Study Group Report 2009: human decompression modelling (VR Technology Limited)
This is the final report on the problem of human decompression modelling, brought to ESGI68 by VR Technology Limited. Click on the link at the bottom to download the full report as a pdf document.

Report coordinator
Vera Hazelwood (Knowledge Transfer Network for Industrial Mathematics)

Executive summary
At present, no decompression algorithm is able to predict safe decompression for all dive scenarios. In practice, empirical adjustments are made by experienced organisations or divers in order to improve decompression profiles for the range of depths and durations needed on any particular dive. Bubble formation and growth in the human body are the fundamental causes of decompression sickness, and it is believed that there is significant scope for incorporating better modelling of these processes into the design of decompression algorithms. VR Technology is a leading supplier of technical dive computers. The company is interested in expanding upon an existing algorithm (the Variable Gradient Model - VGM), which is used to design ascent profiles/decompression schedules and thereby mitigate the risk of decompression sickness in divers. The Study Group took the approach of trying to extend the existing Haldane model to account more explicitly for the formation of bubbles. By extending the model to include bubble dynamics it was expected that some physical understanding could be gained for the existing modifications to some of the parameters. The modelling that occurred consisted of first looking at the Haldane model and then considering a single small isolated bubble in each of the compartments and interpreting the predictions of the model in terms of decompression profiles.

Introduction
The amount of gas which can be dissolved in a liquid is an increasing function of pressure. If a liquid is saturated with gas at a given pressure and the pressure is suddenly reduced there is the potential for the explosive release of bubbles as gas comes out of solution (the so-called champagne bottle effect). This can lead to disastrous consequences for deep sea divers. As divers descend, their blood and tissues can become saturated with dissolved gases at the prevailing ambient pressure which is a linearly increasing function of depth. If divers then try to ascend too quickly it is possible for bubbles to form as the dissolved gas in their blood and tissues is forced out of solution. These bubbles can cause mechanical damage to tissues and can also block capillaries hence starving tissues of oxygen. This phenomena is known as decompression sickness or `the bends' and it is potentially fatal. Divers can minimize the risk of decompression by performing a gradual ascent from depth which allows the dissolved gas content of their blood and tissues to slowly equilibrate with the ambient.

The nature of the problem
VR Technology is a leading supplier of technical dive computers. The company is interested in expanding upon an existing algorithm (the Variable Gradient Model - VGM), which is used to design ascent profiles/decompression schedules and thereby mitigate the risk of decompression sickness in divers. At present, no decompression algorithm is able to predict safe decompression for all dive scenarios. In practice, empirical adjustments are made by experienced organisations or divers in order to improve decompression profiles for the range of depths and durations needed on any particular dive. Bubble formation and growth in the human body are the fundamental causes of decompression sickness, and it is believed that there is significant scope for incorporating better modelling of these processes into the design of decompression algorithms. The question that VR Technology brought to the Study Group was: How can we better model bubbles in the body to design decompression algorithms that work over the entire range of conditions that a diver might experience and to allow for a diver's physiology? How might we estimate the risks to the diver if the decompression requirements of the algorithm are not performed, owing to dive-dependent circumstances?

Numerical solutions
The problem as outlined can be considered as a "forward problem" where the external pressure experienced by the diver is given as a function of time and the ODEs then solved to determine how the partial pressure of Nitrogen in part of the body and the radius of the body vary with time. A Matlab code using the "chebfun" package allowed extremely accurate solutions to be found to these ODEs and these are presented here. The code assumes the diver to be breathing air, so there is a single inert gas, nitrogen, and eight different types of tissue are considered. The numerical predictions are shown in Figures 1 and 2.

Figure 1: Numerical solutions that show how bubble's size and the nitrogen pressure in the compartments change during the ascent from a 50 meter deep dive.

Figure 2: Numerical solutions that show how a bubble grows, and how nitrogen concentrations change, during an immediate ascent (left) and during a staged ascent allowing for decompression (right).

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Download 'VRTechnology-Human decompression modelling.pdf' (291 Kb).