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@MISC{Goldman_,

author = {Saul Goldman and Saul Goldman and S Goldman},

title = {},

year = {}

}

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### Abstract

Goldman S. A new class of biophysical models for predicting the probability of decompression sickness in scuba diving. J Appl Physiol 103: [484][485][486][487][488][489][490][491][492][493] 2007. First published April 19, 2007; doi:10.1152/japplphysiol.00315.2006.-Interconnected compartmental models have been used for decades in physiology and medicine to account for the observed multi-exponential washout kinetics of a variety of solutes (including inert gases) both from single tissues and from the body as a whole. They are used here as the basis for a new class of biophysical probabilistic decompression models. These models are characterized by a relatively well-perfused, risk-bearing, central compartment and one or two non-risk-bearing, relatively poorly perfused, peripheral compartment(s). The peripheral compartments affect risk indirectly by diffusive exchange of dissolved inert gas with the central compartment. On the basis of the accuracy of their respective predictions beyond the calibration regime, the three-compartment interconnected models were found to be significantly better than the two-compartment interconnected models. The former, on the basis of a number of criteria, was also better than a two-compartment parallel model used for comparative purposes. In these latter comparisons, the models all had the same number of fitted parameters (four), were based on linear kinetics, had the same risk function, and were calibrated against the same dataset. The interconnected models predict that inert gas washout during decompression is relatively fast, initially, but slows rapidly with time compared with the more uniform washout rate predicted by an independent parallel compartment model. If empirically verified, this may have important implications for diving practice. compartmental modeling; perfusion-diffusion models; multi-exponential exchange kinetics SCUBA DIVING WITH AIR AS the breathing mixture, involves breathing compressed air. The compressed air is provided to the diver (through a "demand regulator") at ambient pressures. Because of the hydrostatic pressure of water, these ambient pressures will exceed the surface pressure. Consequently, during a dive, more nitrogen will be dissolved in blood and body tissues than is normally dissolved at surface pressures. A diver surfacing rapidly from a dive may have a considerable excess of dissolved nitrogen remaining in the blood and tissues. If the excess is large enough, some of the dissolved nitrogen will come out of solution in the form of bubbles which, if sufficiently extensive, can lead to "decompression sickness" (DCS). Severe forms of DCS can include paralysis or death. Therefore, the rate of the ascent, and/or the depth vs. time profile for the ascent, must be appropriately controlled. Decompression models are highly simplified biophysical representations of the body and those regions or tissues of the body that are relevant to the development of DCS. These models are created in an attempt to capture and mimic the most salient factors that lead to DCS. They can be used to predict the probability of developing DCS [P(DCS)] for any dive and to prescribe ascent procedures that would constrain P(DCS) for a particular dive to an acceptable level. This article describes the properties of probabilistic decompression models in which both perfusion and inter-tissue diffusion of dissolved inert gas influence P(DCS). The reason for including inter-tissue diffusion is that an accumulated body of empirical work has shown that inert gas blood-tissue exchanges, for a variety of gases and tissues, are best accounted for by models that involve a mix of perfusion and external diffusion-driven processes. These external processes include arterio-venous countercurrent exchange (5, 15), diffusion between unequally perfused tissues Originating with Haldane (2), a parallel network of independent, perfusion-limited compartments, of the kind illustrated in Subsequently, Novotny et al. (29) carried out a study that directly tested Kety's assertion. This group attempted to fit the kinetics of 133 Xe washout from dog calf muscle, modeled as an independent parallel network of more than 100 perfusionlimited compartments. The study found the model's predictions of key physiological functions to be very inaccurate. For example, the observed mean tissue transit time of 133 Xe was Address for reprint requests and other correspondence: S. Goldman, Dept. of Chemistry and the Guelph-Waterloo Physics Institute, Univ. of Guelph, Guelph, Ontario, Canada N1G 2W1 (e-mail: goldman@chembio.uoguelph.ca). The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. 