DECO THEORY - BUBBLE MODELS

DECO THEORY - BUBBLE MODELS This page descibes pinciples and theoies about bubble geneation and bubble gowth in the scuba dives body and about the effect of bubble fomation on decompession and decompession sickness (DCS, bends) in scuba diving. Wheeas classical (neo-)haldane theoies ae mainly empiical and only take dissolved gas into account, bubble theoies intend to give a physical explanation of the effects of bubbles on decompession. Bubble theoies take dissolved and fee gas into account. Especially the Vaying Pemeability Model (VPM) and Reduced Gadient Bubble Model (RGBM) give good explanation. Histoy In classic decompession theoy accoding to Haldane and successos a cetain amount of supesatuation of the dives tissue with dissolved inet gas is allowed. The dives tissue is divided in a numbe of hypothetical tissue compatments. A cetain limit (M-value) is associated with each compatment to supesatuation levels of dissolved inet gas in the compatment (tissue tension). This theoy suggests efficient decompession by pulling the dive as close to the suface as possible with constaint that in all tissue compatments the supesatuated tissue tension emains within the limits. By pulling the dive as close to the suface the pessue gadient between the supesatuated tissue tension and the pulmonay (o ateial) gas is maximized. This enhances the elimination of the excess gas in the tissue. This theoy is mainly empiical and based on expeiment. At the moment most diving tables and computes ae based on this theoy. Since the ealy days, diving has become moe sophisticated by diving deepe and longe, the use of othe beathing mixtues, etc. Some tech dives have made thei own adaptations to the decompession schedules by inseting depession stops at geate depth ( deep stops, sometimes called Pyle stops afte Richad Pyle). These dives epot feeling bette when using these deep stops. This suggests that classic decompession theoy fails in some situations and cannot be extapolated to evey diving situation. In ode to gain insight in the pinciples of decompession, foming of bubbles duing decompession has been studied fo the last thee decades. This has esulted in new theoies like the Vaying Pemeability Model (VPM) by Yount et al. and the Reduced Gadient Bubble Model (RGBM). Bubble theoies do not only take into account the dissolved gas (like the Haldane models), but also the fee gas in the dives body. In this chapte we will have a look at some featues of bubble theoy. Lots of mathematics will be pesented. The most impotant equations howeve, will be highlighted. Bubbles and suface tension Conside a small ai bubble in a glass of wate. Fo the moment we neglect the solubility of the ai in wate. The small amount of ai within the bubble is suounded by a suface. The suface consists of wate molecules which ae unbound to one side. An unbound molecule epesents moe enegy than a molecule which is completely suounded by othe wate molecules. A suface tension γ is associated with this suface between ai and wate. The suface tension is the amount of enegy pe unit of suface aea and is expessed in J/m 2 o N/m. 1

