Decompression
Theory
DECOMPRESSION MODELS AND PHASE
MECHANICS
B.R. Wienke
INTRODUCTION
Modeling of decompression phenomena in the human body is, at
times, more of an artform than a science. Some take the view that
deterministic modeling can only be fortuitous. technological
advance, elucidation of competing mechanisms, and resolution of
model issues over the past 80 years has not been rapid. Model
applications tend to be ad hoc, tied to data fits, and difficult
to quantify on first principles. Almost any description of
decompression processes in tissue and blood can be disputed, and
turned around on itself. The fact the decompression takes place
in metabolic and perfused matter makes it difficult to design and
analyze experiments outside living matter. Yet, for application
to safe diving, we need models to build tables and meters. And
deterministic models, not discounting shortcomings, are the
subject of this discourse.
MODERN DIVING
A consensus of opinions, and for a variety of reasons, suggests
that modern diving began in the early 1960s. Technological
achievements, laboratory programs, military priorities, safety
concerns, commercial diving requirements, and international
business spurred diving activity and scope of operation. Diving
bells, hot water heating, mixed gases, saturation, deep diving,
expanded wet testing, computers, and efficient decompression
algorithms signaled the modern diving era. Equipment advances in
open and closed circuit breathing devices, wet and dry suits,
gear weight, mask and fin design, high pressure compressors,
flotation and buoyancy control vests, communications links,
gauges and meters, lights, underwater tools (cutting, welding,
drilling, explosives), surface supplied air, and photographic
systems paced technological advances. Training and certification
requirements for divers, in military, commercial, sport, and
scientific sectors, took definition with growing concern for
underwater safety and well being.
In the conquest and exploration of the oceans, saturation diving
gained prominence in the 1960s, permitting exploitation of the
continental shelf impossible with the short exposure times
allowed by conventional diving. Spurred by both industrial and
military interests in the ability of men to work underwater for
long periods of time, notable habitat experiments, such as
Sealab, Conshelf, Man In Sea, Gulf Task, Tektite, and Diogene,
demonstrated the feasibility of living and working underwater for
long periods of time. These efforts followed proof of principle
validation, by Bond and coworkers (USN) in 1958, of saturation
diving. Saturation exposure programs and tests have been
conducted from 35 fsw to 2,000 fsw.
The development and use of underwater support platforms, such as
habitats, bell diving systems, lockout and free flooded
submersibles, and diver propulsion units also accelerated in the
1960s and 1970s, for reasons of science and economics. Support
platforms extended both diver usefulness and bottom time, by
permitting him to live underwater, reducing descent and ascent
time, expanding mobility, and lessening physical activity. Today,
themselves operating from underwater platforms, remotely operated
vehicles (ROVs) scan the ocean depths at 6,000 fsw for minerals
and oil.
Around 1972, strategies for diving in excess of 1,000 fsw
received serious scrutiny, driven by a commercial quest for oil
and petroleum products, and the needs of the commercial diving
industry to service that quest. Questions concerning
pharmacological additives, absolute pressure limits, thermal
exchange, therapy, compression-decompression procedures,
effective combinations of mixed breathing gases, and equipment
functionality addressed many fundamental issues, unknown or only
partially understood. By the early 1980s, it became clear that
open sea water work in the 1,000 to 2,000 fsw range was entirely
practical, and many of the problems, at least from an operational
point of view, could be solved. Today, the need for continued
deep diving remains, with demands that cannot be answered with
remote, or 1 atm, diver systems. Heliox and trimix have become
standards for deep excursion breathing gases, with heliox the
choice for shallower exposures, and trimix the choice for deeper
exposures in the field.
Yet, despite tremendous advances in deep diving technology, most
of the ocean floor is outside human reach. Breathing mixtures
that are compressible are limiting. Breathing mixtures that are
not compressible offer interesting alternatives. In the 1960s,
serious attention was given to liquid breathing mixtures,
physiological saline solutions. Acting as inert respiratory gas
diluents, oxygenated fluids have been used as breathing mixtures,
thereby eliminating decompression requirements.
MODELS
Most believe that the pathophysiology of decompression sickness
syndrome follows formation of a gas phase after decompression.
