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INTRAVASCULAR MICROBUBBLE BLOOD OXYGENATOR: PRINCIPLES AND LIMITATIONS
by
Ashok Krishnan, B.E.
A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master Of Science
COLLEGE OF ENGINEERING
LOUISIANA TECH UNIVERSITY
November 1994
Contact
ABSTRACT
In this thesis, a theoretical evaluation of a novel intravascular oxygenator is carried out, and principles of developing such a device and its limitations are analyzed.
The intravascular microbubble blood oxygenator (IVjiBOX) uses oxygen microbubbles to oxygenate blood within the vena cava. Carbon dioxide is expected to be removed via the physiological pulmonary pathway. A simple geometry of the device is presented, and physical characteristics are evaluated or assumed.
The phenomenon of bubble production from a submerged orifice is reviewed. A force analysis reveals that surface tension is the dominant force on a microbubble during its production. Optimum forces that cause bubble detachment are also found and a representative model of bubble formation is tested for its appropriateness to the device. Interaction between bubbles is treated in the context of primary and secondary bubbles and coalescence of bubbles.
An introduction to the structure of proteins and its alterability at the blood-bubble interface is provided and followed by some specific cases of blood reaction to bubbles. It is found that there will be no RBC lysis during either bubble dissolution or bubble production, and also that the lungs act as efficient filters for microbubbles.
Although there are no data pointedly suggesting the non-feasibility of the device, two important aspects are not known: 1) a method for producing microbubbles at rates and sizes required for a human, and 2) the total blood reaction to microbubbles. Theoretical and experimental work needs to be carried out on these two areas to make the oxygenator a reality.
APPROVAL FOR SCHOLARLY DISSEMINATION
The author grants to the Prescott Memorial Library of Louisiana Tech University the right to reproduce, by appropriate methods, upon request, any or all portions of this Thesis. It is understood that “proper request” consists of the agreement, on the part of the requesting party, that said reproduction is for his personal use and that subsequent reproduction will not occur without written approval of the author of this Thesis. Further, any portions of the Thesis used in books, papers, and other works must be appropriately referenced to this Thesis.
Finally, the author of this Thesis reserves the right to publish freely, in the literature, at any time, any or all portions of this Thesis.
Author_
Date
GS Form 14 (2/88)
TABLE OF CONTENTS
ABSTRACT……………………………………………….. iii
LIST OF FIGURES…………………………………………. viii
ACKNOWLEDGEMENTS…………………………………….. ix
CHAPTER 1 INTRODUCTION …………………………………. 1
1.1 Gas Exchange in the Human Body………………………… 1
1.2 Gas Transport in the Human Body………………………… 1
1.3 Introduction of Gas into a Liquid………………………….2
1.4 Artificial Oxygenation of Blood…………………………..3
1.5 The Proposed Novel Microbubble Oxygenator………………..5
CHAPTER 2 PHYSICAL CHARACTERIZATION OF DEVICE…………..8
2.1 Introduction…………………………………………8
2.2 Theory…………………………………………… 11
2.2.1 Bubble Requirement……………………………….. II
2.2.2 Bubble Frequency…………………………………. 11
2.2.3 Choice of Process………………………………….. 14
2.3 Device for Use in Humans…………………………….. 15
2.4 Discussion………………………………………… 18
CHAPTER 3 THEORETICAL ASPECTS OF BUBBLE PRODUCTION AND
BEHAVIOR………………………………………21
3.1 Bubble Production from a Submerged Orifice………………..21
3.1.1 Introduction…………………………………..—– 21
3.1.2 Bubble-Production Theory……………………………22
3.1.2.1 Overview of Bubble Production……………………….22
3.1.2.2 Tsuge’s Classification Scheme………………………..24
3.1.3.3 Force Analysis During Bubble Production………………..25
3.1.2.4 Model of Ramakrishna, Kuloor. and Kumar (1969)…………. 29
3.1.3 Application of Theory to IV^BOX………………………30
3.1.4 Applicability of Theory to Device……………………….33
3.1.4.1 Reconsideration of Equations (3.10) and (3.12)…………….35
3.1.4.2 Approaches Towards Bubble Detachment………………..36
3.1.4.3 Discussion of Figures 3.4,3.5,3.6, and 3.7 ……………….39
3.1.5 Discussion of Bubble Production………………………..45
3.2 Bubble-Bubble Interaction……………………………..46
3.2.1 Introduction………………………………………46
3.2.2 Bubble Interaction Theory…………………………….47
3.2.2.1 Interaction Between Primary and Secondary Bubbles………..47
3.2.2.2 Interaction Between Neighboring Bubbles………………..47
3.2.2.3 Bubble Coalescence ……………………………….47
3.2.3 Bubble Interaction in IVuBOX…………………………49
3.2.4 Discussion ……………………………………….49
CHAPTER 4 THEORETICAL ASPECTS OF BUBBLE PRESENCE
IN BLOOD ………………………………………52
4.1 Introduction………………………………………..52
4.2 Correlation with Bubbles from IVUBOX……………………54
4.2.1 Visualization of Control Volume……………………….. 54
4.2.2 Conditions for Bubble Dissolution………………………55
4.2.3 Fluid Classification of Environment……………………..57
4.3 Bubble Dissolution in Liquid……………………………57
vi
4.3.1 Case of Pure Liquid ………………………………..57
4.3.2 Effect of Surface Contamination………………………..59
4.3.3 Effect of Bubble Stabization………………………….59
4.3.4 Effect of Gas Adsorbing Particles……………………….59
4.4 Blood-Bubble Interaction—-…………………………..60
4.4.1 Bubble as a Foreign Substance …………………………60
4.4.2 Bubble-Blood Interaction Study………………………..61
4.4.3 Three-Dimensional Protein Structure…………………….62
4.4.4 Protein Interaction with Interface ……………………….62
4.4.5 Some Specific Blood Reactions…………………………64
4.4.5.1 Complement Activation …………………………….64
4.4.5.2 Platelet Aggregation……………………………….65
4.4.5.3 Macrophage Interaction …………………………….65
4.4.5.4 Albumin Adsorption……………………………….65
4.4.5.5 Cell-Bubble Interaction …………………………….66
4.5 Oxygen Bubble Dissolution in Blood………………………67
4.6 Microbubble Filtration by Lungs …………………………69
4.7 Hemolysis Caused by Bubbles…………………………..69
4.8 Discussion…………………………………………70
CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS……………72
APPENDIX 1 BLOOD COMPONENTS……………………………76
APPENDIX 2 NOMENCLATURE……………………………….78
APPENDIX 3 TABLE OF NOMINAL VALUES ……………………..81
BIBLIOGRAPHY………………-…………………………..82
VITA ………………..*89
vii
list of figures
Figure 2.1. Details of IVu^BOX (not to scale)………………………….9
Figure 2.2. Bubbles on surface of device……………………………. 13
Figure 2.3. Surface plot of Equation (2.12)………………………….. 17
Figure 2.4. Surface Plot of Equation (2.12)………………………….. 19
Figure 3.1. Details of bubble formation from a submerged orifice………….. 23
Figure 3.2. Forces acting on bubble during formation in a cross-flow situation … 26
Figure 3.3. Plot of Equation (3.15) ……………………………….. 37
Figure 3.4. Plot of Equation (3.25) ……………………………….. 40
Figure 3.5. Surface plot of Equation (3.26)………………………….. 41
Figure 3.6. Surface plot of Equation (3.27)………………………….. 42
Figure 3.7. Surface plot of Equation (3.28)………………………….. 43
Figure 4.1. Conceptual visualization of a 10-micron diameter bubble……….. 56
viii
chapter 1
introduction
1.1 Gas Exchange in the Human Body
The body derives its energy during rest solely from aerobic metabolism utilizing oxygen and produces carbon dioxide as a by-product (Guyton 1991). Gas exchange between the atmosphere and the body is carried out by the process of respiration, using the lungs as the gas exchange device.
Respiration has four distinct functional processes (Guyton 1991):
1) pulmonary ventilation, which is the airflow between lung alveoli and the atmosphere;
2) oxygen and carbon dioxide diffusion between alveoli and blood;
3) oxygen and carbon dioxide diffusion between blood and cells; and
4) regulation of the above processes.
In this thesis, a novel device for substituting oxygen-related functions 1 and 2 (pulmonary ventilation and gas diffusion between alveoli and blood) is investigated. The motivation is that insufficient pulmonary ventilation or deficient gas transfer between alveoli and blood can occur because of disease states, trauma, lack of oxygen in the atmosphere or when the regulatory aspects of respiration are not fully functional. Artificial oxygenation is carried out using masks, airway pressure support devices, or oxygenating blood inside or outside the body.
1.2 Gas Transport in the Human Body
Oxygen is used and carbon dioxide produced inside each cell of the body. Gas transport takes place between these cells and the atmosphere, with blood acting as the gas carrier. There are two most important contributors to gas transport (Guyton 1991):
1) hemoglobin present in erythrocytes, and
2) water present in plasma and cells of blood. Most of the oxygen (about 97%) is transported from the lungs to tissues in chemical
combination with hemoglobin (oxyhemoglobin). The remaining 3% is transported by the water of plasma and cells as dissolved gas. The reversible combination of carbon dioxide with water in red blood cells (RBCs) under the influence of carbonic anhydrase accounts for about 70% of carbon dioxide transport Only 7% of the carbon dioxide is transported as dissolved gas in blood. Carbon dioxide reacts with hemoglobin to form carbarnino hemoglobin which accounts for most of the remaining mode of transport
1.3 Introduction of Gas into a Liquid
Artificial oxygenation of blood requires oxygen to be introduced from a non-pulmonary pathway. Since the objective of this thesis is to study a novel and hitherto untried method of oxygenation, methods by which a gas can be introduced into a liquid are listed below, followed by the justification of choosing the intravascular microbubble method. Potential methods for introducing gas into a liquid include the following:
1) gas released by a chemical reaction in the liquid;
2) electrolytic breakdown of chemical species of liquid into its component gases. For example, water can be split into hydrogen and oxygen gases (Khosla et al. 1986);
3) direct entrainment of gas into liquid using baffles or blades, or agitating the liquid to entrap gas around it;
4) encapsulation of gas in a membrane which can later be introduced into the liquid (Mori etal. 1990);
5) saturating a second liquid with gas and then introducing it into the liquid (Bilge et al 1989);
6) diffusion of gas into liquid by exposing it directly to a gaseous atmosphere;
7) exposing gas to a liquid through an intervening gas-permeable membrane (Gaylor 1988); and 8) bubbling gas into liquid using submerged orifices.
1.4 Artificial Oxygenation of Blood
Only a few of the methods mentioned above can be used to introduce oxygen into blood because of the sensitivity of blood to temperature, mechanical impacts, and its chemical and physical environment (Skalak and Chien 1987).
The above-mentioned methods of introducing gas into a liquid are evaluated in the context of a practical and usable artificial blood oxygenator.
1) There has been no report on the oxygenation of blood by employing a chemical reaction that releases oxygen gas. Sensitivity of blood to its chemical environment may be a deterrence to this method.
2) Electrolysis as a method for oxygenation has not been reported. The present state of raicrodevice technology may aid in this technique.
3) Mechanical agitation of blood using baffles or blades will cause excessive cell lysis to occur, and such methods can be discounted for blood oxygenation.
4) Fluorocarbon encapsulation of oxygen in liquid membranes was attempted (Mori et al. 1990), but oxygen transfer efficiency was found to be slightly less than direct bubbling of oxygen into extracorporeal (EC) blood.
5) Oxygen dissolved in a perfluorodecalin fluid was injected into a canine peritoneal cavity (Bilge et al. 1989). There were no conclusive results obtained, but it was found that physiologically large amounts of oxygen could be delivered by this method in an acutely hypoxic animal.
6) Exposing blood to an oxygen-rich atmosphere was previously used to oxygenate blood using disc oxygenators, but this method caused excessive cell lysis and blood reactions (Garby and Meldon 1977).
7) Gas permeable membranes are employed in membrane oxygenators (MO) with two different oxygenation routes possible: extracorporeal (EC) and intravenous (TV). The general configuration for an ECMO is (Skalak and Chien 1987) suction pump to drain blood (if gravity drainage is not used), blood reservoir, blood pump, membrane oxygenator, heat exchanger, and a filter. An IV oxygenator currently in use is the IVOX® developed by Cardiopulmonics Inc. of Utah, which has been used in clinical trials since February 1990 (Jurraann et al. 1992). The following description of the IVOX used for adults is obtained from Mortenson and Berry (1989): it is a small-diameter (OD a 10.8 mm), elongated (lengtha650 mm) fiber oxygenator with 1,200 hollow fibers each of 270 microns ID and 380 microns OD. The surface area of all the fibers combined is 9,318 square centimeters, giving an oxygen transfer capability of 325 ml per minute and a carbon dioxide transfer capability of 490 ml per minute when used in venous blood of normal gas content The oxygenator (in its furled state) is placed into the vena cava through a femoral venotomy and then unfurled. Oxygen is introduced into the lumen of the hollow fibers through a small-diameter tube, and carbon dioxide exits from a gas outlet tube. Subatmospheric pressure is used to pull gas through the device rather than to force gas flow using a positive pressure.
