0010.482519255.00+ .a, @ 1992 Pergamon Press Ltd

Cornput. Biol. Med. Vol. 22, No. 6. pp. 389-406, 1992 Printed in Great Britain

NUMERICAL PHASE ALGORITHM FOR DECOMPRESSION COMPUTERS AND APPLICATION B. R. WIENKE Advanced Computing Laboratory, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. (Received 30 January 1992; in revised form 19 June 1992; received for publication 3 July 1992)

Abstract-Present generation decompression computers employ a simplified algorithm, limiting dissolved gas build-up in tissue and blood according to a method proposed by Haldane 80 years ago. Such a model works well for single dives, but is usually liberal and theoretically incomplete for multiple exposures within 24 hr spans. Using the critical phase hypothesis in a bubble model, we have extended the classical model of Haldane to multi-exposures. This model is discussed, and a decomputer algorithm described for multi-diving. The focus is permissible bubble excess, not just dissolved gas per se, with phase constraints affecting all tissues, fast and slow, and requiring a systematic lowering of repetitive tissue tensions. Deep repetitive and shallow multiday exposures are impacted most by the procedure. Within nucleation theory deeper-than-first dives are also treated. A set of multi-diving fractions, 5, accounting for micronuclei excitation and regeneration, reduced bubble elimination in repetitive activity, and coupled effects on tissue tension, are proposed, with 5 representing a set of multiplicative factors (less than one) applied to critical tissue tensions for multi-exposures. These factors affect repetitive activity over short time spans, deeper-than-previous and continuous multi-day activities, compared to standard computer software, and are easily encoded into existing decompression meters, potentially extending their range and flexibility over exposure regimes. Decompression meters Multi-diving software

Nucleation and bubbles

Separated phase algorithm

INTRODUCTION On the heels of growing interest in underwater science and exploration following World War II, monitoring devices have been constructed to control diver exposure and decompression procedures. Devices, with records of varying success, include mechanical and electrical analogs, and within the past 15 years, microprocessor-based digital computers. With inexpensive microprocessor technology, recent years have witnessed explosive growth in compact digital meters used to control diving activity in the commercial, sport, military, and scientific sectors. All use a simple dissolved tissue gas model proposed by Haldane [l] some 80 years ago. Given the sophistication of many of these underwater decompression meters (decomputers), many feel that broader models can be incorporated into meter function today, increasing their range and flexibility. Although the biophysics of bubble formation, free and dissolved phase build-up and elimination is formidable, and not fully understood yet, contemporary models treating both dissolved and free phases, correlated with existing data, and consistent with diving protocols might extend the utility of diving computers. One such model and its implementation are the focus here. Generically in the industry, such new models are termed bubble mechanical, because they focus on bubbles and their interactions with dissolved gas in tissue and blood. And the combination of a model, set of equations, limit or trigger points, and flow of table or meter implementation is also termed an algorithm. Biophysical model, trigger points, constitutive parameters, meter flow, and application follow directly, while defining equations are summarized in the Appendix; in short, we detail the phase algorithm. By convention, 389

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B. R. WIENKE

and for convenience, pressures and depths are measured in the same units, that is, feet-of-sea-water (fsw), with 1 atm = 33 fsw. MOTIVATION

AND

OPERATIONAL

PROCEDURES

The past ten years, or so, have witnessed a number of changes and additions [l-29] to diving protocols and table procedures, such as shorter non-stop time limits, slower ascent rates, discretionary safety stops, ascending repetitive profiles, multi-level techniques, both faster and slower controlling repetitive tissue half-times, lower critical tensions (Mvalues), longer flying-after-diving surface intervals, and others. Stimulated by the Doppler technology, decompression meter development, theory, statistics, or safer diving concensus, these modifications affect a gamut of activity, spanning bounce to multi-day diving. As it turns out, there is good support for these protocols on operational, experimental, and theoretical grounds, and a comprehensive model addressing these concerns on firmer basis than earlier models is desirable. Developments

Spencer [4] pioneered the use of Doppler bubble counting to suggest reductions in the non-stop time limits of the standard U.S. Navy tables, on the order of a repetitive group or two at each depth in the tables (l-4 fsw in critical tensions), basing recommendations on lowering bubble counts at shorter non-stop time limits. Others have also made similar recommendations over the past 15 years. Smith and Stayton [30] noted marked reductions in precordial bubbles when ascent rates were cut from 60 fsw/min to 30 fsw/min. In similar studies, Pilmanis [14] witnessed an order of magnitude drop in venous gas emboli (VGE) counts in divers making short, shallow, safety stops following nominal bounce exposures at the 100 fsw level, while Neumann et al. [31] recorded comparable reductions in divers making short, but deeper, stops after excursions to 200 fsw for longer periods of time. An American Academy of Underwater Sciences (AAUS) workshop on repetitive diving, recorded by Lang and Vann [32], and Divers Alert Network (DAN) statistics [13] suggest that present diving practices appear riskier under increasing exposure time and pressure loading, spawning development of ancillary safety measures for multi-diving. Dunford et al. [33] noted persistent Doppler scores in divers performing repetitive, multi-day diving, suggesting the presence of VGE in divers, all the time, under such loadings. Ascent rates, safety stops, decompression computers, and altitude diving were also the subject of extensive discussion at workshops and technical forums sponsored by the American Academy of Underwater Sciences and the Undersea and Hyperbaric Medical Society (UHMS), as summarized by Lang and Hamilton [ 111, Lang and Egstrom [12], and Sheffield [34]. Results of discussions culminated in a set of recommendations, folded with standard Haldane [l] table and meter procedures, even for exposures exceeding neither time limits nor critical tissue tensions. Discretionary protocols