103: 484-493, 2007. First published April 19, 2007 doi:10.1152/japplphysiol.00315.2006 6.7 times longer than the value predicted by Kety's model for this system. J Appl Physiol More recently, Doolette et al. found additional inconsistencies associated with the application of parallel perfusionlimited models in sheep (5-7). These workers found that the models, which involved a mix of perfusion-limited and external diffusion-driven processes, provided improved agreement with their data. In addition to these studies, the general idea that DCS-prone tissue is unlikely to be isolated from less susceptible tissues and may be indirectly affected by them has been around for some time. For example, 17 years ago, Vann pointed out the possibility that unsusceptible tissue (such as lipid) may act as a reservoir of dissolved inert gas relative to contiguous, more susceptible tissue and may thereby, indirectly, affect DCS risk by diffusive exchange of inert gas with the susceptible tissue (38). In view of the above, it seems timely to investigate the properties of decompression models consisting of interconnected compartments of differing susceptibilities to DCS and that involve a mix of perfusion and external diffusion. Before proceeding, however, the preexisting work on interconnected decompression models and compartmental, multi-exponential kinetics is briefly outlined. The " In addition, earlier exploratory calculations on compartments that involve two-exponential kinetics have been carried out Models. The probabilistic approach to decompression theory, developed by Weathersby and his coworkers (Refs. 34, The type of interconnected gas distribution models that were studied are illustrated in As discussed in the introduction, models that involve a mix of perfusion and intercompartmental diffusion and whose compartments, as a consequence, display multi-exponential kinetics seem best at accounting for observed inert gas blood-tissue washout curves. A central premise of this work is that DCSprone tissue is not exceptional in this regard. It is also generally accepted that not all regions/tissues of the body are equally susceptible to decompression injury Four interconnected compartmental models will be considered: two with two compartments apiece and two with three compartments apiece. The two-and three-compartment models will be characterized by two-and three-exponential compartmental kinetics, respectively (16, chapter 2). Because of the computational requirements associated with this application, linear kinetics is here assumed. Under linear kinetics, wherein transfer rates are proportional to the first power of the relevant pressure differences, the rate equations can be solved analytically, i.e., both exactly and very rapidly. This is essential for keeping the calculations simple and the computer time requirements low (see APPENDICES A and B, available online at the Journal of Applied Physiology website). This work deals with the influence of a decompression model's compartmental structure on its properties. The issues of how best to define the risk function, or what its basis ought to be, are not addressed. The approach taken was to use a simple risk function of exactly the same functional form for all the models. This was done to ensure that comparisons among the models would reflect differences due only to the form assumed for the gas-distribution part of the model. The risk function used was of the dissolved gas (or single phase) type, developed by Weathersby and coworkers (34, 36). It was where B ϵ (P th Ϫ P fvg ), the subscript i designates the compartment number, r i (t) is the risk per unit time, c i is a proportionality constant, p i (t) is the Henry's law-based dissolved inert 1 "Well-perfused" here implies a relatively large perfusional rate constant (or fractional transfer coefficient) fi0. These constants differ from "flow rate per unit volume," a commonly used measure of perfusion, by a factor involving the gas partition coefficient. As used here, "a relatively well perfused central compartment" implies the following relations between the perfusion-based fi0s: f10 ϾϾ f20; f10 ϾϾ f30. If the compartments were uniformly perfused, in the sense that f10 ϭ f20 ϭ f30, the chemical potential or, equivalently, the Henry's law-based partial pressure of nitrogen would be the same in all the compartments (11, 21). It would rise and fall exactly in unison in all the compartments, and no driving force for intercompartment diffusion would exist. gas partial pressure (11, 21), P 0 (t) is the total ambient hydrostatic pressure, and P th and P fvg are, respectively, the "threshold" and "fixed venous gas" pressures (below, and APPENDIX C, available online at the Journal of Applied Physiology website). For all the interconnected compartmental models (2CM, 2CG, 3CM, and 3CG) i was 1. For the two-compartment, independent, parallel model used for comparison (2CP), i was 1 and 2, since here each compartment carries risk. The term B carries contributions to r i (t) from effects other than those due to the relative excess nitrogen partial pressure: [p i (t) Ϫ P 0 (t)]/P 0 (t). P fvg , which accounts approximately for the influence of the venous gases H 2 O(g), CO 2 (g), and O 2 (g), increases the risk by adding the sum of the partial pressures of these gases to that of N 2 (g). P th , on the other hand, provides an approximate contribution to