Mailing List Archive Mailing List Archive
Folks,
Here is the rest of my transmission on models. Note in the thermodynamic
model,
the staging criteria is not the inhereny unsaturation, but the separated
gas, $ chi $, according to the mass balance equation. A number of models
are included for your interest. Some of this is out of the book --
"Basic Decompression Theory And Application" by myself, and some is
from the new book "Basic Diving Physics And Application" (again me),
available from BEST Publishing Company, 2355 N. Steves Blvd., Flagstaff
Arizona 86004, (800) 468-1055
The original part I sent earlier is appended, alonwith an answer to Hans
Roverud about
deco stop in the thermodynamic model -- please note that under the Hill's
thermodynamic or Behnke oxygen window, times vary at stops until tissue
tensions drop to meet the model criteria (no such thing as 3 minutes to
allow oxygen equilibration or whatever). BE CAREFUL and read what the
criteria require. In the thermodynamic model, the criteria is the
separated phase fraction, and in the oxygen window approach, the criteria
relates to inherent unsaturation.
Bruce Wienke brw@la*.go*
(ENCLOSURE)
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, $ p sub a $, and the instantaneous tissue tension,
$p$. Tissue response is modeled by exponential response functions, bounded
by arterial and initial tensions, $p sub a $ and $p sub i $, satisfying a
perfusion rate equation of the form,
{ partial p } over { partial t } = - lambda ( p - p sub a ) ,
and which take the expilicit form,
p - p sub a = ( p sub i - p sub a ) exp (- lambda t ) ,
lambda = .6931 over { tau } ,
with 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, $p sub i $ and $p sub a $ 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 $ p sub i $ and $ p sub a $, is
maximized
by pulling the diver as close to the surface as possible. Exposures are
limited by requiring that the tissue tensions never exceed $M$, written,
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 8.1. In absolute
units, the corresponding critical gradient, $G$, is given by,
G = Q - P ,
with $P$ ambient pressure, $M$ critical nitrogen pressure, and $Q = M/.79$.
At altitude, some critical tensions have been correlated with actual testing,
in which case,
an effective depth, $d$, is referenced to the absolute pressure,
d = P - 33 ,
where surface pressure, $P sub h $, at altitude, $h$, is given by ($fsw$),
P sub h = 33 exp ( - 0.0381 h ) = 33 over { alpha } ,
alpha = exp ( 0.0381 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.
Consider the following.
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, $a$, 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 beween 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.
One of its more significant features can be seen in Figure 8.4. 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.
The thermodynamic model couples both the tissue diffusion and blood perfusion
equations. Cylindrical symmetry is assumed in the model. From a boundary
vascular zone of thickness, $a$, gas diffuses into the extended extravascular
region, bounded by $b$. The radial diffusion equation is given by,
D { partial sup 2 p } over { partial r sup 2 } + D over r { partial p } over
{ partial r } = { partial p } over { partial t } ,
with the tissue tensions, $p$, equal to the venous tensions, $p sub v $,
at the vascular interfaces, $a$ and $b$. The solution to the tissue diffusion
equation is given in Appendix C,
p - p sub v = ( p sub i - p sub v ) 4 over { (b/2) sup 2 - a sup 2 } sum
from n=1 to inf over { epsilon sub n sup 2 } { J sub 1 sup 2 ( epsilon sub n
b/2) } over { J sub 0 sup 2 ( epsilon sub n a) - J sub 1 ( epsilon sub n b/2 )
} exp ( - epsilon sub n sup 2 Dt ) ,
with $epsilon sub n $ eigenvalue roots of the boundary conditions,
J sub 0 ( epsilon sub n a) Y sub 1 ( epsilon sub n b/2) - Y sub 0 ( epsilon
sub n a ) J sub 1 ( epsilon sub n b/2) = 0 ,
for $J$ and $Y$ Bessel and Neumann functions, order 1 and 0. Perfusion
limiting is applied as a boundary condition through the venous tension, $p sub
v $,
by enforcing a mass balance across both the vascular and cellular regions
at $a$,
{ partial p sub v } over { partial t } = - kappa ( p sub v - p sub a ) -
3 over a gamma D left ( { partial p } over { partial r } right ) sub r=a ,
with $ gamma $ the ratio of tissue to blood gas solubilities, $ kappa $
the perfusion constant, and $p sub a $ the arterial tension. The coupled
set relate tension, gas flow, diffusion and perfusion, and solubility in
a complex feedback loop.
