# Growth of a gas bubble in a perfused tissue in an unsteady pressure field with source or sink

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## Abstract

In the context of decompression sickness, this paper presents analytical formulae and explanations for growth of a gas bubble in blood and other tissues in an unsteady diffusion field with a source or a sink. The formulae are valid for variable (through decompression) and constant (concerning diving stops/at sea level) ambient pressure. Under a linear decompression regime for ambient pressure, the gas bubble growth is proportional to ascent rate, tissue diffusivity and initial tissue tension and inversely proportional to surface tension, initial ambient pressure and the strength of the source/sink parameter \(k\) which gives the conditions for bubble growth. We find that the growth process is noticeably affected by changing *k*-values within a specified range, with no significant effect on the value of the bubble radius when *k* is outside this range. We discuss the effect of the presence of multiple bubbles, and of repetitive diving. Of the three available models for bubble growth, the predicted time to completion is longest in the model by Srinivasan et al. (J Appl Physiol 86:732–741, 1999), where the bubble grows in a steady diffusion field, but shortest in the model we describe for *k-*values closest to the boundaries of the interval \([0. 9 5 8 7,\;\;1.0]\). This is because our model considers the effect of the presence of a source, increasing the bubble growth rate and not taken into account in our previous (2010) model predicting an intermediate timeframe for bubble growth. We believe our new model provides a more accurate and widely applicable description of bubble growth in decompression sickness than previous versions.

## Keywords

Gas bubble Growth Unsteady diffusion Perfused tissue Source/sink Repetitive diving risk## List of symbols

- \(C\)
Concentration of dissolved gas \(\left( {{\text{mol}}\;{\text{m}}^{ - 3} } \right)\)

- \(C_{\infty }\)
Concentration of dissolved gas in the tissue far from the bubble \(\left( {{\text{mol}}\;{\text{m}}^{ - 3} } \right)\)

- \(\Delta C_{0}\)
\(= C_{\infty } - C_{0}\), the concentration difference \(\left( {{\text{mol}}\;{\text{m}}^{ - 3} } \right)\)

- \(D_{\text{T}}\)
Gas diffusion coefficient in tissue \(\left( {{\text{m}}^{2} \;{\text{s}}^{ - 1} } \right)\)

- \(k\)
A constant represents the strength of the source/sink \(\left( {\text{Dimensionless}} \right)\)

- \(P_{\text{a}}\)
Gas partial pressure in arterial blood, Eq. (1) \(\left( {{\text{N}}\;{\text{m}}^{ - 2} } \right)\)

- \(P_{\text{amb}}\)
Ambient pressure \(\left( {{\text{N}}\;{\text{m}}^{ - 2} } \right)\)

- \(P_{\text{atm}}\)
Atmospheric pressure \(\left( {{\text{N}}\;{\text{m}}^{ - 2} } \right)\)

- \(P_{\text{g}}\)
Pressure of the bubble wall \(\left( {{\text{N}}\;{\text{m}}^{ - 2} } \right)\)

- \(\dot{Q}\)
Blood flow per unit tissue volume \(\left( {{\text{s}}^{ - 1} } \right)\)

- \(\Re\)
General gas const \(\left( {{\text{N}}\;{\text{m}}/{\text{mol}}\;{\text{K}}} \right)\)

- \(r\)
The distance from the origin of the bubble \(\left( {\text{m}} \right)\)

- \(R_{0}\)
Initial bubble wall radius \(\left( {\text{m}} \right)\)

- \(R\)
Instantaneous bubble wall radius \(\left( {\text{m}} \right)\)

- \(\dot{R}\)
Instantaneous bubble wall velocity \(\left( {{\text{m}}\;{\text{s}}^{ - 1} } \right)\)

- \(t\)
Time elapsed \(\left( {\text{s}} \right)\)

- \(T\)
Temperature of the gas inside the bubble \(\left( {\text{K}} \right)\)

## Greek symbols

- \(\dot{\alpha }\)
Ascent rate \(\left( {{\text{N}}\;{\text{m}}^{ - 2} \;{\text{s}}^{ - 1} } \right)\)

- \(\alpha_{\text{b}}\)
Gas solubility in blood \(\left( {{\text{s}}^{2} \;{\text{m}}^{ - 2} } \right)\)

- \(\alpha_{\text{t}}\)
Gas solubility in tissue \(\left( {{\text{s}}^{2} \;{\text{m}}^{ - 2} } \right)\)

- \(\sigma\)
The surface tension of liquid surrounding the bubble \(\left( {{\text{N}}\;{\text{m}}^{ - 1} } \right)\)

- \(\tau\)
Tissue time constant, defined by Eq. (3) \(\left( {\text{s}} \right)\)

## Subscripts

- 0
Initial value quantities

- g
Constants and variables corresponding to the gas bubble

- m
Final or maximum value

- T
Constants and variables corresponding to the tissue

## Notes

## References

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