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Department of Physiology, University at Buffalo, State University of New York, Buffalo, New York 14214
ABSTRACT
INTRODUCTION
THEORY AND METHODS
RESULTS AND DISCUSSION
APPENDIX
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Van Liew, Hugh D., and Soumya Raychaudhuri. Stabilized
bubbles in the body: pressure-radius relationships and the limits to
stabilization. J. Appl. Physiol.
82(6): 2045-2053, 1997.
We previously outlined the fundamental
principles that govern behavior of stabilized bubbles, such as the
microbubbles being put forward as ultrasound contrast agents. Our
present goals are to develop the idea that there are limits to the
stabilization and to provide a conceptual framework for comparison of
bubbles stabilized by different mechanisms. Gases diffuse in or out of
stabilized bubbles in a limited and reversible manner in response to
changes in the environment, but strong growth influences will cause the
bubbles to cross a threshold into uncontrolled growth. Also, bubbles
stabilized by mechanical structures will be destroyed if outside
influences bring them below a critical small size. The in vivo behavior
of different kinds of stabilized bubbles can be compared by using plots
of bubble radius as a function of forces that affect diffusion of gases
in or out of the bubble. The two ends of the plot are the limits for
unstabilized growth and destruction; these and the curve's slope
predict the bubble's practical usefulness for ultrasonic imaging or
O2 carriage to tissues.
bubble nuclei; cavitation; decompression sickness; surfactants; surface-active films; ultrasonic imaging
INTRODUCTION THERE ARE SEVERAL REASONS for interest in the behavior
of small, stabilized gas cavities (bubbles) inside the human body. Microbubbles are being developed for enhancement of ultrasonic contrast
of blood vessels and other body structures (5, 9, 13, 19, 20, 22, 30,
31). Some of these bubbles are small enough to traverse capillary beds
and are stabilized by mechanisms that cause them to persist much longer
than a simple bubble of air (23). Also, if microbubbles are permeable
to gases, they could carry appreciable amounts of
O2 from lungs to tissues; it has
been proposed that microbubbles could serve as a substitute for blood
hemoglobin (4, 24). Furthermore, if stabilized bubbles are analogs of
the micronuclei that may be the precursors of decompression sickness
bubbles (18, 34) and the initiation sites for damaging cavitation in an
ultrasonic field (2), their behavior is of interest in diving safety
and medical uses of ultrasound. In all these cases, the way that the
stabilized microbubbles respond to environmental changes is extremely
important. If the change causes the microbubbles to become
unstabilized, they can cease to exist or can grow into large, damaging
bubbles.
Because pressure due to surface tension is a function of bubble size,
it is obligatory for the stabilization mechanism to be a function of
size as well (23). Two general classes of mechanisms can stabilize
bubbles. First, slowly permeating gases (13) remain for relatively long
times in the bubbles, while other more rapidly permeating gases diffuse
in or out according to their concentrations in the environment (22,
23); total pressure inside is greater than that outside because of
pressure due to surface tension. Second, structures at the gas-liquid
interface can serve as stabilizers; examples are surface-active films
(30), surface-active protein that may be denatured (19, 31), and
gelatin (5). We have shown that, despite the differences between the
two classes of stabilizing mechanisms, slowly permeating gases and
structural stabilizers can be characterized by similar mathematical
expressions (23).
The specific objectives of the present paper are to provide two
extensions to our previously published theoretical treatments (4,
22-24): 1) to include the idea
that there are upper and lower size limits to stabilized bubbles and
2) to consider how bubbles
stabilized by various mechanisms can be compared. The crucial aspect of
a structural stabilizer is that it must produce a negative pressure
inside the bubble to counter the tendency for outward diffusion of the
gases inside, especially to counter the strong positive internal
pressure due to surface tension when bubbles are small. The theory
presented below, couched in a mathematical framework, encompasses this
fact and other basic facts that are valid for any stabilized bubble. We
present examples to show how bubbles stabilized by different kinds of
mechanisms may behave.