The thermodynamic trigger point for decompression sickness is the volume
fraction, $ chi $, of separated gas, coupled to mass balance. Denoting
the separated gas partial pressure, $P sub {N sub 2 }$, under worse case
conditions of zero gas elimination upon decompression, the separated gas
fraction is estimated,
chi P sub {N sub 2 } = S sub t ( p - P sub {N sub 2 } ) ,
with $S sub t $ the tissue gas solubility. The separated nitrogen partial
pressure, $P sub {N sub 2 }$ is taken up by the inherent unsaturation, and
given by ($fsw$),
P sub {N sub 2 } = P + 3.21 ,
in the original Hills formulation, but other estimates have been employed.
Mechanical fluid injection pain, depending on the injection pressure, $ delta $,
can be related to the separated gas fraction, $ chi $, through the tissue
bulk modulus, $Y$,
Y chi = delta ,
so that a decompression criteria requires,
Y chi <= delta ,
with $ delta $ in the range, for $Y = 3.7 times 10 sup 4 dyne cm sup -2 $,
0.34 <= delta <= 1.13 fsw .
VARYING PERMEABILITY MODEL
The varying permeability model (VPM) 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.
Inert gas exchange is driven by the local gradient, the difference between
the arterial blood tension, $ p sub a $, and the instantaneous tissue tension,
$p$. Such behavior is modeled in time by mathematical classes of
exponential response functions, bounded by $p sub a $ and the initial
value of $p$, denoted $p sub i $, as before,
p = p sub a + ( p sub i - p sub a ) exp (- lambda t ) ,
with the perfusion constant, $ lambda $, related to the tissue halftime, $ tau
$,
through,
lambda = .6931 over { tau } .
Compartments with 1, 2, 5, 10, 20, 40, 80, 120, 240, 480, and 720 halftimes,
$ tau $, are again employed. While, classical (Haldane) models limit exposures
by
requiring that the tissue tensions never exceed the critical tensions, fitted
to the US Navy nonstop limits, for example, in units of absolute pressure
($fsw$),
Q = 193.3 tau sup -1/4 + 4.110 d tau sup -1/4 .
The varying permeability model, however, limits $G$,
G = Q - P ,
through the phase volume constraint.
A critical radius, $r sub s $, at fixed pressure, $P sub s $, represents
a cutoff for growth upon decompression to lesser pressure. If $r sub s $
is the critical radius at $P sub s $, then the smaller family, $r$, excited
by decompression from pressure, $P$, obeys,
1 over { r } = 1 over { r sub s } + { DELTA P } over 158 ,
and, $ DELTA P = P - P sub s $, measured in $fsw$, with $r$ in $microns$.
At sea level, $P sub s = 33 fsw $, $r sub s = .8 microns$, and $ DELTA P = d$.
Deeper decompressions excite smaller, more stable, nuclei. Nonstop time limits,
$t sub n $, at depth, $d$, satisfy a modified law, that is, $d t sub n sup 1/2
= 400 fsw min sup 1/2 $,
with gradient, $G$, written for each compartment, $ tau $, using the
nonstop limits and excitation inverse radial difference, $ DELTA r sup -1 $,
G = DELTA r sup -1 DELTA G + G sub 0
DELTA r sup -1 = 1 over r - 1 over { r sub s } ,
at generalized depth, $d = P - 33 fsw$. A nonstop exposure, followed by direct
return to the surface, thus allows $G sub 0 $ for that compartment.