8) Oxygenation using bubbles is used in bubble oxygenators (BO). Unlike MOs, BOs have been used only in EC circuits. In an ECBO, bubbles are introduced into a volume of blood, and oxygen transfer takes place while the bubbles rise through the blood. A defoamer placed at the top of the chamber containing blood removes all undissolved bubbles (Skalak and Chien 1987).
1.5 The Proposed Novel Microbubble Oxygenator
Although the EC system has been used for a long time, an IV oxygenator has been developed only recently (Mortenson and Berry 1989). EC procedures require more manpower resources (surgeons, nurses, technicians) and support facilities which only a few medical centers can have (Shapiro and Peruzzi 1993). EC oxygenators, by nature, require additional blood pathways like pumps, filters, defoamers, heat exchangers, and tubes (Skalak and Chien 1987). An intracorporeal support device will reduce the necessity of these resources, support facilities, and equipment
The IVOX avoids many of the drawbacks of EC oxygenators, but it has low oxygen and carbon dioxide transfer rates (Zapol 1992). The remaining gas exchange has to be carried out by the lungs using airway pressure therapy. Recent clinical trials of the IVOX in patients with advanced respiratory distress syndrome (ARDS) have reported a maximum metabolic gas exchange supplementation of 28% (High et al. 1992) and 30% (Conrad et al. 1993).
It has been the hope of the medical community eventually to develop an ideal oxygenating device that will totally substitute for the lungs. The oxygenating component of one such method, the Intravascular Microbubble Blood Oxygenator (IVfiBOX) is proposed in this thesis. It should be noted that a prototype of the device has not been constructed, nor has any experimental analysis been carried out Rather, this is a theoretical study. The IVuBOX is similar to the ECBO in that it uses bubbles for oxygenation, but different in that it does not perform the function of carbon dioxide removal. One of the procedures that can be employed in this case is the one similar to Low Frequency Positive Pressure Ventilation with Extracorporeal Carbon dioxide Removal (LFPPV-ECCO2R) (Gattinoni et al. 1986). Here, the lungs are kept nearly at rest (about 4 breaths per minute) at a low peak airway pressure of between 26 mm Hg to 34 mm Hg, thus buying time for the lungs to heal. Oxygenation is achieved primarily through the resting lungs (called apneic oxygenation), while carbon dioxide is removed in an EC circuit using a membrane lung. The ventilation part of this procedure (LFPPV) can be used to wash out carbon dioxide from the lungs using oxygen gas. The ideal method for carbon dioxide removal, one that avoids bulky ventilation equipment and is as versatile as the IVuBOX itself, would be to use an intranasal tube or a mask and, employing negative pressure, draw out the carbon dioxide.
The principles and limitations of a device that will oxygenate blood within a large vein will be studied in this thesis. Since an oxygenation technique using microbubbles intravascularly has not been explored before, many aspects of a device employing this method will have to be considered:
1) The device dimensions, number of bubble-producing orifices, and bubble production rate per orifice (all of which determine the overall oxygen delivery capacity), and, based on these factors, technology that is apt for fabricating the device and a design plan.
2) The mechanism of microbubble production that details the physics behind formation and detachment of bubbles and the forces that come into play during their lifetime.
3) Behavior of microbubbles after their release from the device, which considers the interaction between a just-released bubble and a newly forming bubble beneath it, and coalescence of bubbles already released.
4) Effect of bubbles in blood which considers aspects like blood reaction in the context of bubbles being foreign particles, gas transfer times from bubble to blood, stress induced by bubble collapse, and passage of microbubbles through lungs.
This thesis addresses the physical, medical, and technological limitations to the successful implementation of the device. Each will be evaluated when the above mentioned aspects are considered, and the information needed to understand these aspects will be identified.
CHAPTER 2 PHYSICAL CHARACTERIZATION OF DEVICE 2.1 Introduction
The goal of the intravascular microbubble oxygenator (referred to as IVpBOX) presented in this thesis is to introduce oxygen microbubbles of uniform size directly into the blood stream. These bubbles will be produced at high rates from numerous orifices present on the device surface. In order to maximize bubble production, the number of bubble-producing orifices, and hence the device surface area, should be maximized, yet it must be able to fit into the vena cava (the largest vein) without obstructing any of the veins emptying into it
There can be many variations in geometry that the device can take, but a simple design given in Figure 2.1 is adopted here since the objective of this thesis is to investigate principles and limitations of the device and not its intricate design details. The design given in Figure 2.1 is essentially a long cylinder closed at one end with gas-conveying pores on its surface. Oxygen gas is introduced into the open end of the cylinder, passes through the gas-conveying tubes pores, and enters the bloodstream as bubbles.
The physical constraints imposed by the device on the number of bubbles that can be produced per second is related to the number of orifices and the device surface area. The number of bubbles required per second is related to the oxygen supplementation needs. The number and size of orifices and their concentration on the device’s surface will dictate the process to be chosen to fabricate the device.
chapter 2 page 9-1 (missing)
Figure 2.1. Details of IVuBOX (not to scale)
This IVuBOX design is for use in an adult This device fits into the vena cava without obstructing vessels emptying into it Device dimensions are OD = 1 cm, ID =0.8 cm, length = 30 cm.
(a) and (b) device and its cross-section.
(c) enlarged area of surface of device showing orifices on it
(d) enlarged area of device surface and wall showing orifices on the surface and tubes in the wall.
NOTE: orifices and tubes will be more numerous in reality, but they are shown reduced in number for the sake of clarity
However, an intravascular bubble oxygenator has not been reported, and some initial assumptions will have to be made. In this thesis, the most critical assumptions made are as follows:
1) bubbles 10 microns in diameter can be produced at a frequency of at least 1,000 bubbles per second;
2) oxygen bubbles 10-microns in diameter will dissolve in venous blood within a specific time; and
3) total blood reaction caused by the bubbles is within acceptable limits.
The first two assumptions (regarding generation and dissolution of bubbles) are addressed in Chapters 3 and 4, respectively. Although the last aspect (blood reaction) is discussed in Chapter 3, the assumption is not justified because of a lack of quantitative knowledge in this field. Therefore, this aspect can be treated as a possible limitation.
Throughout this thesis, a bubble diameter of 10 microns and a bubble production frequency (rate) of 1 kHz are used as examples. The latter is a dependent variable since the value of bubble diameter, once chosen, will lead to the required bubble frequency (as illustrated in this chapter).
The reason for choosing a 10-micron value for bubble diameter is not wholly arbitrary but a combination of the following:
1) The experimental observation by Unkel (1991) showed that oxygen bubbles of this size shrink in saline;
2) this size is comparable to the 8-micron diameter of an RBC lending ease to conceptualization of the control volume (see Chapter 3) around the bubble;
3) the highest deposition rate on a 10-micron bubble is for particles in the 8-micron size range, the significance of which is discussed in Chapter 3; and
4) the value is computationally simple.
2.2 Theory
2.2.1 Bubble Requirement
Since oxygen is supplied by the device in the form of bubbles, the total oxygen supply can be stated in terras of number of bubbles per unit time. If is the oxygen requirement in moles per second, and Mbub is the number of moles of oxygen in one bubble of radius rt,Ub at a temperature of T (in Kelvin), the bubble frequency required for the device as a whole, Nb„b will be
N – Mnq “bub–
bub
(2.1)
The pressure inside a b”v caused by surface tension:
le sura of ambient pressure and excess pressure
PtM = P.
tot ~ ‘a**
where Pa is given by (Sears etal. 1988),
(2.2)
rbub (2.3)
where c is surface tension of the liquid. Using the equation for volume of a sphere, and substituting in the ideal gas law,
Mbub $
bub
R T
(2.4)
and R (in J.mole-^-1), and is specific for oxygen. Equation (2.1) can be written as
xV
bub
bub
(2.5)
2.2.2 Bubble Frequency
The total number of bubbles produced per second will depend on the bubble frequency per orifice and the number of orifices on the device. If the interbubble spacing Sbub (measured from the outer edges of the bubble) and bubble diameter dbub are specified, then area AbUb that can accommodate one bubble (shaded area in Figure 2.2) will be
*bub = Wbub I hub)1 (2.6)
dbub can be stated in terms of the orifice diameter d^ :
ndbc i
where n
“** 1 d^b (2-8)
where nSbub is the ratio of interbubble spacing to bubble diameter. It should be noted that the specification (see Chapter 3) for Sbub is that it should be at least twice dbub* that is, nsbub S 2. The total number of orifices on the device, Norj, will be the ratio of the total device surface area, A^ew to the area containing one orifice. Since the bubbles are produced from orifices, the area containing one orifice is equivalent to the area Abub containing one bubble (shaded area in Figure 2.2). Therefore Norj can be expressed as the ratio of the total device surface area to the area containing one bubble:
N°ri ~ (2-9)
Since each orifice produces one bubble, the total number of orifices on the device determines the total number of bubbles that can be produced from the device at any one instant of time. The total number of bubbles produced from the device will be the product of the bubble frequency per orifice and the total number of orifices on the device:
Nbub ~ B/ie
(missing)
Figure 2.2. Bubbles on surface of device
Shaded portion represents area A0ub>
Bubbles have formed at the orifices, and hence the orifices are hidden below the bubbles.
Substituting Equations (2.5) and (2.9) in Equation (2.10) and rearranging yields
Expressing dbub in terms of rbub in the above equation and simplifying gives
d _ 3RTf (*+*sbub)2 1Mnq
~ — L ^ y + 2a J{2A2)
2.23 Choice of Process
In the above section, two parameters, bubble diameter and ratio of interbubble spacing to bubble diameter, were incorporated into Equation (2.12). The technology used to fabricate the device will be influenced by these two parameters since they determine two key parameters of the process: resolution and aspect ratio. The relevance of resolution and aspect ratio in the fabrication of the device is discussed below:
1) Resolution. The device should be fabricated by a method which can create orifices uniform diameter. If the orifices have varying diameters, bubbles will be produced only from orifices of larger diameters because the surface tension force opposing bubble growth is inversely proportional to orifice diameter (see Chapter 3), resulting in bubbles forming at the largest orifice more easily. Therefore, the resolution of the manufacturing method should be very high, dictating that a microlithographic process be chosen rather than a conventional manufacturing process. However, traditional lithographic processes have lower resolution because of the large wavelength of exposure sources. Since x-rays have much shorter wavelengths (Brodie and Murray 1992), a lithography process using x-rays for exposure should be used.
2) Aspect ratio. An important criterion that the manufacturing method should meet is the aspect ratio. The ratio of lateral dimension to device depth, called the aspect ratio, is very high for the IVuBOX. This ratio becomes critical in the case of the gas-conveying tubes, where the aspect ratio, defined as the ratio of tube length (equal to orifice diameter) to tube diameter, will be high (Figure 2.1 d). In this thesis, the orifice diameter specified is 5 microns and the tube length (i.e. the thickness of the device’s wall) specified is 100 microns. The excess pressure required to blow 10-raicron diameter bubbles in plasma is about 0.28 atmospheres (see Chapter 3, section A.2.a) and the device walls have to be thick enough to support this pressure. Specific force analysis will reveal the actual thickness required, and this is complicated by the fact that the device wall is not a continuous surface but perforated with tubes connecting the orifices to the inside of the device. These finer aspects are not pursued here.
Methods reported in the literature describing nozzle fabrication using conventional lithography (non x-ray) cannot achieve nozzle tip diameters much greater than the nozzle depth. For example, the technique reported by Farooqui and Evans (1992) gives a nozzle tip diameter of 3 microns but a base diameter of 6 microns when the depth is 3.5 microns. However, using deep x-ray lithography Ehrfeld and Munchmayer (1991) have fabricated micronozzles of 2-microns in diameter and 300 microns in height with parallel side walls. In the IVuJBOX, orifice density is high (for example, when dori = 5 microns, ndbo= 2 and nSbub=3, orifice density is 5%) and interorifice spacing is low, the sidewalls have to be parallel so that adjacent tubes do not intersect Collimated and intense x-rays produced by synchrotron rings will make it possible to achieve parallel sidewalls even for depths up to 1,000 microns (Ehrfeld and Munchmayer 1991).
2.3 Device for Use in Humans
The use of IVuBOX is not restricted to humans, and, in fact initial testing will be conducted in animals. The following example considers two cases: total lung oxygenation supplementation in humans and a generic case.
