation 2.4-5 gives and comparison of this equation with 2.1-6 yields K = -r^ (2.4-13) 0 -5 2 Using a value of 2.5 x 10 cm /sec for the diffusive ty of oxygen into -3 salj'ne (,53), we obtain a bowiidary layer thickness of 1.1 x 10 cm. This boundary layer thickness fits the criterion as stated in Equation 2.4-11 for our proposed assuniption quite v.'ell as sliown in Table 2.4-3. 70 TABLE 2.4-3 BOUNDARY LAYER THICKNESS /d\'D PROFILE PARA>IETERS Parameter Value 1.1 X 10 ^ cm f- 1.1 X iO-2 B ^ ■ 4.9 X lO""^ sec ■\ CHAPTER 3 OBSERVATIONS DURING OPEN -HEART SURGERY 2ii Theory of Gas Transfer Through Blood In Chapter 2, oxygen transfer into saline during operation of the bubble oxygenator was discussed, and it was determined that the oxygenator could be best characterized, in fact, as a perfectly mixed stage. This is also true for blood in the oxygenation chamber, but in this ca-^e the oxygen absorption process is more complex ov.'ing to the reactions which take place as discussed in Chapter 1, Section 3. This circumstance increases the complexity of the system and invalidates the second derivation presented in Chapter 2 of Equation 2.1-7 except for the case of first order reactions. Since oxygen and carbon dioxide transport must be accounted for both as dissolved gas and in the chemically bound form, a m.ass balance including all of these species must be written. In matrix form this mass balance is V{(C) - (C^)} = -A[K]{(C) - (C*)} + (R) (3.1-1) v.-'here (C) is the. column matrix of chemically distinct species concentrations at the exit of the oxygenator, (C ) is the colamin matrix of chemically distinct species concentrations at the oxygenator entrance, (C ) is the column matrix of chemically distinct species concentrations in equilibrium: with che gas phase, and (R) is the column matrix of chemical reaction coefficients. We shall dismiiss out of hand all of the off-diagonal elements of the matrix [K] arguing thi't the solution 71 ri is dilute in C0„ and 0.^ , therefore these gases diffuse into the blood as binary 0 -blood and CO^-blood pairs. In these circuinstances , Equation 3.1-1 reduces to I A * 1 . •(3.1-2e) ^.'here C,„ . , C.__ and C,,^^^ are the concentration of oxygen and hbO„ riCO- libCO- carbon dioxide bound to each of these chemical species. Addition of Equations 3.1-2a, 3.1-2b, and 3.1-2c leads to the total oxygen transport (■^O^ - '^O^'tOI ■■ - ~^[\.b\ ■ "^O^^ * Vo,,B!ARY OF DATA TAJCEN DURING OPEN-HEART SURGERY Maximum Blood No. of Model Flow Capacity (liter/min) Operations ILF 1.0 2 2I.F 2,0 5 3LF 3.0 3 6LF 6.0 4 No. of Data Points 3 12 6 12 78 sedative, generally Ne-T.butal, and atropine, is administered, 2. Before surgery is begun, an anesthetic such as Pentothal is administered to the patient. Other anesthetic agents used are halothane, morphine, and nitrous oxide. 3. Surgery is begun by cannulating the fe^ioral artery located in the thigh. This artery serves as the arterial return from the blood oxygenator. 4. Next, the chest cavity is opened and both superior and inferior vena cava are canoulated. These two veins serve as the \enous supply to the oxygenator. 5. The patient is now placed on 60% bypass, i.e., 60% of total blood flow is bypassed through the oxygenator. At this point an anticoagulant, heparin, is administered. 6. The heart is then def ibrillated either by electric shock or by surging cold blood through it. 7. The bypass flow is brought to 100% body perfusion rate and surgical repairs are made, 8. After surgery on the heart is completed, the bypass flow rate is reduced to 60% and the heart is fibrillated by electric shock. 9. The patient is taken off bypass completely and all wounds are closed. Th.e startup and operating procedure for the oxygenator is as follows: 1. The oxygenator, including all tubing, is primed v;ith Ringer's solution and then whole blcod. 2. The blood pumps are sLarLeJ and the oxygen flow valve is 79 opened while the primiiig solution is circulated around the ox^-genator and heat exchanger in a closed loop. This is to insure the initial blood entering the patient's body is saturated with oxygen and at the desired temperature. 4. The blood flow rates are adjusted to 60% desired f].ow as the patient is placed on partial cardiac bypass. Simultaneously a maximumi 5% (by volume) flow rate of halothane is introduced into the oxygen inlet stream. 5. The blood flew rate through the oxygenator is gradually increased to 100% of desired flow. 6. Venous and arterial blood samples are taken at approximate 20-miinute intervals or upon request of the operating surgeon. These samples are analyzed for oxygen and carbon dioxide partial pressure, plasma pH, hematocrit, and plasma bicarbonate concentration. At the tlm.e blood samples are drawn, and blood and oxygen flow rates, as v;ell as temperature, are recorded. 7. The blood temperature is gradually increased to normal body temperature, and bypass flow Is reduced to oO% of normal flow. 8. The patient is removed from bypass system. In addition co these procedures blood hemolysis is also monitored at various time intervals. The. device used to analyze blood samples v;as an Asterup Type /KEl, Estimated accuracy of the Asterup equipm.ent for various measured quantities is rihcvjn in Table 3.2--?.. Bled hc.natocrits vers macsur cd 80 TABLE 3.2-2 ACCUBACY OF EXPERIMENTAL DATA TAt^-EN DURING OPEN-HEART SURGERY Paraineter Range of Data Accuracy 0 Partial Pressure 32-590 .nm t 0.5 mm CO Partial Pressure 16-33.5 mm ±0.02 mm Temperature 2S-37°C t 1.0°C pH 7.2-7.6 ± 0.007 Hematocrit 0.280-0.385 ± 2% 81 by centrifaging two blood saraples dravn in capillary tubes for a period of not less than three minutes and then measuring the volume fraction cf the separated red cells and plasma. The estimated accuracy of these measurements is shown in Table 3.2-2. 3 . 3 Experim.ental Results Data taken during the 16 operations referred to above are listed in Appendix D. Of the 32 data points taken approximately one-half or 15 were at ratios of ox>gen flow to blood flow which exhibited channelling and stagnating flow in the saline sim.ulation experim.ent. These data points were consequently disregarded in a least square fit to calculate m.ass transfer coefficients. As previously stated, a linear least squares fit (56) of the data was made with Equations 3.1-8 and 3.1-13 written in the form I * X = -"— /- = 1.0 + K . ~ • X (3.3-1) °2 (C - C ) °2--^ °B 0 0^ TOT and I * ^ CO CO^-^TOT , X = — -^— -^ ^ 1.0 + K B B- • X (3.3-2) wnere V ^^0 - ^C ^ X = :^. --^— V- (3.3-3) V 2 (C - C ) z 2 and 82 (P - P ) >^ " ~ "^CO ~^ ^v ~ (3.3-4) Both Henry's lav; constant and the mass transfer coefficient are functions of temperature and hematocrit, and both of these varied from data point to data point. The temperature dependence of Henry's law cc^nstants was accounted for by fitting data reported by Sendroy etal. (59), and Davenport (60). For temperatures ranging between 25°C and 37°C, it vas found that the 0 and CO solubilities could be predicted accurately by a = (8.971448 - 0. 02566618. T) • a. „,. (3.3-5) and a^,Q = (9.26475738 - 0. 