Tables 8.2 and 8.3 summarize $t sub n $, $G sub 0 $, $ DELTA G $, and
$ delta $, the depth at which the compartment begins to control exposures.
Table 8.2 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. 160 6.5
70 39. 170 5.8
80 27. 180 5.3
90 22. 190 4.6
100 18. 200 4.1
110 15. 210 3.7
120 12. 220 3.1
Gas filled crevices can also facilitate nucleation by cavitation. The
mechanism is responsible for bubble formation occuring 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 surafce 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 experiments
confirm the existence of $stable$ gas phases, so no matter what the
mechanism, effective surface tension must be zero.
Table 8.3 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
The minimum excitation, $G sup bub $, initially probing $r(t)$, accounting for
regeneration of nuclei over time scales $ omega sup -1 $, is,
G sup bub = { 2 gamma ( gamma sub c - gamma ) } over { gamma sub c r(t)
} = 11.01 over { r(t) } ,
with,
r(t) = r + ( r sub s - r ) [ 1 - exp ( - omega t ) ] ,
$gamma $, $ gamma sub c $ film, surfactant surface tensions, that is,
$ gamma = 17.9 dyne/cm $, $ gamma sub c = 257 dyne/cm $, and
$ omega $ the inverse of the regeneration time for stabilized gas
micronuclei (many days). Nuclei probed depend on depth.
The excitation threshold, $G sup bub $, represents that minimal free-dissolved
gas gradient, just balanced by the surface tension, supporting growth.
Saturation exposures permit, $G sup sat $,
G sup sat = 58.6 DELTA r sup -1 + 23.3 = .372 P + 11.01 .
The relationship for $G sup sat $, deduced from exposure data and given
above, agrees with specific parameterization of the controlling
compartment in critical tension algorithms.
Although the actual size distribution of gas nuclei in humans is unknown,
experiments in gels have been correlated with a decaying exponential
(radial) distribution function, $n$. For a stabilized distribution, $n$,
accommodated by the body at fixed pressure, $P sub s $, the excess number of
nuclei, $ DELTA n $, excited by compression-decompression from new pressure,
$P$, is,
DELTA n = N left ( 1 - r over { r sub s } right ) = N r DELTA r sup -1 ,
assuming the small argument expansion of the exponential function, and
with $N$ a constant. For deep
compressions-decompressions, $ DELTA n $ is large, while for shallow
compressions-decompressions, $ DELTA n $ is small.
The rate at which gas inflates in tissue depends upon both the excess bubble
number, $ DELTA n $, 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 limit point, $ alpha V $, with $ alpha $ a
proportionality constant. Accordingly this requires,
int from 0 to inf DELTA n G dt = alpha V ,
for $V$ the limiting gas volume. Assuming that tissue gas gradients
are constant during decompression, $ t sub d $, while decaying exponentially
to zero afterwards, and taking limiting condition of the equal sign, yields
for a bounce dive,
DELTA n G ( t sub d + lambda sup -1 ) = alpha V .
For nonstop exposures with linear ascent rate, $v$, we have $t sub d = d / v$.
With saturation exposures, the integral must be evaluated iteratively over
component decompression stages, maximizing each $G$ while satisfing the
constraint equation.
In the latter case, $t sub d $ is the sum of individual stage times plus
interstage ascent times, assuming the same interstage ascent speed, $v$.
Employing the above iteratively, and one more constant, $ delta $, defined by,
delta = { gamma sub c alpha V } over { gamma beta r sub s N } = 7500 fsw
min ,
we have,
left ( 1 - r over { r sub s } right ) G ( t sub d + lambda sup -1 ) =
delta { gamma } over { gamma sub c } = 522.3 fsw min ,
from the Spencer bounce and Tektite saturation data.