The upshot of these studies, workshops, discussions, and tests are a set of discretionary protocols, not necessarily endorsed in all diving sectors, but which might be summarized as follows: (1) reduce non-stop time limits a repetitive group, or two, below the standard U.S. Navy limits; (2) maintain ascent rates below 60 fsw/min, preferably slower, and requisitely slower at altitude; (3) limit repetitive dives to a maximum of three per day, not exceeding the 100 fsw level; (4) avoid multi-day, multi-level, or repetitive dives to increasing depths;

Numerical phase algorithm for decompression computers and application (5)

(6) (7) (8) (9) (10)

391

12 hr before flying after nominal diving, 24 hr after heavy diving (taxing, near decompression, or prolonged repetitive) activity, and 48 hr after decompression diving; avoid multiple surface ascents and short repetitive dives (spikes) within surface intervals of 1 hr; surface intervals of more than 1 hr are recommended for repetitive diving; safety stops for 2-4 min in the lo-20 fsw zone are advisable for all diving, but particularly for deep (near 100 fsw), repetitive, and multi-day exposures; do not dive at altitudes above 10,000 ft using modified conventional tables, or linear extrapolations of sea-level critical tensions; in short, dive conservatively, remembering that tables and meters are not bendsproof. wait

Procedures such as those above are prudent, theoretically sound, and safe diving protocols. Ultimately, they can all be linked to free phase and bubble models. BIOPHYSICAL

MODEL

Validation is central to diving, and significant testing [2-171 of non-stop and saturation diving schedules has transpired. In between, repetitive (more than one dive in a 12 hr period), multi-level (arbitrary depths throughout the course of a single dive), deeperthan-previous (second repetitive dive deeper than first), and multi-day (repetitive dives over days) diving cannot claim the same benefits, though some ongoing programs [8,9] are breaking new ground. Application of just dissolved gas models in latter cases possibly has witnessed slightly higher decompression sickness (bends) incidence than in the former ones, as discussed in newsletters [13], workshops [ll, 121, and technical forums. Some hyperbaric specialists also suggest higher incidence of rash (skin bends) under repetitive loading. While statistics are not yet conclusive, they raise some concerns theoretically addressed by considering both dissolved and free phase gas build-up and elimination in broader based bubble models. Such models often focus on the amount of free phase precipitated by compression-decompression, and contain dissolved gas models as subset. In limiting the amount (volume) of free phase in time, they must also limit the growth rate. Certainly bubble growth [l&25] can be partially addressed through reduced tissue supersaturation, or effectively, tissue tensions, particularly as they drive bubble excitation and growth beyond permissible levels. We consider permissible bubble excesses and tissue tensions within critical phase hypotheses [19]. A repetitive criterion, underscoring reduction in tissue tension, can be systematically developed. Need for reductions arises because of lessened degree of bubble elimination over repetitive intervals, compared to long intervals, and the need to reduce bubble inflation rate through smaller driving gradients. Deep repetitive and spike exposures feel the greatest effects of gradient reduction, but shallower multi-day activities are impacted. Bounce diving enjoys long surface intevals to eliminate bubbles within the critical phase hypothesis, while repetitive diving must contend with shorter intervals, and thus reduced time for bubble elimination. Theoretically, a reduction in the bubble inflation driving term, namely, the tissue gradient or tension, holds the inflation rate down. The concern is bubble excess driven by dissolved gas, and such a model is called the reduced gradient bubble model [26]. Gas dynamics

Inert gas exchange is driven by the local gradient, the difference between the arterial blood tension and the instantaneous tissue tension, as described in the Appendix. Such behavior is modeled in time by mathematical classes of exponential response functions, bounded by arterial blood and initial tissue tensions. These multi-tissue functions are well known in Haldane application, tracking both dissolved gas build-up and elimination symmetrically in hypothetical tissue compartments.

392

B. R.

WIENKE

5 min

01

0

1

I

I

1

I

I

!

I

15

30

45

60

75

90

105

120

absolute pressure --

fsw

Fig. 1. Critical tensions. Critical tensions, M, are fitted linear functions of absolute pressure, P, as depected for various tissue compartments (5, 10, 20, 40, and 120 min). Faster compartments permit greater amounts of dissolved nitrogen, slower compartments less. During a dive, tensions in compartments must stay below the curves above in the Haldane approach. At sea level, they can be reduced to the approximate form, M= 193.35-“4+4.11dr-“4> for depth, d (fsw), and T the tissue compartment

half-life (min).

Tissue compartments with 1, 2,5, 10, 20,40, 80, 120,240,480, and 720 min half-lives, r, are a realistic spectrum, according to inert gas washout experiments, and are independent of pressure. The initial and arterial tensions represent extremes for each stage, more precisely, the initial tension of the present and the arterial tension at the beginning of the next stage. Classical models limit exposures by requiring that tissue tensions never exceed maximum values (denoted M), fitted [20] to the U.S. Navy decompression data as seen in Fig. 1.

-

22

-

P

r

Fig. 2. Bubble pressure balance. The total gas pressure within an air bubble equals the sum of ambient pressure, P, plus effective surface tension pressure, 2ylr, according to, P+T=

PO,+ PNz+ PH*O+ PC@&,

with y the surface tension, r the bubble radius, and Pi partial pressures of constituent gases (j= oxygen, nitrogen, water, or, carbon dioxide). At small radii, surface tension pressures are large, while at large radii surface tension pressures are smaller. Effective surface tension is the difference between Laplacian (thin film) tension and skin (surfactant) tension. Stabilized nuclei exhibit zero effective surface tension, so that total gas pressures and surrounding tissue tensions are equal. When nuclei are destabilized (bubbles), any gradients between free and dissolved gas phases will drive the system to different configurations, that is, expansion or contraction until a new equilibrium is established.