Yet, the physiological evolution of the gas phase is poorly
understood. Bubble detection technology has established that
moving and stationary bubbles do occur following decompression,
that the risk of decompression sickness increases with the
magnitude of detected bubbles, that symptomless, or silent,
bubbles are also common following decompression, and that the
variability in gas phase formation is likely less than the
variability in symptom generation. Taken together, gas phase
formation is not only important to the understanding of
decompression sickness, but is also a crucial model element in
theory and computation.
Bubbles can form in tissue and blood when ambient pressure drops
below tissue tensions, according to dissolved-free phase
mechanics. Trying to track free and dissolved gas buildup and
elimination in tissue and blood, especially their interplay, is
extremely complex, beyond the capabilities of even
supercomputers. But safe computational prescriptions are
necessary in the formulation of dive tables and digital meter
algorithms. The simplest way to stage decompression, following
extended exposures to high pressure with commensurate dissolved
gas buildup, is to limit tissue tensions. Historically, Haldane
first employed the approach, and it persists today in modified
form.
History
Tables and schedules for diving at sea level can be traced to a
model proposed in 1908 by the eminent English physiologist, John
Scott Haldane. He observed that goats, saturated to depths of 165
feet of sea water (fsw), did not develop decompression sickness
(DCS) if subsequent decompression was limited to half the ambient
pressure. Extrapolating to humans, researchers reckoned that
tissues tolerate elevated dissolved gas pressures (tensions),
greater than ambient by factors of two, before the onset of
symptoms. Haldane then constructed schedules which limited the
critical supersaturation ratio to two in hypothetical tissue
compartments. Tissue compartments were characterized by their
halftime, tau . Halftime is also termed half-life when linked to
exponential processes, such as radioactive decay. Five
compartments (5, 10, 20, 40, 75 minutes) were employed in
decompression calculations and staged procedures for fifty years.
Some years following, in performing deep diving and expanding
existing table ranges in the 1930s, US Navy investigators
assigned separate limiting tensions (M-values) to each tissue
compartment. Later in the 1950s and early 1960s, other US Navy
investigators, in addressing repetitive exposures for the first
time, advocated the use of six tissues (5, 10, 20, 40, 80, 120
minutes) in constructing decompression schedules, with each
tissue compartment again possessing its own limiting tension.
Temporal uptake and elimination of inert gas was based on
mechanics addressing only the macroscopic aspects of gas exchange
between blood and tissue. Exact bubble production mechanisms,
interplay of free and dissolved gas phases, and related transport
phenomena were not quantified, since they were neither known nor
understood. Today, we know much more about dissolved and free
phase dynamics, bubbles, and transport mechanisms, but still rely
heavily on the Haldane model. Inertia and simplicity tend to
sustain its popularity and use, and it has been a workhorse.
To maximize the rate of uptake or elimination of dissolved gases,
the gradient, simply the difference between arterial and tissue
tension, is maximized by pulling the diver as close to the
surface as possible. Exposures are limited by requiring that the
tissue tensions, never exceed limits (called M-values), for
instance, written for each compartment in the US Navy approach
(5, 10, 20, 40, 80, and 120 minute tissue halftimes, tau , as M =
M sub 0 + DELTA M d, with, M sub 0 = 152.7 tau sup -1/4 , and,
DELTA M = 3.25 tau sup -1/4 , as a function of depth, d, for
DELTA M the change per unit depth. Obviously, M, is largest for
fast tissue compartments ( tau small), and smallest for slow
tissue compartments ( tau large). Fast compartments control short
deep excursions, while slow compartments control long shallow
excursions. Surfacing values, M sub 0 , are principal concerns in
nonstop diving, while values at depth, DELTA M d , concern
decompression diving. In both cases, the staging regimen tries to
pull the diver as close to the surface as possible, in as short a
time as possible. By contrast, free phase (bubble) elimination
gradients, as seen, increase with depth, directly opposite to
dissolved gas elimination gradients which decrease with depth. In
actuality, decompression is a playoff between dissolved gas
buildup and free phase growth, tempered by body ability to
eliminate both. But dissolved gas models cannot handle both, so
there are problems when extrapolating outside tested ranges.
In absolute pressure units, the corresponding critical gradient,
G = Q - P, is related to ambient pressure, P, and critical
nitrogen pressure, M, with, Q = 1.27 M. In bubble theories,
supersaturation is limited by the critical gradient, G. In
decompressed gel experiments, Strauss suggested that G approx. 20
fsw at ambient pressures less than a few atmospheres. Other
studies suggest, 14 <= G <= 30 fsw, as a range of critical
gradients (G-values). In diffusion-dominated approaches, the
tissue tension can be limited by a single, pressure criterion,
such as, M = 709 P / P + 404 .