THEORY AND METHODS In recent publications (4, 22-24), prediction of the behavior of
stabilized microbubbles in blood depended on an important basic
assumption: that the size of stabilized bubbles can vary in response to
changes in the bubbles' surroundings; stabilization does not imply a
fixed size or rigid structure. This assumption is supported by the
observation that the ultrasonic signal from a bubble stabilized by an
albumin coating decreased when pressure was applied and then returned
to the initial level when pressure was released (15). For present
purposes, we add another assumption: that the stabilizing mechanism may
fail at some small size or rupture due to overexpansion. For most
examples, we drew inspiration from surface films or surfactant
monolayers, a specific type of structural stabilizer. Other structural
stabilizers will be analogous to surface films in that they counter
surface tension but may differ in particulars.
An important related assumption is that the stabilized bubbles are
permeable to diffusion of gases, so that diffusive exchanges will occur
when there are gas partial pressure differences between inside and
outside. The concept that gases diffuse in or out to change the
contained volume when stabilized bubbles vary in size contrasts with
the idea put forward by Yount and co-workers (32) that stabilized
bubbles become impermeable when they are very small. It seems unlikely
that a single layer of molecules, particularly if they are largely
lipid in makeup, could prevent gas diffusion, even if densely packed. A
monolayer could conceivably block water diffusion into the bubble and
may slow permeation of some gases (3), but gases with high lipid
solubility would have their diffusion enhanced by a lipid monolayer.
Stabilized bubbles with multiple layers have been described (20), but
if the layers are mainly lipid, permeability may not be appreciably
affected.
We take advantage of a simplifying assumption so that we can use radius
to characterize size: that stabilized bubbles always remain spherical.
This is probably not valid for at least some kinds of bubbles; e.g.,
bubbles stabilized by shells of denatured protein are said to change
shape when small (7), and bubbles stabilized by surfactant have been
observed to become nonspherical when the surfactant molecules are
forced close to each other (10). The assumption of sphericity and other
simplifications mean that the results of our calculations are
approximate; this does not detract from our goal of exploring the basic
characteristics of stabilized bubbles.
The partial pressures of gases inside a bubble are influenced by
hydrostatic pressures (23). The well-known Laplace-Young equation gives
the hydrostatic pressure (P) exerted by surface tension on the contents
of a spherical bubble
where
R is bubble
radius.
(1)
Glossary
 |
Exponent defined by Eq. 2, dimensionless |
 Pabs |
Pressure difference for bubble absorption, kPa |
 |
Surface tension, 50 dyn/cm = 50 kPa · µm |
 |
Constant, ~3.1416 |
 |
Reduction of surface tension due to a surface-active film, dyn/cm |
| A | Area per stabilizer element on a bubble surface, µm2 |
| Ac | Minimal area per structural element before crushing occurs, µm2 |
| Ao | Actual area that a stabilizer element occupies on a bubble surface, µm2 |
| b | Constant, 2 k T/Ac,
kPa · µm
|
| c | Scaling constant for pressure exerted by an elastic sphere, kPa/µm |
C |
Constant that relates P to a function of bubble radius;
units depend on the function
|
| k | Boltzmann's constant, 1.3805 × 10 23 J/°K
|
| n | Number of structural elements in a mechanical stabilizer |
P |
Pressure generated by surface tension on a bubble, kPa |
| P |
Pressure generated by a bubble stabilizer, kPa |
| Pz | Sum of all hydrostatic pressures and partial pressures that could affect diffusive exchanges of a bubble, other than those that depend on bubble radius, kPa |
| Pzc | Pressure at the crushing radius R*c, kPa |
| Pzg | Pressure at the critical growth radius R*g, kPa |
| R | Bubble radius, µm |
R |
Unstressed radius of an elastic sphere, µm |
| R* | Stable radius of a bubble, µm |
| R*c | Crushing radius of a bubble, µm |
| R*g | Critical growth radius of a bubble, µm |
| Ro | Asymptotic minimal radius of a bubble, µm |
| R*u | Stable radius of bubble when Pz = 0, µm |
| T | Temperature, °K |
A stabilization mechanism exerts a counterpressure
(P) against the tendency of
surface tension and other forces to cause outward diffusion of the
bubble's gaseous contents. A major point for this paper is that the
behavior of a bubble is determined by behavior of
P as a function of radius; each
different stabilizer will have its own
P function. To provide examples that have a quantitative basis, we relied on previous theory about bubbles stabilized by an ideal surfactant (23) for our starting point.