Consider the specific case of a 70-kg human at rest M,^ in this case will be 1.67 x 10″4 moles s-1 (Guyton 1991). If it is required to supplement the oxygenation capacity of the lungs (100% of Mreq) totally, a device to fit into the vena cava, which is the largest vein, can be used. The sizes that the rVuBOX can take are highly variable since size will depend on factors like bubble frequency and orifice spacing, as discussed in Chapter 1. However, the geometry of the device will closely follow the dimensions of the vena cava. Mortenson and Berry (1988) provide the dimensions and geometry of an IVOX that is implanted in the vena cava of an adult, and these are used as a guide in the present study. For the geometry of Figure 2.1a, if the OD of the device D0d is 1 cm and length Ldev 30 cm,
&d*v ~ 71 I>oJ L
Bfitq I 1077s-1 « 1 kHz
Figure 2.3 illustrates the operating range when the oxygenation requirement is total lung supplementation as the upper limit and 50% supplementation as the lower limit The area of the device in the figure varies from 94 cm2 (the value obtained above) to 188 cm2. This Adev and Mreq combination gives a range of 270 hertz to 1,080 hertz for Breq.
To obtain values of Mreq , Adev, and Bfoq at other points (which corresponds to the use of device in other animals, percentage of oxygen supplementation, or choice of bubble frequency), the following ranges are considered:
(missing)
The plot shows the operating range of the device for a 70-kg human. The maximum in terms of both molar oxygen requirement and bubble frequency is indicated by the shaded oval. As the area of the device is increased, the bubble frequency reduces dramatically.
Mreq: 0.25 x 10~* moles s”1 to 2.5 x 1(H moles s”1 which corresponds to 15% to 150% of normal oxygenation needs as mentioned above.
AdeV: 20 x Ifr4 m2 to 200 x 1(H ra2 which corresponds to 20% to 200% of the Adcv • Bfteq: 75 s”1 to 7500 s-1 which corresponds to 7.5% to 750% of the bubble frequency.
The surface plot of these ranges is illustrated in Figure 2.4. This plot is given here only as a guide for further work on the device and is not further elaborated upon further.
During its operation, the device should deliver oxygen only in quantities that are required by the body and in quantities that can be fully used by the body. If oxygen bubbles introduced into blood are in excess of requirements, or usage, or can dissolve in blood without over-saturating it, they may not dissolve. Even if the bubbles do dissolve, there will be a subsequent reformation of oxygen bubbles elsewhere in the blood vessel network. This is a situation similar to decompression bubbles commonly found during resurfacing after deep-sea diving (Guyton 1991).
Although under-oxygenation will not present the problems mentioned above for the case of over-oxygenation, an oxygenator must perform its function of maintaining proper oxygen levels in blood. The control of over- and under-oxygenation is therefore important, and a control system that addresses this aspect has to be developed and integrated with the device. However, the development of such a control system is not treated in this thesis and is left as a topic for future research.
2.4 Discussion
The intravascular oxygenator will have dimensions that will provide maximum surface area without obstructing any blood vessel, and the device design chosen in this thesis has a length of 30 cm, OD of 1 cm, and ID of 0.8 cm. Because of the high aspect ratio and resolution requirements of the orifices and gas conveying tubes, an x-ray
(missing)
operating range for 70-kg human requiring
Figure 2.4. Surface Plot of Equation (2.12).
This generic plot shows pictoriaUy the direct relationship between oxygen requirement and surface area of device, and the inverse relationship between bubble frequency per orifice and area of the device. The shaded oval in the central region of the plot represents the operating range for an adult human.
uthographic process should be used to fabricate the device. Equation (2.12) describes the bubble frequency in terms of interbubble spacing, bubble radius, molar oxygen requirement, and device surface area. This equation can be used to obtain optimum performance from the device. A control system that addresses the aspects of over- and under-oxygenation has to be integrated with the microbubble device, but such a system is not addressed in this thesis.
In this chapter, it has been assumed that a particular orifice diameter will produce bubbles of predictable diameter at a determinable frequency. Simple as it may seem, this assumption is very important and far-reaching because the process of bubble formation and detachment is highly complex. Hence, the theoretical aspects of bubble production and behavior will be considered next
CHAPTER 3
THEORETICAL ASPECTS OF BUBBLE PRODUCTION AND BEHAVIOR
3.1 Bubble Production from a Submerged Orifice
3.1.1 Introduction
Assurance of size and frequency of bubbles produced by the IVfxBOX can be guaranteed only if the theory behind the production of bubbles is known. The production of bubbles from a submerged orifice is a complex phenomenon. Although there has been a huge volume of publications since the turn of the century on this subject, a universally applicable theory to predict bubble production does not exist Reports appearing in the literature tend to be based on a particular set of experimental conditions. Since there are numerous reports in the literature on bubble production, it would be easier to classify their regimes of production and refer to a particular group of literature that is relevant to the IViiBOX. Such a classification scheme was adopted and experiments carried out within each of the regimes of the classification by a group of researchers in 1969, starting with Raraakrishna et al. (1969). This work studied in bubble formation under constant gas flow conditions and was followed by reports on bubble formation under constant pressure conditions by Satyanarayan et al. (1969) and bubble formation under intermediate conditions by Khurana and Kumar (1969). Tsuge (1986) improved this classification scheme (Le. bubble formation under constant flow, constant pressure, and intermediate conditions) by dividing gas-flow rates into three ranges for each of these conditions. Tsuge’s scheme is adopted in this thesis to characterize bubble formation. Such an approach will aid further work to be carried out on the IVpBOX without having to test every report on bubble production appearing in the literature for its applicability to the device.
3.1.2 Bubble-Production Theory 3.1.2.1 Overview of Bubble Production
This review is adapted from Tsuge (1986) and Rabiger and Vogelpohl (1986). Assume a capillary tube of radius r0 submerged in a liquid (Figure 3.1 a) through which gas is passed. When gas flow starts, gas pressure pushes any liquid that had entered the tube because of the capillary phenomenon. The gas-liquid interface is pushed until it reaches the capillary tip (Figure 3.1 b), but the area of interface remains constant It should be noted that most papers assume the nozzle to point away from the gravitational field.
Although there are various forces acting on the bubble during its production (e.g., hydrostatic and viscous), only surface tension force is considered in this description because of its much greater magnitude. This effect is proved in a later section on force analysis. Surface tension is a phenomenon associated with the boundary surface between a liquid and a solid or gas (Sears et al. 1988). When the liquid is in contact with a substance, the interface behaves as if it were under tension. The force that causes this effect is called the surface tension force, and it is defined as a force per unit length. The surface tension of a liquid decreases with increasing temperature and solute concentration (Jamialahmadi and Muller-Steinhagen 1992).
Excess pressure on the gaseous side balances the force caused by surface tension, a, of the liquid (provided the capillary is perfectly wetted), and this excess pressure corresponds to (2 o~) / ra. For the case of a 5-raicron radius orifice submerged in plasma, the excess pressure caused by surface tension will be approximately 0.28 atmosphere.
(missing)
neck
g
(c)
(d)
Figure 3.1. Details of bubble formation from a submerged orifice
(a) gas-pressure forcing liquid out of tube
(b) hemispherical shape reached for the liquid gas interface
(c) neck formation starting
(d) neck closing.
Note: these figures are based on Tsuge (1986).
When the gas-liquid interface is pushed to the capillary tip, a point of inflexion is attained since the maximum interface size that can be maintained by surface tension force for the given capillary radius has been reached. Any pressure that exceeds this will cause the interface to expand, and it will continue expanding (as long as there is gas flow into the bubble, but the pressure does not necessarily have to continue increasing) until some imbalance causes the bubble or a part of it to break away. In the case of small bubbles, this separation is caused by the process of necking (Figure 3.1 c and d) where the neck size depends on gas-flow rate. Bubble detachment occurs as soon as the neck collapses and the detaching bubble will surge forward causing a wake which will affect the formation of the next bubble. This phenomenon, when coupled with the bubble formation phenomenon becomes highly complex (Tsuge 1986), and is beyond the scope of this thesis.
3.1.2.2 Tsuge’s Classification Scheme
Tsuge (1986) has classified models of bubble formation from submerged orifice into three groups. This classification is somewhat dependent on the chamber volume (Antoniadis et al. 1992; Tipton et al. 1992) which is the volume of the chamber that contains the orifice. However, in the literature, the classical description of a chamber is that it is present immediately beneath the orifice, and that the length of the tube connecting the orifice and chamber is comparable to the size of the orifice (see Antoniadis et al. 1992). Although there is no chamber just below the orifice in the IV|iBOX, (see Figure 2.1), it does not present a problem for the classification of bubble production from the IVuBOX because it tits the constant flow criteria of Tsuge (1986). Criteria for constant pressure and intermediate conditions do not apply here but can be found in Tsuge (1986). Tsuge’s classifications are listed below:
A) Constant flow condition, where the gas flow rate through the orifice is constant irrespective of bubble volumes or production rate, but the bubble gas pressure varies with time. The condition for constant flow is
-5 > 106 using cgs units (3.1)
where Lq is the length of the nozzle.
B) Constant pressure condition, where the gas-flow rate is time-variant such that the pressure at the orifice remains constant
C) Intermediate condition, where both the gas-flow rate and pressure vary with time.
Tsuge (1986) has also classified gas flow into three ranges using a functional relationship called the gas-flow rate number Nw.
Nw-*L4-l{£y» (3.2)
where UQ is gas superficial velocity through the orifice, g is acceleration caused by gravity, and pL is density of liquid phase. The numerical ranges for Nw are
I. Nw < 1 Uniform bubbles are formed.
n. 1 < Nw < 16 Bubble volume increases with increase in Nw. ID. Nw > 16 Bubbles break down after detachment from orifice.
3.13.3 Force Analysis During Bubble Production
The force acting on a bubble during its production is the resultant vector sum of forces generated by several different physical phenomena. Forces acting on a bubble that is forming at a submerged orifice are listed below (Figure 3.2): 1) Buoyancy force Fb. This force arises because of the difference between bubble and liquid densities, causing the bubble to be displaced away from the gravitational field. Buoyancy force is proportional to the volume of the bubble and the difference in densities
(missing)
Figure 3.2. Forces acting on bubble during formation in a cross-flow situation
between the two phases. These relationships can be found in elementary physics
textbooks (eg.. Sears et aL 1988).
FB = ( QirUt > < f * ) g (33)
2) Hydrostatic force Fh- The pressure field around a bubble exerts a force everywhere normal to its surface (Sears et aL 1988), where
Fg = ambient pressure x surface area of bubble (3.4)
3) Surface tension force F©. When a gas, liquid, and solid are in contact with each other, a force which tends to minimize the surface area of the liquid acts along the interface of the three phases. This is called surface tension force and is given by (Ramakrishna et aL 1969, but see Sears et al. 1988 for specific details)
F0 = 2 x re a cos 6 (3-5)
where a is the surface tension of the liquid, 2* rQ is the length along which this force acts. aod 0 is the contact angle between the two phases. For a perfectly wetted orifice, 8=0* giving cos 9= 1, and this is the assumption made in most of the literature.
4) Force caused by gas momentum Fj^ (Tsuge 1986). Gas flowing through an orifice wiB strike the liquid surface at the orifice, thus exerting a force on it. This force is proportional to the gas density pg, gas superficial velocity through the orifice and volumetric gas-flow rate QD through the orifice. Gas superficial velocity can also be represented as volumetric flow rate per unit cross-sectional ansa of the orifice. Therefore,
Fm = (3.6)
5) Force caused by Uquidviscosit>‘Fvrcerharze:al. 1991). Outward bubble growth will be resisted by the viscosity of liquid pp. and this force is given by (Ramakrishna et aL 1969),
Fv = 6jc rbfip Vlip (3.7)
where V^p is the growth velocity of the bubble tip.
6) Force caused by liquid co-current flow Fcf (Tsuge 1986). a bubble forming in flowing liquid will experience a force proportional to the liquid velocity V^u, given by
f=cD(ii^)(^) I §
where Co is the drag coefficient of the bubble in plasma. The drag coefficient is given by (Gerhartetal. 1991)
The Reynolds number Re is given by (Gerhart et aL 1991):
Re = Q, VW^* (3-8 “)
Using values of plasma density, and 8 cm/s as an approximate value for blood velocity in the vena cava (Guyton 1991), Re = 0.667, and using this value in Equation (3.8 a), CD = 39.5.
If die design of Figure 2.1 is used, the liquid flow direction cannot be considered co-current (see Tsuge 1986) anymore since die blood flow is perpendicular to the nozzle orientation. This will be a cross—flow situation provided mere is no turbulence at the device surface. However, turbulence is bound to occur around bubble-producing areas, and the direction of blood cannot be stated with certainty in a turbulent situation such as this. Therefore, the direction of the vectorial force Fcf cannot be stated with certainty. The minimum contribution towards bubble detachment occurs when the liquid flows in the opposite direction of bubble formation (counter-current flow), in which case the bubble is actually prevented from detaching. The maximum contribution towards bubble detachment caused by liquid flow occurs when the liquid flows from beneath the bubble in the direction of bobble tip movement, which is a co-current flow. The magnitude of the force caused by counter-current flow will be the same as that for the co-current situation, but the direction will be reversed. In later sections of this chapter, efforts are directed towards finding out the possible ways by which a bubble can be caused to detach from an orifice. Since the maximum vectorial force contributing towards bubble detachment is for co-current flow, only mis force is evaluated here.