Blood rich, well-perfused, aqueous tissues are usually thought to be fast (small tau ), while blood poor, scarcely-perfused, lipid tissues are thought to be slow (large tau ), though the spectrum of halftimes is not correlated with actual perfusion rates in critical tissues. As reflected in relationship above, critical parameters are obviously larger for faster tissues. The range of variation with compartment and depth is not insignificant. Fast compartments control short deep exposures, while slow compartments control long shallow, decompression, and saturation exposures.
Bulk Diffusion Model
Diffusion limited gas exchange is modeled in time by a sum of
exponential response functions, bounded by arterial and initial
tissue tensions. However, instead of many tissue compartments, a
single bulk tissue is assumed for calculations, characterized by
a gas diffusion constant, D. Tissue is separated into
intravascular (blood) and extravascular (cells) regions. Blood
containing dissolved inert and metabolic gases passes through the
intravascular zone, providing initial and boundary conditions for
subsequent gas diffusion into the extravascular zone. Diffusion
is driven by the difference between arterial and tissue tensions,
according to the strength of a single diffusion coefficient, D,
appropriate to the media. Diffusion solutions, averaged over the
tissue domain, resemble a weighted sum over effective tissue
compartments with time constants, alpha sub 2n-1 sup 2 D,
determined by diffusivity and boundary conditions, with alpha sub
2n-1 = (2n - 1) pi / l for tissue thickness, l.
Applications fit the time constant, K = pi sup 2 D / l sup 2 , to
exposure data, with a typical value employed by the Royal Navy
given by, K = 0.007928 min sup -1 , approximating the US Navy 120
minute compartment used to control saturation, decompression, and
repetitive diving. Corresponding critical tensions in the bulk
model, M = 709 P / P + 404 , fall somewhere between fixed
gradient and multitissue values. At the surface, M = 53 fsw,
while at 200 fsw, M = 259 fsw. A critical gradient, G = P ( 493 -
P ) / ( P + 404 ), also derives from the above. Originally, a
critical gradient, G, near 30 fsw was used to limit exposures.
Such value is too conservative for deep and bounce exposures, and
not conservative enough for shallow exposures. Hempleman
introduced the above relationship, providing the means to
parameterize bounce and saturation diving.
Bulk models are attractive because they permit the whole dive
profile to be modeled with one equation, and because they predict
a t sup 1/2 behavior of gas uptake and elimination. Nonstop time
limits, t sub n , are related to depth, d, by the bulk diffusion
relationship, d t sub n sup 1/2 = C, with approximate range, 400
<= C <= 500 fsw min sup 1/2 , linking nonstop time and
depth simply through the value of C. For the US Navy nonstop
limits, C approx. 500 fsw min sup 1/2 , while for the Spencer
reduced limits, C approx. 465 fsw min sup 1/2 . In the
Wienke-Yount model, C approx. 400 fsw min sup 1/2 .
Multitissue Model
Multitissue models, variations of the original Haldane model,
assume that dissolved gas exchange, controlled by blood flow
across regions of varying concentration, is driven by the local
gradient, that is, the difference between the arterial blood
tension and the instantaneous tissue tension. Tissue response is
modeled by exponential functions, bounded by arterial and initial
tensions, and perfusion constants, lambda , linked to the tissue
halftimes, tau , for instance, 1, 2, 5, 10, 20, 40, 80, 120, 180,
240, 360, 480, and 720 minute compartments assumed to be
independent of pressure.
In a series of dives or multiple stages, initial and arterial
tensions represent extremes for each stage, or more precisely,
the initial tension and the arterial tension at the beginning of
the next stage. Stages are treated sequentially, with finishing
tensions at one step representing initial tensions for the next
step, and so on. To maximize the rate of uptake or elimination of
dissolved gases the gradient, simply the difference between
arterial and tissue tensions is maximized by pulling the diver as
close to the surface as possible. Exposures are limited by
requiring that the tissue tensions never exceed M = M sub 0 +
DELTA M d, as a function of depth, d, for DELTA M the change per
unit depth. A set of M sub 0 and DELTA M are listed in Table 1.