In terms of the depth at which a compartment controls the exposure,
the radii of nuclei excited as a function of controlling halftime, $ tau $,
in the range, $12 <= d <= 220 fsw $, are fitted,
1 - DELTA r sup -1 = r over { r sub s } = .9 - .43 exp ( - .0559 tau ) ,
with halftime measured in minutes. For large $ tau $, $r$ is close to
$r sub s $, while for small $ tau $, $r$ is on the order of .5 $r sub s $.
REDUCED GRADIENT BUBBLE MODEL
Within the reduced gradient bubble model (RGBM), according to Wienke, one
extends the critical phase criterion to repetitive diving, that is, the phase
integral over multiexposures, by writing,
sum from j=1 to J left [ DELTA n G t sub { d sub j } + int from 0 to t sub
j DELTA n G dt right ] <= alpha V ,
with index, $j$, denoting each dive segment, up to total, $J$, and,
$t sub j $, surface interval after the $j sup th $ segment. Particular
$G$ are general, and not necessarily the set derived for bounce and saturation
diving.
For the inequality to hold, that is, for the sum of all growth rate
terms to total less than $ alpha V $, obviously each term must be less than
$ alpha V $. Performing the indicated operations yields a revised criterion,
sum from j=1 to J DELTA n G bar ( t sub {d sub j} + lambda sup -1 ) <=
alpha V ,
with the important property,
G bar <= G .
Because of the above constraint, the approach is termed a reduced gradient
bubble model.
The terms $ DELTA n G$ and $ DELTA n G bar $, differ by effective bubble
elimination during the previous surface interval. To maintain the phase
volume constraint during multidiving, the elimination rate must be
downscaled by a set of bubble growth, regeneration, and excitation factors,
cumulatively designated, $ xi $, such that,
G bar = xi G .
A conservative set of bounce gradients, $G$, can be employed for multiday
and repetitive diving, provided they are reduced by $ xi $.
Three bubble factors, $ eta $, reduce the driving gradients to maintain the
phase volume constraint. The first bubble factor, $ eta sup reg $, reduces $G$
to
account for creation of new stabilized micronuclei over time scales, $ omega
sup -1 $,
of days,
eta sup reg = exp ( - omega t sub cum ) ,
7 <= omega sup -1 <= 21 days ,
for $t sub cum $ the cumulative (multiday) surface interval. The second bubble
factor, $ eta sup exc $, accounts for additional micronuclei excitation
on deeper-than-previous dives,
eta sup exc = = { ( DELTA n ) sub prev } over { ( DELTA n ) sub pres } = { ( rd
) sub prev } over { (rd) sub pres } ,
for excitation radius, $r$, at depth, $d$, and the subscripts referencing
the $previous$ and $present$ dives. Obviously, $ eta sup exc $ remains
one until a deeper point than on the previous dive is reached. The
third bubble factor, $ eta sup rep $, accounts for bubble growth over
repetitive exposures on time scales, $ chi sup -1 $, of hours,
eta sup rep = 1 - left ( 1 - { G sup bub } over { G sub 0 exp ( - omega t
sub cum ) } right ) exp ( - chi t sub sur ) ,
10 <= chi sup -1 <= 120 minutes ,
0.05 <= {G sup bub } over {G sub 0 } <= 0.90 ,
according to the tissue compartment, with $t sub sur $ the repetitive surface
interval.
In terms of individual bubble factors, $ eta $, the multidiving fraction, $ xi
$,
is defined at the start of each segment, and deepest point of dive,
xi = eta sup reg eta sup rep eta sup exc
with surface and cumulative surface intervals appropriate to the preceeding
dive segment. Since $ eta $ are bounded by zero and one, $ xi $ are similarly
bounded by zero and one. Corresponding critical tensions, $M$, can be
computed from the above,
M = xi G + P ,
with $G$ listed in Table 8.3 above. Both $G$ and $ xi $ are lower bounded
by the shallow saturation data,
G <= G sup bound = .303 P + 11 ,
for $P$ ambient pressure, and similarly,
xi <= xi sup bound = { .12 + .18 exp (- lambda sub bound 480 ) } over {
.12 + .18 exp (- lambda sub bound tau ) } ,
lambda sub bound = .0559 min sup -1 .