Figue 1: In equilibium the intenal pessue in the bubble is equal to the sum of the ambient pessue and the skin pessue due to the suface tension A system will always ty to minimize enegy. Suface tension tends to minimise the bubble s suface. Hence, a bubble tends to collapse. Howeve, collapsing a bubble deceases its volume. This will incease the gas pessue in the bubble (Boyle s law), until equilibium is established: the intenal pessue compensates the suface tension. The intenal pessue due to the ambient pessue and suface tension is given by the Laplace equation: P in = P amb + P suf = P amb + 2γ (1) γ P in P amb P suf Radius of the bubble in m Suface tension in joule/m 2 of N/m. The suface tension of wate at 273 K is.73 N/m. Pessue inside the bubble in N/m 2 =1 5 ba Ambient pessue in N/m 2 =1 5 ba Pessue due to the suface tension in N/m 2 =1 5 ba Fom this equation we lean that the smalle the bubble, the highe the pessue inside. You can expeience the adius dependency of the pessue by tying to blow a balloon (bubble pinciples pefectly apply to a balloon up to the point whee the balloon explodes). To get the fist blow of ai into the balloon (small adius) is a hell of a job, wheeas it becomes easie if the balloon becomes lage. Bubbles and diffusion When we have a bottle of bee things get a bit moe complicated (usually the opposite holds, but when we look at the bubbles it might be). Bubbles in bee contain Cabon Dioxide. Thee is also Cabon Dioxide in solution in the bee. Cabon Dioxide can diffuse fom the solution into the bubble o vice vesa, depending on the patial pessue of the Cabon Dioxide in solution and in the bubble. If we assume that the bubble consist of only Cabon Dioxide, the Cabon Dioxide pessue in the bubble is given by equation 1 and depends on the adius of the bubble. We define the patial pessue of the Cabon Dioxide in solution in the bee to be P t (if we egad the bottle of bee as a pimitive model fo a dive, we could call it tissue tension ). If the bottle is closed, the patial pessue of the Cabon Dioxide in solution P t is in equilibium with the ambient pessue P amb, which is the pessue in the bottle. If we assume thee is only Cabon Dioxide gas in the (closed) bee bottle, the bee is satuated with Cabon Dioxide and P t will be equal to P amb (we can neglect hydostatic pessue). The pessue in the bubble P in will be highe than P t due to the suface tension. Gas fom within the bubble will diffuse into solution and the bubble will collapse. So evey bubble will collapse eventually due to this gadient P in P t. This is why in a closed bottle of bee thee ae no bubbles and thee is no foam. Howeve, if we open the bottle things will be diffeent. The ambient pessue will dop, wheeas the value of P t emains the same, at least fo the moment. In this case P t is lage than P amb : the bee is supesatuated with Cabon Dioxide. 2

Given an ambient pessue P amb and the patial pessue P t of the Cabon Dioxide in solution, thee is a citical bubble adius min at which the pessue inside the bubble P i n equals P t. The citical adius can be found by substituting P in by P t in equation 1: min = 2γ P t P amb (2) Fo bubbles which size exceeds this citical size the pessue P in in the bubble is smalle than the patial pessue P t of the Cabon Dioxide in solution. Cabon Dioxide will diffuse fom solution into the bubble. The bubble will gow. Fo bubbles smalle than the citical size, the opposite holds: gas fom the bubble diffuses into solution and the bubble shinks until it collapses completely. Bubbles at the citical size ae in equilibium, though it is an unstable equilibium. This is depicted in Fig. 2. Figue 2: Bubble gowth dependens on the adius So evey bubble with a adius lage than min will stat to gow. When we look at ou opened bottle of bee we see bubbles becoming visible and heading fo the suface, whee they fom foam. If you scutinize a bubble you ll see that it gows duing ascent. Its diamete might have doubled o tipled when it aives at the suface. You might think this is due to Boyle s law. Howeve it takes an ascent of seveal metes fo a bubble to double its diamete. The gowth of the bubble is due to the diffusion descibed above. As an example, we can calculate citical adii fo Spa Baisat Soda (6.4-8. g/l Cabon Dioxide). The pessue in the bottle specified by Spa is shown in next table (dependent on tempeatue). The patial pessue P t of the Cabon Dioxide in solution is oughly that value. If we open the bottle the ambient pessue P amb dops to 1 ba, wheeas the patial pessue P t emains at the high value. Using equation 2 we can calculate the citical adius min. 3

Tempeatue ( C) Pessue (ba = 1 5 P a) min (µm) 15 3.73 2 3.75.53 25 4.5.42 3 5.3.34 35 6.29 4 7.24 The Vaying Pemeability Model Accoding to pevious chapte, in a supesatuated situation any bubble exceeding a citical size min will gow (and will disappea by floating to the suface) and any bubble smalle than this size will collapse. In a nomal non-supesatuated situation, min appoaches infinity. Any bubble will collapse. So we do not expect any bubbles aound afte a while. You might expect that if no initial bubbles ae aound, thee is no bubble to gow on supesatuating the liquid. The tensile stength of wate is estimated on 1 atm, making immense supesatuations possible, befoe bubbles (voids) ae ceated. If no initial bubbles would be pesent in the wate making up the dive, a dive could easily dive to a kilomete depth and pop up to the suface without any poblems. In pactice, this is not the case. Bubbles fom on modest decompession as low as 1 atm. Hee comes in the Vaying Pemeability Model (VPM). The VPM was initially defined by Yount et al. [2] in ode to give a quantitative explanation on the fomation of bubbles in decompessed gelatin [1] (as model fo dives tissue). Late on, they showed this model can be used to calculate dive tables as well [3], [4]. In next paagaphs we will have a look at the gelatin theoy. Late on we will apply the theoy to diving. Figue 3: Skins of vaying pemeability ae the base of the VPM The gelatin expeiments Expeiments on gelatin have been pefomed, by David Yount and othe eseaches [1]. The advantage of gelatin ove wate is that any bubble appeaing duing decompession gets tapped and won t flow to the suface. In this way they can be obseved and counted. Yount applied the udimentay pessue of Figue 4 to gelatin samples: Gelatin samples wee made at ambient pessue Pamb=P of 1 atm. The samples wee apidly compessed in a 1 4