7) Force caused by inertia of bubble F\. Once the bubble starts to grow, the inertial force caused by this movement will cause the bubble to continue moving away from the orifice and later detach. The momentum of a bubble is the product of its growth velocity and virtual mass, and the time derivative of this momentum represents inertia. The inertial force is given by (Ramakrishna et al. 1969)
12 x 8) Wake force. Because of the complex nature of force caused by the wake of the previous bubble, very few reports are available on this subject (see, for example, Tsuge 1986). The wake forces are not calculated here because it is felt that the forces involved are highly variable after a steady state (corresponding to the time after many bubbles have been released) is attained. The wake forces are very sensitive to pressure fields in the neighborhood of the bubble, and, in the IVuJBOX, the neighborhood is again full of bubbles being formed or detached. Experimental work will have to be carried out to clarify the nature and magnitudes of these forces.
3.1.2.4 Model of Ramakrishna, Kuloor, and Kumar (1969)
This concise model has been widely referred to and also has been confirmed experimentally by Bowonder and Kumar (1970). They used a maximum bubble frequency of about 100 Hertz and a minimum bubble size of 1.3 mm. Ramakrishna et aL’s (1969) paper is used here as a representative sample of models that could possibly be adapted to the IVuBOX and to test its suitability for such a purpose.
Bubble formation is assumed to consist of an initial expansion stage and a later detachment stage. During the expansion stage, the bubble base remains attached to the orifice tip; while in the detachment stage, the bubble base moves away from the tip, the bubble being connected to the orifice through a neck. The final bubble volume Vp is the sum of volumes in the initial stage Ve and the later detachment stage Vp. For the initial stages the equation is derived by making a force balance on the bubble using forces caused by buoyancy, viscous drag, surface tension, and inertia, and is given by
192 * g + 2 H I QLV’ + 8<>LrE (3.10) The nozzle orientation for which the above equation was derived is shown in Figure 3.2. Since, in the device design of Figure 3.1, the nozzles are present all around the device, the effect of changing the nozzle orientation has to be considered, and this is discussed in a later section.
3.13 Application of Theory to IVuBOX
Using Equation (3.1) to test for constant flow conditions in the IVu,BOX, it is found that for Lq = 100 microns, or 0.01 cm, and do = 5 microns, or 0.0005 cm (values chosen for the geometry of Figure 2.1a):
% = 1.6 x 1011 *> 106
do
and. therefore, bubble formation can be assumed to occur at under constant gas-flow conditions.
Using the value of plasma density pp and U0=2.87 x 104 microns/second (computed later in this section) in Equation (3.2),
Since Nw< 1, the region of bubble formation for the IVuBOX is A-I as per Tsuge’s classification.
In order to analyze the forces on a bubble quantitatively, the case of 10-micron diameter oxygen bubbles being released into a large vein at a frequency of 1,000 bubbles per second for orifices of 5-raicron diameter will be considered (neglecting bubble interaction). Design choices that lead to this representative bubble size and frequency have been dealt with elsewhere in this thesis. It is to be noted that this is a section on a force analysis which is used to evaluate the magnitudes of the respective forces, and it is not intended to be a derivation of a force balance where even the direction of the forces has to be considered. The forces on the bubble during production are listed here:
1) Buoyancy force Fr. Using the value of the density of plasma in Equation (3.3) yields FB = 53xlO-12 N.
2) Hydrostatic force Fh- Pressure in a large-diameter vein is essentially atmospheric (Guyton 1991), and, since the device will be placed in a large vein, the ambient pressure under which the device operates can be assumed to be atmospheric (1.013 x 10s Nm-2). Using this value, Fh = 3.18 x 10-5 N.
3) Surface tension force F0. Since the device will blow bubbles into plasma (and not whole blood: see Chapter 3), the surface tension of plasma is used, which gives Fo=1.10 x 10-* N.
4) Force caused by gas momentum FM. The gas-flow rate and gas superficial velocity will have to be calculated first The gas-flow rate through each orifice Q0 will be
Nw 1 1.51 x 10~5
Qo = bubble production rate per orifice x volume of 1 bubble
Qc= BfaX | Jt r3 (3.11)
which gives Q0 = 5.24 x 105 p.3 s-1. Using this value and the density of oxygen at 1 atra (assuming that the density of oxygen remains constant under the operating conditions of the device), FM = 1.91 x 10″16 N.
5) Force caused by liquid viscosity Fv Since the bubble radius and bubble growth velocity vary with time, only an approximate Fv is computed here for comparing magnitudes of the different forces. The bubble radius used will be a final radius of 5 microns. Bubble production rate is 1000 s*1; therefore, one bubble of 10-micron diameter is formed every 0.001 second. The bubble tip velocity would then be
V. = 10 * IP”6 ffl y. | iq-2 ^-i
* 0.001 s Using Equation (3.7) gives Fy = 1.13 x 10~9 N.
6) Force caused by liquid co-current flow Fcf Using an approximate value of 8cm/s for velocity of blood flow in the vena cava (Guyton 1991) in Equation (3.8), FCf= 1.02 x 10-* N.
7) Force caused by bubble inertia Fr. Using value of flow rate Q obtained for the case of Fm and the density of plasma in Equation (3.9), Ft. = 2.05 x 10-13 N.
Using the same values of Q, pp, do, o* and \i as used for the force analysis and substituting in Equation (3.10) and simplifying gives
vi = 2.055 x 109 + 1.925 x 10s V*_ + 1.22 x 108 vt (3.12) Solving Equation (3.12) for Ve gives VE = 1.22 x 108 m3 and using equation for volume of sphere, rbub = 308(x.
3.1.4 Applicability of Theory to Device
Gas-flow rate through the nozzle was found to be constant because the length of the nozzle is much greater than its diameter. Also, the flow rate number Nw falls well into the range required for satisfying constant flow assumptions of Tsuge. For the IVuBOX to be classified as A-I, an important requirement to be met is that bubbles be formed quasi-statically, i.e., slow formation of single bubbles. In the IVuBOX, bubbles are formed at high frequencies (in the kilo hertz range), requiring that bubbles form in quick succession, and bubble interaction is not included in any of Tsuge’s classifications. Still, this is the closest classification that can be made for the IVuBOX. Other models for constant flow conditions include those of Wraith and Kakutani (1974) and Wraith (1971). However, these models are not applicable to the IVuBOX because some forces have been ignored. Wraith (1971) ignores the surface tension and liquid viscosity forces, and Wraith and Kakutani (1974) ignore the force caused by liquid viscosity. If, at a later stage in the development of the device, it is found that bubble production from the device cannot be treated as being characteristic of the A-I group, then a new classification should be arrived at based on experimental work.
Comparing the magnitudes of forces acting on the bubble (in Newtons) and ignoring their directions of action (which is not the case in later sections of this chapter),
fb fh ‘• Fo ‘■ FM ‘• Fv ‘■ FCF : Fj 5.3xlfr12 : 3.2xlfr5: l.lOxlO”6: 1.9U10″15 : l.BxlO”9 : 1.02*10-* :2.05xl0~13
it can be seen that F0, Fh > Fb, Fm, Fv, Fcb Fl • Since the bubble is very small, it can be assumed that force Fh acts equally at all points on the bubble and is balanced by the gas pressure inside the bubble (this has been assumed by most models). The latter pressure can be accounted for by the gauge pressure since gauge pressure can be calibrated to include atmospheric pressure.
The numerical values obtained from the force analysis provide a good picture of the forces that dominate during bubble production. However, this is a analysis of individual forces, and the interaction between these forces was not considered. Bubble-production models satisfy the need for such a force balance and enable the prediction of bubble sizes. It can be concluded from the above force analysis that the surface tension force has the highest magnitude and contributes greatest to the prevention of bubble detachment From Figure 3.2, it can be seen that the forces contributing to bubble detachment from the orifice are as listed:
1) Fi, the inertial force of the bubble (which causes the bubble to continue moving away
from the orifice);
2) Fm, the force caused by gas momentum (which strikes the tip of the bubble);
3) Fcf, the force caused by flowing liquid striking the bubble surface (note that the nozzle orientation has a great effect on this force); and
4) Fb, the buoyancy force which tries to force the bubble farther away from the earth’s gravity (and since the only place to which the bubble is attached is die orifice, the buoyancy force aids in bubble detachment). In the force analysis, it was seen that the buoyancy force is six orders of magnitude smaller than the surface tension force and three orders of magnitude smaller than forces caused by viscosity, both of which pre vent bubble detachment Although die nozzle orientation will affect the buoyancy force, this change in the net buoyancy force caused by a different nozzle orientation will again be much lesser in magnitude than the forces caused by surface tension and viscosity. Therefore, in die context of the IVuBOX, nozzle orientation has very little effect on either bubble formation or bubble detachment It must be noted that this applies only when the bubbles are in the 10-raicron diameter range, and the assumption is invalid for bubbles of larger diameters. At larger diameters, buoyancy becomes the dominant force for bubble detachment, and nozzle orientation becomes important
3.1.4.1 Reconsideration of Equations (3.10) and (3.12)
It should be again emphasized that nozzle orientation is important for larger bubbles. In the following analysis, the simple case illustrated in Figure 3.2 is considered because it is required of the device to produce bubbles that are small (in the 10-micron range) and parameters like bubble frequency and orifice size are later varied so as to achieve smaller bubbles. A large part of this analysis is devoted to these aspects.
Equation (3.12) predicts the bubble diameter to be 616 microns when using an orifice of 5-micron diameter and the gas-flow rate required by the IVuBOX. This is an unacceptable value since the diameter of bubbles required to be produced by the device is 10 microns. From the solution Ve = 1.22 x 108 u,3, it can be seen that values of the first and second terms of RHS of Equation (3.12) are negligible compared to the third term,
dependent only on the third term. Therefore, Equation (3.10) can be rewritten
as
8 Ql e
(3.12 a)
Simplification yields
If Equation (3.12
8 Ql
is used, then the diameter of the bubble
which is close to the value of 61(
of 616 microns obtained using the full form of Ramakru et al.’s Equation (3.8). The value of parameters specified in solving the initial Equation (3.10) were Q0 = 524 us”1 and d^ = 5 \i. For this set of parameters, Equation (3.10) predicts a bubble diameter of 608 microns, and also that the viscous and inertial forces are not significant for bubble production when the bubbles are in this size range. However, it is possible that the parameters specified restrict Equation (3.10) to predict large bubble sizes only, and this choice of parameters causes the equation to neglect the viscous and inertial terras. To resolve this, the analysis has to be carried out in a backwards direction, whereby the desired bubble size is specified and the resulting parameters are obtained. In order to carry out such an analysis, the values of acceleration caused by gravity, plasma viscosity, plasma density, and plasma surface tension are substituted into Equation (3.10), giving
V| = 4.83xl0″3 Go + 2.87×10-7 go V* + 2.18xl0~5 d0 v| (3.13) Using Equation (3.11) to express the flow rate per orifice Q0 in terras of bubble frequency and bubble volume, substituting in Equation (3.13), and rearranging gives
d0 = 1.92 x 105 r3 + 1.5 x 103 Bj^ r4 – 3.42 x 10~2 B^ r2 (3.14) If the required bubble radius is 5 microns (i.e. 5×10″* m), the above equation reduces to d0 = 2.4xl0″n – 9.35xl0″19 B^q – 8.55xl0~13 Bfreq (3.15)
The plot of Equation (3.15) is given in Figure 3.3. From this plot, it can be seen that, as the bubble frequency increases, there has to be a drastic reduction in the orifice diameter. Beyond a bubble frequency of 25 Hertz, the orifice diameter falls to levels which cannot be physically realized.
3.1.4.2 Approaches Towards Bubble Detachment
In the previous analysis, the following two cases were found: 1) the bubble sizes predicted by the equations are too large when parameters like orifice diameter, bubble frequency, and gas-flow rate are specified; and
(missing)
10 15 20
bubble frequency in Hertz
Figure 3.3. Plot of Equation (3.15)
Backward solution of the bubble-model equation of Ramakrishna et al. (1969) for the specific case of a 10-micron diameter bubble in plasma. The set of values of bubble frequency and orifice diameter for which 10-micron bubbles will be formed is given by the plot above. The plot shows that, in order to produce 10-micron diameter bubbles, the orifice diameter don has to decrease drastically with increasing bubble frequency.
2) when the bubble size is specified and a backward solution is carried out, the bubble frequency and orifice radius predicted by the equations are too low.