A set of repetitive, multiday, and excitation factors, $ eta sup rep $,
$ eta sup reg $, and $ eta sup exc $, are drawn in Figures 8.5, 8.6, and 8.7,
using
conservative parameter values, $ chi sup -1 = 80 min $ and $ omega sup -1 = 7
days$.
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 $ chi sup -1 $ (bubble elimination halftime) and $ omega sup -1 $ (nuclei
regeneration halftime) will tend to decrease $ eta sup rep $
and increase $ eta sup reg $. Figure 8.5 plots $ eta sup rep $ as a function of
surface interval in minutes for the 2, 10, 40, 120, and 720 minute tissue
compartments, while Figure 8.6 depicts $ eta sup reg $ as a function of
cumulative
exposure in days for $ omega sup -1 $ = 7, 14, and 21 $days$. 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.
Excitation factors, $ eta sup exc $, are collected in Figure 8.7 for exposures
in the range
40-200 $fsw$. 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. Figure 8.7 depicts
$ eta sup exc $ for various combinations of depths, using 40, 80, 120, 160,
and 200 $fsw$ as the depth of the first dive.
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 VPM and RGBM.
Bubbles shrink or grow according to a simple radial diffusion equation
linking total gas tension, $p sub t $, ambient pressure, $P$, and surface
tension, $ gamma $, to bubble radius, $r$,
{ partial r } over { partial t } = { DS } over r left ( p sub t - P - { 2
gamma } over r right ) ,
with $D$ and $S$ the diffusivity and solubility. Bubbles grow when surrounding
gas tensions exceed the sum of ambient plus surface tension pressure, and vice
versa. Higher gas solubilities and gas diffusivities enhance the rate.
Related bubble area, $A$, and volume, $V$, changes satisfy,
{ partial A } over { partial t } = 8 pi r { partial r } over { partial t }
{ partial V } over { partial t } = 4 pi r sup 2 { partial r } over { partial
t } ,
The corresponding molar current, $J$, can be computed from Fick's law, assuming
an ideal gas,
J = - { DS } over { RTl } left ( p sub t - P - { 2 gamma } over r right )
,
with $R$ the universal gas constant, $T$ the temperature, and $l$ the effective
bubble thickness (diffusion barrier). The molar flow rate is just the molar
current times the interface area, that is,
{ partial n } over { partial t } = J A ,
for $n$ the number of gas moles, and $t$ the time. From the ideal gas law,
the change in pressure and volume of the bubble, due to gas diffusion, is
thus given by,
{ partial ( PV + 2 gamma r sup -1 V ) } over { partial t } = R { partial ( nT
) } over { partial t } ,
for $V$ the bubble volume. Obviously, the above constitute a coupled set,
solvable for a wide range of boundary and thermodynamic conditions connecting
$P$, $V$, $r$, $n$, and $T$.
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. As seen in Figure 8.8, the model has been coupled to
statistical likelihood, correlating bubble size with decompression risk, a
topic discussed in a few chapters. 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 representating
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.
FOLKS:
The following enclosure describes elements of the inherent unsaturation
(biological)
and oxygen window (diving related to inherent unsaturation). Let
me point out here that the oxygen window approach of taking up
the inherent unsaturation does not prevent bubble formation nor does
it eliminate later phase separation. Bubbles are an enigma all unto
themselves -- and must be coupled to dissolved phase transfer carefully.
The thermodynamic approach does not really take bubbles directly into
consideration. Similarly, the oxygen window and inherent unsaturation
were related to diffusion gradients across bubble interfaces in the above, as
well as their use in the original Hill's concept of phase equilibration
as a staging mechanism in the thermodynamic model.