Numerical phase algorithm for decompression

computers and application

393

c-

Fig. 3. Bubble gas diffusion. A bubble in hydrostatic equilibrium will grow or contract according to its size and any relative gradients between free gas in the bubble and dissolved gas in surrounding tissue. Gradients are inward if tensions exceed bubble gas pressures, and outward if free gas pressures exceed tensions. A critical radius, r,, separates growing from contracting bubbles for any given set of pressures and tensions. The critical radius depends on the total tissue, ambient pressure, P, and effective surface tension, y. 2Y rc=Po*+PN*+PHZO+PCOZ-P~

with p, the tissue tension of the gas species (j = oxygen, nitrogen, water, and carbon dioxide). Bubbles with radius r > r, will grow, while bubbles with radius r < rc will contract for fixed y. Some stabilized gas micronuclei in the body can always be excited into growth, or contraction, by pressure changes (compression-decompression), and structural changes in effective surface tension, y, in the same processes, further affecting behavior.

Bubbles, which are unstable, might grow from instantaneously stable, micron size, gas nuclei which resist collapse due to permeable skins [23] of surface-activated molecules (surfactants), or possibly reduction in surface tension [16] at tissue interfaces. Figures 2 and 3 portray the interplay between surface tension, tissue tension, ambient pressure, internal bubble pressure (always greater than ambient because of surface tension), diffusion gradients, composition, and bubble radius for micronuclei destablized into growth, or contraction, by changes in pressure, surface tension, and combinations of both. For modeling purposes, it is’not important whether performed micronuclei exist for minutes or months, just their presence is assumed. In uiuo studies of these micronuclei have been inconclusive, but their presence has been demonstrated in blood serum and egg albumin [21,22]. If families of these micronuclei persist, they vary in size, surfactant content, tissue location, effective surface tension on excitation to growth, and number density. Micronuclei are probably small enough to pass through the pulmonary filters, yet dense enough not to float to the surfaces of their environments, with which they are in both hydrostatic (pressure) and diffusion (gas flow) equilibrium. Compression-decompression is thought to excite them into growth, by compromising skin integrity. Such a model Table 1. Bounce time limits Depth

Time limit

Depth

d (fsw)

(min)

30 40 50 60 70 80 90

250.0 130.0 73.0 52.0 39.0 27.0 22.0 18.0 15.0 12.0

100 110 120

CBH22-6-B

-

Time limit

d

t

(fsw)

(min)

130 140 150 160 170 180 190 200 210 220

9.0 80 7.0 6.5 5.8 53 4.6 4.1 37 31

394

B. R. 320

I

I

WIENKE I

I

I 7

260 240 % 3

200

E M

160

7d j”

120

‘C 0

60 40 0 0.3

0.4

0.5

0.6

excitation radius --

0.6

0.7

0.9

microns

Fig. 4. Critical gradients. Permissible gradients, G, are plotted as a function of excitation radius, r, for various tissue compartments (2, 10, 40, 120, 720 min), and obviously vary implictly with depth, d. The deeper the dive, the smaller the excitation radius, and the faster the tissue compartment which controls the exposure. The corresponding critical tension, M, in the dissolved gas Haldane model is given by, M=G+P,

at absolute pressure, P.

of skin behavior, called the varying permeability model (VPM), was proposed by Yount [23], Strauss [X3], and co-workers, and extended in the reduced gradient bubble model (RGBM) by Wienke [26,27]. Measured nuclear radii are on the order of 1 micron or less, and their number density in bio-media decreases exponentially with increasing radius. Critical supersaturation gradients and phase volume

Any set of non-stop time limits can be plugged into tissue equations, and ensuing maximum tensions across all compartments and depths assigned as the critical tensions, M. Corresponding critical gradients, G, are obtained by subtracting off ambient pressure, P. This is the standard way to extract M, or G, from the non-stop limits. However, our approach, as described in the Appendix, is to employ the phase volume constraint consistently to extract the critical gradients, G, from a set of non-stop time limits slightly more conservative than those of Spencer [4]. Our non-stop limits, t, at depth, d, satisfy the bulk diffusion law, dt”* = 400 fsw min1’2(465 fsw min1’2is the Spencer constant). And it is interesting that the phase volume constraint, non-stop time limits, bulk diffusion law, and resulting set of G bootstrap each other self-consistently to yield a Table 2. Operational gradients Half-life t (min) 2 5 10 20 40 80 120 240 480

Threshold depth 6 (fsw) 190 135 95 65 40 30 28

Surface gradient G0 (fsw)

Gradient change AC

151.0 95.0 67.0 49.0 36.0 27.0 24.0 23.0 22.0

0.518 0.515 0.511 0.506 0.468 0.417 0.379 0.329 0.312

Numerical phase algorithm for decompression

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computers and application

1.0 0.9

LEGE:ND 2 min ..,., K. min.~. ____4UE!_~..... _!.g.g -_---

0.0 0.7 0.6 0.5 0.4 0.3 0.2 I

/

1

I

I

40

00

120

160

200

0.1

surface

interval

--

minutes

Fig. 5. Repetitive reduction factors. Within the phase volume constraint, bubble elimination periods are shortened over repetitive diving, compared to bounce diving. Therefore, a gradient reduction factor, qrcp, proportional to the difference between maximum and surface bubble inflation rate, is employed to maintain the separated phase volume below a limit point deduced from data in the VPM and RGBM. Repetitive fractions are plotted for variouis tissue compartments (2, 10, 40, 120, 720min) for surface intervals up to 200min. Faster compartments are impacted the most, but a11fractions relax to one after a few hours.

set of critical tensions, M, close to the standard Spencer set employed today in Haldane (dissolved gas) models [20]. Because the time limits used here are conservative, extracted critical parameters are conservative. The non-stop time limits are given in Table 1. Figure 4 plots G as a function of minimum bubble excitation radius, r, for the 2, 10,40, 120, and 720 min tissues, with surface gradient, Go, extracted at r = r,, = 0.8 microns. The surfacing gradient, GO, allows direct surface ascent. Compartment sea level parameters, Go and AG, with G = Go + AGD at depth, d, are listed in Table 2, alongside the threshold depth, 6 at which the compartment begins to control exposure.