026607855.T) • a^^^ ^^-^ '(3.3-6) where T is the temperature of the blood in degrees Kelvin. The variation of these constants with respect to hematocrit was also reported by Sendroy and Da^'o.nport as ^38'C = "re ' " "^ ^'p^^ ~ ^^ (3.3-7) where c* is the solubility of 0„ or CO^ in the red cell ,:ind -x is re ■'2 2 p the solubility of these gases in plaima and H is the volume fraction of red cells in whole blood. For oxygen, values of 0.0253 cc (STP) /cc-atir, and 0,0209 cc (STP) /cc-atm, v;ere reported for a and 0. , respectively. re p For carbon dioxide, values of 0.423 cc (STP) /cc-atm and 0.509 cc (STP/cc-atm, were reported for a and a , respectively. ic P 83 A correction for the variation in mass transfer coefficient due to fluctuations in hematocrit nas made based on the diffusivity correction used by Bradley (17). Bradley, noting that blood was a suspension of red cells in plasma, drew an analogy for oxygen and carbon dioxide diffusion into blood to electrical conduction in a suspension of noninteracting ellipsoids (61). From this analogy he suggested that the diffusivity of gases in whole blood could be related to the diffusivity of the same gases in plasma by the equation %'"o 1 + 0.49H where D, is the diffusivity of the gas in whole blood and D is the b P diffusivity of the gas in plasma. Now, we have already noted that the mass transfer coefficient is simply K = f (3.3-9) where L is the length of the diffusion path. Substituting Ecuaticn 3.3-9 into 3.3-8 gives the desired result \ = 1 - H(0.65) (3 3_,Q) K i ■•:- 0.49H P It v;a3 assumed further, that the relationships developed for gas h'l/ldup volume as a function of the ratio of the gas flow rate to the liquid flow rate ir. the saline experiment was valid for the blood experiments . Using Equations 3.3-1 and 3.3-2 and correcting for temperature 84 and hematocrit variations, it vas found that the best least squares fit of the 17 data points analyzed predicted an oxygen mass transfer coefficient of and a carbon dioxide mass transfer coefficient of ^CO^.B • f^= 15.96 .in-^ A graph based on these results of percent oxygenation versus residence time T^ is shown in Figure 3.3-1. The standard of deviation for K was t. 21%, and the standard deviation for K. was _ 73.3%. The large standard of CO^.B deviation in the case of K should not be surprising since very CO^ > B small changes in carbon dioxide partial pressure correspond to a relatively large change in total carbon dioxide concentration. Since data reported v: > vD cn Q c: 6 [2 E 11 LTl o u ■ • •r-l . >^ >< X o o c (U o C '•^^ . u O a; •H 11 U-) 4J u '■H P. rt UJ %J (x: o 0) tn s J3 o H < r-l n o 1 0 n .-1 • cq o o rH o o o ( s) uoT-Ba>r:jB;^ pronp?t^ 92 c o •H 4-1 u ■J 4-J C3 '-0 "O 01 a 3 'J 35 0.35 0.30 0.25 0.20 0.15 0.1 T = 29°C T - 33°C T = 37''C X J_ T Her^atocrit = 0.363 = 34.5 mm o37 = venous P Venous P 37' 0^ 40.0 mm pll = 7.36 V /V =2.3 Go Blood _L 2.0 2.2 2.4 2.6 2.8 Blood Flow Rate (liter/rain) 3.0 Figure 3.3-4. Effect of Temperature on Oxygen Ab-'^orptlon in the 3LF Bubble Oxygenator. 93 O II S 0) e s o CM 0 r- O r- .--J -U n u >-o o ^ O p^ ,1-1 ro O 'J) 'JJ r- PQ 3 3 •> o o 11 -^ c a CM a 'D ,:::: O , > > a..> / o o o 11 •H g U V •r-l ■H O •iH 4J P- o (0 oo 1) o rH 13 o o P3 M-l G ■ O U O J iJ :-4 aj a) M 01 o 3 lA I 3 to ( S) 'JOXaBlPlPS p3onp9>j 94 in saturation is dv.e to the shift in the equilibrium between oxygen- bound hemoglobin and oxygen tovjard increased saturation at lower temperatures. To a lesser extent the increase in relative saturation is due to the increase in solubility of oxygen in blood at lower temperatures . figures 3.3-6 through 3.3-9 show the effects of blood flow rate and the ratio of gas flow rate to blood flow rate on arterial oxygen partial pressure for each of the oxygenators. Figures 3.3-10 through 3.3-13 show the influence of these same parameters on arcerial carbon dioxide partial pressure. As can be seen, an optimal value of the ratio of oxygen to blood flow rate is obtained by all of the oxygenators, For the 1-liter bag this ratio is 8.5, for the 2-liter bag it is 3.6, for the 3-liter bag 1.9, and for the 6-liter bag it is 1.7. For ratios greater than these, the holdup volume remains constant until a gas flow rate of about 7-8 liters/min is reached at which point the effective holdup volume decreases with the onset of bubble coalescence. Figures 3.3-14 through 3.3-17 show the effect of venous oxygen partial pressure on the arterial oxygen partial pressure. The small slope of these curves at lovv venous partial pressures is due to the fact that a large amount of oxygen added to the blood in this pressure range combines with hemoglobin and thus does not concribute to increase the partial pressure of dissolved elemental oxygen. 3 . 4 Conclusions and F.ecomm.endations In summ.ariiing the results of the blood oxygenator experiments, the UEin features of oxygen and carbon dioxide transport are as follows: 95 B 200.0 (U a (0 0) (U >4 ■rl p^ 100.0 •H V /v 0^ Blood = 6.0 0.0 V /v O' Blood ± f 1 Hematocrit -- 0.345 37° P;:' = 34.5 mm 50.0 mm pH = 7.34 Temperature = 31°C 0.2 0.4 0.6 0.8 Slood Flov7 Rate (llter/min) 1.0 Figure 3 3-6. The Effect of 0 to Blood Flow Rate Ratio on Arterial 0 Partial Pressu in the ILF Bubble Oxygenator. 96 100.1 e e. 0) u oi tn g; u u •H M < 80, 60.0 - 40.0 20.0 0.0 1 1 1 ■ 1 Kematocrit = 0.345 37° 1 2 37° Pq =50.0 mm ^0^ pH = 7.34 — =6,0 Temperature = 31°C Blood ^ — Zir^~^^~— — - ____ - '"^ -.0 1 . - Blood 1.0 1.4 1.6 1.8 Blood Flow Rate (liter/min) 2.0 Figure 3 . 3--7 The Effect of 0., to Blood Flow Rate Ratio on Arterial 0^ Partial Pressure i:i the 2LF Bubble Oxygenator. 97 100 . 0 6 6 u (0 U 30.0 60.0 o^ 40.0 CO •rl 0) 4-1 20.0 2.0 V /V = 2 . ■^ 0^ Blood Hematocrit ~ 0.345 37° Q - O P~' = 50.0 mm pH = 7.34 Temperature = 31°C V- /V„. , = 0.67 0 Blood J.. J.. 2.2 2.4 2.6 2.8 Blood Flow Rate (liter/min) 3.0 Figure 3.3-8. The Effect of 0 to Blood Flow Rate Ratio on Arterial 0„ Partial Fre^su in the JLF Bubble Oxygenator. 98 100.0 g 80.0 01 u CO ?j 60.0 •■a •H u ^40.0 O •H U S 20.0 u < V /V 0^ Blocd 2.1 V /V , - 0.7 0 nlood JL Hematocrit = 0.345 37° Pq =50.0 mm pH =7.34 Temperature = 31°C -L. 3.0 4.0 5.0 Blood Flow Rate (liter/inin) 6.0 Figure 3.3-9. The Effect of 0 to Blood Flow Rate on Arterial 0 Partial Pressure in the 6LF Bubble Oxygenator . 99 B u 7i CO u i-1 u te Pk O O U 40.0 30.0 20.0 - L 0.0 HeKatocrit = 0.363 37° ^CO^ ^ 34.5 .. Pq'' = 50.0 mm 2 pH = 7.34 Temperature = 31°C ^'o/^Blood= 1-0 0 Blood 0.2 0.4 0.6 O.S __L„ 1.0 Blood Flow Rate (liter/min) Figure 3.3-10. The Effect of 0 to Blood Flo-.-; Ratio on Arterial CO Partial Pressure in the ILF Bubble Oxygenator. 100 i (U u 3 03 03 Q) U •H Co O •H 0) •u <: 25.0 L 24.0 23.0 - 1.0 ! -" 1 1 ■■- 1 Hematocrit = 0.363 ~' '^ -f^ 37'= P^' = 34.5 rrjn 37° P^ = 50.0 iran "2 pH = 7.34 Tempeirature = 31°C — - y- y - / V /v O' Blood -- 1.0 • ^0 Blood ■"^ — / 1 L-_ . .1. 1 1 1.2 1.4 1.6 1.8 Blood Flow Rate (liter/min) 2.0 Figure 3.3-11. The Effect of 0 to Blood Flow Ratio on tiie Arterial CO Partial Pressure in a 2LF Bubble Oxygenator, 101 i 25.0 u 3 CO CO a) 5-1 p4 cj ■r-l 4J rt 24.0 p. o H •H m u < 23.0 ._ ^o/^'Blood -'•' 2.0 2.2 2.4 2.6 2.3 ^ y V /V =2.5 0^ Blood — 1 _ , I ,,. I._ 1 ... . 1 3.0 Blood Flow Rate ( liter /min) Figure 3.3-12. The Effect of 0.^ Lo Bl^od Flov; l-atio on the Arterial CO^ Partial Pressure in the 3LF Bubble Oxygonator, 102 25.0 s s U 3 ui «l cu u p^ CO 24.0 03 O u (a •H a; u < 23.0 3.0 V /V = 0.67 0„ Blood V /V„^ , = 6.10 0„ Blood J.. _t 4.0 5.0 6.0 Slood Flow R,-ate (liter /rain) Figure 3.3-13. The Effect of C to BlcoH Flow Ratio on the Arterial C0„ Partial Pressure in the 6LF Bubble Oxygenator. 103 3C0.0 E 200.0 - u 3 w o u u a. O .100.0 0.0'- 1 y -T- - 1 1 !■■ ■ Hematocrit = 0.363 \ 37°r \ Temperature = 31°C \ pH = 7.34 \ \ \ \ 37° X /~~ ? = 80.0 mm X. 2 \ \^^^ P,^^ - 60.0 mm \^ ^^--1_____^ 1 , ,1 ... "^"~~>~^^ — . ~^ 1 / 37° P;" = 50.0 mm ^2 0.2 0.4 0.6 0.8 1.0 Blood Flow Rate (liter/min) Figure 3.3-14. Che Effect of Venous 0 Partial Pressure on Oxygen Absorption in the ILF Bubble Oxygenator. 104 200.0 0) u 3 ui U) (U Ph tri •H •u CO CM O C8 •H i-l Q) •U < 100.0 0.0 -37< -J J Hematocrit: - 0.363 37° P^^^ = 34.5.u. Temperature = 31°C p- =80 pH = /.34 2 mm V^^/V^^^^^ =-3.0 .1_- .J 1.8 1.0 1.2 1.4 1.6 Blood Flow Rate (licer/h.in) 2.0 Figure 3.3-15. The Effect of Venous 0„ Partial Pressure on Oxygen Absorption in the 2LF Bubble Oxygenator. 105 100.0 I u CO m -l •J 80.0 60.0 .0.0 20.0 0.0 X Hematocrit = 0.363 37° Venous P = 34.5 n.m Temperature = 31°C pH =7.36 V /V =2.3 ^0^' Blood 37° P;: = 80.0 mm 37° P;: =60.0 mm ^2 37° Pq = AO.O nm _L J_ X 2.0 2.2 2.4 2.6 2.8 Elood Flow Rate (2iter/mln) 3.0 Figure 3.3-16. Effect of Venous O^ Partial Pressure on Oxygen Absorption in the 3LF Bubble Oxygenator. 106 1 1 ■ 1 i 1 — - "~ i \0 • o o II II II •H ro O 3 O - Hematoc Venous Tempera pH = 7. •> / - / - _ / o / / i o / O o "" 1 / vO -j- o " / II II * r^ CN ' o o ^^ r^ CM r^ rsi no/ 0-) O / n o / . J_, / 1 ' O MD 00 o c u o o 0) cd u G 3 Q) 05 to •J) >. nj X ^-N M o c P-i •.^ QJ o g i-l M • ^^^ <"j ^ ua u •H ^ a) U 3 aJ '•-i X) •H n) t— 1 P-. 