We needed an expression that can account for the quantitatively different responses to stress that may occur with different kinds of
stabilizing structures. We devised Eq. 2 to satisfy this need; APPENDIX
A shows the relation of Eq. 2 to ideal surfactant films and bubbles of slowly
permeating gas.
|
(2) |
involves the
reciprocal of a function of radius instead of the simple reciprocal of
radius. If the stabilizer is a surface-active film, the
C constant is
proportional to the number of molecules in the film. The
Ro represents the minimal radius of the sphere made up of the aggregate of the elements of the mechanical stabilizer, and ( + 1) is an arbitrary
exponent. If and
Ro are both zero,
Eq. 2 is the formula for a simple, ideal "gaseous" surface-active film
(23). Our introduction of the hypothetical and
Ro parameters
allows us to anticipate the behaviors of real, nonideal surfactants
that may be found in nature. Because the function of the stabilizer is
to counter surface tension, the value of the right side of
Eq. 2 must be >2/R when
R is small; it follows that the value
of for a stabilized bubble can be any number more positive than
2.0. Our examples using Eq.
2 are intended to portray some
possibilities for real stabilizers; the purpose of the examples is to
explore the general properties of different possible stabilizer
mechanisms.
The hydrostatic pressures and partial pressures that affect the
diffusive exchanges of a bubble can be categorized as positive absorptive pressures (Pabs) that give a tendency for shrinkage or as
negative absorptive pressures that give a tendency for growth. The sum
of all pressures that relate to Pabs is
|
(3) |
and
P terms are functions of bubble
size, as seen by Eqs. 1 and 2. The term
Pz symbolizes the sum of all
influences that are independent of bubble size. Blood pressure is an
example of a positive absorptive hydrostatic
Pz. Another type of positive Pz is the gas concentration
difference due to the inherent unsaturation caused by
O2 metabolism, the so-called
O2 window (25), for a bubble in
tissue or venous blood. The gas supersaturation that occurs in tissues
of a diver who ascends from depth (27) is an example of a negative,
size-independent concentration difference; the dissolved gas in the
tissue is temporarily supersaturated, relative to the ambient pressure,
so it tends to diffuse into a bubble.
Computations and plotting. We provide graphic examples developed using Microsoft QuickBasic on a Macintosh Classic computer. The plots assume surface tension to be 50 dyn/cm, a value reported for surface tension of blood (29).
RESULTS AND DISCUSSION
The three panels of Fig. 1 illustrate the
effects of changes in Pz for one
particular bubble. Figure 1A is the
fundamental balance-of-forces diagram introduced in another publication
(23). The uppermost dashed trace shows the absorptive pressure due to surface tension as a function of bubble radius. If unopposed, surface
tension would have little effect when radius is large and, at small
radii, would greatly elevate the hydrostatic pressure inside, causing
outward diffusion of all constituent gases. The lowermost dashed trace
in Fig. 1A shows
P, the growth pressure exerted
by a stabilization mechanism; if it were unopposed, the stabilizer
would produce a growth tendency of about 50 kPa when radius is 2 µm and would exert very strong growth pressures at smaller radii.
Pabs)-vs.-radius plots for 1 particular stabilized bubble. Dashed curves of pressure generated by surface tension on a bubble
(P) and pressure generated by
a bubble stabilizer (P) are
the same in A-C
(P curve drawn from
Eq. 1,
P curve from Eq. 2
for C = 400 kPa · µm3 with
= 0 and Ro = 0). A: curve labeled Sum (from
Eq. 3) is the resultant of 2 oppositely
acting pressures caused by surface tension and by stabilizer. 
,
Stable radius; 
, point where stabilization mechanism fails; 
,
maximum of Sum curve. B: addition of a
size-independent absorptive influence (horizontal dashed trace) moves
Sum curve upward from where it was in
A and decreases stable radius.