Also, caution to those of you diving with the thermodynamic protocol.
Dropout at 30 $fsw$ (the tail of the Haldanian curve) can be hazardous
especially for shallow saturations -- recall that experiments (like Tektite)
in shallow saturation diving clearly indicate the need for decompression
for depths as shallow as 20 $fsw$.
Hans Roverud, please note that there is no fixed time for a stop in the
thermodynamic model (like the 3 $minutes$ you suggest in your letter to me),
but that $( p - P -B )$, the phase gradient must satisfy the thermodynamic
constraint on the separated gas fraction, $ chi $, under THERMODYNAMIC MODEL
above.
(ENCLOSURE)
GAS TRANSFER NETWORKS
The pulmonary and circulatory systems are connected gas transfer networks, as
Figure 6.1
suggests. Lung blood absorbs oxygen from inspired air in the alveoli (lung air
sacs), and releases carbon dioxide into the alveoli. The surface area for
exchange is enormous, on the order of a few hundred square meters. Nearly
constant values of alveolar partial pressures of oxygen and carbon dioxide
are maintained by the respiratory centers, with ventilated alveolar volume
near 4 $l$ in adults. The partial pressure of inspired oxygen is usually
higher than the partial pressure of tissue and blood oxygen, and the partial
pressure of inspired carbon dioxide less, balancing metabolic requirements
of the body.
Gas moves in direction of decreased concentration in any otherwise homogeneous
medium with uniform solubility. If there exist regions of varying solubility,
this is not necessarily true. For instance, in the body there are two
tissue types, one predominantly aqueous (watery) and the other (lipid),
varying in solubility by a factor of five for nitrogen. That is, nitrogen
is five times more soluble in lipid tissue than aqueous tissue. If aqueous
and lipid tissue are in nitrogen equilibrium, then a gaseous phses exists
in equilibrium with both. Both solutions are said to have a nitrogen tension
equal to the partial pressure of the nitrogen in the gaseous phase, with
the concentration of the dissolved gas in each species equal to the product
of the solubility times the tension according to Henry's law. If two nitrogen
solutions, one lipid and the other aqueous, are placed in contact, nitrogen
will diffuse towards the solution with decreased nitrogen tension. The
driving force for the transfer of any gas is the pressure gradient, whatever the
phases involved, liquid-to-liquid, gas-to-liquid, or gas-to-gas. Tensions and
partial
pressures have the same dimensions. The volume of gas that diffuses under any
gradient is a function of the interface area, solubility of the media, and
distance traversed. The rate at which a gas diffuses is inversely proportional
to the square root of its atomic weight. Following equalization, dissolved
volumes of gases depend upon their individual solubilities in the media.
Lipid and aqueous tissues in the body exhibit inert gas solubilities
differing by factors of roughly five, in addition to different uptake
and elimination rates. Near $standard$ temperature and pressure (32 $F sup o $,
and 1 $atm$), roughly 65% of dissolved nitrogen gas will reside in aqueous
tissues, and the remaining 35% in lipid tissues at equilibration, with the
total weight of dissolved nitrogen about .0035 $lb$ for a 150 $lb$ human.
The circulatory system, consisting of the heart, arteries, veins, and
lymphatics, convects blood throughout the body. Arterial blood leaves the
left heart via the aorta (2.5 $cm$), with successive branching of arteries
until it reaches arterioles (30 $microns$), and then systemic capillaries (8
$microns$)
in peripheral tissues. These capillaries join to form venules (20 $microns$),
which in turn connect with the vena cava (3 $cm$), which enters the right
heart. During return, venous blood velocities increase from 0.5 $cm/sec$ to
nearly 20 $cm/sec$. Blood leaves the right heart through the pulmonary
arteries on its way to the lungs. Following oxygenation in the lungs, blood
returns to the left heart through the pulmonary veins, beginning renewed
arterial circulation.