E

0.8

32

0.7

b

0.6

B -T

0.5

2

0.4 0.3

2

4

cumulative

6

a

interval

10

--

12

14

days

Fig. 6. Regeneration reduction factors. Micronuclei are thought to regenerate over adaptation time scales of many days, contributing to existing pools of gas seeds not yet eliminated in diving activity. A factor, q”s, accounting for creation of new micronuclei, reduces permissible gradients by the creation rate, thus maintaining the phase volume constraint over multi-day diving. Multi-day fractions are plotted for 7, 14 and 21 day regeneration times. Shorter regeneration times impart greater multi-day penalties.

396

B. R. WIENKE

0.4

0.3 0

40

120

60

successive

depth

--

160

200

fsw

Fig. 7. Deeper-than-previous diving activity stimulates smaller bubble seeds into growth according to the VPM and RGBM. Scaling gradients by the bubble excess on the deepest point of the previous dive to the bubble excess on the present dive, t,r’“, also maintains the phase volume constraint for multi-exposures. Excitation fractions are plotted for a series of deeper exposures, following initial dives to 40, 80, 120, and 160 fsw. Shallow initial excursions, followed by deep dives, induce the largest reductions in permissible gradients on deeper-than-previous next dives.

We know from Doppler measurements of moving bubbles in the pulmonary arteries that free gas phases exist in the body following decompressions. After a time, Doppler bubble counts drop, suggesting that bubbles are safely eliminated by the body, though slower than dissolved phases in the same tissues. Implicit in a bubble model approach, then, is the viable assumption that free gas (bubbles) continuously leaves the body. A permissible bubble excess, called An, represents the difference between the actual bubble number and the amount safely eliminated by the body. In satisfying the critical phase volume constraint over time, the integral of the product of the bubble excess and supersaturation gradient must remain below a critical value over all time. Yount and Hoffman [21] correlated data for bounce and saturation exposures with the approach, and Wienke [26] extended the approach to repetitive diving. Multi-diving gradients and fractions

The repetitive criterion acts as a constraint on multi-dives, with repetitive growth rate held less than the bounce growth rate. Reduction in growth rate is effected by a reduction in permissible tissue tension, systematically determined from a set of bubble multipliers, 6, defined at the outset of each dive segment, with 0 I Lj5 1. These factors, in multiplying the bounce set, G, impart shorter non-stop repetitive time limit, a penalty in Table 3. Condensed repetitive reduction factors (n’“‘) Surface interval between successive dives within 8 hr (ioT:22”6 1”4;_4:;2 Beyond 480 (min) (min) (min) (min)

(;z;;) Half-life (min) 2 5 10 20 40 80 120 240 480

0.80

0.90

0.95

0.97

1.0

0.87

0.94

0.96

0.98

1.0

0.96

0.98

0.99

1.0

0.97

0.99

1.0

1.0

Numerical phase algorithm for decompression

397

computers and application

Table 4. Condensed regeneration reduction factors (r,P) Half-life r (min)

2-3 (Days)

2-480

0.96

Consecutive days of diving without 24 hr surface interval Beyond 14 4-5 (Days) (&is) (D”$) ti$$ (ky!) (Days) 0.92

0.88

0.84

0.86

0.82

0.81

effect, decreasing with e obviously. The multi-diving set of critical gradients is denoted G, and relates to the bounce set, G, simply by, G = (G. Of course, knowing dive profiles in advance, we could iterate model equations over time, until a worthy set of G obtained self-consistently, satisfying all constraints, and that would be an optimal set. In unplanned (free style) repetitive diving, we cannot perform this exercise, so we must then conservatively estimate Zjahead of time, based on previous exposure history, surface intervals, non-stop limits, and permissible tensions for bounce exposures. As surface time intervals decrease, appropriate ,$ should get smaller, and staging approach saturation limits as repetitive frequency increases. As surface time intervals increase, 5 should get larger, and staging approach bounce limits as surface interval increases. In between, behavior depends on total elapsed time, total surface interval, tissue compartment, and profile. Considering interpolating behavior, a checklist of properties of 5, correlating with operational diving practice, is worthwhile;

(1) 5 equals

one for a bounce dive, remaining less than one for repetitive dives within some time interval; (21 6 decrease monotonically with increasing exposure time; (3) 5 increase monotonically with increasing surface interval time; (4) 5 scale faster tissue compartments the most; (5) l decrease with depth of dive segment; dives the most; (6) 5 scale deeper-than-previous 6 change with every dive segment, but only within any dive segment when a (7) greater depth is reached; (8’) 5 decrease with micronuclei regeneration; (9) time constants controlling 5 are linked to bubble elimination time scales, x-l, and micronuclear regeneration time scales, w-l. In a bubble model, three factors need consideration, namely, excitation on deeperthan-previous dives, repetitive bubble removal rate, and micronuclei regeneration. Each suggests a factor, 7, detailed in the Appendix with 5 the product of the three 7, used to downscale phase volume growth rate under repetitive loading, through the bubble gradient, G. Consistent with recent workshops, reports, industry consensus, and flyingafter-diving recommendations, reduction factors, 77and 5. relax to 1 following any 24 hr surface interval of non-diving. Tissue tensions and bubble excesses would tend to equilibrate with ambient pressure on those time scales. In application, penalty for multiTable 5. Condensed excitation reduction factors (q’“‘)

(fsw)

Surface interval between present deeper-than-shallowest dive within 16 hr O-80 80-240 240-480 480-960 Beyond 960 (min) (mm) (min) (min) (min)