C: 3 different Sum curves correspond
to different Pz values (dashed horizontal line segments). 
, "Critical growth" point where,
because of a Pz of 20 kPa,
bubble is on the verge of growing irreversibly. See
Glossary for other definitions.
The radius is stable where the Sum curve, the resultant of
P and
P influences, crosses the axis
for zero Pabs in Fig. 1A (at 2 µm). At the crossing, there is a positive slope so that if the bubble
were slightly larger than its stable size, pressure due to surface
tension would be greater than the negative pressure due to the
stabilizer; there would be a positive pressure inside the bubble, which
would cause gas inside to diffuse out until the bubble had shrunk down
to the stable size. Conversely, if radius were between 1.5 and 2 µm
in Fig. 1A, the bubble would grow to
the stable radius. Every stabilized bubble, no matter what the
mechanism, must have a positive slope on a balance-of-forces diagram,
such as that illustrated in Fig. 1A.
Stabilization requires that the
P must be larger in magnitude
than P when radius is small.
Within that constraint, P
curves can have different shapes, as discussed below.
In addition to the point where the Sum curve crosses the axis for zero
Pabs, two other important features in Fig.
1A are the point at which the Sum
curve reaches a maximum (the square, which will be discussed just below
in connection with Fig. 1C) and the
left-hand end of the P curve.
The represents the minimum size for the bubble's stabilization; we
arbitrarily assign a minimum radius of 1.5 µm for this particular
bubble; at smaller radii, the Sum curve does not exist because of
failure of the stabilization mechanism. This is discussed further, in
connection with Fig.
2A.
, crushing radius; note that it is directly
above on P curve; ,
critical growth radius. Inset:
counterclockwise rotation of either of solid curves of
panel A gives a plot of radius vs. pressure.
B: information from
panel A replotted on a
radius-vs.-Pz plot, with added
points for unstressed radius (
) and for a bubble in venous blood
(x).
Figure 1B shows the effect of adding a
third pressure, a Pz of +30 kPa
(arrow), to the P and
P pressures of Fig. 1A. The addition moves the Sum curve
up by 30 kPa over the entire range of radii. Comparison with Fig.
1A shows that the point where the
Sum curve crosses the axis has moved to the left by ~0.3 µm. If a
negative Pz had been added, the
Sum curve would have been lowered instead of raised. Adding or
subtracting various Pz values simply translates the Sum curve up or down, and the at the maximum of the Sum curve will be at the same radius, no matter what the Pz. It is shown next that when the
maximum of the Sum curve is brought below the axis for zero Pabs,
the bubble cannot be stable; it grows irreversibly.
Irreversible growth. The three solid
Sum curves on Fig. 1C show effects of
three different Pz values. A
negative Pz acts in the same
direction as the stabilization mechanism; both tend to cause growth.
When Pz is negative enough to
bring the maximum of the Sum curve below the axis for zero Pabs, a
previously stabilized bubble grows into a relatively large,
unstabilized bubble that can do damage in the body, as in decompression
sickness (21, 27). For the curve with
Pz of 45 kPa in Fig.
1C, there is no stable radius so the
bubble grows irreversibly at all radii. The stabilized bubble with
Pz of 20 kPa has a unique
point (); Pz for that point is
the critical Pz for irreversible
growth, Pzg. When a bubble is located at that point, the bubble would grow if, by
chance, Pz became a little more
negative or the radius increased a little. When the critical growth
pressure is exceeded, the rate of growth that occurs increases as size
increases because of the negative slope of the Sum curve to the right
of the curve's maximum. As outlined before (21), when a small bubble
is in an environment that fosters growth, there is a positive-feedback loop, whereby increased radius due to growth causes decrease of pressure due to surface tension, which encourages further growth. There
is also a positive-feedback loop involving surface area of the bubble;
as gases diffuse into the bubble, surface area increases, which, in
turn, increases the amount diffusing under a particular gas
concentration gradient.