At altitude, some critical tensions have been correlated with
actual testing, in which case, an effective depth, d = P - 33, is
referenced to the absolute pressure, P, with surface pressure, P
sub h = 33 exp. ( -0.0381 h ), at elevation, h, and h in
multiples of 1,000 ft. However, in those cases where critical
tensions have not been tested, nor extended, to altitude, an
exponentially decreasing extrapolation scheme, called similarity,
has been employed. Extrapolations of critical tensions, below P =
33 fsw, then fall off more rapidly then in the linear case. A
similarity extrapolation holds the ratio, R = M/P, constant at
altitude. Estimating minimum surface tension pressure of bubbles
near 10 fsw, as a limit point, the similarity extrapolation might
be limited to 10,000 ft in elevation, and neither for
decompression nor heavy repetitive diving.
Models of dissolved gas transport and coupled bubble formation
are not complete, and all need correlation with experiment and
wet testing. Extensions of basic (perfusion and diffusion) models
can redress some of the difficulties and deficiencies, both in
theory and application. Concerns about microbubbles in the blood
impacting gas elimination, geometry of the tissue region with
respect to gas exchange, penetration depths for gas diffusion,
nerve deformation trigger points for pain, gas uptake and
elimination asymmetry, effective gas exchange with flowing blood,
and perfusion versus diffusion limited gas exchange, to name but
a few, motivate a number of extensions of dissolved gas models.
The multitissue model addresses dissolved gas transport with
saturation gradients driving the elimination. In the presence of
free phases, free-dissolved and free-blood elimination gradients
can compete with dissolved-blood gradients. One suggestion is
that the gradient be split into two weighted parts, the free-
blood and dissolved-blood gradients, with the weighting fraction
proportional to the amount of separated gas per unit tissue
volume. Use of a split gradient is consistent with multiphase
flow partitioning, and implies that only a portion of tissue gas
has separated, with the remainder dissolved. Such a split
representation can replace any of the gradient terms in tissue
response functions.
If gas nuclei are entrained in the circulatory system, blood
perfusion rates are effectively lowered, an impairment with
impact on all gas exchange processes. This suggests a possible
lengthening of tissue halftimes for elimination over those for
uptake, for instance, a 10 minute compartment for uptake becomes
a 12 minute compartment on elimination. Such lengthening
procedure and the split elimination gradient obviously render gas
uptake and elimination processes asymmetric. Instead of both
exponential uptake and elimination, exponential uptake and linear
elimination response functions can be used. Such modifications
can again be employed in any perfusion model easily, and tuned to
the data.
Thermodynamic Model
The thermodynamic approach suggested by Hills, and extended by
others, is more comprehensive than earlier models, addressing a
number of issues simultaneously, such as tissue gas exchange,
phase separation, and phase volume trigger points. This model is
based on phase equilibration of dissolved and separated gas
phases, with temporal uptake and elimination of inert gas
controlled by perfusion and diffusion. From a boundary (vascular)
zone of thickness, gases diffuse into the cellular region.
Radial, one dimensional, cylindrical geometry is assumed as a
starting point, though the extension to higher dimensionality is
straightforward. As with all dissolved gas transfer, diffusion is
controlled by the difference between the instantaneous tissue
tension and the venous tension, and perfusion is controlled by
the difference between the arterial and venous tension. A mass
balance for gas flow at the vascular cellular interface, a,
enforces the perfusion limit when appropriate, linking the
diffusion and perfusion equations directly. Blood and tissue
tensions are joined in a complex feedback loop. The trigger point
in the thermodynamic model is the separated phase volume, related
to a set of mechanical pain thresholds for fluid injected into
connective tissue.
The full thermodynamic model is complex, though Hills has
performed massive computations correlating with the data,
underscoring basic model validity. Considerations of free phase
dynamics (phase volume trigger point) require deeper
decompression staging formats, compared to considerations of
critical tensions, and are characteristic of phase models. Full
blown bubble models require the same, simply to minimize bubble
excitation and growth.