40-80

0.80

0.90

0.95

0.98

1.0

g0:;;; (52 10) 120- 160 (tr5)

0.77

0.87

0.94

0.97

1.0

0.73

0.86

0.93

0.97

1.0

0.70

0.85

0.92

0.96

1.0

Depth difference AP

Beyond 160 (tr2)

398

B. R. WIENKE

Display ofmvestatus Fig. 8. Decompression

meter schematic. Decompression

microprocessors consist of a pressure chip with ROM and RAM, power source, and pixel display screen.

transducer, internal clock, analog to digital signal converter,

diving results in systematic reductions in non-stop time limits, depending times, and repetitive frequency, as determined by 6. MODEL

on depths,

IMPLEMENTATION

Standard microprocessor technology is appropriate for meter implementation of the model equations and fundamental parameters. Tabular parameters include non-stop time limits, tissue compartments, critical gradients, and nuclei regeneration and elimination time scales. Other parameters are related to this set, assuming an exponentially decreasing number density in nuclei radius. All relations are detailed in the Appendix. Parameters

Model non-stop time limits are indicated in Table 1, certainly a conservative set, consistent with the present tendency to reduce non-stop limits. Values are close to both the Buhlmann [2] and Spencer [4] non-stop time limits. Corresponding gradients, G, are tabulated in Table 2, with G = Go + AGd at depth d. Compartments begin to control bounce exposures at the threshold depths, 6, listed. A set of repetitive, multi-day, and excitation factors, prep, qreg, and vex’, are drawn in Figs 5-7 for conservative parameter values, x-l = 80 min and 0-l = 7 days. Clearly, the repetitive factors, prep, relax to one after about 2 hr, while the multi-day factors, l;lreg, Table 6. Multi-diving fractions (120/10, 01120, 120/10 for 3 days) t(min) 2 5 10 20 40 80 120 240 480

6,

6

63

&

&

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.95 0.95 0.95 0.95 0.96 0.96 0.97 0.98 1.00

0.93 0.93 0.93 0.93 0.94 0.94 0.95 0.97 1.00

0.88 0.88 0.89 0.89 0.89 0.90 0.92 0.96 1.00

0.86 0.86 0.86 0.87 0.87 0.89 0.91 0.95 1.00

h

0.82 0.83 0.84 0.85 0.86 0.88 0.90 0.94 1.00

399

Numerical phase algorithm for decompression computers and application Table 7. Non-stop

time limit comparison for 3 day (hazardous) (12000, o/120, 120/10) Day 1 dive 1 dive 2 (min) (min)

Decomputer Orca Delphi Orca Skinny Dipper Beuchat Aladin SuuntolSea Quest SME-ML Dacor MicroBrain ProPlus Sherwood Source ScubaPro DC1 1 RGBM

10 10 8 10 8 12 6 11

repetitive

Day 2 dive 4 dive 3 (min) (mm) 10 10 8 10 8 l? 6 8

10 10 8 10 8 9 6 10

schedule

Day 3 dive 5 dive 6 (mm) (min) 10 10 8 10 8 12 6 6

10 10 8 10 8 9 6 7

10 10 8 10 8 9 6 6

continue to decrease with increasing repetitive activity, though at very slow rate. Increases in x-l (bubble elimination half-time) and o -’ (nuclei regeneration half-time) will tend to decrease prep and increase qTeg.Figure 5 plots prep as a function of surface interval in minutes for the 2, 10,40, 120, and 720 min tissue compartments, while Fig. 6 depicts qreg as a function of cumulative exposure in days for w -’ = 7, 14 and 21 days. The repetitive fractions, prep, restrict back to back repetitive activity considerably for short surface intervals. The multi-day fractions get small as multi-day activities increase continuously beyond 2 weeks. Excitation factors, vex’, are collected in Fig. 7 for exposures in the range 40-200 fsw. Deeper-than-previous excursions incur the greatest reductions in permissible gradients (smallest p”“‘) as the depth of the exposure exceeds previous maximum depth. Figure 7 depicts 7”’ for various combinations of depths, using 40, 80, 120, 160 and 200 fsw as the depth of the first dive. A less conservative, and time-averaged, compilation of q, relaxing to 1 within the surface intervals noted, and directly amenable to tabular meter implementation, appears in Tables 3, 4, and 5. In these cases, x-l = 20 min and 0-l = 21 days. Meter configuration As schematized in Fig. 8, a microprocessor decompression computer consists of a power source, pressure transducer, analog to digital signal converter, internal clock, microprocesor chip with RAM (random access memory) and ROM (read only memory), and pixel display screen. Some 3-9 V is sufficient power source to drive the computer for a couple of years, assuming about 100 dives per year. The pressure transducer records ambient pressure, and sends a voltage reading to the analog to digital converter. Converting the voltage to a depth, the analog to digital converter sends the signal to the microprocessor chip. The ROM contains the program (step application of equations), all constants, and queries the

Table 8. Non-stop time limit comparison for single day (hazardous) repetitive schedule (147/S, 0160, 147/S, 0160, 14715)

Decomputer Orca Delphi Orca Skinny Dipper Beauchat Aladin SuuntolSea Quest SME-ML Dacor MicroBrain ProPlus Sherwood Source ScubaPro DC1 1 RGBM

Day 1 dive 1 (min)

dive 2 (min)

dive 3 (min)

400

B. R. WIENKE Table 9. Comparative non-stop time limits Depth d (fsw) 30 40 50 60 70 80 90 100 110 120 130

Workman t (min)

Spencer t (min)

200 100 60 50 40 30 25 20 15 10

225 135 75 50 40 30 25 20 15 10 5

Buhlmann t (min) 290 125 75 54 38 26 22 20 17 15 11

Wienke t (min) 250 130 73 52 39 27 22 18 15 12 9

transducer and clock, while the RAM maintains storage registers for all dive calculations, which are ultimately sent to the display sceen. Calculations are updated every 25 set, approximately. Their are now about 25 dive computers commercially marketed, all employing the Haldane algorithm. They are limited for the reasons discussed, and the phase algorithm described can be encoded within most units. APPLICATION