1-5 CN VO 0) O 4-1 0) i3 01 rC k; o u S c C o a) ■H r-i > U^ r^ '--) b •d o •H o 4-1 o .J Cu .-1 o ^ o m 0) o ■ U-l w 0 (C) - (g X >_ 0 t = 0 (4.1-2) (C) = (C.) X = «- t > 0 o — Using the Bclc.-'.mann similar Lty transformation n = -— - (4.1-3) r. rr zv t Ill Disc Surface Bulk 0 Cone. Blood-0 Interface Figure 4,1-1. 0 Transfer on a Blood Film. 112 we obtain the result dn and the boundary conditions become (c) = (c/ n = 0 (4.1-5) (c) =-- (c) n = " o We Lave reduced the system of second order partial differential equations with one initial and two boundary conditions to a system of second order ordinary differential equations with two boundary conditions, In component form, Equat:ion 4.1-5 becr^mes dC . n d^ . -2ri ---- = y D.. — ^-1 + R. (4.1-6) dri ." ij 2 1 j=l -' dn Recognii.ing that the cross terms, D.. (i / j) are small for dilute solutions. Equation 4.1-6 can be simplified to dC^. d'^C. -2r, ---- = D. . ri + R. (4.1-7) dn 11 , 2 1 dn Ak this point the conditions on tho problein must be further specified to elimiriate the reaction term R. and to utilise the assumption of local equilibvLi-m. If a chemical species is denoted by the subsc-ript k and total amcunt of this species per un? t volume in both reacted and unreacted forms is summed in Equation 4.1-6, the results are M dn 113 and I S,.^ = 0 The result that the .summation of the reaction rates involving the component k must be equal to zero is due to the conservation of mass, not the assumption of local equilibrium. The assumption of local equilibrium is now invoked using the equilibrium relationships I \t,= i.^^^S-.,C^,....,C^,T,?) (4.1-9) i.e., the amount of component k per unit volume is a function of tV:e concentration of component k, the concentrations of all the components reacting with component k, the temperature, and the pressure. At constant temperature and pressure we obtain dVc^^ df,, dC . -^= UP"- -r^ -(4.1-10) dr, V '^C , dri 1 1 Furtherm.ore , viewing the inverse relationship of Equation 4.1-9, the concentration of k bound to any one component is a function of the concentraticjn of component k bound to the remaining species and the total concentration or 'Mk '" S/-i'^j'^ '••■•'Is&'r.p) (4.1-11) Therefore, d^C,^ i^G„ dC. dC. 3G„ d^C . TT ^ I \ TcT^T Tn' dfn '- 4 IcT TT" (..1-.2) dn 1 J 1 j 1 1 dri 114 Substituting Equations ■'t.1-11 and '4. 1-10 into Equation 4.1-8 gives, the final result "V qC, V rC. dn ,; MM 3^G^, dC. dC. 5G,, d^C. y _ Jl 1 __] , r _H 1 ^ oC Sr dn rln 4 h ' 9 . oC . 3C . dri dn J 1 J i 1 dn (4.1-13) Equation 4.1-13 is the starting point for any analysis of thin film mass transfer on a disc. In theory, it can be solved for a wide range of functional forms of the nonlinaaricies G,. and f , bv numerical M M techniques, but the cost of computation becomes prohibitive if more than a few coupled components are considered. Consequently, it is advantageous to model processes as simply as possible, retaining the important physical characteristics of the system and deleting those features which have only minor influence. There are many limiting cases of Equation 4.1-13 which are of practical significance, one of which will be demonstrated in the discussion of the disc oxygenator in the next section. A very sim.ple limiting case arises \vhen the dif f usivities of all the components are equal. In this case Equation 4.1-8 becomes d y C. -^n M Mk dn d^ 1 C MM , 2 an Mk (4.1-14) This, of course, is a linear differential equation v.'hich has the simple solution M where 115 I Cmv^-) = 1" C M ■'Mk M 'Mk ind I Sfv^O) = I C; ■^Ik n -*■ "= n = 0 M '"^ M Assumine that a solution of C . as a function of n can be found, the mass transfer in the bulk liquid phase can now be determined. Writing a mass balance across one ideal mixing stage (see Figure 4.1-1) we obtain the equation 2 - v(.C out C^^) = CO 2T-r(C^(r,x) - C°''^)dxdr (4.1-16) R, vjhere V = volume flow rate of liquid through the ideal mixing stage C. = inlet concentration of component i X 0°'"^^ ^-^ concentration of component i in the bulk liquid pliase in i , . . ._ the mixing stage C.(r,x) = concentration of component i re-entering the bulk phase after 1 revolution around the disc and uj = the angular rotational speed of the disc- It should be noted that C. is a function of both radial position on 1 the disc and of the depth below the surface of the film. The radial position, r, can be r-lated tc the time parameter, t, in Equauion 4.1-3, Referring to Figure 4.1-2, the length of time t that an element of a liquid film at radius r is exposed to the gas phase is given by uhe equation t = f ^1 2 arcs in I — 2rT ±] Ii6 (U > o o ■-{ •H Lm • J o M IS o (U 6 o > +.1 CO Pi ds )4 o u « c 00 o u •H Q CD OO u u 0) o a -H 4J cfl I o en CM I 0) VI 3 bO •H in 117 whers R is the radius to Xvliich the dibc is inmersed in the bulk phase. Thus, if C. is known as a function of n, it can be calculated as a function of x and r, and the analytical or numerical integration of Equation 4.1-16 can be carried out. It should also be noted that C^ ■> C?""^- as X ^ 6 (4.1--17) where f is sjir.e finite penetration thickness. Incorporating Equation 4.1-17 into 4.1-16 gives the final result ^2 6 oj 2-rC^(r,x)dxdr •(- vc';^ (,out 1 i ,„2 „2. If these stages are arranged in series, calculations can be done for each stage in stepwise fashion with the inlet concentration for the i+lth stage being set equal to the outlet concentration of the ith stage. 4 . 2 Analytical Results — Computer SiiTiula ti on Blood oxygenation in the disc oxygenator is an interesting example of diffusion-controlled mass transfer with chemical reaction in a thin film. The chemical reactions involved in uhis process were outlined in Chapter 1, Section 3. The equilibrium relationship used for oxygen binding is given in Equation j.3-6. Thus, the 'iquation for total oxygen transport obtained by substituting Eq-.-ation 4 . 3--6 into Equation 4.1-8 is 118 9 d P. 2 2 dn + D HBO. d^HBO, dn (4.2-1) Since the hencglobin molecule is large and has a high molecular weight, its diffusivity is small; consequently, mass transport of oxygen due to the diffusion of oxyhemoglobin is negligible. Setting the last term in Equation 4.2-1 equal to zero gives the desired result dP -2r, a^ + 3S 0„ 9P, 0, "dhT a D °2°2 d^P, dn (4.2-2) with the boundary conditions P = P^"' 0^ 0^ P^ = P. °2 ^^2 r, = 0 Equation 4.2-2 can be solved numerically as a function of n if the diffusi^'ity and solubility of oxygen are known. We have already discussed the solubility of oxygen in blood as a function of hematocrit and temperature in Chapter 3, and we have pointed out that the diffusivity of both carbon dioxide and oxygen can be represented by Equation 3.3-8 as a function of hematocrit. It renains then to calculate the dif fusivities of these two gases in plasma as a function of temperature. Spaeth (63) showed that the ratio of the diffusivity of oxygen in plasr.ia to the diffi;sivity in v/ater was 0.58 ac all temperatures studied. Using data taken from Bradley (17), roust (64), and Perry (5j), the diffusion ccef f iciei:t of oxygen in water can be represented by 11^ D ., . = -677.2838 x 10~^ + 4.A588A6 x 10 ^ • T - 7.3C7692 x 10 ^T^ ^2'V where T = temperature (°K) 2 D^ ,, n ^ difCusivity (cm /sec) Thus (4.2-3) \,HB = \,n^o • ^°-^s> ^'-'-'^ Reliable data on the diffusivity of carbon dioxide in blood piasraa are not available; consequently, as Mackros (l9) also assumed, the ratio of carbon dioxide diffusivity to oxygen diffusivity in blood v.'as set equal to tb.e ratio of these diffusiviti.es in water. Again correlating existing data (17), (19), (52), it was found that this ratio is predicted by the empirical equation R - 1.0169 + 6.94355985 x 10~^ • (310 - T) - 3.54312 x 10~"^(310 - T)^ (4.2-5) wh.ere T = temperature (°K) and thus ^CO^,HB •= ^O^.Hb/^ ^'-'-'^ Having calculated all pertinent physical constants. , only the equilibrium equation for carbon dioxide remains to be incorporated into the equation for CO^ transfer to compleiely define the problem. Unf ortu'-ately Equation 1.3-10 is applicable only over a restricted pressure range v;hose minimum value is greater than 30 