If it were possible to detach the bubbles before they grew to larger sizes, then the device can perform its expected oxygenation requirements. Bubble detachment can be enhanced by increasing the forces that cause bubble detachment (liquid co-current flow, bubble inertia, buoyancy, and gas momentum) or decreasing those forces that prevent bubble detachment (surface tension and liquid viscosity). Since surface tension is the greatest force that prevents bubble detachment (as seen in the force analysis section), forces that enhance bubble detachment are analyzed with respect to the surface tension force.
Since the properties of plasma cannot be changed, the values of surface tension, density, and viscosity of plasma are substituted into Equations (3.3), (3.5), (3.6), (3.8),
and (3.9) to give, respectively,
FB = 5.14 x 103 d3bub (3.16)
Fa = 0.22 d0 (3.17)
FM = 0.46 x B*\bub (3.18) do
FCF = 1.55 x 104 V2 d2bub (3.19)
Fi- 20BU4ub (3.20) Four separate cases for which bubble detachment can occur are
Fo | FB or 0.22 dQ = 5.14 x 103 d3bub (3.21)
B2 d6
Fa = FM or 0.22 d0 1 0.46 x ***** (3 22)
<%
Fa = Fcp or 0.22 da = 1.55 x 104 V| d2bub (3.23)
Fa | F, or 0.22 d0 = 20 Bj^ d*hub (3.24)
Simplifying Equations (3.21), (3.22), (3.23), and (3.24) gives, respectively,
d0 = 2.34 x 104 d\ (3.25)
do = 1.28 d2bub Bfreq (3.26)
d0 = 7.05 x 104 V£ d2bub (3.27)
* I 90’91 BU 4ub (3.28)
The plot of Equation (3.25) and the surface plots of Equations (3.26), (3.27), and (3.28) are given in Figure 3.4, Figure 3.5, Figure 3.6 and Figure 3.7, respectively. From these figures, a range of values of blood velocity, bubble frequency, and bubble diameters, and the corresponding orifice diameters that will cause detachment can be found. By using these sets of values, bubble detachment can be forced to occur.
3.1.4.3 Discussion of Figures 3.4,3.5,3.6, and 3.7
When the forces that balance surface tension are equated individually, Equations (3.25), (3.26), (3.27), and (3.28) resulted, whose graphical representations are given by
detachment for the intravascular oxygenator.
If the buoyancy force alone is relied upon to disengage the bubble by counteracting the surface tension force, it is found from Figure 3.4 that the orifice diameter has to be extremely small (in the picometer range). Such orifice sizes are physically unrealizable, and therefore, the use of buoyancy force alone to detach bubbles is not feasible.
When the surface tension force and force caused by gas momentum are equated, the resultant equation has three variables: orifice diameter, bubble diameter and bubble frequency. When a range of values of the three variables to be plotted is chosen, it
(missing)
4 5 6 bubble diameter in microns
10
Figure 3.4. Plot of Equation (3.25)
The effect of buoyancy force in causing bubble detachment is shown here. Surface tension force is balanced by the buoyancy force, and the corresponding set of values of orifice diameter and bubble diameter
(missing)
Figure 3.5. Surface plot of Equation (3.26)
This plot shows the relationship between the orifice diameter, bubble diameter, and bubble frequency when only the gas momentum is considered for bubble detachment The variable parameters are orifice diameter, bubble diameter, and bubble frequency.
(missing)
bubble diameter dbub m microns
15 0.
blood velocity V”l in m/s
(missing)
Figure 3.6. Surface plot of Equation (3.27)
The effect of blood flow velocity on bubble detachment by counteracting the force caused by surface tension is described by this plot. The blood velocity is varied from the normal value of 0.08 m/s to 1.2 m/s (i.e. a 15-fold increase), and the corresponding values of orifice diameter and bubble diameter are given.
Figure 3.7. Surface plot of Equation (3.28)
This plot describes the effect of bubble inertia in causing bubble detachment by counter-acting the surface tension force. Values of bubble frequency and bubble diameter and the corresponding values of orifice diameter for which bubble detachment will occur are given here.
becomes necessary to choose a very high range for the bubble frequency (in the kilo Hertz range) so that the resultant orifice diameter remains within a range that is physically realizable. Although an alternate method of achieving acceptable orifice diameters would have been to increase the bubble diameter, such an approach has its limitations imposed by the maximum acceptable bubble size that can be introduced into the body. From Figure 3.5, it can be seen that the bubble radius will have to be much greater than 50 microns to achieve bubble detachment Because of these limitations, it can be concluded that bubble detachment using the force caused by gas momentum is not possible.
Equation (3.28) and its accompanying surface plot Figure 3.7, pertain to the situation whereby only the bubble inertia is used for detachment. Again, high ranges are chosen for the bubble diameter and bubble frequency so as to keep the orifice diameter within physically realizable dimensions.
Equation (3.27) describes the situation whereby the velocity of blood is used to detach bubbles from the device. The normal velocity of blood in the vena cava is 0.08 m/s (Guyton 1991). From Figure 3.6, it can be seen that, if the blood velocity is increased to a much higher value than normal, bubble detachment will occur for bubbles in the 10-micron range. Also, it is evident that for orifice diameters less than 5 microns, bubble detachment will not occur for any value of bubble diameter. This seems to be the most attractive among methods employable for bubble detachment because the orifice diameter and bubble diameters (see Figure 3.6) are much closer to values required by the device. However, the question of possible methods by which blood velocity can be increased in the vicinity of the device remains unanswered. A simple approach would be to constrict the vena cava starting from the lower end of the device so that there is a concomitant increase in blood velocity after this point as per the conservation of mass requirement (see, for example, Welty et al. 1969). The conservation of energy requirement necessitates a drop in pressure after the constriction ends, and this aspect has to be addressed during the design of the device.
3.1.5 Discussion of Bubble Production
Bubble production from a submerged orifice is a complex phenomenon, and, to simplify its study, the bubble production regime of the IVuBOX was classified using Isuge’s (1986) scheme as A-I. Here, the gas flow rate is constant, and the dimensionless gas-flow rate number Nw is much less than unity.
The bubble formation model of Ramakrishna et al. (1969) was used to predict bubble sizes when the gas-flow rate, bubble frequency and orifice diameter were specified. This resulted in a bubble size of 608-micron diameter. Using a modified form of the bubble equation and employing a backwards computation method whereby the bubble radius is specified, a range of bubble frequency and orifice diameter was obtained. This gives a choice of parameters that can be used to produce 10-micron diameter bubbles. The model in its original form does not incorporate all the forces that will occur during bubble production. Forces caused by liquid flow, bubble wake, and the changing pressure fields in the neighborhood of the bubbles have to be incorporated into the model so that it can accurately predict bubble sizes and frequencies.
Since a force analysis revealed that the force caused by surface tension has the greatest magnitude, the four forces that cause bubble detachment (buoyancy, gas momentum, bubble inertia, and liquid flow) were equated to the surface tension force and analyzed. The resulting plots indicate that the buoyancy, gas momentum, and bubble inertia terms are not good candidates for enhancing bubble detachment because of the high ranges of bubble frequency and bubble diameter required so as to keep the orifice diameter within physically realizable limits. It was also found that an increase in blood velocity did provide a means of bubble detachment and that such a method is better suited to the device.
3.2 Bubble-Bubble Interaction 3.2.1 Introduction
During or after their production, two or more bubbles can interact with each other in a variety of ways. This interaction can be between primary and secondary bubbles from the same orifice or between bubbles being produced from neighboring orifices or bubbles that have already been produced. The end products of such an interaction need to be known to predict the resulting bubble characteristics, the most important for the IVuBOX being bubble size. Studies on bubble-bubble interaction have been few; the majority of these deal with bubble coalescence.
The effect of salts in inhibiting bubble coalescence was studied initially by researchers like Marrucci (1969), Marrucci and Nicoderao (1967) and Lessard and Zieminski (1971). The equations detailing electrolyte concentration and their effect on bubble coalescence formulated by earlier researchers were modified by Prince and Blanch (1990b) and Sagert and Quinn (1978) to give specific transition concentrations. Experimental results of Marrucci and Nicodemo (1967) and Lessard and Zieminski (1971) agree well with the modified equation of Prince and Blanch (1990b). The effect of organic compounds on bubble coalescence was studied by Jamialahmadi and Muller-Steinhagen (1992), Sagert and Quinn (1987), and Keitel and Onken (1982), although the effect of specific proteins on coalescence has not yet been reported.
3JL2 Bubble Interaction Theory
3.2.2.1 Interaction Between Primary and Secondary Bubbles
As the gas-flow rate increases, interaction between the primary bubble (which is in
the process of detachment from the neck) and the secondary bubble (which is being
formed simultaneously at the orifice) becomes significant (Deshpande et al. 1992). The
degree of interaction increases with increasing gas-flow rate and decreasing orifice
diameter (Deshpande et aL 1992). Secondary bubbles can also be formed when the
constricted neck severs and one end of the neck impinges on the already departed bubble
causing liquid entrainment into the departed bubble which, in turn, disintegrates the
departed bubble (Rabiger and Vogelpohl 1989).
3.2JIJ2 Interaction Between Neighboring Bubbles
During their formation, two adjacent bubbles can interact and thus alter their production characteristics. The condition for interaction between bubbles is that the inter-orifice spacing be greater than the final bubble diameter that would be obtained from a single, isolated orifice (Solanki et aL 1992).
3.2.2.3 Bubble Coalescence
Bubble coalescence can take place during bubble formation or after bubble release from the device. When two bubbles come in contact with each other, there is a possibility that they will adhere and coalesce. The process of coalescence occurs by the thinning and final rupture of a thin liquid film between the bubbles (Marrucci 1969). Salts retard the thinning of the film between two bubbles and thus inhibit bubble coalescence, provided the salt is above a certain concentration (Marrucci 1969; Lessard and Zieminski 1971; Sagert and Quinn 1978; Prince and Blanch 1990b). The intervening film between two bubbles thins abruptly if the concentration of the dissolved salt is less than a particular value called the transition concentration given by (Prince and Blanch 1990b):
C, = 1.18 | ( Sff | *, T ( fc )-> (3.29)
where B is the retarded van der Waal’s force, a is surface tension in gs~2, T is temperature in Kelvin, Rg is gas constant, rb is bubble radius in cm,
nj is the number of ions formed upon dissolution of salt, 5a /5c is the surface tension gradient
The above equation is arrived at by balancing the forces caused by pressure difference between the liquid of the film (the intervening film between the two bubbles) and the liquid outside the border of the film. This force caused by pressure difference is matched at any instant of time by a difference in surface tension between the surface liquid in the film and the surface outside the film. The rate of thinning is determined by the balance of these two forces. Separating out the rb term and rearranging Equation (3.29)
c’ * i (3.29 a)
r\
where k I 1.18 | ( B o)h Rg T ( ^ )~2 (3.29 b)
oc
k contains terms that are constant for a given salt solution. Equation (3.29) has been verified experimentally (for bubble radius in the 0.18 cm to 0.205 cm range), and this equation gives a better fit to experimental observations than the work of Marrucci (1969). From Table 1 of Prince and Blanch (1990a), when r^ = 0.18 cm, Ct = 0.15 moles/liter.
Therefore,
k I 0.15 x (0.18)* “”j* cm* or * 1 6.36 * 10~2 “”ft cm* /iter /jfer
3.2.3 Bubble Interaction in IVuBOX
1) It is not known whether there will be any interaction between primary and secondary bubbles in the IVuBOX. Hence, a qualitative evaluation cannot be made here. However, Unkel (1991) performed experiments to study the dissolution of microbubbles in 0.9% saline and later observed that these bubbles do not coalesce after a collision but merely bounce off each other.
2) A value of orifice spacing can be chosen after trials are conducted on a prototype of the device and after considering factors like total device area available. In the IVuBOX design presented in this thesis, specifications for orifice spacing are two bubble diameters, and this is minimum spacing to prevent bubble interaction.
3) Bubbles released from the IVuBOX have an initial radius of 5 microns. As they shrink (which is discussed in Chapter 4), the probability of coalescence becomes lower because of the inverse relationship between bubble radius and transition concentration. Considering a 5-micron radius bubble, the transition concentration can be found using Equation (3.29 a) to be Q = 2.84 moles/liter.
3.2.4 Discussion
Interaction between primary and secondary bubbles can be manipulated by varying gas-flow rates. However, this might become a limiting factor in that there is a minimum required gas-flow rate (depending on body oxygen need); in an effort to prevent bubble interaction, the gas-flow rate cannot be reduced below a certain level. A balance between the final bubble diameter acceptable and the gas flow rate requirement for oxygen supplementation will have to be found.
Lowest orifice spacing should be two bubble diameters although just one bubble diameter spacing will prevent bubble interaction because single bubble spacing can cause bubbles with a bimodal diameter distribution to be produced (Solanki et al. 1992).