Figue 4: Pessue schedule applied to the gelatin samples by Yount Accoding to the VPM, in aqeous media like wate and gelatin stable gaseous cavities ae pesent. They ae called nuclei. Radii ange fom a few 1/1 m up to aound 1 m. Any nucleus in wate lage will flow to the suface and disappea. Wheeas an odinay bubble with these adii would collapse unde nomal conditions (no supesatuation), these nuclei appea to be exceptionally stable and have a long life. Yount poposed this stability is due to an elastic skin made up of sufactant, as shown schematically in Figue 3. Sufactant consists of (hydophobic) suface active molecules, which ae aligned. Duing the compession stage, these skins ae pemeable fo gas up to a pessue of aound 8 atm. Diffusion though the skin takes place. The pessue P in of the gas in the nucleus is equal to the dissolved gas tension P t in the suounding liquid. Above this pessue, the skin becomes impemeable. Upon decompessing (educing the ambient pessue) the skins ae egaded pemeable. The skin gives ise to a suface compession Γ which opposes the egula suface tension γ of the wate/ai suface, as shown in Figue 5: P in + 2Γ = P amb + 2γ (3) Figue 5: Pessues acting on the suface of the bubble The skin tension Γ is not constant but anges fom to a maximum, which is called the cumbling compession. The idea is that small vaiations of the size of the nucleus can be suppoted by vaying the distance between the molecules in the skin. This gives ise to vaying Γ. This situation is descibed by equation 3 and is efeed to as the small-scale situation. In this equilibium situation and in the pemeable egion, due to 5

diffusion the intenal pessue P i n is equal to the tension P t. In the samples (no hydostatic pessue, 1% Nitogen) P t equals Pamb. So P in = P t = P amb. In this situation Γ equals γ, accoding to equation 3. Upon compessing and decompessing, vaiation of the size of the nucleus becomes to lage to be suppoted by vaying distances between molecules. Sufactant molecules have to be expelled fom o taken up into the skin in ode to compensate fo the aea decease esp. incease of the nucleus. This is schematically shown in Figue 6. The skin is suounded by an amount of sufactant, which is not pat of the skin. This amount acts as a esevoi, taking up o supplying sufactant molecules fom o to the skin. The esevoi molecules ae not aligned and cannot suppot a pessue gadient. Γ takes its cumbling value in this lage-scale situation. Yount poposes two deivations of the VPM [2]: one fom a themodynamic point of view and one fom a mechanical point of view. Figue 6: The lage-scale situation: vaiation in the size of the nucleus esult in expelling molecules fom the skin In the oiginal sample thee is a initial distibution of nuclei with adii distibuted accoding to some function f( ). (The in efes to the initial situation). On applying the pessue schedule, it is assumed that all nuclei with a adius lage than some minimal initial adius min will gow into bubbles. The numbe of bubbles N that occu is given by the integation of f( ) fom min to infinity. N = min f( )d (4) Applying this theoy to a dive, it might be assumed that the seveity of Decompession Sickness (DCS) might be elated to this numbe of bubbles, which occu afte decompession. Hence, min becomes an indication fo the seveity of DCS. It is assumed that no nuclei ae extinguished o ceated duing application of the pessue schedule. Futhemoe it is assumed that the odeing of nuclei is peseved: if one nucleus is lage than an othe one, this is still tue afte a pessue change (odeing hypothesis). At the end of the pessue schedule thee is a new distibution of adii g( f ) and a new adius min f min ends up as a nucleus with adius f min calculations is 1. To define the allowed numbe of bubbles by defining min 2. To find a elation between f and (and hence between min f 3. To find the elation based on min f above which all nuclei will gow into bubbles. Note: a nucleus with adius afte application of the pesue schedule. The aim of next VPM and min ) which govens the bubble fomation on decompession. 4. To calculate the esulting esticting elations fo the pessue schedule, given the value of min and hence, the numbe of esulting bubbles afte application of the pessue schedule. 6

