Christian R. Gutvik

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.

A Dynamic Two-Phase Model for Vascular Bubble Formation During Decompression of Divers

Accumulated inert gas during a dive and subsequent reduction of ambient pressure may lead to formation of gas bubbles, which is the initial cause of decompression sickness (DCS). Decompression procedures are used to get divers safely up from depth, and traditionally, the algorithms are evaluated against clinical symptoms of DCS. However, this approach has several weaknesses. The symptomatology of DCS is very diffuse and there are ethical concerns evaluating procedures through provoking DCS on the test subjects. In recent decades ultrasonic Doppler and imaging to detect venous gas emboli (VGE) have been used as additional tools to evaluate decompression procedures. A statistical correlation between VGE and DCS has been shown and the method is more sensitive than clinical manifestation. This paper suggests a dynamic mathematical model for VGE. We have used a physiological approach in the model derivation with VGE as a measurable endpoint. We propose that the underlying physiological and physical mechanisms of the model can be better validated with such an objective quantitative measurement method. Two simulation examples are given to illustrate the properties of the model and why there is a potential of improving the consistency of controlling bubble formation, and consequently, the risk of getting DCS.

Venous gas embolism as a predictive tool for improving CNS decompression safety

A key process in the pathophysiological steps leading to decompression sickness (DCS) is the formation of inert gas bubbles. The adverse effects of decompression are still not fully understood, but it seems reasonable to suggest that the formation of venous gas emboli (VGE) and their effects on the endothelium may be the central mechanism leading to central nervous system (CNS) damage. Hence, VGE might also have impact on the long-term health effects of diving. In the present review, we highlight the findings from our laboratory related to the hypothesis that VGE formation is the main mechanism behind serious decompression injuries. In recent studies, we have determined the impact of VGE on endothelial function in both laboratory animals and in humans. We observed that the damage to the endothelium due to VGE was dose dependent, and that the amount of VGE can be affected both by aerobic exercise and exogenous nitric oxide (NO) intervention prior to a dive. We observed that NO reduced VGE during decompression, and pharmacological blocking of NO production increased VGE formation following a dive. The importance of micro-nuclei for the formation of VGE and how it can be possible to manipulate the formation of VGE are discussed together with the effects of VGE on the organism. In the last part of the review we introduce our thoughts for the future, and how the enigma of DCS should be approached.

Barrier function nonlinear optimization for optimal Decompression of divers

This paper is based on a comprehensive dynamic mathematical model (Copernicus) of vascular bubble formation and growth during and after decompression from a dive. The model is founded on the statistical correlation between measurable Venous Gas Emboli (VGE) and risk of severe Decompression Sickness (DCS) where VGE has been shown to be a reliable and sensitive predictor of decompression stress. By using the Copernicus model the diving decompression problem can be formulated as a nonlinear optimal control problem, where the objective is to minimize the total ascend time subject to constraints on the maximum bubbles volume in the pulmonary artery. A recent study reveals that the optimal solution can be obtained by solving the optimization problem with some equality constraints. Inspired by which, a simpler approach using barrier function is proposed in this paper, through which we achieve a more efficient and robust numerical implementation. The paper also studies the effect of ascent profile parameterization.

Optimal Decompression of Divers [Applications of Control]

Decompression modeling of divers is a research field that is over 100 years old and since the beginning has been studied mainly as a clinical and biomedical problem. The topic is largely unexplored by the technological sciences using methods and theories for modeling and control. This article outlines a structure where the process of bubble formation is modeled as a nonlinear dynamic model and then used to design a state estimator and model-based predictor. Procedures are then calculated using explicit MPC. We further discussed dive-computer implementations using approximate explicit solutions. Finally, we showed practical differences for divers using computers that implement this approach. When future advances in sensor technology are made, the present structure can be further developed to include more feedback control of the estimator and optimal control formulation.

Optimal Decompression Through Multi-parametric Nonlinear Programming *

Abstract Recently, a comprehensive dynamic mathematical model named Copernicus has been established to discover the mechanism of the vascular bubble formation and growth during and after decompression from a dive. The model uses Venous Gas Emboli (VGE) as a measurement and connects it to the risk of severe Decompression Sickness (DCS). Being validated by a series of diving tests, Copernicus model is believed to be suitable and efficient to predict DCS hence generate optimal decompression profiles for the divers. This paper is based on the Copernicus model and presents a nonlinear model predictive control approach, where multi-parametric nonlinear programming is used to construct an explicit solution for the ease of implementation on a typical low-cost diving computer.

Use of heart rate monitoring for an individualized and time-variant decompression model

Individual differences, physiological pre-conditions and in-dive conditions like workload and body temperature have been known to influence bubble formation and risk of decompression sickness in diving. Despite this fact, such effects are currently omitted from the decompression algorithms and tables that are aiding the divers. There is an apparent need to expand the modeling beyond depth and time to increase safety and efficiency of diving. The present paper outlines a mathematical model for how heart rate monitoring in combination with individual parameters can be used to obtain a customized and time-variant decompression model. We suggest that this can cover some of the individual differences and dive conditions that are affecting bubble formation. The model is demonstrated in combination with the previously published Copernicus decompression model, and is suitable for implementation in dive computers and post dive simulation software for more accurate risk analysis.

Approximate Explicit Nonlinear Receding Horizon Control for Decompression of Divers

This paper is based on a comprehensive dynamic mathematical model (Copernicus) of vascular bubble formation and growth during and after decompression from a dive. The model is founded on the statistical correlation between measurable venous gas emboli (VGE) and risk of severe decompression sickness (DCS) where VGE has been shown to be a reliable and sensitive predictor of decompression stress. By using the Copernicus model the diving decompression problem is formulated as a nonlinear optimal control problem, where the objective is to minimize the total ascend time subject to constraints on the maximum bubbles volume in the central venous pool. A recent study reveals that the optimal solution can be obtained by solving the optimization problem with equality constraints. Inspired by which, a simpler approach using barrier function is proposed in this paper, through which we achieve a more efficient and robust numerical implementation. To reduce the complexity of the nonlinear optimization problem this paper also studies the decompression profile parameterization and its effect. Furthermore, by applying multi-parametric nonlinear programming technique, an approximate explicit solution to the nonlinear optimization problem is obtained, which makes the practical implementation on a typical low-cost diving computer possible.

Erratum to: Venous gas embolism as a predictive tool for improving CNS decompression safety

Erratum to: Eur J Appl PhysiolDOI 10.1007/s00421-011-1998-9An incorrect citation was included in the originalpublication:Vince RV, Chrismas B, Midgley AW, McNaughton LR,Madden LA (2009) Hypoxia mediated release of endo-thelial microparticles and increased association ofS100A12 with circulating neutrophils. Oxid Med CellLongev 2:2–6The correct reference should be:Vince RV, McNaughton LR, Taylor L, Midgley AW,Laden G, Madden LA (2009) Release of VCAM-1 asso-ciated endothelial microparticles following simulatedSCUBA dives. Eur J Appl Physiol 105:507–513