mm,. Since one of the bcundaiy 120 conditions for '.he CO transport equation is ^CO. = 0 n = 0 Equation 1.3-11 cannot be used. In its place, the Heuderson-Hasselbalch equation for the equilibrium of bicarbonate ion with carbon dioxide has been used. This equation is based on the two reactions which take place in the plasma and CO^ + H^O T- H^CO^ li^CO^ Z H + KCO^ (4.2-7) The corresponding e.quilibriura equations for these two reactions are K, = [CO^]VA^O] and K 1 [H,C03] [H2CO3] (4.2-8) ^ [h"^] [HCO3] Combining Equations 4.2-8 gives the desired result ^ '' [h'^][KC03] (4.2-9) Furthermore since the concentration of water is essentially constant, [K^O] can be incorporated into the term K 'K , giving the result pa = pK + log [HCO3] ] 3; 13 (4.2-10) , C0„ CO. J vaers pK = log 121 [H2O] (4.2-11) I ^^1^2 J Thus, the concentration of bicarbonate Ion in the plasT.a is given by the equation t^^^°3^ ^ ^CO/CO^ '' (pK-pK) (4.2-12) and the total concentration of carbon dioxide in plasir.a is given by [CO 1 = a P (1 + IqP^'P^) (4.2-13) Equation 4.2-13 involves the assumption that the amount of undissociated carbonic acid is negligible. The value of pK at 37°C is 6.1 (65), and temperature corrections have been calculated for temperatures to as low as 25 "C (65). It has been reported (67) that the concentration cf carbon dioxide in whole blood is related to the concentration of carbon dioxide in plasma by the relationship ^^°2^T0T, Blood [COJ TOT, plasma 1.2 (4.2-14) Substitution of Equations 4.2-13 and 4.2-14 into 4.1-8 gives dP --l-/-U^.O<^-P«,--/A=,, n 2 2 dn 2 HP 2 - "^ "^CO^ d^[HCO ] — 2^^°Hco: — -- an (4.2-15) Although there is less justification for assuming negligible bicarbonate transfer than for assuming negligible oxyhemoglobin :ransfer, this assumption v.-ill be made since values of D„p(^- are not currently 122 available. EliT?.inating the last term in Equation 4.2-15 by this assumption leads to 2 n I u I.N '^^'-Oo ^ ^CO. ^^^[l.-10^P^^-P^^]---^=D^„ ^-^ I"- an C0„ , 2 2 an (4.2-16) with the boundary conditions CO, - 0 n = 0 = P int CO. Equation 4.2-16 is a linear differential equation the solution of which is ,(pH-pk) 1 +_ip "1.2 D CO, (4.2-17) The solution for the oxygen concentration profile is more complicated owing to the nonlinearity of r.he differential equation. A computer program was written to solve Equation 4.2-2. The technique used in the program to solve this equation is the Adams Bashforth predictor-corrector method for numerical integration wliich has been written in a conv>;n:ent subroutine package by Dr. T. C. Eallock, Departm(=!nt of Electrical Engini-.eriug, University of Florida. A plot of oxygen concentration profiles versus n at different initi-1 0 partial pressures is shov^n in Figure 4.2-1. It should be noted that che concentration gradients steepen, and the penetration thickness decreases, ^s the initial partial pressur^a decreases due to 123 --r I u -* o n O • n o II 1! g S U ^ c 13 ^J Pi c CI Q) o U d ■H o --^ c3 u •H 4J u ■H V a c >> 01 M (T! w J ^^ '4-i S a o u •~^ u CO 'J T3 iz m C '-M 3 M-l O w cq a; 0) r; -C o u .3 ■H aanss^a,^ -[-ett.iUiJ uagXxQ 124 the ncnlinearity introduced into the differential equation by oxygen binding to herriOglobin. The effect of tei-perature is sh.o;%'n in Figure A. 2-2; the advantage of operating at a lower temperature is illustrated clearly. The carbon dioxide partial pressure profile is shown in Figure 4.2-3. Since the differential equation describing this process is linear, the profile is independent of initial partial pressure. The temperature effects are shown in Figvire 4.2-4. with the concentration profile across the diffusion layer specified, Equation 4.1-18 can be employed to calculate the bulk concentration of the ith stage for both CO and oxygen. Typical results for a series of 5 stages are shown in Figures 4.2-5, 4.2-6 and 4.2-7. These reveal that the largest amount of transfer takes place in the first few stages, an effect which is caused by the reduction in the difference between bulk phase concentration of gases and the concentration of these gases at the blood-gas interface, and the corresponding reduction in concentration gradients of these components across the diffusion layer in succeeding stages. It should be noted that the inlet concentration of the oxygen in blood entering the disc film was set equal to the known inlet concentration of the blood entering the ith atage rather than "he unknown bulk concentration of the ith stage. If the differential equations for oxygen transport were liiiear, this assuinptiuu would not be necessary since the concentration profile is independent of initial concentration when written in dirnensionless form. Since Equation 4.2-2 is nonlinear, however, this approximation is made to reduce coniputaticnal time re(,ulred to solve the problem. Moreover, as the increase in concentration of oxygen 125 c o •r-( U u JJ ■a 0) u CU Pi 0.06 0.05 O.OA 0.03 0.02 0.01 T Hematocrit = 0.34 37° 37° P, =55 nun pH = 7.4 V„, , - 50 cc/sec Blood - 37' 'C Hematocrit = 0 34 p37» . CO. 25 mm 37° 55 mm 0, \' , , = 50 cc/sec Blooa Stage Figure 4.2-2. Tee Effect of Temparature on 0 Absorption in the Disc Oxygenator. L26 4J ■H C CM •H O Cm 0) U 3 CO CO (U u ■ri U U OS o u T3 (U CJ 3 T3 (U OS 1.0 0.8 - 0.6 - 0.2 0.0 Figura 4.2-3. Carbon Dioxide Boundary Layer Profile. 127 •a o o PQ O o o p-( H CO o G o u CM o 0.422 0.420 0.418 ~ 0.416 ._ 0.414 - 0.412 0.410 I — Stage Figure 4.2-4. The Effect of Trmpei-ature on CO, Qesorption in the Disc Oxygenalor, 128 u 3 u en dj u p^ to •rt U U c 0) 60 o 50 1 1 1 1 Hematocrit = 0.34 37° P = 25 mm Temperature - 30°C — 40 " ^ — -^ pH =7^JiO_____- — 30 — ^ 20 — — 10 1 1 1 1 — Stage Figure 4.2-5. 0 Partisl Pressure as a Function of Stage I>o. for a Blood Flow of 40 cr/sec. 129 i 0) u J} en 4-1 c (U >> X o 50 40 30 20 10 Hematocrit = 0.84 „37' CO, - 25 mm Temperature = 30°C pH - 7.40 Stfgs Figure 4.2-6, 0 Partial Pressure as a Function of Stage No. for a Blood Flc;; of jC cc/scc. 130 3 0) M CD U C6 ■H 4-1 !-i Pj c o 50 — 1 1 1 1 Hematocrit =0.34 37° P"^^' = 25 mm Temperature = SO'^C - 40 — pH = 7.40 — 30 f - 20 — — 10 — 1 J ...L .. I .... — Stage Figure 4.2-7 0„ Partial Pressure as a Function of Stage No. for a Blood Flow of 75 cc/sec. 131 across any one stage is smal] , this assumption should noc lead to serious errors. 4.3 Conclusion and Recommend ations The model proposed for the diffusion-controlled transport gives reasonable estimates of blood oxygenation in disc oxygenators. The method of solving the pertinent nonlinear equations by numerical integration gives a satisfactory solution, but the computation time required to solve the equation is excessive. This problem is further aggravated by the fact the boundary values are given at two different values of n, one of which is infinite. Consequently, an initial guess of the derivative at n = Q was made to transform the problem into a one-point boundary value problem instead of a two-point boundary value problem. The numerical integration v.-as carried out aiid the asymptotic value as n -+ « was checked against the second boundary condition. If the values did not match, the initial derivative was adjusted accordingly and the process was repeated. tt is suggested that a variable integration step size be introduced into the subroutine involving the Adams-Bashf orth numerical integration. This addition to the program could save a significant amount of time now wasted by the stringent requirement of constant seep size o--'er the entire integration range. It would also be helpful if a more accurate method of predicting and correcting the initial derivative (outlined in Appendix A) could be found. It is recommended that oxygenation data either from the operating room, in_ _viv_o experiments with animals, or in vitro 132 expei'lTients be taS:en and analyzed to test the validity of the proposed model. It is also suggested that this ir.odel be applied to bio- oxidation of V7aste water by the BIO-DISC developed by Allis-ChalT.ers for secondary uasire treatment. The chemical kinetics of this system will have to be developed for tliis application. Finally, it is suggested that a m.embrane oxygenator design incorporating tlie principle of the disc oxygenator be investigated as a means to oxygenate blood for long perfusion times. CHAPTER 5 GAS TRANSPORT IN LIQUID-FILLED LUNGS 5. 