Total electrolyte concentration in blood plasma is 745 mg per 100 ml. For the sake of simplicity and illustration, assume all of this electrolyte is NaCl. Then, the molar concentration of NaCl in blood will be 0.127 moles/liter. The concentration required to prevent coalescence of 5-micron radius bubbles was found to be 2.84 moles/liter. When these two values are compared, it can be seen that the salt concentration in blood is not enough to prevent coalescence. This result does not necessitate the requirement of smaller bubbles (or abandoning development of the device) for the following reasons:
1) Organic compounds can reduce bubble coalescence as reported by Jamialahmadi and Muller-Steinhagen (1992), Sagert and Quinn (1987), and Keitel and Onken (1982). There have been no reports on the effects of proteins on coalescence.
2) From Figure 2.2, it can be seen that die bubbles will be separated by more than two bubble diameters. The probability of these bubbles meeting each other depends on the time for complete dissolution and turbulence of blood. These factors have been studied by Prince and Blanch (1990b) for air sparged bubble columns. However, it is premature and inappropriate to use this study to predict bubble coalescence in blood when using the IVjiBOX since there are many unknowns and uncertainties like turbulence in the device vicinity and time for complete dissolution of the bubble. Since these studies are statistical, proper results can only be obtained by conducting experiments with bubbles of sizes and frequency used in the actual device.
3) The surface around the gas bubbles will be electrically charged because of polarizing forces (see Chapter 4), causing repulsion between two bubbles approaching each other (Keitel and Onken 1982; Jamialahmadi and Muller-Steinhagen 1989). Thus the first step in bubble coalescence will be inhibited, although this cannot be stated in quantitative terms until experiments are carried out
The theory of bubble production and bubble behavior in a non-reactive liquid with no surface active ingredients and ignoring gas diffusion was considered in this chapter. However, bubbles produced by the IVpBOX will be released into blood, and the effects of formed elements and other constituents of blood on the bubbles have to be considered. The next chapter deals with these aspects.
CHAPTER 4
THEORETICAL ASPECTS OF BUBBLE PRESENCE IN BLOOD
4.1 Introduction
Knowledge of the history of bubbles released into blood is essential for the FVjiBOX to be approved for use in humans. Bubbles released into blood from the IVpBOX will have an unique (immediate) environment which will be very different from the environment of bubbles with sizes in the centimeter range. This situation arises because of the comparable sizes of the microbubbles released from the device and the formed elements of blood. To put tins in perspective, the control volume of a microbubble released into blood is described first, followed by statements regarding microbubble dissolution in blood and the appropriate fluid classification of the bubble environment
Understanding the bubble’s history during and after its production will enable the prediction of three important aspects that will determine the feasibility and design considerations of the device:
1) total reaction caused by the presence of bubbles in blood;
2) approximate location of released bubbles in the body’s blood vessel network; and
3) damage caused to formed elements in blood because of the mechanical impact of bubbles during the process of production or during their collapse (dissolution).
These three aspects are quantitatively dependent upon the time taken for producing the bubble and time taken for complete dissolution of the bubble in blood. This time factor depends on the gas transport characteristics between blood and the bubble. To simplify the study of gas transport between a bubble and the control volume of blood around it the system is split into three separate aspects:
1) dissolution of the bubble in a simple liquid and changes caused by surface contamination, bubble stabilization, and gas adsorbing particles;
2) characteristics of bubble—blood interface; and
3) dissolution of the bubble in whole blood.
The subject of bubble dissolution in liquid-gas solutions has been studied for a long time. A majority of publications, particularly earlier ones like Epstein and Pies set (1950) and Yang and Yeh (1966), generally dealt with bubble dissolution in pure liquids. Factors like the presence of chemical reaction (Shah, 1972), properties of the liquid (Grand et al. 1992) and presence of solvents are considered in fewer reports.
Surface active impurities like proteins can form a layer around the bubble, changing mass transport characteristics leading to partial or total stabilization of the bubble. Since the subject of microbubble dissolution in blood is not well researched, the treatment of this subject will proceed from the simplest case of a bubble dissolving in a liquid without any surfactant Any mathematical treatment in expectation of accurate or truly representative results will be futile. This is because blood is a complex fluid containing many “impurities,” where higher order chemical reactions take place (for example, combination of oxygen from bubble and hemoglobin present in RBCs is a second-order reaction) with the added complexity of blood motion to be considered. Also, blood has components which can be activated and cause cascade effects and further activation, denaturation, polymerization (of fibrin strands), aggregation and a host of other chemical and physical reactions. All of these components primarily affect the bubble surface, and it is through this layer that gas transport has to take place.
The characteristics of die interface between a bubble and blood has been widely reported (for example, see articles in me same journal volume of the paper by Vroman ct al. 1971). However, a general theory which quantitatively describes the amount of reaction products of blood components with respect to the radius of the bubble and time of exposure to the interface is not available which prompts the study of bubble-blood interaction to be undertaken in a generic manner. In this thesis, the study of interaction starts with a discussion of the possible ways that blood can react to the presence of a bubble, and continues with the structure of proteins in the context of its alterability at the bubble surface and is followed by some specific examples.
Dissolution of oxygen bubbles in degassed whole blood and plasma was reported in a series of papers by (Yang 1971; Yang et al. 1971a, 1971b). There have been no publications on this topic since then.
Bubbles that do not completely dissolve will proceed towards the lungs. The capacity of the lungs to filter out these bubbles has been studied by Butler and Hills (1979,1985) who found a range of bubble diameters that could be fdtered. Applicability of these findings can be judged once the diameter reduction of a bubble with time is known.
4.2 Correlation with Bubbles from IVuBOX 4.2.1 Visualization of Control Volume
The concept of control volume is used here to visualize the immediate environment into which 10-micron diameter oxygen bubbles are released by the IVuBOX. Control volume of a bubble (in the context of die IVuBOX) is defined here as the volume surrounding a bubble that is needed for all oxygen in the bubble to be completely taken up by RBCs present in that volume (Figure 4.1).
All the hemoglobin in 100 ml of whole blood will combine with 20 ml of oxygen (Guyton 1991) when blood is 100 % saturated. In a vein, the average oxygen saturation is 75%; therefore 100 ml of whole blood in a vein can combine with 5 ml of oxygen. The number of moles in 5 ml of oxygen is
5 ml x density of Q2 Smt molecular weight of 02
The above expression gives Msmi = 2.03 x 10″4 moles. The number of moles of oxygen
in a 10-micron bubble is found using Equation (2.4) to be 2.63 x I0″14 moles. Since 100
ml of whole blood contains 5.2 x 1011 RBCs, the number of RBCs that each bubble can
combine with will be
NRBC = 2-63xl0’1Amoles_Ax ?’2xl0U or Nnsc-67 “SBC 2.03 x 10″4 moles
Since each 100 ml of whole blood contains 5.2 x 1011 RBCs, the 67 RBCs will be
contained in 1.29 x 10-8 ml of whole blood, or 1.29 x 104 u\3. Adding this to the volume
of the bubble (Vbub Bj 524 |i3), total volume occupied by the bubble and the required 67
RBCs will be 1.34 x 104 uA Visualizing this volume as a cube, the sides of the cube will
be approximately 24 microns each. Since each milliliter of whole blood contains
2.8 x 108 platelets, the number of associated platelets will be about 4 in this control
volume. The number of all other formed elements will be much less than 1 (see
Figure 4.1).
4.2.2 Conditions for Bubble Dissolution
Before an analysis of bubble dissolution in blood is conducted, specific conditions unique to an intravascular oxygenator that are to be met should be identified. During use of the IVjiBOX, it is required mat the bubble introduced into blood not grow but dissolve within a short period of time. This period of time is not specifically defined, but the following restricting conditions can be stated:
(missing)
Figure 4.1. Conceptual visualization of a 10-micron diameter bubble.
Bubble is situated in a cubic volume of whole blood. Each side of the cube is about 24 microns. There are about 67 RBCs and 4 platelets in this volume.
1) the bubble should dissolve to a size that will not obstruct smaller vessels and also dissolve within the time that the protein adsorption on its surface does not exceed dimensions that will cause occlusion of smaller vessels;
2) the bubble should dissolve fast enough so that the total reaction (in terms of protein denaturation, platelet aggregation, complement activation, etc.) caused by all bubbles that are being continuously introduced into blood does not exceed acceptable levels at any time. The device, ideally, should be able to function indefinitely without violating this condition.
It will be observed that the above two conditions cannot be specified independently of each other—both conditions have to be satisfied concurrently.
4.2.3 Fluid Classification of Environment
When studies of bubbles that are approximately the same size (or smaller than) as an RBC are conducted, it should be remembered that the Casson fluid property of blood will not be applicable; rather the Newtonian property of plasma will have to be assumed because the control volume around the bubble will be filled with plasma (and also with formed elements but present as separate entities) and not whole blood (see, for example, Shima and Tsujino 1978).
4.3 Bubble Dissolution in Liquid 4.3.1 Case of Pure Liquid
Epstein and Plesset (1950) have derived equations which predict the dissolution time of bubbles in pure liquids. They start with the classical Fick’s diffusion equation and incorporate a surface tension term and modifications for a moving boundary (corresponding to the dissolving bubble’s reduction in radius). The resulting equation is similar to the one used by Yang (1971), except that there is no chemical reaction taking place, and it is solved using a familiar solution in heat conduction (this method of solution is also adopted by Yang 1971). Epstein and Plesset (1950), using their solution, calculated times for dissolution of bubbles in pure water. A 10-micron radius air bubble present in water was predicted to dissolve in 1.17,1.46,1.96,2.99 and 6.63 seconds when the ratio of dissolved concentration of air in solution to the concentration for a saturated solution was 0,0.25,0.50,0.75, and 1.00, respectively. In experiments carried out by Unkel (1991) in 0.9% saline, oxygen bubbles of initial radius in the 12-micron range completely dissolved in 17 seconds when the dissolved concentration in saline of oxygen was 19.3% and 39.3%, and in 18 seconds when the dissolved concentration was 57.4%. Although these results are at variance with results reported by Epstein and Plesset, they might not be comparable because of different experimental conditions, notably the use of saline and probable presence of surfactants in the water used in Unkel’s (1991) work, as opposed to the case of pure liquids.
Ward and Tucker (1975) state that a critical radius exists for a bubble in a liquid with two component gases (they used oxygen and nitrogen) and arrive at this conclusion by using the thermodynamic theory of liquids and correlating it with the condition for homogenous nucleation of bubbles. They have established the existence of a critical radius by photographically observing two bubbles in water -one above the critical radius (111 microns) and one below the critical radius (85.7 microns from their photograph, but 97.1 microns from their text description) and found that the former grew while the latter shrunk. The bubble diameter chosen for the IV^BOX is 10 microns and can be expected to dissolve rapidly provided the bubble does not become stabilized, which is possible because of changes that might occur at the surface layer. A positive aspect is that these bubbles are present in a liquid that can chemically combine with the gas in the bubbles.
4.3.2 Effect of Surface Contamination
The gas in the bubble can be considered to be diffusing through a surface contamination of a particular thickness, and a model which takes this into account should be used (Engelking 1985). Studies considering this factor have been carried out, among others, by Engelking (1985) (for microbubbles) and Arefmanesh et al. (1992). Traces of surface active impurities present in the fluid adsorb at the gas-liquid interface, increase the drag, and lower the heat and mass transfer rates (Sangani 1986; Raymond and Zieminski 1971; Gal-Or and Waslo 1968).
4.33 Effect of Bubble Stabilization
In their attempts to generate stabilized microbubbles in sea water, Johnson and Cooke (1981) introduced microbubbles of 40- to 100-micron diameters into sea water and found mat most of them dissolved. The dissolution rate became rapid after a 10-micron diameter limit was reached, and the bubbles that dissolved completely did so in less than 10 seconds. However, some of the bubbles stopped dissolving and became stabilized by surface films formed during the bubble size reduction process. All the stabilized bubbles had a diameter less than 13.5 microns, with some even in the sub-micron range. This study has implications for the IVuBOX in that, although the dissolution of the bubble produced by it will be fast because of surface tension force, some bubbles may dissolve to smaller sizes and men become stabilized. These stabilized bubbles can accumulate in the blood and cause reactions as discussed elsewhere.
4.3.4 Effect of Gas Adsorbing Particles
Vinke et al. (1992), studying slurry reactors, report that gas adsorption can be enhanced as a result of the adhesion of gas-adsorbing particles to die gas-liquid ii The stability factor for particle—bubble attachment is dictated, among other things, by the electrical charge interaction between the particle and bubble, and the ratio of particle size to bubble size (Edzwald et al. 1991). From the results of their work, it can be seen that the lowest particle deposition rate on a 10-micron bubble surface is for particle sizes in the 1-micron range, and the rate gradually increases below and above this point The highest deposition rate is for particles of 8-raicron size, which incidentally is the largest dimension of an erythrocyte. Therefore, there is a possibility of faster bubble dissolution when erythrocytes get deposited on the bubble surface caused by a facilitation-like effect (discussed in section E below).