Themodynamic equilibium Fom a themodynamic point of view the left-hand side of equation 3 epesents the skin pessue P S : P S = P in + 2 (5) Γ has been eplaced by the lage-scale value. Similaly the ight hand tem of equation 3 epesents the esevoi pessue P R : P R = P amb + 2γ (6) In the lage-scale situation tanspot of sufactant is not descibed by setting P R equal to P S but by the equiement that the electochemical potential in the skin and esevoi ae equal. The electochemical potential ξ is given by ξ = µ + kt ln (ρ) + pv + Ze ψ (7) ξ Electochemical potential µ Pue chemical potential k Bolzmann contant T Absolute tempeatue in K ρ Molecula concentation o numbe density p Static pessue v Active volume occupied by one sufactant molecule Ze Effective chage of one sufactant molecule ψ Electostatic potential In the esevoi we have and in the skin we have ξ R = µ R + kt ln (ρ R ) + P R v + (Ze ψ) R (8) ξ S = µ S + kt ln (ρ S ) + P S v + (Ze ψ) S (9) Requiing ξ R is equal to ξ S and substituting P S and P R by the values in equation 5 esp 6 esults in: P in + 2 β = P amb + 2γ (1) in which β = 1 v [ kt ln ρ ] R + (µ R µ S ) + (Ze ψ) R (Ze ψ) S ρ S (11) Equations 1 and 11 can be used to calculate the changes in adii afte applying a pessue step. We have a look what happens when applying the pessue schedule of Figue 4 to the sample. At the beginning of the pessue schedule P amb = P. The pessue of the gas in the nucleus is Just befoe compession equation 1 is: P in = P t = P amb = P (12) P + 2 β = P + 2γ (13) Afte the pessue ise to the pessue P whee the skins becomes impemeable equation 1 eads: 7

P + 2 β = P + 2γ (14) Substacting equation 14 fom 13, assuming β = β and ewiting it a bit esult in: [ 1 2( γ) 1 ] = P P (15) We continue to compess apidly fom P to P m. Since the skin is not pemeable now, the pessue in the nucleus vaies with its adius accoding to Boyle s law (P V = P 4/3π 3 = constant): P in = P t 3 3 (16) is the coe- In this equation is the adius of the nucleus at the beginning of the impemeable pocess. Pt sponding dissolved gas tension, which is equal to P. So afte the compession to P m we have: ( ) 3 P + 2 β m = P m + 2γ (17) m m m Assuming β m = β = β, subtacting equation 17 fom 14 and ewiting a bit we end up with: ( 1 2 ( γ) 1 ) [ ( ) ] m = P m P 3 + P 1 m (18) We now have elations (equation 15 and 18) between the adius m of the nucleus afte compessing and the adius pio to compession. The satuation phase (P amb = P s ) that follows satuates the liquid so that finally the dissolved gas tension P ts = P s. We might expect that the adius of the nucleus inceases to its oiginal value (see the the note in the Mechanical Equilibium section). Howeve, this has not been obseved. So we assume s = m (19) In fact the adius estoes quite slowly, but fo the moment equation 19 holds. Pio and afte decompessing (which is fully pemeable) equation 1 eads: esp. P s + 2 s β s = P s + 2γ s (2) P s + 2 f β f = P f + 2γ f (21) Assuming β f = β s (not equal to β m, subtacting equation 21 fom 2 and ewiting a bit esult in: ( 1 2 ( γ) 1 ) = P f P s (22) f s So we now have a elation between all adii of the nucleus duing the entie pofile. A nucleus with adius ends up as a nucleus with adius f though a numbe of stages (, m, s ) defined by the elations 15, 18, 19 and 22. We now define the citeion fo bubble fomation, which is given by the Laplace equation 1: P in P amb = P s P f 2γ f min (23) 8