Reduced Gradient Bubble Model
The reduced gradient bubble model (RGBM), developed by Wienke,
treats both dissolved and free phase transfer mechanisms,
postulating the existence of gas seeds (micronuclei) with
permeable skins of surface active molecules, small enough to
remain in solution and strong enough to resist collapse. The
model is based upon laboratory studies of bubble growth and
nucleation, and grew from a similar model, the varying
permeability model (VPM), treating bubble seeds as gas
micropockets contained by pressure permeable elastic skins
Inert gas exchange is driven by the local gradient, the
difference between the arterial blood tension and the
instantaneous tissue tension. Compartments with 1, 2, 5, 10, 20,
40, 80, 120, 240, 480, and 720 halftimes, tau , are again
employed. While, classical (Haldane) models limit exposures by
requiring that the tissue tensions never exceed the critical
tensions, fitted to the US Navy nonstop limits, for example. The
reduced gradient bubble model, however, limits the
supersaturation gradient, through the phase volume constraint. An
exponential distribution of bubble seeds, falling off with
increasing bubble size is assumed to be excited into growth by
compression-decompression. A critical radius, r sub c , separates
growing from contracting micronuclei for given ambient pressure,
P sub c . At sea level, P sub c = 33 fsw , r sub c = .8 microns,
and DELTA P = d. Deeper decompressions excite smaller, more
stable, nuclei.
Within a phase volume constraint for exposures,
a set of nonstop limits, t sub n , at depth, d, satisfy a
modified law, d t sub n sup 1/2 = 400 fsw min sup 1/2 , with
gradient, G, extracted for each compartment, tau , using the
nonstop limits and excitation radius, at generalized depth, d = P
- 33 fsw. Tables 2 and 3 summarize t sub n , G sub 0 , DELTA G ,
and delta , the depth at which the compartment begins to control
exposures.
Table 2. Critical Phase Volume Time Limits.
______________________________________________
depth nonstop limit depth nonstop limit
d(fsw) t sub n (min) d (fsw) t sub n (min)
30 250 130 9.
40 130. 140 8.
50 73. 150 7.
60 52. 1606 5
70 39. 1705 8
80 27. 1805 3
90 22. 1904 6
100 18. 2004 1
110 15. 2103 7
120 12. 2203 1
Gas filled crevices can also facilitate nucleation by cavitation.
The mechanism is responsible for bubble formation occurring on
solid surfaces and container walls. In gel experiments, though,
solid particles and ragged surfaces were seldom seen, suggesting
other nucleation mechanisms. The existence of stable gas nuclei
is paradoxical. Gas bubbles larger than 1 micron should float to
the surface of a standing liquid or gel, while smaller ones
should dissolve in a few seconds. In a liquid supersaturated with
gas, only bubbles at the critical radius, r sub c , would be in
equilibrium (and very unstable equilibrium at best). Bubbles
larger than the critical radius should grow larger, and bubbles
smaller than the critical radius should collapse. Yet, the Yount
gel experiments confirm the existence of stable gas phases, so no
matter what the mechanism, effective surface tension must be
zero.
Table 3. Critical Phase Volume Gradients.
__________________________________________________________________________
halftime threshold depth surface gradient gradient change
tau (min) delta (fsw) G sub 0 (fsw) DELTA G
________________________ ____________ ____________
2 190 151.0 .518
5 135 95.0 .515
10 95 67.0 .511
20 65 49.0 .506
40 40 36.0 .468
80 30 27.0 .417
120 28 24.0 .379
240 16 23.0 .329
480 12 22.0 .312
Although the actual size distribution of gas nuclei in humans is
unknown, these experiments in gels have been correlated with a
decaying exponential (radial) distribution function. For a
stabilized distribution accommodated by the body at fixed
pressure, P sub c , the excess number of nuclei excited by
compression-decompression must be removed from the body. The rate
at which gas inflates in tissue depends upon both the excess
bubble number, and the supersaturation gradient, G. The critical
volume hypothesis requires that the integral of the product of
the two must always remain less than some volume limit point,
alpha V , with alpha a proportionality constant. A conservative
set of bounce gradients, G bar , can be also be extracted for
multiday and repetitive diving, provided they are
multiplicatively reduced by a set of bubble factors, eta sup rep,
eta sup reg , eta sup exc , all less than one, such that G bar =
eta sup rep eta sup reg eta sup exc G.
These three bubble factors reduce the driving gradients to
maintain the phases volume constraint. The first bubble factor
reduces G to account for creation of new stabilized micronuclei
over time scales of days. The second factor accounts for
additional micronuclei excitation on deeper-than-previous dives.
