Parameter estimation of the copernicus decompression model with venous gas emboli in human divers


Decompression Sickness (DCS) may occur when divers decompress from a hyperbaric environment. To prevent this, decompression procedures are used to get safely back to the surface. The models whose procedures are calculated from, are traditionally validated using clinical symptoms as an endpoint. However, DCS is an uncommon phenomenon and the wide variation in individual response to decompression stress is poorly understood. And generally, using clinical examination alone for validation is disadvantageous from a modeling perspective. Currently, the only objective and quantitative measure of decompression stress is Venous Gas Emboli (VGE), measured by either ultrasonic imaging or Doppler. VGE has been shown to be statistically correlated with DCS, and is now widely used in science to evaluate decompression stress from a dive. Until recently no mathematical model has existed to predict VGE from a dive, which motivated the development of the Copernicus model. The present article compiles a selection experimental dives and field data containing computer recorded depth profiles associated with ultrasound measurements of VGE. It describes a parameter estimation problem to fit the model with these data. A total of 185 square bounce dives from DCIEM, Canada, 188 recreational dives with a mix of single, repetitive and multi-day exposures from DAN USA and 84 experimentally designed decompression dives from Split Croatia were used, giving a total of 457 dives. Five selected parameters in the Copernicus bubble model were assigned for estimation and a non-linear optimization problem was formalized with a weighted least square cost function. A bias factor to the DCIEM chamber dives was also included. A Quasi-Newton algorithm (BFGS) from the TOMLAB numerical package solved the problem which was proved to be convex. With the parameter set presented in this article, Copernicus can be implemented in any programming language to estimate VGE from an air dive.

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    The notation used in the article is \(\tilde{}\) (tilde) for measurements and \(\hat{}\) (hat) for model estimates.


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This study has been supported by UWATEC AG, Switzerland and by the Norwegian Petroleum Directorate, Norsk Hydro, Esso Norge and Statoil under the “dive contingency contract” (no 4600002328) with Norwegian Underwater Intervention (NUI). Thanks to R. Y. Nishi for providing the data set from DCIEM on the square profile dives.

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Correspondence to Christian R. Gutvik.

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Gutvik, C.R., Dunford, R.G., Dujic, Z. et al. Parameter estimation of the copernicus decompression model with venous gas emboli in human divers. Med Biol Eng Comput 48, 625–636 (2010).

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  • Diving
  • Vascular bubbles
  • Nonlinear optimization
  • Ultrasound
  • Decompression sickness