Thee is no efeence to Γ o in this equation. We assume the skin of the nucleus to be pemeable. So the skin does not estict bubble fomation: gas simply flows though the skin and foms a gas shell outside the skin. If the skin should not be pemeable as has been poposed by othes some teaing stenght o teaing tension Γ = γ T is intoduced. The bubble foming equation becomes: By combining equation 15, 18, 19, 22 and 23 we find the VPM equations. Fo the eve-pemeable egion P m P : P in P amb = P s P f 2 (γ + γ T ) f min (24) P ss min = 2γ γ min + P cush γ (25) Fo the pemeable-impemeable-pemeable situation P m > P we find: [ P min ss = 2γ γ ( ) ] c γ 3 γ + P min m P (26) m In these equations we have the cushing pessue P cush = (P amb P t ) max = P m P (27) and the supesatuation pessue P ss min = (P t P amb ) max = P s P f (28) Equation 26 can be witten as: The paametes and B ae defined as: P min ss = 2γ γ + γ (P P min ) + γ 1 ( ) 1 + (P m P ) (29) B = m (3) 2 ( γ) B = ( ) (31) P m + 1 + m Mechanical equilibium The othe way the VPM is deived is by looking fom a mechanical point of view. Changes in nuclea adius can be calculated by the equation poposed by Love, which eads (in the VPM fom, [2]): 2(Γ γ) 2 = P in P amb (32) In the pemeable egion of the VPM, P in emains constant and equal to P t. Hee P in is. Fo lage-scale vaiations in the pemeable egion of VPM equation 32 eads 2( γ) 2 = P amb (33) 9

In the impemeable egion, P in is given by equation 16. Diffeentiated it eads: Fo the lage-scale vaiations in the impemeable egion, equation 32 eads P amb = (3P 3 ) t (34) (2 ( γ) + 3P t 3 2 4 ) 2 = P amb (35) Togethe with the Laplace equation 23 and the assumption of equation 19, equation 33 and 32 can be used (by integating) to deive the VPM equations 25 and 26: equation 15 and 22 can be obtained by integating equation 33, equation 18 can be obtained by integating equation 35. The deivation is given in [2]. Note: Duing the compession phase P in is zeo in equation 32(24). Assuming the pessue schedule takes place in the pemeable egion, integation of 32 fom P amb = P to P m esults in: ( 1 2 ( γ) 1 ) = P m P (36) m Duing the satuation phase that follows P amb is zeo in equation 32. Integation of 32 fom P i n = P to P s esults in: ( 1 2 ( γ) 1 ) = P P s (37) s m Adding equation 37 to 36 esults in s =, assuming P m = P s (in fact in the non-pemeable situation o a situation in which P m P s we could deive the same esult, though it takes some moe deivation). This suggests that the nucleus is fully estoed to the oiginal size duing satuation. The effect of the cushing is lost in this situation. Howeve, this is in shap disageement with expeiment. Futhe indication is given by special cases in which a pessue spike is pesent at the stat of the schedule (Figue 4), so that P m < P s. In this cases the bubble count only depend on P ss and P cush and not on P s! Hence, the assumption defined by equation 19 is made: s = m. Equilibium consideations In this section we will conside some implications fom the equilibia discussed above. Rewiting equations 13, 14, 17, 2 and 21 gives us: β = 2 ( γ) (38) β = 2 ( γ) (P P ) (39) β m = 2 ( γ) m ( ( ) ) 3 P m P m (4) β s = 2 ( γ) s (41) β f = 2 ( γ) f (P f P s ) (42) (We ecall ou assumptions that β = β m = β and β s = β f.) Accoding to equation 11 β is independent of adius. Howeve, accoding to equation 38 β appeas to be a function of. Assuming that β is constant fo all nuclei at P amb = P we obtain an emakable pediction that inceases with inceasing : 1