Crushing. A bubble stabilized by a mechanical structure can be expected to behave in a discontinuous manner at small radii; a strong positive Pz may cause the bubble to reach its minimum size, beyond which it changes drastically or ceases to exist. Items in support of this possibility are the idea that micronuclei, the putative precursors of decompression-sickness bubbles, can be inactivated or "crushed" by high pressure (26); observations of "buckling" or "crumbling" of surface-active films (Ref. 1, p. 115; Ref. 17); observations of fissioning of bubbles into several smaller bubbles (33); and irreversible loss of ultrasonic signal from several types of microbubble contrast agents when pressure is applied to them (28).
The in Fig. 1A showed the radius
and pressure combination at which the bubble's stabilization mechanism
fails. We envision that there will be failure of the stabilizing
molecules in most kinds of mechanical structures; we call the limiting
small size the "critical crushing" radius. A
Pz of +50 kPa will bring the Sum
curve of the bubble depicted in Fig. 1 to 1.5 µm, the critical radius
for bubble destruction or crushing; this is illustrated by the top
solid curve in Fig. 2A.
When there is no longer a stabilization mechanism because of crushing, we envision that the bubble's gas contents rapidly diffuse out under the influence of surface tension. Hyldegaard et al. (12) observed that once a bubble in animal tissues had disappeared, application of growth influences did not make it reappear. On the other hand, Liebermann (14) observed that a residue left after apparent disappearance of free, unstabilized bubbles in water gave rise to new bubbles if the liquid was decompressed; this suggests that some sort of stabilized micronuclei remained in the residue. Bubbles stabilized by slowly permeating gas (13) may be an exception to the idea that all bubbles can be crushed; if the gas does not change phase to a liquid, such bubbles would not be expected to fail cataclysmically but will be absorbed more rapidly when Pz is large.
Limits to stabilization. A
Pz of 20 kPa brought the
Sum curve for the particular bubble shown in Figs. 1 and 2 to one limit of stability, where the bubble would grow irreversibly (lower solid
curve in Fig. 2A) and addition of a
sufficiently large positive Pz
(+50 kPa) moved the Sum curve up to the critical point where crushing
would occur (top solid curve in Fig.
2A). The Sum curves for all possible
stabilized sizes must lie between the two solid curves shown in Fig.
2A. Because the two curves in Fig.
2A are for the same bubble, which has
a particular
C, they are
identical except that they are translated vertically from each other by differences in Pz. The
inset shows either of the two solid
Sum curves of Fig. 2A rotated 90°
counterclockwise. Such a pressure-vs.-radius display is analogous to
stress-vs.-strain diagrams for other materials. The display is made
practical by using Pz for the
pressure axis, as we have done in Fig.
2B.
on the curve in Fig. 2B. If
subjected to Pz of 89 kPa, the
particular bubble illustrated here would be crushed in the tissue
capillaries or venous blood. To have persistent bubbles that can
recirculate in a person breathing pure
O2, it would be necessary to have
a bubble that is stabilized by a mechanism with a stronger resistance
to crushing.
Note that because displays such as Fig.
2B are for the condition of stability,
the bubble is in diffusive equilibrium with its surroundings for all
points on the curve. After a rapid change in the environment, there
will be a transient phase in which the bubble leaves the
radius-vs.-Pz curve. Diffusive
readjustment of contents will bring the bubble back to rest on the
curve.
Varieties of stabilizers. Figures 1 and 2 show how a single bubble with a particular stabilization
mechanism behaves as environmental conditions change. In what follows,
we focus on distinguishing between different stabilization mechanisms.
We hope to gain insights into possibilities for various kinds of real
bubbles by simulating the behaviors of hypothetical bubbles having
various characteristics; to do so, we vary the parameters in
Eq. 2.
Figure 3A,
drawn for the specific kind of stabilizer that is dealt with in
APPENDIX B, shows that larger
C moves the
stabilizer curve to the right and down. Figure
3B shows that larger
C gives larger
bubbles that require less negative Pz for unstabilized growth and
less positive Pz for crushing; in
Fig. 3B, larger bubbles are more fragile than small bubbles: they are less resistant both to growth and to crushing. Higher C gives a
steeper slope at Pz = 0.
to a function of bubble
radius (C);
dashed curves were drawn from Eq. 2 with = 0 and
Ro = 0. A: location of stabilizer curves for
different values of
C. , Crush
points for an arbitrary choice of 140 kPa · µm for
b in Eq. B11.