4.4 Blood-Bubble Interaction 4.4.1 Bubble as a Foreign Substance
The human body attempts to resist any substance which it deems to be foreign, be it a toxin, bacteria, or a dust particle (Guyton 1991). A bubble inside the blood will also be treated as a foreign particle, and the body can respond by the action of any of the following:
1) leukocytes and cells of tissue macrophage system
2) lyzosomes
3) basic polypeptides
4) complement complex
5) natural killer lymphocytes
6) antibodies.
Only substances with a molecular weight of more than 8,000 can generally act as antigens (Guyton 1991). There is the possibility that oxygen, with a molecular mass much less than 8,000, can elicit an immune response by acting as a hapten (a hapten first combines with a substance that is antigenic, such as a protein, and the combination will elicit an immune response). But the very fact that oxygen normally present in blood as dissolved gas does not cause immune reactions can discount this possibility. Another factor that can cause immune reactions is the gas-bubble surface of a bubble since it has properties (like surface energy) different from the bulk of blood which can alter the chemical and physical make-up of substances adsorbing at the surface.
4.4.2 Bubble-Blood Interaction Study
The classical approach used to investigate the effects of a foreign particle in blood has been to view it on a macro scale. In the macro approach, the actual particle is not viewed under magnification and analyzed for any changes; rather a volume of blood is chemically or physically analyzed for absence or presence of certain components/sub—components or changes in physical properties. A macroscopic approach is inappropriate for studying changes caused in blood if the size, number, and composition of the bubbles are not known since these are the attributes to which blood responds.
In a micro approach presented here, the bubble surface has to be viewed for physical and chemical changes and adsorption of molecules/particles. The microscopic approach to studying effects of bubbles produced by the IV|iBQX in blood has to start with a conceptual understanding of the sizes, shape, content, and number of the different components of blood with respect to a spherical bubble of 10-micron diameter (illustrated in Figure 4.1). This kind of a study will pinpoint alterations of components at the surface and the range of impact of the surface.
Reports in the literature generally do not use the microscopic approach to study bubble effects. However, when the bubble is placed in perspective with regard to its size and neighborhood (Figure 4.1). the bubble-blood study can be carried outeven using the macroscopic studies reported, albeit with a loss of details at the micro level.
Blood has numerous components that will react with a bubble, all with complicated and not fully understood characteristics. Many of these components have been discovered only recently (the component iC3 is a good example). Studies of these components and their mechanism of reaction with bubbles is ongoing. This thesis will not catalogue each of these proteins separately or analyze its effects on bubbles; rather, the general structure of proteins is summarized below, after which follows a description of some specific bubble—protein interactions reported in the literature. Blood components relevant to this thesis are listed in the appendix.
4.43 Three-Dimensional Protein Structure
The following description of the structure of proteins is obtained from Branden and Tooze (1993). The functional properties of proteins depend upon their three-dimensional structures. The primary structure of the protein is its amino acid sequence in the polypeptide chain. Different regions of the sequence form local regular secondary structures, and the tertiary structure is formed by packing these structural elements into compact globular units. The final protein may have several polypeptide chains arranged in a quaternary structure. By forming such tertiary and quaternary structures, amino acids far apart in the sequence are brought close together in three dimensions to form an active site.
4.4.4 Protein Interaction with Interface
The genetic code specifies twenty different amino acid side-chains. There are three kinds of amino acids depending on the chemical nature of the side-chains: hydrophobic, charged, and polar. Most of the amino acids in the interior of a water-soluble protein have almost exclusively hydrophobic side-chains. Therefore, the interior of proteins is usually hydrophobic and the surface hydrophilic (Branden and Tooze 1991). Protein molecules are rapidly adsorbed at an air-aqueous interface (Vroraan et al. 1971), and this is one of the first events to occur at the interface (Escudero et al. 1994; Chuang et al. 1978).
Denaturation of plasma proteins can occur upon contact with surface polarizing forces at the blood-gas interface (Pekna et aL 1993). A series of physiocheraical stresses and effects are induced by the exposure of globular plasma protein molecules to an interface between the aqueous phase of plasma and gaseous phase of the bubble (Aberg et al. 1982). hi the boundary layer of such an interface, operating over a distance of 40-100 A, strong electrokinetic forces are created that tend to exert a polarizing influence upon dipolar molecules which enter the boundary zone. A dipolar molecule will orient itself with its polar elements (hydrophilic) in the aqueous (plasma) component and its non-polar, oppositely charged elements protruding from the surface into the gaseous atmosphere. The coiled folded protein molecule, with its polar groups external and its non-polar groups internal, must alter its physical sub-molecular structure in order to orient in the polarizing influence of the force field. The weaker stabilizing bonds, mainly hydrogen bonds, are broken fust The specific force of a bond as related to the polarizing force of the interface will determine the degree of bond breakage and molecular unfolding. Tune of exposure in the boundary layer, temperature, pressure, and molecular concentration also influence the degree of alteration/denaturation (Vroraan et al. 1971).
Interfacial denaturation results in a decrease in solubility of the plasma proteins, with a decrease in water binding capacity, an increased exposed (non-polar) lyophobic constituents of the molecule, an increase in polarity, and an increase in reactive side groups (Vroman et al. 1971).
Such molecules in the process of denaturation are available for adsorption to the surface of erythrocytes, cyloraicrons, and platelets. Subsequent in terra olecular aggregation of the abnormal protein coatings of the particulate elements in blood may then occur, with the formation of agglutinated cells by cross-linking or complexing of partially unfolded protein molecules on adjacent cells. Such aggregation has been observed in flowing as well as stagnant blood (Vroraan etal. 1971). As a physiochemical process, protein denaturation can be reversible or irreversible. Denaturation is considered reversible if, on restoring rcversibly the original conditions, the native conformation along with other properties is regained (Lapanje 1978).
The primary factors mat trigger the body’s response to bubbles in blood are not fully understood, but those reported in the literature are discussed below.
4.4.5 Some Specific Blood Reactions 4.4.5.1 Complement Activation
The gas-liquid interface can activate plasma complements along the alternate pathway (Thorsen etal. 1993; Bergh etal. 1993; Ward etal 1986; Ward etal. 1987;Pekna et aL 1993), the third component of complement, C3, playing a central role in the complement cascade (Pekna et aL 1993). C3 is modified at the interface forming iC3, with this modified complement having C3b—like properties. C3b strongly activates phagocytosis by macrophages and neutrophils (Guyton 1991). Additionally, after C3 undergoes conformational changes at the interface producing iC3 and its fragments, it binds to erythrocytes (Pekna et aL 1993). The iC3 molecule is also susceptible to cleavage by regulatory complement proteins and fibrinogen (Nilsson et al. 1992).
4.4.5.2 Platelet Aggregation
Interactions between gas bubbles and blood platelets cause reduction in circulatory platelets. Although bubbles cannot be considered classical agonistic, platelet aggregation induced by bubbles is irreversible, slow and similar to the primary aggregation caused by classical agonists (Thorsen et al. 1993). When platelets bind to the denatured proteins at the interface, their receptor sites for fibrinogen may be exposed. This exposure may cause fibrinogen adhesion to the platelets at the interface, and this fibrinogen will again cause platelets from the bulk of the liquid to adhere to it, thereby causing platelet aggregation (Ugarova et aL 1993).
4.4.53 Macrophage Interaction
Shanbag et aL (1994) have reported on the effect of particle size, composition and surface area on macrophage interaction using titanic and polystyrene spheres with diameters of 2 microns and leas. Their results indicate that smaller particles (<0.45 microns) elicited lower levels of macrophage response than the larger particles (1.76 microns mean) elicited. This result is relevant in the IVuBOX because bubbles will not dissolve instantaneously in blood, and some of the bubbles may become temporarily stabilized by protein layers around them. These proteins might become irreversibly ckrnatured, and macrophages win attempt to phagocytize them. This is not a problem of immediate concern since, for every 100 bubbles released, there are only 8 leukocytes. However, this problem becomes of concern when there is an accumulation of undissolved bubbles in blood.
4.4.5.4 Albumin Adsorption
Albumin might have the most pronounced attraction for apolar molecules, so, adsorbed to an apolar surface, the hydrophobic sites of albumin needed for antialbumin to attach itself would be unavailable (Vroman et al. 1971). Although the concentration of albumin is the highest among proteins in plasma (both by weight and number), it is not known which protein will be the first to adsorb onto the bubble surface or what percentage of the bubble surface will be covered with albumin after a certain time. This effect becomes even more unpredictable since the bubbles are situated in non-stationary blood. Chuang et al. (1978) report that an increase in the flow rate (from 0 to 10 ml per minute) caused an increase in maximal albumin absorption. Shear stress influences both protein adsorption and subsequent cell adhesion (Chuang et al. 1991). So, the results reported in works where stationary bubbles were used have to be carefully judged before being applied to the IVuBOX.
4.45.5 Cell-Bubble Interaction
In their studies of interaction between insect cells and 50-micron bubbles, Bavarian et al. (1991) report that, if there is a collision between a bubble and a cell, the two adhere to each other and do not detach. In a continuing study, Chalmers and Bavarian (1994) also found that all bubbles introduced into a suspension of cells had cells attached to them. They also report on cell destruction stating that it is caused by shear stresses produced by the destruction of smaller bubbles. In similar studies, Lu et al. (1992) and Cherry and Hulle (1993) arrive at the same conclusion of Bavarian et al (1991). These studies may explain why RBC lysis is greater in bubble oxygenators than in membrane oxygenators, since bubble oxygenators contain defoamers which act as bubble breakers. An advantage of using the IV|xBOX is that bubble breakers are not required since the bubbles are expected to dissolve completely in blood and not rise or accumulate at any point, thus considerably reducing cell lysis.
4.5 Oxygen Bubble Dissolution in Blood
The study of oxygen diffusion and its chemical reaction with hemoglobin in the cells was carried out by Forster (1969). The partial differential equations used to describe the combined process of gas diffusion and chemical reaction were solved analytically for the case of oxygen bubbles in blood by Yang (1971). The resulting equation in diraensionless form relating oxygen bubble radius with time is
i /dV _ 2 a* (1 + 2C.) (1-JT) 2 o-*(l + 2C) a* + (1-C.) K j 3 (1-C.) 3 (1-C.)2 o* + /?*
= 2 Cj (1 – C. )t* (4.1)
In the above equation, all terms are dimensionless and given by
R*=R/R<, where R is the instantaneous bubble radius at any time t and Ro is the initial
bubble radius;
a* = (2 a)/(Ro Pg) where a is the surface tension of the liquid (in this case plasma), Pg is the total pressure exerted by the gas on the bubble surface and is the sum of surface tension and atmospheric pressures (see Chapter 3);
Coo = Coo / Cs where Cm is the initial content of dissolved oxygen in blood which in this case will be the venous content of oxygen, 15 ml /100 ml of blood. Cs is the dissolved content of oxygen when blood is 100% saturated, which is 20 ml /100 ml of whole blood (Guyton 1991);
C*p = Q I pg where pg is the density of oxygen, 1.33 x 10~3 g/cm-3 at 30 *C and atmospheric pressure (CRC Handbook 1981);
t* = fp t/Re2 where fp is the diffusion coefficient of oxygen in plasma, 9 x 10~* cra^1 (Tanasawa et al. 1971).
4.6 Microbubble Filtration by Lungs
It is known that venous gas embolism is much less hazardous than arterial gas embolism (Butler and Hills 1985). Bubbles in the arterial system can obstruct the capillaries of the heart and brain before they enter the venous system.
The bubbles in the venous system have to pass first through the lungs, and the lungs act as a very efficient filter for microbubbles (Butler and Hills 1979, Butler and Hills 1985). In case an instability is reached in the rVjiBOX (probably during startup or cessation of bubbling) causing it to generate bubbles of sizes higher than normal, the lungs can filter these bubbles, with a lower cutoff diameter of 22 microns (Butler and Hills 1979). Even though the lungs act as extremely efficient filters of microbubbles, it is very important to note that the filtration capability of the lungs nearly vanishes when pulmonary vasodilators are used or when the lungs are overloaded with extremely large bubbles (Butler and Hills 1979). Overloading in this work on canine subjects was a bolus injection of 30 ml, of air into the right atrium. The threshold value of 0.30 ml. kg^min”"1 (corresponding to 0.94 raillimoles per minute for a 70-kg human adult) has been reported by Butler and Hills (1985) above which air bubbles will not be filtered by the lungs. Since they conducted experiments on canine models using air bubbles of a specific size (100 micron), their results may vary significantly for other bubble sizes and must be used with caution.
4.7 Hemolysis Caused by Bubbles
When a bubble dissolves in a liquid, it sets up stress fields caused by the inward movement (towards the bubble center) of the bubble surface with reduction in bubble volume (Yang 1974). If the velocity of dissolution of the bubble is greater than 2,000 cm/s, RBC lysis can occur (Yang 1974). A similar situation arises in the case of a bubble during its production from an orifice— cell lysis can occur because of the outward movement of the growing bubble surface.