AND

COMPARISONS

For illustration, we apply the algorithm to both repetitive and bounce diving, and contrast effects. Results and trends are representative of a broad class of algorithmic impacts on no-decompression diving regimens. Multi-day, repetitive diving At first application, consider two repetitive dives a day, 120 fsw for 10 min, separated by a 2 hr surface interval, over three consecutive days. This profile, extended to three repetitive dives a day, has produced decompression sickness in three out of four cases [28], so it is interesting. Employing a bubble (tissue) surface tension of 8.3 fsw, a conservative bubble mechanical factor, taking w-l= 14 days and x-l =40min, the algorithm reduces the permissible gradients in each tissue compartment, on each segment of the six dives, according to Table 6, which lists t at the start of each multi-day, repetitive segment. Reductions in gradients approach 20% in the fast compartments, and 15% in the slower ones, on the last dive. On the first day, reductions in the fast compartments are near 5% on the second dive, and near 10% on the second dive of the second day. Smaller reductions, by a few per cent, are seen in the slow compartments. Exposures in the 120 fsw range are controlled by the 10 min compartment, with 11 min the non-stop limit on the first dive (c = 1) from Fig. 5. By the sixth dive, the second dive of the third day (6=0.82), that non-stop time limit drops to 7 mn. Heavy, multi-day, repetitive diving are penalized the most, and if deeper-than-previous exposures are attempted, additional restrictions are also imposed. Table 10. Comparative surfacing critical tensions (MO) Half-time r (min) 5 10 20 40 80 120

Workman MO(fsw) 104 88 72 58 52 51

Spencer MO(fsw) 100 84 68 53 51 49

Buhlmann & (fsw) 102 82 65 56 50 48

Wienke MI (fsw) loo-70 81-60 67-57 57-49 51-46 48-45

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Table 7 contrasts various commercial computer non-stop limits at each segment of the same multi-day profile against the RGBM. Application of 5 results in a decreasing sequence of non-stop time limits. Non-stop time limits were obtained following pressure chamber tests of commercial computers. A somewhat deeper repetitive profile, three 147 fsw exposures for 5 min with 1 hr surface intervals, is also hazardous according to Edmonds [ll]. Decomputer performance is contrasted with the RGBM in Table 8. Again note systematic reductions in the non-stop limits for repetitive diving within the RGMB. Numbers are rounded of course, but depend on surface intervals. Present computers offer varying limits, but the same limits for each repetitive segment. Bounce diving

For comparison, Table 9 lists non-stop limits according to the classical Workman [3], and more recent Spencer [4], Buhlmann [2], and Wienke [26] algorithms. Further reduction in time limits would seem to play off optimality against safety. Limits much below the Spencer, Buhlmann, and Wienke times would restrict repetitive diving, but at the expense of bounce diving. Statistics compiled by Gillian [29] suggest that divers using conservative time limits (Buhlmann) have compiled an enviable track record, an incidence of decompression sickness below 0.01% in combined table and meter usage, and many regard such an incidence rate as acceptable. A more natural way to restrict repetitive and multi-day diving, than reducing bounce time limits, is suggested by the RGBM, employing the critical phase volume trigger point, as described, whereby requisite reductions in supersaturation gradients translate to systematic reductions in permissible tensions on sucessive dives, but do not restrict non-stop time limits on bounce dives. For simple operational comparison, Table 10 lists corresponding maximum (critical) surfacing tensions (M,) in the Workman, Spencer, Buhlmann, and Wienke algorithms, with the first three fixed, Haldane model values and the last variable, bubble model (RGBM) limits. Note in Table 10 that critical tensions in the latter three algorithms are smaller, by some l-4fsw, compared to the Workman (U.S. Navy) values, effectively shortening the non-stop time limits a group, or two, within U.S. Navy tables. SUMMARY Repetitive, deeper-than-previous, multi-day, and multi-level diving potentially present problems for the Haldane model which might be lessened in impact by a systematic reduction in critical gradients, or tensions, consistent with bubble mechanics and the phase volume limit. Reductions are based on possible excitation and regeneration of micronuclei, and bubble inflation rates, and not only dissolved gas build-up. A model, called the reduced gradient bubble model, described and applied to marginal repetitive and multi-day profiles, requires systematic reductions in gradients and tensions, and hence non-stop time limits, across individual dive segments. Overall, the approach is conservative. Bounce limits are shorter than the U.S. Navy limits, and are close to the Buhlmann set. The corresponding critical tensions, M,,, are also conservative when applied to repetitive diving, and the fixed set of critical tensions experience reductions over repetitive, multi-day, and deeper-than-previous activity, in a manner consistent with both free and dissolved gas dynamics. Parameters in the model can be correlated with any multi-exposure data set. Parameters employed herein correlate with conservative non-stop time limits and the shallow saturation schedules, as mentioned. In between, the Edmonds, Leitch and Barnard hazardous repetitive profiles can be used to delimit remaining model parameters. Acknowledgements-We thank colleagues and friends for their help and advice in this analysis, including Tom Kunkle (LANL) and David Yount (University of Hawaii) for discussions of bubble experiments, in vivo and in vitro, Doug Toth, Jim Dexter, and Richard Bonin (SCUBAPRO Industries) for meter implementation studies,

402

B.

R.

WIENKE

involving both hardware and software. Special thanks to Charles Lehner and Ed Lanphier (University of Wisconsin) for updates on the etiology of decompression sickness, and access to their BIOTRON data on central nervous system (CNS) bends incidence in goats experiencing deep decompressions, in profiles ranging bounce to saturation.