B:
radius-vs.-Pz curves for the 3 cases of panel A. See
Glossary for other definitions.
The nature of the stabilization mechanism determines the shape of a curve of stabilizer pressure as a function of bubble size. Our examples so far have used Eq. 2 with a (
+1) exponent of 1.0 and
Ro asymptote of
zero; these choices approximate the behavior of a bubble stabilized by
either a slowly permeating gas or by a "gaseous" type of
surfactant film (23). We presume that we can approximate curves for
other types of surfactants or mechanical structures by varying the exponent or the
Ro asymptote; a
few examples are presented in Fig. 4.
in
Eq. 2 (with C = 400 kPa · µm3,
Ro = 0; crushing
arbitrarily set at Pz = 23.4 kPa).
C and
D: variation of
Ro in
Eq. 2
[with C = 400, = 0, crushing occurs when (Rc*2 R2o) is less than a
certain value, arbitrarily set at 1.5 µm2]. See
Glossary for other definitions.
Figure 4, A and B, displays effects of alterations in the value of the
parameter to simulate variations in the mechanical properties of stabilizers. When is large, the
P curves are steeper (Fig.
4A), the
radius-vs.-Pz curves are more
horizontal where they cross the axis for zero
Pz (Fig.
4B), and the bubbles are more
resistant to unstabilized growth ( in Fig.
4B). We cannot anticipate the effect
of different values on the crushing phenomenon, so we arbitrarily
chose to set the crushing point at a constant Pzc. However,
it may be that the steeper curves in Fig.
4A are also more resistant to
crushing; if so, the curves for larger values should have the crush
point at greater negative Pabs in Fig.
4A and greater positive
Pz in Fig.
4B.
Because Ro
characterizes the aggregate surface area of elements of the mechanical
stabilizer, larger
Ro indicates
larger elements. An increase in
Ro moves the
asymptotes to the right and causes divergence of the stabilizer curves
from each other at negative Pabs values (Fig.
4C). The consequences are that there
is little variation of the Pz for
unstabilized growth but there is variation in the crushing
Pz (Fig.
4D). Bubbles with large
Ro are less
resistant to crushing.
Note that the kinds of diagrams shown in Figs. 3 and 4 may be crude
approximations for real bubbles. The
P curves may not fall smoothly
to the left as shown in Figs. 3A,
4A, and
4C if there are changes in the state
of the material in the structure. For example, if a surfactant film
goes from "gaseous" to "liquid" to "solid" states
(Ref. 1, p. 129-133), each state may have its own smooth
curve, which is joined to the curve for the next state at a transition
point. Other complexities are likely. For example, a bubble that
collapses like a football bladder when its volume decreases would have
different radii in different aspects, and one of the radii would change
much more than others when volume decreases.
Coherent elements. Some bubbles may be
stabilized by elements or molecules that tend to cohere to each other,
such as denatured proteins (7, 15, 19, 28, 31). Such materials can be expected to exhibit the property of elasticity, which would result in a
Pabs-vs.-radius plot that rises without leveling off, as depicted in
Fig. 5A.
As in Figs. 1, 2, 3, 4, the Sum curve is the sum of contributions from
the P curve and the
P curve. For this case, the
P curve was drawn by a formula inspired by properties of elastic sheets:
P = c
[R (R2/R)].
This formula incorporates the idea that force per unit length exerted
by an elastic sheet that covers a sphere is proportional to change of
area from an unstressed area (with radius = R) and was derived by analogy
with Eqs. A1-A6 in
APPENDIX A. For Fig.
5, we arbitrarily assigned the scaling constant
c to be 10 kPa/µm and
R to be 4 µm. Although the
Sum curve has no maximum, irreversible growth can occur by a different
mechanism. The discontinuity on the right end of the
P curve of the elastic,
coherent material in Fig. 5A
represents failure of the stabilizer because of excessive stretch,
analogous to explosion of a balloon when it is overinflated. In Fig.