risk reduction that arises from the existence of a threshold depth (d th ). Here, d th refers to the deepest depth from which rapid decompression to P 0 ϭ 1 atm will never cause DCS, even when the time spent at that depth is sufficient to ensure saturation; i.e., 1 day or more. It was here found to be 11.7 fsw (APPENDIX C), where fsw represents "feet of sea water." The values used for P fvg , P th , and B were 0.192 atm, 0.213 atm, and 0.021 atm, respectively. The value of P fvg was taken from Ref. 34, and P th was determined from d th as described in APPENDIX C. The value 0.021 atm for B was used in the r i (t) expressions of all five models. In addition, as described in APPENDIX C, a different value of B, stemming from a different choice made for d th was used in supplementary trial calculations. These calculations were carried out to assess the sensitivity of the interconnected models with respect to the value used for d th or B. The main result was that the qualitative properties of the interconnected models do not depend on a specific value of d th or B. METHODS Calibrations. This is an initial demonstration paper. Its purpose is not to present new models ready for application. Rather, its purpose is to compare the properties and the potential usefulness of models with the structure illustrated in An important test of the potential usefulness of a model is its "robustness." A robust model is one that is relatively insensitive to the regime in which it is applied. For example, if model x, which is calibrated against a low-risk square profile dataset, is subsequently found to accurately predict the P(DCS) values of a high-risk square profile dataset, then model x demonstrates some robustness. The robustness stems from the applicability of same parameter values of model x in the two different regimes. Clearly, this is a useful property. If this same model x also accurately predicts the P(DCS) values of a very low-risk multilevel profile dataset, then model x is that much more robust and useful. Although robustness here is suggestive of a model's capturing the salient features of the underlying physiology, this is not necessarily the case and, in any event, is irrelevant to the present purpose. We therefore require both a dataset consisting of profiles of a particular kind (for purposes of calibration), and additional data that can be used to check the quality of the calibrated model's extrapolations. If the extrapolations are to be meaningful, the differences between the two sets of data must be both well defined and significant. Since dive profiles can be classified both by their type and by the degree of risk they pose, meaningful extrapolations include those between profiles of the same type but with different degrees of risk and those between profiles that differ both by type and by the degree of risk they pose. There is of course a trade-off involved here. By restricting the calibration dataset to profiles of a particular kind, one necessarily reduces the size of the dataset from what it might otherwise be. This usually has the effect of reducing the accuracy of the fitted parameters. For the parallel models, it also resulted here in an inability to distinguish the 3CP model from the 2CP model. However, as explained below, neither of these drawbacks resulted in a major sacrifice. Specifically, it will be shown that two other 3CP models, "USN93" (Ref. 35 and references therein), and "EE1(nt)" (34), which are very similar to the 3CP model initially considered here, but were calibrated using large, mixed-profile datasets, produced extrapolations that were qualitatively similar to those of the 2CP model (see Furthermore, as shown by the entries in The limitations inherent in the reduced size of the calibration dataset were therefore knowingly accepted, in exchange for the means to assess the quality of a model's extrapolations. In view of the purpose of this work, the benefit was judged to outweigh the cost. The calibration dataset consisted of 725 square profile dives using air, for which the overall average P(DCS) was ϳ0.11. It is described in greater detail below and in Thus two different extrapolations are possible. With target 1, the extrapolation is from square profiles of low to high risk, to square profiles of very high risk. With target 2, the extrapolation is from square profiles of low to high risk, to a mix of extremely low-risk profiles of different types. The parameter estimates for all five models were obtained by a calibration to the dataset described in Maximum likelihood The general p i(t) expressions for the interconnected models have not previously been derived. They are obtained by analytical integration of the underlying coupled rate equations (APPENDIX A and Eq. A2). The general form of the result is given by Eq. A3, and its working form for this application is given by Eq. A10. The way this equation is used to obtain P(DCS) values for the interconnected models at arbitrary points on a dive profile is outlined in APPENDIX B, section 1a. Statistical functions. A modern version of 2 testing, based on evaluating the incomplete gamma