REFERENCES 1. A. E. Boycott, G. C. C. Damant and J. S. Haldane, The prevention of compressed-air illness, 1. Hygiene 8.342-443 (1908). Sickness. Springer, Berlin (1984). 2. A. A. Buhlmann, DecompressionlDecompression 3. R. D. Workman, Calculation of decompression schedules for nitrogen-oxygen and helium-oxygen dives. USN Experimental Diving Unit Research Report, NEDU 6-65, Washington (1965). 4. M. P. Spencer, Decompression limits for compressed air determined by ultrasonically detected blood bubbles, J. Appl. Physiol. 40, 229-235 (1976). 5. H. R. Schreiner and R. W. Hamilton, Validation of decompression tables. Undersea and Hyperbaric Medical Society Publication 74 (VAL), Bethesda (1987). 6. A. R. Behnke, The application of measurements of nitrogen elimination to the problem of decompressing divers, USN Med. Bull. 35, 219-240 (1937). 7. T. D. Kunkle and E. L. Beckman, Bubble dissolution physics and the treatment of decompression sickness, Med. Phys. 10, 184-190 (1983). 8. E. D. Thalmann, Phase II testing of decompression algorithms for use in the US navy underwater decompression computer. USN Experimental Diving Unit Report, NEDU l-84, Panama City (1984). 9. E. D. Thalmann, Air-N,O, decompression computer development. USN Experimental Diving Unit Report, NEDU 8-85, Panama City (1986). 10. F. P. Farm, E. M. Hayashi and E. L. Beckman, Diving and decompression sickness treatment practices of Hawaii Sea Grant Report UNIHIamong Hawaii’s diving fisherman. University SEAGRANT-TP-86-01, Honolulu (1986). 11. M. A. Lang and R. W. Hamilton, @rot. Am. Acad. Underwater Sci. Dive Comput. Workshop, University of Southern California Sea Grant Publication, USCSG-TR-01-89, Los Angeles (1989). 12. M. A. Lang and G. H. Egstrom, Proc. Am. Acad. Underwater Sci. Biomech. Safe Ascents Workshop. Diving Safety Publication AAUSDSP-BSA-01-90, Costa Mesa (1990). 13. R. D. Vann, J. Dovenbarger, C. Wachholz and P. B. Bennett, Decompression sickness in dive computer and table use, DAN Newsletter 3-6 (1989). 14. D. N. Walder, Adaptation to decompression sickness in caisson work, Biometeor. 11, 350-359 (1968). 15. A. A. Pilmanis, Intravenous gas emboli in mn after compressed air ocean diving, Office Of Naval Research Contract Report, NOOO14-67-A-0269-0026,Washington (1976). 16. B. A. Hills, Decompression Sickness. John Wiley, New York (1977). 17. H. V. Hempleman, Further basic facts on decompression sickness. Investigation into the decompression tables, Medical Research Council Report, UPS 168, London (1957). 18. D. E. Yount and R. H. Strauss, Bubble formation in gelatin: a model for decompression sickness, J. Appl. Phys. 47,5081-5089 (1976). 19. T. R. Hennessy and H. V. Hempleman, An examination of the critical released gas concept in decompression sickness, Proc. R. Sot. London B 197, 299-313 (1977). 20. B. R. Wienke, Tissue gas models and decompression computations: a review, Undersea Biomed. Res. 16, 53-89 (1989).

21. D. E. Yount and D. C. Hoffman, On the use of a bubble formation model to calculate diving tables, Aviat. Space Environ. Med. 57, 149-156 (1986). 22. D. E. Yount. On the evolution generation, and regeneration of gas cavitation nuclei, J. Acoust. Sot. Am. 71, 1473-1481 (1982). 23. D. E. Yount, Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei, J. Acoust. Sot. Am. 65, 1431-1439. 24. D. E. Yount, C. M. Yeung and F. W. Ingle, Determination of the radii of gas cavitation nuclei by filtering gelatin, J. Acoust. Sot. Am. 65, 1440-1450 (1979). 25. D. E. Yount, E. W. Gillary and D. C. Hoffman, A microscopic investigation of bubble formation nuclei, 1. Acoust. Sot. Am. 76, 1511-1521 (1984). 26. B. R. Wienke, Reduced gradient bubble model, Int. 1. Biomed. Comput. 26, 237-256 (1990). 27. B. R. Wienke, Bubble number saturation curve and asymptotics of hypobaric and hyperbaric exposures, Int. .I. Biomed. Comput. 29, 215-225 (1991). 28. D. R. Leitch and E. E. P. Barnard, Observations on non-stop and repetitive air and oxynitrogen diving, Undersea Biomed. Res. 9, 113-129 (1982). 29. B. C. Gilliam, Evaluation of decompression sickness in multi-day repetitive diving for 77,680 sport dives, S. Pac. Under. Med. Sot. J. 22, 24-30 (1992).

30. K. H. Smith and L. Stayton, Hyperbaric decompression by means of bubble detection. Office of Naval Research Contract Report. NOOO1469-C-0402,Washington (1978). 31. T. S. Neuman, D. A. Hall and P. G. Linaweaver, Gas phase separation during decompression in man: ultrasound monitoring, Undersea Biomed. Res. 3, 121-130 (1976). 32. M. A, Lang and R. D. Vann, Proc. Am. Acad. Underwater Sci. Repetitive Diving Workshop. American Academy of Underwater Sciences Diving Safety Publication, AAUSDSP-RDW-02-92, Costa Mesa (1992). 33. R. G. Dunford, C. Wachholz, K. Huggins and P. B. Bennett, Doppler analysis of sport diver profiles: a second look, Undersea Biomed. Res. 19, 70 (1992). 34. P. J. Sheffield, Flying After Diving. Undersea and Hyperbaric Medical Publication 77 (FLYDIV), Bethesda (1990).

