Relationship between two different functions derived from diffusion-based decompression theory.

Hempleman's diffusion-based decompression theory yields two different functions; one is expressed by a simple root function and the other by a complex series function. Although both functions predict the same rate of gas uptake for relatively short exposure times, no clear mathematical explanation has been published that describes the relationship between the two functions. We clarified that (1) the root function is the solution of the one-dimensional diffusion equation for a semi-infinite slab, (2) the series function is an applicable solution for a finite slab thickness, (3) the parameter values of the root function can be used to determine the parameter values of the series function, and (4) the predictions of gas kinetics from both functions agree until an adequate amount of diffusing inert gas reaches the boundary at the opposite end of the finite slab. The last point allows the use of the simpler root function for predicting short no-stop decompression limits. Experience dictates that the inert gas accumulation for a 22 min at 100 feet of seawater (fsw) dive is considered safe for no-stop decompression. Although the constraint, Depth square root of Bottom Time = 100 square root of 22, has been applied as an index to determine either the safe depth or bottom time (given the other) for no-stop decompression, it should not be applied more broadly to dives requiring decompression stops.

Exercise during decompression reduces the amount of venous gas emboli.

To determine the effects of moderate, intermittent exercise during decompression on the Doppler detectable amount of venous gas emboli (VGE), 29 healthy male volunteers performed 44 wet (8 degrees +/- 2 degrees C) dives to 45 msw (450 kPa) for 30 min with standard air decompression. During compression and the bottom period, all subjects were inactive; during decompression, 28 remained inactive, 11 performed leg exercise, and 5 did arm exercise. Intermittent exercise was controlled at approximately 50% of each subject's arm or leg aerobic capacity. At 30-min intervals after surfacing, subjects were monitored with a Doppler ultrasonic bubble detector. The Doppler scores were used to calculate the Kisman Integrated Severity Score (KISS). The KISS were log transformed (with zeroes being equivalent to log 0.01) and analyzed with a one-way analysis of variance. No significant differences (P < or = .05) between mean KISS scores after arm or leg exercise were observed, thus these data were pooled and compared to those of the inactive controls. The mean pooled KISS after exercising during decompression were significantly lower than those of the inactive controls. Moderate, intermittent exercise during decompression apparently reduces the amount of Doppler-detectable VGE after diving. The incidence rate of decompression sickness in both groups was not significantly different (P < 0.05).

Calibration of a bubble evolution model to observed bubble incidence in divers.

The method of maximum likelihood was used to calibrate a probabilistic bubble evolution model against data of bubbles detected in divers. These data were obtained from a diverse set of 2,064 chamber man-dives involving air and heliox with and without oxygen decompression. Bubbles were measured with Doppler ultrasound and graded according to the Kisman-Masurel code from which a single maximum bubble grade (BG) per diver was compared to the maximum bubble radius (Rmax) predicted by the model. This comparison was accomplished using multinomial statistics by relating BG to Rmax through a series of probability functions. The model predicted the formation of the bubble according to the critical radius concept and its evolution was predicted by assuming a linear rate of inert gas exchange across the bubble boundary. Gas exchange between the model compartment and blood was assumed to be perfusion-limited. The most successful calibration of the model was found using a trinomial grouping of BG according to no bubbles, low, and high bubble activity, and by assuming a single tissue compartment. Parameter estimations converge to a tissue volume of 0.00036 cm3, a surface tension of 5.0 dyne.cm-1, respective time constants of 27.9 and 9.3 min for nitrogen and helium, and respective Ostwald tissue solubilities of 0.0438 and 0.0096. Although not part of the calibration algorithm, the predicted evolution of bubble size compares reasonably well with the temporal recordings of BGs.

Prediction of decompression illness using bubble models.

The method of maximum likelihood was applied to models of bubble formation and evolution against data involving decompression illness (DCI). Equilibrium and non-equilibrium gas kinetic models were tested under the constraint of a finite tissue volume. The equilibrium model (leq), where the internal gas of a bubble is in partial pressure and mechanical equilibrium with the gas dissolved in tissue, assumed formation of a bubble upon any gas supersaturation. The non-equilibrium model (neq), where mechanical equilibrium is maintained but the exchange of gas between the bubble and the tissue is governed by a rate constant, assumed formation of a bubble at the metastable equilibrium state which requires a specific degree of gas supersaturation. In addition, another version of bubble evolution based on the diffusivity of gas in tissue (vl) was tested under similar finite volume constraints. Model parameters included liquid surface tension, the gas exchange rate constant, gas solubility, and the tissue time constant. The risk of DCI was based on the bubble radius (R) raised to powers ranging from 0 to 6. The data included 2,023 man-dives in 630 different dive profiles of air and nitrox gas mixtures with depth ranging from 1.75 to 7.09 bar and bottom time ranging from 2.8 to 300.2 min. There were 97 occurrences of DCI and 27 occurrences of marginal symptoms. Predictions of the neq and vl models were quite similar and suggested that the tissue primarily responsible for bubble formation leading to DCI in the present analysis has a perfusion rate of about 4.0 ml blood.100 ml-1.min-1. The best fit of the data for a single compartment of 10(-4) ml vol was obtained with the leq model and a risk based on R4, and an estimated time constant of 95.6 +/- 9.8 min.