5B, an added
Pz of 50 kPa lowers every
point on the Sum curve so that the stable radius is at the point of the
discontinuity in the stabilizer curve. After the stabilization
structure ruptures and no longer provides a counterpressure, there are
only two forces acting on the gases in the bubble: surface tension is
small because the size of the bubble is large, and the negative
Pz that was used to induce the
present state is relatively large. The sum of these puts the bubble in
the growth region, so it grows (arrow in Fig.
5C).
, Stable radius. Stabilization may fail due to crushing (left ) or overexpansion
(right ).
B: addition of a large negative Pz lowers the sum curve and
increases stable radius to where it is overstretched and destroyed
(). C: after stabilizer disappears, bubble grows irreversibly (arrow). D:
Pz-vs.-radius plot of bubble shown
on panels A-C.
, Unstressed radius; , rupture points; compliance is more uniform
over range of Pz than with Figs. 3 and 4.
Figure 5D is a plot of radius as a function of Pz for the elastic stabilizer, analogous to Fig. 2B. The
on the left represents the point
at which the structure ruptures, producing an unstabilized bubble that
grows. We envision that this sort of elastic membrane will crush or
rupture at high positive pressures in the same way as a
surfactant-stabilized bubble, so that it ceases to serve as a
stabilizer at some small radius ( on
right).
Applications of radius-vs.-Pz curves.
Knowledge of how bubbles behave when they are stabilized by a
particular mechanism would allow practitioners to choose the bubble
with appropriate characteristics for a particular purpose. Also,
specific characteristics can be sought in the formulation of bubbles by
new techniques or by the use of new materials. It might be desirable,
for example, to have bubbles that would not be crushed at all under
physiological conditions or, alternatively, an application might call
for bubbles that would be crushed by arterial blood pressure. A
radius-vs.-Pz curve conveniently
summarizes the characteristics of different kinds of stabilized
bubbles; the critical growth radius, the unstressed radius, the
crushing radius, and the bubble's compliance all appear on one simple
graph. Data to characterize a particular bubble on a
radius-vs.-Pz curve could be
obtained by microscopic observations of it during slow changes of
external pressure while dissolved gases in the liquid environment are
controlled.
We next cite three cases where distinguishing between different types
of bubbles may be desirable. First, if bubbles are used to augment
O2 carriage by blood (4), the
amount of O2 that is unloaded at a
particular partial pressure depends heavily on the bubble's tendency
to change size as well as on the local
PO2 (24). Therefore, a highly
compliant bubble is desirable if one wants to unload large amounts of
O2 in tissue at a given partial pressure. The curves in Fig. 3B
indicate that bubbles with larger C generally
exhibit greater compliance (change of radius per change of pressure).
In contrast, noncompliant bubbles, such as those with higher exponent illustrated in Figs. 4A and
4B, would be less effective in
O2 delivery.
Second, it remains to be seen which stabilizer characteristics give the
best signal when bubbles are used for ultrasonic contrast (16, 35):
large elements, many elements, or large compliance. In some
circumstances, the ultrasonic signal due to a bubble is proportional to
the sixth power of radius (35), so stabilizing mechanisms that give
rise to large bubbles offer far more enhancement of a given signal than
mechanisms that make for smaller bubbles.
The third case concerns the idea that the precursors of
decompression-sickness bubbles are permanent or semipermanent small bubbles or gaseous "micronuclei" (11, 21, 34) stabilized by
structures of the kind discussed in this paper. If so,
Eqs. B8 and B11 indicate that large
micronuclei are both more easily destroyed by crushing and more easily
transformed into bubbles. Exposure of a diver to high environmental
pressure is thought to crush micronuclei (26). If so, the crushing
would delete some of the population of micronuclei, so there would be
fewer available to be forced into irreversible growth to cause damaging bubbles by the subsequent decompression (18). Furthermore, it would be
desirable to determine the characteristics of the stabilizing mechanisms of decompression-sickness precursors; perhaps it will be
possible to develop techniques to interfere with their transformation into damaging bubbles.
ACKNOWLEDGEMENTS
We are greatly indebted to Mark E. Burkard for insightful suggestions and criticisms.