1 Introduction The problei.i of gas transport in liquid-filled lungs was originally brouf;at to our attantion by Dr. J. H. Modell, Chairman of the Department of Anesthesiology at the University of Florida and nerribers of his staff. In their attempts to provide adequate oxygen and carbon dioxide exchange in canine lungs, they foiuid that the f luorccarbons PID :,nd FX-SO provided adequate oxygen transfer but did not provide efficient carbon dioxide elimination from the blood. Compari.ig their data with a diffusion model similar to Kylstia's (A6), they suggested that the limited CO^ transport could be due to the small differences in partial pressures of carbon dioxide between the blood and the f luorocarbon entering the lung; thus the correspondingly small diffusion flux resulted in only a small net mass transfer of carbon dioxide . .-.fter investigating t'ne problem and determining the flow rote of fluorccarbon in the lungs (apprQx:''r.;ately 1.8 liters per minute) and the total volume of the lungs (.approximately 0.8 liters), it was suggested that the mode of mass transfer of dissolved gases throu'^h the liquid-filled lungs could be by convc.ctive mixing as opposed to moleculai diffusiun. To 'est these two hypcth.esyses , a series of experir.i£nts was propo^-ed . In brief, the experiments consisted of filling a dog's luiigs with solution containing a knov;n dye ':oncentr3ticn and -hen breathing the dog using the same solution but v;ith hi^,her dye 1.33 134 concentration for a specified number of respirations. After the specified number of respirations, the liquid would then bi drained from the li\ng= and the concentration of tlie dye measured as a function of volume drained. It was anticipated that from the resulting dye conceatration profiles as a function of lung volum.e, one could discrim.inatfi between the diffusion and convective transport models. 5.2 Theory of Diffusion In setting up a theoretical model of the transient response in the lungs to a step change in dye concentration at the lung entrance during liquid breathing, there are two diffusion-controlled cases to be considered. In both cases, the lung as a transport unit will be arbitrarily separated into two parts: 1) a region in which convective transport Jom.inatas and 2) a region in which diff.ision dominates. In the first limiting case, it will be assumed that in the region in v.-hich transfer is dominated by convection, the motion of the fluid is described by plug flow. In the second limiting case, it will be assum.Gd that the r-^gion in which transfer is domdnated by convection, the liquid is perfectly mixed. In the iirst cas'', Iha concentration of the dye at the boundary of the dif fusioa-Gcminated region is the same as the inlet concentration, the problem, is reduced to transient diffusion of a dye into a finite slab with a .step in dye conc-'entration at som.e initial time, t = 0, at one of the boundaries of the slab. Writing the continuity equation for such a system, v;e obtain 135 3t = -V (5.2-1) where C = the coi:cent ration of the dye and J = mass flux of the dye. Substituting Pick's diffusion law for J into Equation 5.2-1 gives the result 1^ = -V . (Cv) + DV^C (5.2-2) As we are interested in the liiy.itlng case of diffusion-controlled transport, it will be assumed that the fluid velocity is equal to zero, and, furthermore, that the problem is one-dimensional, i.e., the concentration across any cross-sectional area of the slab is constant. With these assumptions. Equation 5.2-2 becomes = D oX 'C 2 (5.2-3) with the boundary conditions = C, c = c. X = 0, t > 0 t = 0 The solution to Equation 5.2-3 is straightforward; and it can be written as C - C, ^0-S y -A - (2n-!-l)TT n-o • sir. exp (2n+l)7T (2a-M)TT I Dt 9. \ot - ^^D I (5.2-4) 136 V7he.re C = concentration of the dye C = concentration of the dye at the entrances to che lungs C = concentration of the dye initially in the lungs = length of the diffusion path V = voluxc cf the slab (in this case the voluir.e of the lungs) V = voluir.e of the lung at which diffusion becon-ies the dominant n:ode of mass transfer V = total volume of the lungs, tot The dye concentration profiles which would result in the lungs if such a tv.'o-regime raass transfer process occurred are shown in Figure 5.2-1, v;here it has been assumed that the volume of the convection- dominated region is one-half the total volum.e of the lungs. It should be noted that the parameters plotted in Figure 5.2-1 are dimensionlcss quantities <*> = (2n+l):T Dt Therefore, these curves are independent of the magnitude of the diffusivity and diffusion length of the system. In the second case, the concentration C V7ill be a function of timiC, since, with each breath, the newly inspired ] iquid will mix with the liquid already in the convection-dominaued region. At the end of each inspiration, the concentration in this region will be perfectly mixed, consequently, C will be 1 init 2 R 0 ~ ' "v '+"v„ 1 z 137 1.0 o u o u o •H 4J ■U c 0) o c: o u CO 0) 0) C5 o •H CO 0) 0.8 0.6 - 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.0 0.1 0.2 0.3 O.A 0.5 0.6 0.7 0.3 0.9 1.0 Volume Fraction (V/V ) Dif fusion-Con Crollecl Model of liquid Figure 5.2-1, Bre-thing -'ith Plus Ylow. 138 where C. . = concentration of the dye m the convective region at init , , . . , to ■^ the start of an inspiration C = concentration of dye in the inspired liquid V - volune of liquid in the convective region at the beginning of an inspiration V„ - tidal volurr.e of the lung" Since the process of diffusion is relatively slow, it will be assumed that the concentration of the perfectly nixed region reaches the limiting value of the inlet concentration bafore significant diffusion takes place. Granting this to be the case, Equations 5.2-3 and 5.2-4 are applicable again, and the solution can be represented graphically as shewn in Figure 5.2-2, wh.ere once again reduced parameters have been plotted. In making these calculations it has also been assumed that the volume of the convection-dominated region is equal to one-half of the total lung volume, and the tidal volume was set equal to one- third the total volume of the lung. It should be noted that the ].ung has been modelled as a rectangular geometry. Other gecm.etrifs , such as spheres and cylinders, have been used W'h.ich may reflect th.e geometry of the lung m.ore realistically, but it should be emphasized that, regardless of the geometrical conf iguratioii . we. slinuJd still observe the same qiial.itative behavior in ^ - ^1 V ^D the experiments, namely, the sharp rise in .-; — '-'r ^^ ~\j ~ v^ — ' ''"" 0 1 tot tot there exiscs a region within the li'ng in v.'hich m.asG transport is diffusion- controlled . It should be recalled from Chapter 1, that the tidal volume Is defined as the volume inspired or expired during one breath. 139 o 1.0 o o 1 1 C_) U Q)«8 G o •H u U) 5-1 J (-• r-< UJ o c: o.s o u w en 0) ,H o 0.4 •H 03 e «J e ■rA a 0.2 Reservoir Concentration 1 4 Insp. 0.0 3 Insp . -0.2 2 Insp. -0.4 -0.6 I InSD. -0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.3 0.9 1.0 VoluiT:e Fraction (V/V ) o i-igure 5.2-2. Diffusion-Controlled Model of Liquid Breathing with Perfect Mixing. 