The third bubble factor accounts for bubble growth over
repetitive exposures on time scales of hours. Clearly, the
repetitive factors, eta sup rep , relax to one after about 2
hours, while the multiday factors, eta sup reg , continue to
decrease with increasing repetitive activity, though at very slow
rate. Increases in bubble elimination halftime and nuclei
regeneration halftime will tend to decrease eta sup rep and
increase eta sup reg . The repetitive fractions, eta sup rep ,
restrict back to back repetitive activity considerably for short
surface intervals. The multiday fractions get small as multiday
activities increase continuously beyond 2 weeks.
Deeper-than-previous excursions incur the greatest reductions in
permissible gradients (smallest eta sup exc ) as the depth of the
exposure exceeds previous maximum depth.
Tissue Bubble Diffusion Model
The tissue bubble diffusion model (TBDM), according to Gernhardt
and Vann, considers the diffusive growth of an extravascular
bubble under arbitrary hyperbaric and hypobaric loadings. The
approach incorporates inert gas diffusion across the
tissue-bubble interface, tissue elasticity, gas solubility and
diffusivity, bubble surface tension, and perfusion limited
transport to the tissues. Tracking bubble growth over a range of
exposures, the model can be extended to oxygen breathing and
inert gas switching. As a starting point, the TBDM assumes that,
through some process, stable gas nuclei form in the tissues
during decompression, and subsequently tracks bubble growth with
dynamical equations. Diffusion limited exchange is invoked at the
tissue-bubble interface, and perfusion limited exchange is
assumed between tissue and blood, very similar to the
thermodynamic model, but with free phase mechanics. Across the
extravascular region, gas exchange is driven by the pressure
difference between dissolved gas in tissue and free gas in the
bubble, treating the free gas as ideal. Initial nuclei in the
TBDM have assumed radii near 3 microns at sea level, to be
compared with .8 microns in the RGBM.
As in any free phase model, bubble volume changes become more
significant at lower ambient pressure, suggesting a mechanism for
enhancement of hypobaric bends, where constricting surface
tension pressures are smaller than those encountered in
hyperbaric cases. For instance, a theoretical bubble dose of 5 ml
correlates with a 20% risk of decompression sickness, while a 35
ml dose correlates with a 90% risk, with the bubble dose
representing an unnormalized measure of the separated phase
volume. Coupling bubble volume to risk represents yet another
extension of the phase volume hypothesis, a viable trigger point
mechanism for bends incidence.
SATURATION CURVE
The saturation curve, relating permissible gas tension, Q, as a
function of ambient pressure, P, for air, sets a lower bound, so
to speak, on decompression staging. All staging models and
algorithms must collapse to the saturation curve as exposure
times increase in duration. In short, the saturation curve
represents one extreme for any staging model. Bounce curves
represent the other extreme. Joining them together for diving
activities in between is a model task, as well as joining the
same sets of curves over varying ambient pressure ranges. In the
latter case, extending bounce and saturation curves to altitude
is just such an endeavor.
Models for controlling hypobaric and hyperbaric exposures have
long differed over range of applicability. Recent analyses of
very high altitude washout data question linear extrapolations of
the hyperbaric saturation curve, to hypobaric exposures, pointing
instead to correlation of data with constant decompression ratios
in animals and humans. Correlations of hypobaric and hyperbaric
data, however, can be effected with a more general form of the
saturation curve, one exhibiting the proper behavior in both
limits. Closure of hypobaric and hyperbaric diving data can be
managed with one curve, exhibiting linear behavior in the
hyperbaric regime, and bending through the origin in the
hypobaric regime. Using the RGBM and a basic experimental fact
that the number of bubble seeds in tissue increase exponentially
with decreasing bubble radius, just such a single expression can
be obtained. The limiting forms are exponential decrease with
decreasing ambient pressure (actually through zero pressure), and
linear behavior with increasing ambient pressure. Asymptotic
forms are quite evident. Such general forms derive from the RGBM,
depending on a coupled treatment of both dissolved and free gas
phases. Coupled to the phase volume constraint, these models
suggest a consistent means to closure of hypobaric and hyperbaric
data.
DECOMPRESSION MODELS AND PHASE MECHANICS
B.R. Wienke
Advanced Computing Laboratory
Los Alamos National Laboratory
Los Alamos, N.M. 87545