FOOTNOTES
Address for reprint requests: H. D. Van Liew, Dept. of Physiology, 25 Sherman Annex, State Univ. of New York at Buffalo, SUNY, Buffalo, NY 14214.
Received 3 May 1996; accepted in final form 11 February 1997.
APPENDIX A
Relationship of Eq. 2 to Surface Films
Equation A1 describes the behavior of various nonideal gaseous surfactants on a planar surface, such as on a Langmuir trough (Ref. 6, p. 228)
|
(A1) |
, is related to the
number n of molecules or elements in the film. Because is the difference of surface tension between pure
solvent and solvent containing the surfactant (Ref. 6, p. 218), it has
the nature of a surface tension, so the Laplace-Young equation can
characterize the counterpressure on a spherical bubble due to the
surfactant
|
(A2) |
|
(A3) |
|
(A4) |
We group constants in Eq. A4 into a
single constant
C
|
(A5) |
constant
is proportional to the number of stabilizer elements in the interface
(23).
Substitution of the
C constant
into Eq. A4 yields
|
(A6) |
|
(2) |
is 0, Eq.
2 equals Eq. A6. If both and
Ro are zero,
Eq. 2 is appropriate for bubbles stabilized by a slowly permeating gas (23)
and by the ideal surfactant discussed in APPENDIX
B.
APPENDIX B
Mathematical Characterization of Stable Radii
In what follows, we develop equations that are applicable to bubbles stabilized by ideal gaseous surfactants and by slowly permeating gases (23) and are approximations for bubbles stabilized by other kinds of surfactant. This simplified case is convenient for mathematical manipulations; it uses a simplified form of Eq. 2 in which both Ro and are zero
|
(B1) |
By using Eq. 1 and Eq. B1, Eq. 3 can be written as a function of radius
|
(B2) |
Pabs = 0, so there is no net diffusion of gas between the bubble and
its surroundings
|
(B3) |
|
(B4) |
Critical radius for irreversible
growth. We next amplify on the contention, illustrated
in Fig. 1C and
2A, that the maximum of a Sum curve on
a Pabs-vs.-radius plot is associated with an upper limit to the
radius for a stabilized bubble. The limiting large size is brought
about when addition of a strong growth pressure (negative
Pz) brings the maximum of the
Sum curve to the zero Pabs axis. The smallest pressure at which the
bubble is forced from a stable state into a state of unstable growth
can be called the "critical growth pressure"
(Pzg ), and the size of the bubble at that pressure can be called the "critical growth radius"
(R*g).
The maximum of the Sum curve is found by equating the first derivative
of Pabs as a function of R
(Eq. B2) to zero
|
(B5) |
Pabs = 0) and at the maximum defined by Eq. B5
|
(B6) |
times larger than the unstressed radius.
Putting Eq. B6 into
Eq. B3 yields an expression for
critical growth pressure as a function of
C
|
(B7) |
require less
Pz to cause irreversible growth
than bubbles with small
C.
The dependence of critical growth radius on imposed
Pz, seen as in Fig.
3B, is given by solving
Eq. B6 for
C and putting the result into Eq. B3
|
(B8) |
) in the cases shown in Figs. 3B,
4B, and 4D trace out paths in which
Pz to cause growth is inversely
related to radius.
Critical radius for crushing. We assume that a spherical, structurally stabilized bubble is crushed when its surface area is reduced to some particular small area at which the structural elements are pressed together maximally
|
(B9) |
The relation between
C and the
crushing radius is found by putting Eqs.
A5 and B9 together and
combining constants
|
(B10) |
curves of Figs.
3A,
4A, and
4C; we made arbitrary choices for
Ac to give the
crushing radii for the Figs. 3A,
4A, and
4C.
We enter Eq. B10 into Eq. B3 to characterize the pattern of the filled diamonds in Fig. 3B
|
(B11) |
/R*c).
The absorptive force of surface tension aids the external
Pzc in the
crushing phenomenon. As predicted by Eq.
B11, the crushing points trace out paths inversely
related to radius in Figs. 3B and
4D .
Recombination of variables shows that
Pzc is
inversely proportional to the square root of
C
|
(B12) |
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