140 5.3 Theory of Irrperfect Convective Mix ing As opposed to the diffusion models developed in the previous section, concentration profiles can occur in the lung, or any mixing vessel, oving to imperfect mixing. Such cases arise in flow patterns which lie between the limiting cases of plug flow and perfect mixing; these are treated in general, by one of two analyses: 1) the system is modeled a series of perfectly mixed stages; 2) the system is treated as a dispersion problem for wliich relation of the form of Equation 5.2-2 is developed but with the diffusivity replaced by a dispersion coefficient. We have chosen to m.odel the lung as a system of perfectly mixed stages in order to avoid any confusion which might develop due to the similarity between the equations for diffusion-controlled and conr'ective mass transport. The dispersion m.odel is a plausible alternative description of convective mixing in the lung, and should be investigated in the future. In the imiperfect mixing model, the initial assumiption is made that the lung can be divided into n perfectly mixed stages in series with a fixed number of alveoli attached to each stage as shown in Figure 5.3-1. These stages co"ld be of different volumes but, for the sake of simplicity, it will be assumed that they are all of equal size; thus, the volume of the ith stage can be written as V V = --^-- (5.3-1) n Writing a mass balance across the ith stage for the dye component at 141 2^ to U n E— >■ yf / 3 O 10 Pi H W3 U o to 0) •H 0) CO to O I CI to •H 142 some ci-.r.e t after a step change in dye concentration has been introduced into the lung entrance gives the results 1 ) during inspiration dC. V -.-—- = V. C. - V.C. - vC. (5.3-2) dt 1-]. 1-1 111 where C. = concentration of the dve in the ith stage 1 ' ^ V. = volume flow rate of the liquid out of the ith stage V - volume flow rate of liquid into the alveolar sacs connected to the ith stage 2) during expiration dC. V -^~ = ^'■M^'^■u.^ + ve. - V.C. (5.3-3) at 1+1 1+1 1 11 where e. = concentration of the dye in the alveolar sacs connected to the ith stage. Both Equations 5.3-2 and 5.3-3 assume that the flow rates of liquid into and out of the alveolar are the same in all stages and, furthermore, tl;at all volum.e flow rates are constant. If at the beginning of an inspiration or expiration the concen- tration of the dye concentration in the ith stage is C. , Equations 5.3--2 and 5.3-3 can be written, respectively, in matrix form as ^f-- [A](C) "^^ (5.3-A) (C) = (C)^ t = 0 inJ ^~ = [Kj(C) + [B](e) (5.3-5) dt 143 (C) = (C)q t - 0 [A] is an n+1 by n+1 matrix \jhose elements are defined by A. . - 0 i = l,n4-l (V^ + v) i = ?.,n+I A . . - - „ 11 V (5.3-6) A. • . = ^ i = 2, n+1 1-1,1 V and A. . = 0 for all i and j not specified in previous '•^ equations. [K] is an n x n matrix whose elements are defined by __ V. 11. . = - ~ i = 1 , n V (5.3-7) ^+l.i=¥^ -''''-' K. . = 0 for all i and j not specified m previous equations. [B] is an n X n matrix whose elements are defined by [E] = ^ [I] (5-3-8) Th'3 oclutions to Equations 5.3-5 and 5.3-6 for the Inspiration part of the breat?iing cycle are (C) - cxp{[A]t}(C)^ (5.3-9) and for the exjii ration part 144 (C) = 0 '^''^^^-'((C) .+ [K] ^B](e)) - [K] \b) (e) (5.3-10) The dye concentration profile can be determined after any arbitrary number of respirations by solving Equations 5.3-9 and 5.3-10 in step- wise fashion. For the first inspiration, after a step change in inlet dye concentration, the initial concentration profile is essentially constant and equal to some known value (C) . For the first and succeeding expirations, the initial concentration profile is set equal to the final value of the concentration of dye in each stage at the end of the previous inspiration. After the first inspiration, the initial con- centration of the dye for each inspiration is set equal to the final value obtained during the previous expiration. A graphical representa- tion of resulting dye concentration profiles is given in Figure 5.3-2. 5.4 Transient Dye Penetration in the Lung Experimental Procedure In this section, the experimental apparatus and procedure used to introduce a step change in dye concentration and observe its penetra- tion into the lung as a function of tine will be discussed. Two sets of experiments V;ere performed using two different liquids, normal saline and fluorocarbon FX-SO, to liquid-breaths dogs. The sxperimental procedure in each case v/as the same, except for the number of respirations completed before draining the lungs. In the case cf FX-80, only one expari.riient was completed because of the scarcity of fluorocarbon dye. The e.\perimental setup used in all experiments is shown in Figure 5.4--1. It consisted cf a one-liter liquid reservoir, an 145 o I I o u X en C di e •r-l Q l.C 0.8 0.6 0.4 0.2 Tidal Volume = 300 cc Lung Volume = 771 cc Rcopiration Rate = 3 mm 20 Resp. 0.2 0.4 0.5 0.8 1.0 Volume Fraction of the Lung Figure 5.3-2. Response of the Lung to a Step Change in Dye Cone, for CSTR Limiting Cos 3. 146 s •H u a. X w u •H o > (U to OJ Pii •H ^;^ •©- c •H 4-1 nj QJ C3 I •M cr •r-l ^J 0) 4= ^-1 O Cfl •u to u 60 o a nJ 4J C E •H U OJ a. I 0) 3 Pi, 147 endotracheal tub*^ v:ith a bypass drain and a 6 to 8 kilogram dog. The experimental procedure used in all experiments was as follows: 1. A live dog was placed on the operating table and anesthetized with Nembutal (administered in dosages of 0.25 rag/Kg). 2. The dog was intubated with an endotracheal tube which v;as connected to the liquid reservoir. 3. Potassium chloride v;as administered in a dosage of 10 milliliters in order to defibrilate the heart. 4. The reservoir and the dog's lungs were filled ;.-ith liquid (either saline or FX-80) and the lungs were respired by raising and lowering the reservoir until all trapped air was rem.oved. 5. '.slxen the lungs were completely filled with liquid, 300 cc of saline (350 cc of f luorocarbon) were withdrawn, and the tubing connection to the reservoir v/as clamped. 6. The reservoir was emptied and refilled with saline (or f luorocarbon) containing a knovm dye concentration. 7. 300 cc of the dyed saline (350 cc of the dyed f luorocarbon) was introduced into the lung through the endotracheal tube. 8. 300 cc of saline (330 cc of f luorocarbon) was expired from the lung. 9. Steps 6, 7 and S were repeated until desired num.ber of respirations had been achieved. 148 10. Cn final expiration, the lung vas drained in 50 cc '. aliquots until no more liquid could be drawn out of the lungs by gravity. 11. Each 50 cc aliquot of .lung fluid was then analyzed for dye concentration. Each respiration of the dog's lungs required about 20 seconds, 10 seconds for inspiration and 10 seconds for expiration. Approxinately 40 seconds were required to replace the dye sol\ition in the reservoir after each breath in the saline experiment, so that the breathing rate was approximately 1 respiration per minute. From experience gained in the saline experiments, we were able to reduce the time required to exchange reservoir liquid to approximately 15 seconds in the fluorocarbon experiment. Thus the breathing rate xvas increased to 1.7 respirations per minute in the fluorocarbon experiment. In both fluorocarbon and saline experiments, dye concentrations v.'ere measured photometrically with a Klett-Suramerson Photoelectric Color meter. A. green filter with transmission in the wavelength range of 500 to 570 millimicrons was used in m.easuring the concentration of Dupont potamine copper blue dye in the sa.line experiment. A red filter with transmission in the \;avelength r.'_nge of 5''0 to 700 milli- microns V7as used to measure the concentration of the fluorocarbon dye produced by Allied Chemical Corporation in the fluorocarbon experiment. Tn all cases, 10 cc samples were taken for color anaJ.ysis from each 50 cc alfquot dravm from the lung. 149 5.5 Results and Conclusions Figure 5.5-1 shows the experiraental dye. concentration profile observed in our experiments after one inspiration. Also shown are data taken by Vest (A5) using CO , 0 and albunin. It should be eiuphasized that, as shown in Figure 5.5-1, the dyes in our e::periment, as well as West's, penetrated to the extremity of the lungs. This observai:i.on is a strong indication that diffusion is not the rate- cor.trolling mechanism of mass transport through th^. bulk of the lung. Furuhermore, one cannot match the change in dye profiles as a function of number of respirations (Figure 5.5-2) with the diffusion curves shown in Figure 5.2-1 for th-;n the results of the maCching are ■'nconsistent with experimental obseivations . If a value of the diffusi n resistance — is estimated from the concentration profile affar one breath, the diffusion model predicts that the dye concentration through out the lung will be equal to the inlet dj'e concentration after ten breaths. If a value of — is estiinated from the concentration profiles after cen or twenty breaths, the diffusion model predicts that the dye concentration in tl-.e Ii;ng beyond the total vol'ii.ie will be essentially zero immedia';ely after the first breath. Neither of these conclusions is consistent w:i "in uur experimental observations. Comparing the dye penjcration curve as a function of the number of respirations with t)e con'.'ective mixing model, it can be seen that if the number of j,ixing stages is set equal to 20 , the experimental results C'lrelate welJ. v.'ith the micdel . In comparing the experimental data with ths r'.odel, it was assumed that during draining the lung, the 150 1 . 