function (P) was used. The functions and Q represent the number of degrees of freedom and the complement of P [i.e., (1 Ϫ P)], respectively. The relationship between these quantities is shown in Eq. 2 (30). Q is a very useful statistical function. It represents the probability or, equivalently, the "level of confidence" at which a model can be taken to be consistent with the dataset against which it is being compared (30). The advantage of using Q, as opposed to traditional 2 look-up tables, is that, unlike the latter, the former provides the result of a consistency check as a definite value. For example, with look-up tables, the result of a consistency check might take the form "the model is consistent with the dataset at the 95% confidence level but not at the 99% confidence level." Using Q, the result of the same consistency check would take the form "the model is consistent with the dataset at the 96.2% confidence level." The widespread availability of low-cost, high-speed computing has rendered the calculation of Q by Eq. 2 trivial. Look-up tables were constructed to avoid having to calculate Q (from 2 and ) at a time when this calculation was nontrivial. RESULTS Six models were initially considered and fitted to the dataset in The i s in The fairly broad confidence intervals entered for some of the rate constants in The plots in The Hill multi-species model is the best available basis for checking these extrapolations. It is based on combining human air saturation data with high-risk, scaled, pig and rat air saturation data (24). The mixing of the scaled, high-risk animal data with the human data is believed to have produced a more reliable predictive curve in the very high-risk saturation regime than can be obtained from human saturation data alone (24). It is seen that both the 3CM and 3CG models are in agreement with the Hill multi-species predictions at depths of 40 and 50 fsw, whereas the 2CM, 2CG, and 2CP models are not. These models' predictions are too low. Also, it is clear that the curves for the 3CM and 3CG models each have the correct shape, whereas those for all the other models shown do not. The 2CP model's curve has the same (incorrect) shape as those of the USN93 and EE1(nt) models, which are included in The results of extrapolating the five models to a very different regime are given in This is a very demanding test of model robustness, since the extrapolation is to a regime characterized by both a much lower level of risk and profiles of a different type from those in the calibration dataset. In view of this, it is seen that the 3CG model performs remarkably well, the 3CM model's performance is fair, whereas the 2CP, 2CM, 2CG, and EE1(nt) models all perform poorly. As indicated previously, the EE1(nt) model is a 3CP model that was calibrated elsewhere For the two-compartment mammillary (2CM) and two-compartment general (2CG) models, f20 is 0 and 0.2f21, respectively. For the three-compartment mammillary (3CM) model, f20 and f30 are both 0. For the three-compartment general (3CG) model, f20 ϭ 0.2f21, and f30 ϭ 0.2f31 (further details in APPENDIX A). 2CP, two-compartment parallel. using a large mixed-profile dataset. The very low values of Q for both the 2CP and EE1(nt) models stem from these models' significant overestimation of P(DCS) for these profiles. Specifically, the P(DCS) predictions from the 2CP and EE1(nt) models for the entire dataset (phase 1 ϩ 2A ϩ 2B) were distributed around Ϸ0.06 and Ϸ0.22, respectively. In contrast, the corresponding 3CM and 3CG models' predictions were distributed around Ϸ0.03 and Ϸ0.01, respectively. The empirical P(DCS) confidence intervals for the entire dataset can be rigorously calculated only if it is assumed that all the dive sets in the dataset had the same profile (3, 10). Under this assumption, the empirical 95% binomial confidence interval (for 1 incident in 565 identical dive sets) is 0 -0.01. This can be taken as a rough estimate of the actual P(DCS) confidence interval for the entire dataset. The 2CM and 2CG models performed erratically, with high, seemingly good Q values for the phase 1 and phase (1 ϩ 2B) datasets, but values of zero for the phase (1 ϩ 2A ϩ 2B) dataset. The latter stem from these models' P(DCS) predictions of exactly 0.0 for the one dive set (in phase 2A) that actually produced a hit. This causes 2 to become infinite and Q to become zero (see Eq. 2). The entries in The 3CM and 3CG model results are in fairly good agreement with the USN93D P(DCS) predictions, considering the much more limited calibration dataset used here. For each of these models, 15 of 17 P(DCS) predictions are in accord with the corresponding USN93D values, in the sense that the correspond- 2) 6.6 (2.9-9.3) 3.6 (.9-6.5) 100 25 2.1 (1.5-2.8) 3.6 (.9-6.5) 4.6 (0-7.3) 4.4 (.1-6.9) 7.0 (2.5-10) 3.6 (.8-6.7) 110 20 1.9 (1.4-2.6) 2.6 (.3-5.3) 3.7 (0-6.3) 3.5 (0-5.7) 6.9 (1.7-11) 3.0 (.3-6.7) 120 15 1.7 (1.1-2.3) 1.3 (0-3.8) 2.2 (0-4.6) 2.0 (0-3.9) 6.2 (0.6-10) 2.1 (0-5. The entries in parentheses are 95% confidence interval estimates, and the entries for USN93D are from Ref. NEW BIOPHYSICAL MODELS FOR SCUBA DIVING