Assessment of inter-rater agreement on the grading of intravascular bubble signals.

Transcutaneous Doppler ultrasonic bubble detectors are widely used in decompression research. However, interpretation of the complex acoustic signals from the bubble detectors involves a degree of subjectivity, and the comparability of grades assigned by different raters must be assessed. Hypothetical data were used to determine an appropriate method for evaluating the comparability of Doppler raters and to illustrate the limitations of many nonparametric statistics. Two sets of real data were then used to evaluate this procedure, the first from a training exercise carried out by Kisman and Masurel (1978, unpublished) and the second from a test tape that was independently scored by five Defence and Civil Institute of Environmental Medicine Doppler technicians. The results were analyzed by a two-stage approach. First, they were entered into contingency tables and checked for large disagreements, a tendency for one rater to grade higher than the other, and the degree of variability. Second, the results were analyzed with the nonparametric weighted kappa statistic. These studies have led to a practical, efficient method for the evaluation of Doppler raters.

Maximum likelihood analysis of air and HeO2 dives.

The method of maximum likelihood analysis was applied to data consisting of 1,949 man-dives, of which 1,041 were on air and 908 were on HeO2 mixtures. These dives represented a wide range of bottom time and depth combinations, and had an overall incidence of decompression sickness (DCS) of 4.64%. Several models, based on single exponential gas uptake in either one or two compartments, were tested for predicting the incidence of DCS. The criterion for defining the risk of DCS was based on the concept of potential gas volume (i.e., the volume of a bubble that could form and be in equilibrium with the remaining gas dissolved in solution). This criterion takes into account the solubilities of the gases in solution, but can be adjusted to account only for the partial pressures of the gases. The best model for the prediction of DCS was found for two compartments where the kinetics (time constants) and not the gas solubilities of nitrogen and helium were distinguished from each other. Results using the best prediction model with the present data suggests the following: 1) most of the risk of DCS occurs after surfacing; 2) most of the risk occurs in the "slow" compartment (approximately 420 min time constant); and 3) nitrogen contributes about twice as much as helium to the risk of DCS for HeO2 dives.

Relative decompression risk of dry and wet chamber air dives.

The difference in risk of decompression sickness (DCS) between dry chamber subjects and wet, working divers is unknown and a direct test of the difference would be large and expensive. We used probabilistic models and maximum likelihood estimation to examine 797 dry (and generally resting and comfortable) and 244 wet (and generally working and cold) chamber dives from the Defence and Civil Institute of Environmental Medicine, supplemented with 483 wet (working, cold) dives from the Navy Experimental Diving Unit. Several analyses considered whether dry and wet data were distinguishable using several models, whether models obtained from one set of exposure conditions would correctly predict the occurrence of DCS in the other condition, and whether a single wet-dry risk difference parameter was different from zero. Although the two conditions may not produce identical risks, immersion appears to change relative risk of DCS by less than 30% and certainly involves less than a doubling of DCS risk. Uncontrolled differences in exercise and temperature stresses unavoidably complicate interpretation. Several methods are presented to extrapolate results from dry-test subjects in decompression trials to expected at-sea performance.

Use of the maximum likelihood method in the analysis of chamber air dives.

The method of maximum likelihood was used to evaluate the risk of decompression sickness (DCS) for selected chamber air dives. The parameters of two mathematical models for predicting DCS were optimized until the best agreement (as measured by maximum likelihood) corresponding to the observed DCS incidents from a series of dives was attained. The decompression data used consisted of 800 man-dives with 21 incidents of DCS and 6 occurrences of marginal symptoms. The first model investigated was based on a nonlinear gas exchange in a series arrangement of four compartments. The second model was based on a monoexponential gas exchange in a parallel arrangement of two compartments. The overall statistical success in describing the 800 man-dives was quite similar for the two models. Predictions of safety for dives not part of the original data differed for the models due to differences in gas kinetics. For short, no-decompression dives, the series arrangement of compartments predicted a lower incidence of DCS. These predictions were more consistent with the outcome of subsequent testing than were predictions of the parallel compartment model. Predictions of the series arrangement model were also similar to those of a single-compartment, two-exponential model that was evaluated with over 1700 man-dives by the U.S. Navy.