0 G 0.8 0.6 0.4 0.2 0.0 I 0.0 0.2 0.4 0.6 Tracer 0.8 1.0 Figure 5.5-1. Concentration Profiles After 1 I--spiraticn. 151 1.0 5 o o I o u Q CO J3 Q) O •H tn S •H Q 0.6 0.4 0.2 Tidal Voluma = 300 cc Lung Volume = 7 71 cc Respiration Rate = 3niin 0.2 0.4 0.6 0.8 Volume Fraction of Lung 1.0 Figure 5.5-2. Results of Saline Liquid-Breathing Experiment. 152 equations given in Section 3 were valid until the tidal volume had; been removed. After this point it was assumed that the lung was drained in plug flovv. These assumptions were based upon the fact that initially the volume flow rate at which the lung was drained was approximately equal to that obtained during respiration. As lung draining was continued, the drain rate decreased; thus reducing con- vective mixing. Figure 5.5-3 shows the concentration profile in f luorccarbon- filled lungs after 3 respirations. Fitting these data to the convective mixing model, it was determined that 10 perfect mixing stages described the transport process with good accuracy. The fewer mixing stages for the f liiorocarbon experiment as compared to the saliiie experiment are probably due to the shorter time required to change reservoir fluid between eacli breath. This reduction in time between inspirations most likely increases the agitation (mixing) of fluid in the lung with the r?3ult more dye will penetrate into the extremity of the lungs in a given period of time. The conclusion we have reached from these experiments is that in both saline-- and f luorocarbcn~-f illed lungs, the mode of mass transfer through the bulk of tl;e lung be it dye, oxygen, or carbon dioxide, is by convective oiixing. This is not to say that oxygen and c.'rbon dioxide transport into ;:he blood is not dif f usion-ccntrolled . In fact, it is expected unat, in a s.r.all layer immediacely adjacent to the alveolar v;alls, gas transport is diffusion-controlled. V.Tiat must be clarified is that in calculating the rate of oxygen and rr-rbcn dioxide exchange in liquid-filled lungs, both diffusional and coiT.ective mixing must be 153 o C o 1 ; r- o o 0) >^ Q CO OJ t-l o •H M fl (U s 0.0 0.2 0.4 0.6 0.8 Volume Fraction of the Lung Figure 5.5-3. Results of Fluorocarbon (FX-80) Breathing Experiment. 154 accounted for; diffusion alona cannot describe accurately tlie physical phencr.ena which occur. Our recommendations for further research in this area are as follows: 3 . Extensive liquid breathing experinents with dyes or other tracer materials should be carried out with f luorocarbon, varying respiration rates and tidal volumes to determine the effects these pararaetcrs have on convective mixing (i.e., the number of mixing stages). 2. The amount of oxygen and carbon dioxide exchange that can be accomodated by convective mixing should be estimated and compared v.'ith experimental data. 3. In experiments perform.ed in 1 above, a continuous reservoir exchange system should be provided to supply dye solution of a constant concentrat: on without stepping the experiment after each breath to replace reservoir fl^iid. 4. A mechanical device should be employed in all breathing experrmants to provide both constant tidal volume and respiration rate during liquid breathing. APPENDICES APPENDIX A COMPUTER SnaJLATION In t?.is appendix, the computer programs used in various simulations are listed. An attempt has been made to document the program and instructions are supplied in each of them. Subroutines hava been supplied wherever possible to permit easy substitution of relationships to broaden the scope of problems to V7hich the programs can be applied. The program.s in the order in which they appear are: 1. Bubble Oxygenator 2. Disc Oxygenator 3. Convective Model for Liquid Breathing. The program, Bubble Oxygenator, simulates the operation of tlie Miniprime Bubble Oxygenator. The program, Disc Oxygenator, simulates the operation of the disc oxygenator. 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II >■ 1-1 :j X U- Cj ,' Jj ■-- CJ C- ^' ^ LJ 1 — * w— < >-H CJ •n ^- 0 tM .— 1 •JD I IX.' O II 5, :i. a. 1. D. m — ii II — c ;ij (M "5 — • -^ 00 II (M il O CJ -^ -^ 2. ■-' ^ O— ' 00— _J "J3 O i-^ il — •• •" ^ II ^ '-^ I— CJ I! X — I O O U. — > J — _) Lj U. LU ^ '_) a. —1 c^ ci >-■ >- (.3 >- _j o >— • a: oj •A o OJ rsi .\PPENDIX B SOME OBSERVATIONS ON MEMBRANE OXYGENATORS B_. 1 One-dimensional Laminar Flow Model This research study was intended to provide a basis for advanced design of blood oxygenators, including membrane oxygenators. Although ve have not presented a membrane design, preliminary studies are currently underway. The comments that we make here are views and conclusions which were drawn from the literature survey on m.embrane oxygenators. We will restrict these comments to design principles pertinent to mfimbrane oxygenator designs, a more complete understanding of which would, in ovir opinion, be of great assistance to those using these devices. The reader is invited, therefore, to "take the wheat and throw away the chaff." As (Jiscussed in Chapter 1, Section 4, several membrane models have been proposed for blood flow in tubes and between parallel plates made of niembiane material. These models are, In one v:ay or another, boundary layer problems. In the case of laminar flow in tubes, it is usi.ally assumed that transport in the radial direction is diffusion- cjncrolled, and ttiat crnnsport in the axial direction is by convective mixing,. WritTng n-ie flux of coinponent i in the general foL'm leads to N. = vC. f J. (B.1-1) -.1-1 1 and the two assumptions stated above require v. = 0 and J. =0 (B.1-2) - 1 - 1 191 192 The partial differential equation and boundary conditions for this; problem are -r-^ = D.7 C. - V ot 11- (C.v) + R. i~ 1 (B.1-3) and z = 0 ~ir r-R, r=R, where j . - diffusion flux of component i m the blood in the radial ~ir , . . Qirection J. = diffusion flux of component i in the membrane ~ir and R^ = internal radius of the tubing R. = rec.ction rate of component i. The velocity component in the z-direction, v (r) , can be found by solving tlie equation of motion ('^1) ; for a N'ewtonian fluid the solution is where AP = - pressure drop across the length of the tube |j = -viscosity of the blood L = length of the tube. Non-Newtonian fluids give comiewhat different \'elocity profiles. With v specif J ed. Equation B.].-3 can be solved num.erically if not analytically. 193 These r.