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APPENDIX:

computers and application

MATHEMATICAL

403

MODEL

Inert gas exchange is driven by the local gradient, the difference between the arterial blood tension,p,, and the instantaneous tissue tension, p. Such behavior is modeled in time, f, by mathematical classes of exponential response functions, bounded by p. and the initial value of p, denoted p,, P =p. + (P, -PA exp(--W,

(1)

with the perfusion constant, I, related to the tissue half-time, t, through, ),=-

0.6931 5

c-4

The differential equation satisfied by equation (1) is the well-known rate law, linking instantaneous change to instantaneous value by 1, as bulk cooling. nuclear decay, etc. Compartments with 1,2,5,10,20,40,80. 120,240,480, and 720 half-lives, r‘, are employed. The tensions, p, and pa, represent extremes for each stage, or more precisely, the initial tension and the arterial tension at the beginning of the next stage. Classical models limit exposures by requiring that the tissue tensions never exceed the critical tensions, fitted to the U.S. Navy non-stop limits, for example, in units of absolute pressure (fsw), M=193.3r~i’4+4.110dr~i’4.

(3)

Such fit, with tm”4 dependence, is generic to present critical tensions, and the corresponding critical gradient, G, is given by, with P ambient pressure, G=M-P.

(4)

A VPM critical radius, r,, at fixed pressure, PO, represents a cut-off for growth upon decompression to lesser pressure. If r, is the critical radius at PO, then, the smaller family, r, excited by decompression from pressure, P. obeys 1 1 AP -=r,+=, r with AP= P- POmeasured in fsw, and r in microns. At sea level, P0=33fsw, r,=O.S microns, and AP=d. Deeper decompressions excite smaller, more stable, nuclei. Our non-stop limits, t, at depth, d, satisfy a modified law, that is dr”*=400 fsw min”‘, with the bounce gradient, G, written for each compartment, r, using the non-stop limits and excitation inverse radial difference, AT-‘, from equation (5), G=Ar-‘AG+GO AT-

I=‘_‘, r

r0

at generalized depth d = P - 33 fsw. A non-stop exposure, followed by direct return to the surface, thus allows Go for that compartment. The minimum excitation, Gbub, mitially probing r(t). accounting for regeneration of nuclei over time scales W-‘, is, Gb”b

=

NYC -=- - Y) 11.Ol y7(0 44 ’

with. r(t)=r+(rO-r)[l-exp(-wt)],

(8)

y, yc film, surfactant surface tensions, that is, y= 17.9 dyne/cm, yc = 257 dyne/cm, and o the inverse of the regeneration time for stabilized gas micronuclei (many days). Nuclei probed depend on depth according to equation (5). The excitation threshold, Gbub, represents that minimal free-dissolved gas gradient, just balanced by the surface tension, supporting growth. Saturation exposures permit, G”‘, G”‘=58.6Ar-‘+23.3=0.372P+11.01.

(9)

The relationship for G”‘, deduced from exposure data and given by equation (9), agrees with specific parameterization of the controlling compartment in critical tension algorithms. Although the actual size distribution of gas nuclei in humans was unknown, experiments in gels have been correlated with a decaying expontial (radial) distribution function n. For a stabilized distribution, n,

404

B. R. WIENKE

accommodated by the body at fixed pressure, Pa, the excess number of nuclei, An, excited by compressiondecompression from new pressure, P, is, with equation (5) tracking change in radius, r,

An=N

1-i

=NrAr-‘,

(

1

assuming the small argument expansion function, and with N a VPM constant, not important to development here. For deep compressions-decompressions, An is large, while for shallow compressions-decompressions, An is small. The rate at which gas inflates in tissue depends upon both the excess bubble number, An, 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, aV, with a a proportionality constant. Accordingly this requires, cz

I

AnGdt=aV.

(11)

0

for V the limiting gas volume. Assuming that tissue gas gradients are constant during decompression, td, while decaying exponentially to zero afterwards, and taking limiting condition of the equal sign, yields for a bounce dive, AnG(td+ym’)=aV.

(12)

For non-stop exposures with linear ascent rate, u, we have td= d/u. With saturation exposures, the integral must be evaluated iteratively over component decompression stages, maximizing each G while satisfying equation (11). In the latter case, tdis the sum of individual stage times plus interstage ascent times, assuming the same interstage ascent speed, o. Employing equation (11) iteratively, and one more constant, 6, defined by,

ycav = 7500 fsw min,

(13)

G(td+l-‘)=ot=522,3fswmin,

(14)

d=

-

rS@ we have,

from the Spencer bounce and Tektite saturation data. In terms of equation (5), and the depth at which a compartment excited as a function of controllng half-life, t, in the range, 12 sds

controls the exposure, the radii of nuclei 220 fsw, are fitted,

l-Ar-‘=:=0.9-0.43exp(-0.0559r).

(15)

with half-life measured in minutes. For large t, r is close to rO, while for small t, r is on the order of 0.5r0. We extend the critical phase criterion to repetitive diving, that is, the integral of equation (11) to multiexposures, by writing, AnGtd,+

‘I

I

AnGdt II

1

SaV,

(16)

with the index j denoting each dive segment, up to a total of J, and t, the surface interval after the jth 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 in equation (16) to total less than aV, obviously each term must be less than aV. Performing the indicated operations yields a revised criterion, I

c

An@td,+l-‘)SaV,

(17)

with the important property,

Because of the above constraint, the approach is termed a reduced gradient bubble model (RGBM). The terms AnG and AnG differ by effective bubble elimination during the previous surface interval. To maintain the

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computers and application

405

phase volume constraint during multi-diving, the elimination rate must be downscaled by a set of bubble growth, regeneration, and excitation factors, cumulatively designated, 5, such that, G=