# 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

- Chappell MA, Payne SJ (2006) A physiological model of the release of gas bubbles from crevices under decompression. Respir Physiol Neurobiol 153(2):166–180CrossRefGoogle Scholar
- Gernhardt ML (1991) Development and evaluation of a decompression stress index based on tissue bubble dynamics. University of PennsylvaniaGoogle Scholar
- Gutvik CR, Brubakk AO (2009) A dynamic two-phase model for vascular bubble formation during decompression of divers. IEEE Trans Biomed Eng 56(3):884–889CrossRefGoogle Scholar
- Huber C, Su Y, Nguyen C, Parmigiani A, Gonnermann H, Dufek J (2014) A new bubble dynamics model to study bubble growth, deformation, and coalescence. J Geophys Res Solid Earth 119:216–239CrossRefGoogle Scholar
- Lide David R (2005) CRC handbook of chemistry and physics. CRC Press, Boca RatonGoogle Scholar
- Mohammadein SA, Mohamed KG (2010) Concentration distribution around a growing gas bubble in tissue. Math Biosci 225(1):11–17CrossRefGoogle Scholar
- Mohammadein SA, Mohamed KG (2014) Growth of gas bubbles in the tissues with convective acceleration. Int J Biomath 7(6):1450072CrossRefGoogle Scholar
- Mohammadein SA, Mohamed KG (2016) Growth of a gas bubble in a steady diffusion field in a tissue undergoing decompression. Math Modell Anal 21(6):762–773CrossRefGoogle Scholar
- Mohammdein SA (2014) The concentration distribution around a growing gas bubble in a bio tissue under the effect of suction process. Math Biosci 253:88–93CrossRefGoogle Scholar
- Muth CM, Shank ES (2000) Gas embolism. N Engl J Med 342(7):476–482CrossRefGoogle Scholar
- Papadopoulou V, Evgenidis S, Eckersley RJ, Mesimeris T, Balestra C, Kostoglou M, Tang M, Karapantsios DT (2015) Decompression induced bubble dynamics on ex vivo fat and muscle tissue surfaces with a new experimental set up. Colloids Surf, B 129:121–129CrossRefGoogle Scholar
- Srinivasan RS, Gerth WA, Powell MR (1999) Mathematical models of diffusion-limited gas bubble dynamics in tissue. J Appl Physiol 86:732–741CrossRefGoogle Scholar
- Srinivasan RS, Gerth WA, Powell MR (2000) A mathematical model of diffusion limited gas bubble dynamics in tissue with varying diffusion region thickness. Respir Physiol 123:153–164CrossRefGoogle Scholar
- Srinivasan RS, Gerth WA, Powell MR (2003) Mathematical model of diffusion-limited evolution of multiple gas bubbles in tissue. Ann Biomed Eng 31:471–481CrossRefGoogle Scholar
- Tikuisis P, Gault KA, Nishi RY (1994) Prediction of decompression illness using bubble models. Undersea Hyperb Med 21:129–143Google Scholar
- Van Liew HD, Burkard ME (1993) Density of decompression bubbles and competition for gas among bubbles, tissue and blood. J Appl Physiol 75:2293–2301CrossRefGoogle Scholar
- Vann RD, Butler FK, Mitchell SJ, Moon RE (2011) Decompression illness. Lancet 377(9760):153–164CrossRefGoogle Scholar