-.cdels implicitly -lake use of the assumption of a stagnant layer of fluid immediately adjacent to the tube v:all, and they describe the motion of the fluid quite veil as long as there is no secondary and turbulent flow. It is Equations B. 1-1 and B.1-2 which point to the limitations of membrana oxygenators. Noting that the axial velocity is zero in the vicinity of the wall, it should be obvious that gas transport at the wall is diffusion-controlled. This is the rate-limiting step in the tranppoi t of gas in the blood phase. Furthermore, if the membrane does not offer significant resistance to mass transfer, diffusion of gas into the blood near the mem.brane wall is the rate- limiting step for the entire process. If secondary flow or turbulence occurs (which is desirable to increase gas exchange), either a more complicated form of the equation of motion has to be .solved or a new model has to be proposed. For secondary flow due to coiling of tubes, Mackrcs dg) has chosen the first option. An attempt to m.odel turbulent transfer as a CSTR has also been made (67), but, unfortunately, the mechanism cf transfer proposed does not appear to be realistic. It is this model that we will discuss in hopes that a more realistic approach to turbulent transp07"t wi] 1 r^asult. B. 2 The CSTR Model Consider a length of tubing in which blood flow i.s turbulent. If uha tube i-; relatively long, i.e., the ratio of lengih to diameter is large, che mixing of constituents due to turbulent m.otion can be . series of CSTR's a.s ^hown in Figure B.2-1. Gincc the 194 a u -o B 0) S fl •H . o o rH I 3 [i. o [^ •T3 O O i-H pq 195 stage-tc-stage calculations for such a series of perfectly mixed absorbers were discussed in Chapter 4 and are applicable co this situation, wa will consider only one stage here. The r:.odel presented by r;ille et al. (57) for absorption of gases through membranes is called a surface renewal model. As shovm in Figure B.2-2, blood enters a perfectly mixed reservoir, and is immediately mixed wich the blood already in the container. Absorption is m.odeled by assuming that a thin film of blood moves to the m.embrane-blood interface, becomes saturated with oxygen, and is then replaced by a new film. If the residence tir.e of the filmi is S, the rate of gas transfer is d(VC.) V^C. __„L_ . .XJ-. .'B.2-1) dt e ' ' vh.ere VC . = m.ass of component i transferred X V - volume of the film C. - equilibrium concentration of component i in the blood. The concept of a renewable surface is an idea taken from direct- contact absorbers. In these absorbers gas is brought into direct contact with a well-mixed liquid, and since by mixing, the materials at the gas- liquid interface are being replaced constantly, the surface renewal model is an acceptable description. It must be rem.embered that at the interface in a membrane oxygenator a thin film of stagnant blood clings to tha membrane; consequently, it is never replaced. For such a film, a stagnant bo'indary layer model would be a much m.ore realistic descr-' pl"ior>. of the 196 O S Pi EH CO U C3 •H (U 60 C ^ CJ 05 03 O ^J O 4-1 to C! % X o (U c VI ,o 6 QJ CM I CM pq Qi U M ■t-l O O pq 197 phenomenon occurring in a well-mixed stage. In this case, the rate of mass transfer would be given by the equation d (VC . ) -^ = -KA(C. - C?) (B.2-2) dt i i. instead of Equation B.2-1. Finally, it should be noted that the diffusivity of carbon dioxide is low in Silastic which is currently considered the best membrane material. This fact leads to the possible consequence of transport limitations occurring in the membrane rather than in the blood. In any design, the consequences of carbon dioxide transport limitation in the membrane must be considered to m.aintain an acceptable respiratory ratio (ratio of oxygen to carbon dioxide exchange). APPENDIX C GAS EXCHANGE IN ACR BREATHING , Our conin'.ents on air breathing parallel our discussion of the iTieinbrane oxygenator. Once again, we have not expended a large effort to pursue the study of mechanisms of gas exchange in air-filled lungs, and again, our observations and interpretations are limited to areas already investigated but which merit further discussion. As might be expected, the mechanisms proposed for gas exchange during air breathing have been diffusion and convective mixing models. It is the role of diffusion that needs to be clarified. Most models of gas exchange with which we are familiar treat the lung as a single perfectly mixed stage. At the alveolar walls it is assumed that a thin stagnant layer is present in which m.ass transport is diffusion-controlled. For such a sj'stem the rate of oxygen or carbon dioxide transport across the lung can be described by dC. V(G. - C.) - K.A(C. 1 _ __i_ 1 1 1 dt " ■■■ t Vdt '^i) (C-1) where C. 1 C. 1 bulk concentration of the ith com.ponent in the lungs conce.ntracion of the ith component in the gas entering the lungs the concentration of the ith coi,:pcnent that v.-culd be in equilibrium with the coacentraticn of the same component in the blood capillaries V --■ initial volume of the lungs 198 199 V = volume flew rate of gas intio the lungs K = mass transfer coefficient 1 A = surface area available for gas transfer across the alveolar v.'a lis. Although many refinements to it have been irade, Equation C-1 is the starting point for convection i:^ixing models for gas transport in the lungs. The model which leads to it reflects accurately the transport phenomena taking place in the lung. Attempts to use a diffusion model for gas transport in the lung have not met with any considerable success. A cypical case is a study made by La Force and Lewis (5l) • These investigators modelled the lungs as a diffusion-controlled transport unit. A computer program was developed to calculate oxygen concentration as a function of distance from the lung entrance and in terms of time elapsed after a step change in the entrance oxygen concentration had occurred. The conclusions chey drew from the resulting concentration profiles are quite revealing. Among the variables of interest is the length of the diffusion path. To consistently match computer results with experimental data, Conley (.-oncluded that a "churib tack" model should be used to describe the lung. In t'-.a thu~b tack model, it is assum.ed that gas diffuses through a relatively long, narrow entrance, thence into almost innumerable branches in a s'.iort dioLHace. Such a model predicts an extrem.ely sViort diffusion path, which i -; clearly the case, but it once again misses the correct interpretation of the physical phenomenon occurring, namely, the diffusion-controlled region is confined to a thin stagnant layer at :he ab'eolar walls and this gives rise to the apparent short diffusion path 200 In terpis of cal.culational convenience, diffusion-controlled models such as the one presented by Conley may provide reasonable results in terns of prediction of oxygen and carbon dioxide transport in the lungs, but these models provide little insight into the actual workings of the lungs. Consequently, they are of miniir.al long term interest . APPENDIX D EXPERIMENTAL DATA The data taken for the three different types of experiments perforraed in this study are listed as follows: 1. data taken during air bubble measurements are shown in Table D-1; 2. data taken during the saline simulation of blood oxygenators are shown in Table D-2; 3. data of actual blood oxygenation parameters during open- heart surgery are shown in Table D-3; 4. data taken during the saline liquid-breathing experiment ara shown in Table D-4; 5. data taken during the fluorocarbon liquid-breathing experiment are shown in Table D-5 . 101 202 TABLE D-1 DATA TAKEN DURING BUBBLE MEASUREMENT EXPERIMENT Salirie Flow Rate = 1.4 iii:er/min Air Flow Rate =5.9 liter/min Distance Major Diameter Minor Diameter From the Sparger Bubble No. (mm) (mm) 1.83 (cm) lA 1.83 9.0 2A 2.39 2.39 9.0 3A 2.18 2.18 9.0 4A 3.45 2.39 9.0 5A 3.20 2.54 9.0 6A 3.66 3.05 9.0 7A 3.05 2.54 9.0 8A 3.56 3.05 9.0 9A 3.66 2.08 9.0 lOA 3.71 1.43 9.0 llA 2.29 2.29 9.0 12A 3.96 1.63 9.0 ■3A 4.67 2.29 9.0 i4A 1.83 1.83 9.0 15A 3.45 1.93 9.0 IB 3.05 2.29 9.0 2B 2.64 2.49 9.0 3B 3.05 2.29 9.0 4B 3.15 2.90 9.0 53 4.47 4.06 9.0 63 3.15 2.54 9.0 73 2.79 2.54 9.0 8B 3.61 3.25 9.0 93 2.59 2.13 9.0 lOB 2.08 1.52 9.0 IC 3.66 3.05 20.0 2C 3.61 2.54 20.0 3C 3.30 3.30 20.0 4C 4.01 3.81 20.0 5C 3.81 2.84 20.0 6C 2.13 2.13 20.0 7C 2.03 2.03 20.0 8C 3.66 3.56 20,0 9C 2.13 2.13 20.0 IOC 2. ,19 2.08 20.0 lie 3.66 2.90 20.0 12C 3.66 2.90 20. C 203 TABLE D-1 (Continued) Major Diameter Bubble No. (mm^ 13C 3.81 lie ^.06 15C '^.06 ifaC '^•'^2 17C ^.06 18C 3.05 19C 4.06 ID 4.67 2D 4-57 3D 3.30 4D 2.64 5D 2.64 6D 3.61 7D 3.56 BD 2.18 9D ^-^1 lOD 2.49 IID 5.59 12D 5.08 Distance Minor Diameter From the Sparger (mm) (cm) 2.79 20.0 2.54 20.0 2.95 20.0 3.56 20.0 2.39 20.0 2.54 20.0 3.56 20.0 3.45 30.5 4.52 30.5 2.79 30.5 2.03 30.5 2.28 30.5 3.30 30.5 2.54 30.5 2.18 30.5 3.40 30.5 2.79 30.5 2.79 30.5 1.78 30.5 204 CO %! 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