Blood has distinct components to accomplish many functions. Plasma is the
liquid part, carrying nutrients, dissolved gases (excepting oxygen), and
some chemicals, and makes up some 55% of blood by weight. Red blood cells
(erythrocytes) carry the other 45% by weight, and through the protein,
hemoglobin, transport oxygen to the tissues. Enzymes in red blood cells also
participate in a chemical reaction transforming carbon dioxide to a bicarbonate
in blood plasma. The average adult carries about 5 $l$ of blood, 30-35%
in the arterial circulation (pulmonary veins, left heart, and systemic
circulation),
and 60-65% in the venous flow (veins and right heart). About 9.5 $ml$ of
nitrogen are transported in each liter of blood. Arterial and venous tensions
of metabolic gases, such as oxygen and carbon dioxide differ, while blood and
tissue tensions of water vapor and nitrogen are the same. Oxygen tissue
tensions
are below both arterial and venous tensions, while carbon dioxide tissue
tensions exceed both. Arterial tensions equilibrate with alveolar (inspired
air) partial pressures in less than a minute. Such an arrangement of tensions
in the tissues and circulatory system provides the necessary pressure head
between alveolar capillaries of the lungs and systemic capillaries pervading
extracellular space.
Tissues and venous blood are typically unstaurated with respect to
inspired air and arterial tensions, somewhere in the vicinity of 8-13%
of ambient pressure. That is, summing up partial pressures of inspired gases
in air, total venous and tissue tensions fall short in that percentage range.
Carbon dioxide produced by metabolic processes is 25 times more
soluble than oxygen consumed, and hence exerts a lower partial pressure
by Henry's law. That tissue debt is called the biological $inherent$
$unsaturation$,
or $oxygen$ $window$, in diving applications
OXYGEN WINDOW
Inert gas transfer and coupled bubble growth are subtly influenced by
metabolic oxygen consumption. Consumption of oxygen and production of
carbon dioxide drops the tissue oxygen tension below its level in the lungs
(alveoli), while carbon dioxide tension rises only slightly because carbon
dioxide is 25 times more soluble than oxygen. Figure 6.2 compares the
partial pressures of oxygen, nitrogen, water vapor, and carbon dioxide
in dry air, alveolar air, arterial blood, venous blood, and tissue (cells).
Arterial and venous blood, and tissue, are clearly unsaturated with respect
to dry air at 1 $atm$. Water vapor content is constant, and carbon dioxide
variations are slight, though sufficient to establish an outgradient between
tissue and blood. Oxygen tensions in tissue and blood are considerably below
lung oxygen partial pressure, establishing the necessary ingradient for
oxygenation and metabolism. Experiments also suggest that the degree
of unsaturation increases linearily with pressure for constant composition
breathing mixture, and decreases linearily with mole fraction of inert gas
in the inspired mix. A rough measure of the inherent unsaturation, $ DELTA sub
u $,
is given as a function of ambient pressure, $P$, and mole fraction, $f sub { N
sub 2 } $,
of nitrogen in the air mixture, in $fsw$
DELTA sub u = ( 1 - f sub {N sub 2 } ) P - 2.04 f sub { N sub 2 } - 5.47 .
Since the tissues are unsaturated with respect to ambient pressure at
equilibrium,
one might exploit this $window$ in bringing divers to the surface. By
scheduling the ascent strategically, so that nitrogen (or any other inert
breathing gas) supersaturation just takes up this unsaturation, the
total tissue tension can be kept equal to ambient pressure. This approach
to staging is called the zero supersaturation ascent.
Bruce Wienke brw@la*.go*
Catch you all later.
Navigate by Author:
[Previous]
[Next]
[Author Search Index]
Navigate by Subject:
[Previous]
[Next]
[Subject Search Index]
[Send Reply] [Send Message with New Topic]
[Search Selection] [Mailing List Home] [Home]