From the calculation of force on bubble due to viscosity of liquid (Chapter 3), the bubble tip velocity during bubble growth was found to be 1 cra/s. Comparing this with the jet velocity required for hemolysis to occur, the bubble tip velocity is three orders of magnitude smaller. Therefore, erythrocyte lysis caused by bubbles striking them during bubble—formation can be discounted.
To find the possibility of hemolysis resulting from liquid jets caused by bubble
dissolution, the solution of Equation (4.1) for a 5-micron radius bubble in blood can be
used. The dissolution time in that case was found to be 1.48 seconds, and the resulting
liquid jet velocity Vjet will be
Y _ 5 micron J 1.48 seco,
This is clearly lower than what is required to cause hemolysis. 4.8 Discussion
The control volume into which bubbles will be released by the device was depicted in terras of the volume surrounding a single bubble and the number of formed elements in that volume. The conditions for bubble dissolution are specified in terms of the for complete dissolution of bubbles. The theoretical prediction of the time for complete dissolution of a 10-micron air bubble in water with dissolved air was in the range 1.17 to 6.63 seconds, depending on the dissolved content of air in water (Epstein and Plesset 1950). The time for complete dissolution of 12-micron oxygen bubbles in 0.9% saline
was found from experimental observations to be less than 18 seconds (Unkel 1991). Although there has been no report on the experimental observation of microbubble dissolution in blood, computation based on the theoretical work of Yang (1971′ time for complete dissolution of a 10-micron oxygen bubble in blood to be 1.48 seconds. The highly reduced time for bubble dissolution in blood can be attributed to the high affinity of blood for oxygen. However, allowance should be made for the modified surface layer of the bubble since it is through this layer that oxygen should traverse to reach the RBCs. The reduced or enhanced time factor for oxygen dissolution through this modified surface is not known, nor is the total reaction caused by the presence of bubbles in blood.
It was found that hemolysis will not occur during either bubble production or bubble dissolution, and that bubbles that do not dissolve completely will be filtered by the lungs. Theoretical and experimental work on bubble dissolution and blood reaction to bubbles has to be carried out to find out the feasibility of using the device in humans.
chapter 5
conclusions and recommendations
An intravascular microbubble oxygenator will be free from many of the drawbacks extracorporeal bubble and membrane oxygenators have and also from some of those which intravascular membrane oxygenators have. However, an intravascular bubble oxygenator has not been reported, and this thesis lays the groundwork for further study on developing such a device.
The design criteria of the device includes the following:
1) aspect ratio and resolution requirement;
2) size and frequency of microbubble generation; and
3) time for bubbles to dissolve completely.
These design criteria are not entirely independent of each other, and the latter two (size, frequency and dissolution time) are influenced by various factors and are discussed in this chapter.
A device of this nature will have high aspect ratios, and the resolution deliverable by the process should be high. An x-ray lithography process has both of these attributes and, therefore, can be adopted in the prototype and production stages. Details of the device design can be manipulated using Equation (2.12) to obtain optimum performance.
In the theoretical analysis of the principles of fabricating and operating an intravascular microbubble oxygenator, some questions have been left unanswered. The most important of these questions arc given below:
1) the possibility of producing microbubbles at sizes and frequency required, for example, 10-micron bubbles at 1 kHz; and
2) microbubbles of a particular size, say 10-micron, will dissolve within a time which will not cause unacceptable blood reaction and also not occlude smaller vessels.
The first of these issues, i.e., the possibility of microbubble production with specifiable characteristics, cannot be answered unless experiments are conducted in the regime of interest to the IVuBOX. Theoretical modelling of bubble production from the device can be carried out based on experimental results obtained. Force analysis on a 10-micron bubble showed that among the viscous, inertial, buoyancy, and surface tension forces, the surface tension force is much greater than the others. Forces induced by the wake of the previous bubble are not included since it is a very complex phenomenon and there are no accurate descriptions of this effect However, in the course of modelling bubble production from the device, all forces (including wake force and force induced by neighboring bubbles) must be considered. Bubble detachment will occur when the forces preventing bubble growth and detachment (surface tension and liquid viscosity) become lesser than the forces aiding bubble detachment (buoyancy, bubble inertia, gas momentum, and liquid flow). When bubble buoyancy, bubble inertia, gas momentum, and liquid flow velocity were compared individually to the surface tension force, it was found that there is a trade-off between between bubble sizes and the combination of orifice size and bubble frequency. For all the cases (except that of liquid flow), die bubble frequency and bubble diameter have to be very high so that the orifice diameter has dimensions that are physically realizable. Blood velocity can be increased so as to detach bubbles and thus produce bubbles of acceptable sizes. From Equation (3.27), if the blood velocity is increased ten-fold, then 10-micron bubbles can be
produced from 5-micron orifices. Therefore, among all the methods considered, increasing blood velocity around the device seems to be the best choice. Increase in blood velocity can be achieved by using a constriction in the lower end of the device.
Models found in the literature have to be modified suitably to model the device adequately. Analysis of Ramakrishna et al.’s (1969) paper revealed shortcomings in the form of non-inclusion of some forces. The force caused by liquid velocity and forces occurring because of the wake of the previous bubble have to be included in any model so that resulting bubble sizes can be accurately predicted. Therefore, in its present form, their equation is not ideal for modelling bubble production from the device.
Interactions between bubbles produced from the device can be considered in terms of primary and secondary bubbles. Two primary bubbles will not interact if they are separated by at least one bubble diameter, and this detail is incorporated into the design. Primary and secondary bubble interaction is not fully understood and will have to be confirmed experimentally. The coalescence of two secondary bubbles cannot be determined unless practically observed, but it can be stated at this point that the salt concentration of blood by itself is not sufficient to prevent coalescence. Other factors like organic compound concentration, bubble surface characteristics, and surface charge may be decisive aspects in preventing or accelerating coalescence. The conceptual visualization presented in Figure 4.1 will aid in the understanding of physical dimension factors that might alter coalescence behavior.
The other issue left unanswered pertains to bubble dissolution and blood reaction. The bubble-blood surface contains blood components with highly modified properties differing from the bulk of the liquid. But the reaction or alteration produced by this surface is not reported in quantitative terms (nor correlated to bubble sizes) in the
literature. Therefore, it is not possible to extrapolate available reports to fit of 10-micron sized bubbles which is the need in this study. It is best to use a microscopic approach suggested in Chapter 4 to study blood reaction caused by bubble presence.
Time for complete dissolution of a 10-micron oxygen bubble in whole blood was theoretically found to be 1.48 seconds. In a comparison of this result to the case where a bubble (of nearly equal size) would be dissolving in the absence of a chemical reaction, the time for complete dissolution in a pure liquid is less than 6.63 seconds (Epstein and Plesset 1950), and in 0.9% saline is 18 seconds (Unkel 1991), both for worst case situations. Although the accelerated dissolution can be explained as being caused by the reactive nature of blood with oxygen, factors like bubble surface layer coating (which can decelerate gas diffusion across it) and red blood cell facilitation (which can accelerate bubble dissolution by a facilitation effect) will also have to be considered. Again, this dissolution time is best determined experimentally.
Considering the above evaluation of the device feasibility, it appears that, although there are many details unavailable as yet, there is no reason to discontinue further work on the device. Any available data point to the feasibility of such a device being used clinically, and there are no data that pointedly refute this possibility. Therefore, it is recommended that work on the IYfiBOX continue and that investigations be carried out separately and concurrently based on Chapters 3 and 4.
APPENDIX 1
BLOOD COMPONENTS Formed Elements (Guyton ch.32)
Formed elements in human male blood relevant to this thesis are:
Erythrocytes: 5.2 x 1012 / liter (mean diameter 7.5 micron, thickness at outer edge 1.9
microns, thickness at center I micron)
Leukocytes: 7000 / micro liter
Platelets: 2.3 x 105 / micro liter (2.4 micron diameter round/oval disc) Blood Proteins (Altraan 1961)
Proteins constitute 8% by weight in plasma with sizes 100 A – 700 A Albumin 4.8 g / 100 ml, molecular weight 69,000 Globulins 2.5 g / 100 ml, molecular weight 90,000 to 156,000 Fibrinogen 0.3 g /100 ml The Complement System:
| Complement | Molecular Weight | Concentration in Serum (\l$f raL) |
| Clq | 410 | 70 |
| Clr | 90 | 35 |
| Cls | 85 | 30 |
| C2 | 117 | |
| C4 | 206 | 600 |
| C3 | 190 | 1200 |
| C5 | 180 | 85 |
| C6 | 120 | 60 |
The Complement System (continued):
| Complement | Molecular Weight | Concentration in Serum (jxg/ mL) |
| C7 | 120 | 55 |
| C8 | 150 | 55 |
| C8 | 150 | 55 |
| C9 | 79 | 60 |
| properdin | 190 | 25 |
| properdin factor B | 100 | 225 |
| properdin factor D | 25 | 1 |
| CI esterase inhibitor | 105 | 275 |
| C3b inacti-vator | 105 | 275 |
| C4 binding proteins | 560 | 8 |
| B1H | 150 | 500 |
APPENDIX 2
NOMENCLATURE
| Abub | area that can accommodate one bubble (units defined locally) |
| Adev | surface area of device (units defined locally) |
| B | retarded van der Waals force |
| bubble production frequency per orifice in hertz | |
| CS | dissolved content of oxygen in blood when it is fully saturated |
| ct | transition concentration in moles |
| Coo | initial content of dissolved oxygen in venous blood |
| c* | non-dimensionalized gas parameter, defined in text |
| c\. | non-dimensionalized gas parameter, defined in text |
| dbub | bubble diameter (units defined locally, usually microns) |
| do | orifice diameter (units defined locally) |
| Dod | outer diameter of device (units defined locally) |
| FB | buoyancy force in Newtons |
| Fcf | force caused by liquid co-current flow in Newtons |
| FH | hydrostatic force in Newtons |
| Fi | force caused by inertia of bubble in Newtons |
| Fm | force caused by gas momentum in Newtons |
| Fv | force caused by liquid viscosity in Newtons |
| Fa | surface tension force in Newtons |
| g | acceleration caused by gravity in ms~2 |
| i | constant for a particular salt solution, defined in text |
| Ldev | length of device (units defined locally) |
| Ln | nozzle length in centimeters |
| Mbub | number of moles of oxygen in one bubble |
| Mreq | oxygen requirement in moles per second |
| ni | number of ions formed upon dissolution of salt |
| “sbub | ratio of interbubble spacing to bubble diameter |
| Nbub | bubble production rate of the device in Hertz |
| Nori | total number of orifices on the device |
| Nrbc | number of RBCs in control volume |
| NW | dimensionless gas-flow rate number |
| P amb | ambient pressure in Pascals |
| p« | total pressure exerted by gas on bubble surface (units defined locally) |
| Plot | pressure inside a bubble in Pascals |
| Pa | excess pressure caused by surface tension in Pascals |
| Q | gas-flow rate (units defined locally) |
| Qo | gas-flow rate through orifice (units defined locally) |
| fbub | bubble radius (units defined locally, usually microns) |
| r0 | orifice diameter (units defined locally) |
| R | gas constant in Joules.raole~,K~1 |
| Ro | initial bubble radius (units defined in text) |
| R* | dimensionless bubble radius, defined in text |
| Sbub | interbubble spacing (units defined locally) |
| | | non-dimensionalized time parameter |
| T | temperature in Kelvin |
| U0 | superficial gas velocity through orifice (units defined locally) |
V volume of bubble (units defined locally)
Ve volume of bubble at the end of first stage (units defined locally)
Yj velocity of liquid jet (units defined locally)
V&jnd liquid velocity (units defined locally)
Vgp velocity of bubble tip (units defined locally)
5a /5c surface tension gradient in Nm-1raole-1
p diffusion coefficient of oxygen in plasma in craV-1
fi viscosity of liquid phase
pi density of liquid phase in kgm-3
Pa density of gas phase in kgm-3
a surface tension force in Nm_1
o* non-dimensionalized surface tension, defined in text
Appendix 3
81
table of nominal values
Q 20 ml of oxygen per 100 ml of whole blood (Guyton 1991) 15 ml of oxygen per 100 ml of whole blood (Guyton 1991)
R 8.314 Joule, raole”1^1 (Sears et al. 1988)
t body temperature 310 K (Guyton 1991)
Vl average blood velocity in the vena cava, 8 cra/s (Guyton 1991)
p 9 x 10-6 cmV1 (Tanasawa et al. 1971)
|ip plasma viscosity at body temperature, 1.2 x 10-3 Nsm-2 (Altman 1961)
pp density of plasma at body temperature, 1027 kgm-3 (Altman 1961)
pg oxygen density (37 -C, atmospheric pressure), 1.33 kgm-3 (CRC Handbook 1981)
a plasma surface tension at body temperature, 6.99 x 10~2 Nm-1 (Airman 1961)
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