= γ + β 2 (43) Anothe consideation stems fom mechanical equilibium: Small scale equilibium is given by equation 3: P in + 2Γ = P amb + 2γ (44) All popeties of the skin and the esevoi ae incopoated in the small scale skin compession Γ. The equation can be obtained fom 1 by setting: 2Γ = 2 β (45) Substituting the β values of equation 38 to 42 in 45 esults at the espective ambient pessue values P amb = P, P, P m, P s and P f in: Γ = γ (46) Γ = γ (47) Γ m = γ m (48) Γ s = γ (49) Γ f =? (5) A plausible small scale/mechanical equilibium citeion fo bubble fomation is that Γ f is less than o equal to zeo. This esults in: 2 ( γ) s 2 f (51) Substituting this in equation 22 esults in the Laplace equation 23 as used fo the themodynamic deivation. Equation 46 to 5 shows that duing the compession Γ inceases. Duing satuation Γ elaxes to its value pio to compession γ, keeping m constant. Duing decompession Γ dops to, the point at which bubble fomation just stats. Consequences of the VPM elations Plotting P ss vs. P cush Most conveniently, equation 25 is plotted as Pss min vs. P cush. In these plots, Pss min P cush pais esulting in the same numbe of bubbles (and hence, the same DCS mobidity) fom staight lines. Dive vs. gelatin The VPM oiginally was developed to quantitatively explain bubble fomation in gelatin duing decompession [2]. The ultimate goal was to gain undestanding of decompession sickness. To apply VPM to a diving situation it fist was suggested that decompession sickness (DCS) symptoms wee elated to the numbe of bubbles. Say, sevee symptoms occu at a numbe NDCS of bubbles in some tissue. Given the adial distibution f ( ), equation 4 defines a min. If all nuclei with a adius equal o lage than this adius gow into bubbles, we end up with NDCS bubbles (and some bad DCS). Given a dive to some depth esulting in an ambient gadient P cush, equation 25 gives the maximum allowed gadient Pss min esulting in the NDCS bubbles 11

The VPM elations The VPM elations 25 and 26-28 define the maximum allowed gadient between the ambient pessue and the tissue tension. In othe wods: it defines the minimum allowed ambient pessue P amb, given the tissue tension P t. In a diving situation it defines the depth the dive is allowed to ascend given the tissue tension. The elation should be applied to each tissue compatment of the dive. The initial compession (defining P cush ) is impotant fo Pssmin. Duing this stage nuclei ae cushed to a smalle size, making them less active in bubble fomation. The secet lies in equation/assumption 19, which states that no egeneation of the bubble size takes place duing satuation. It implies that a descent duing a dive should be as quick as possible, the deepest pat of the dive should be at the stat of the dive and deepe dives should pecede shallowe dives in a epetitive dive situation. These facts have been empiically found duing a centuy of decompession eseach. Not a 1% Nitogen satuation dive The deivation of the VPM assumed 1% Nitogen and fully satuated gelatin. If we apply the equations to a non-satuating diving situation in which the Nitogen faction is less than 1% (fo example ai, containing 79% Nitogen), the VPM equations 25 and 26-28 can be egaded as a consevative estiction to the dive pofile. Applying VPM to diving In this section we will apply the VPM to a diving situation and descibe a method to geneate diving tables. Wheeas the VPM theoy of pevious sections applies to a special situation of fully satuated gelatin in a 1% Nitogen atmosphee, situations duing diving ae diffeent. The assumption that the seveity of DCS is popotional to the absolute numbe of bubbles leads to vey safe diving tables, not coveing all of the conditions of moden diving tables and often leading to unacceptable long decompession peiods. The VPM was efomulated, as descibed in this section, to fit it with conventional diving tables. Conventional diving tables wee egaded as valid measuements. We will follow the deivation of Yount [3]. Duing this deivation we assume only one inet gas. Late on we will place emaks on using moe inet gasses (Timix, etc). Anothe assumption is the dive takes place in the pemeable egion of the VPM. The efomulated VPM The deivation of the theoy below is based on a numbe of moe o less ad hoc assumptions. The most impotant assumptions concen the elationship between decompession symptoms and the amount of fee gas (bubbles) in the dives tissue: 1. Thee is an amount of bubbles N safe which can be toleated by the dives body, independent of all cicumstances (like tissue tension, degee of supesatuation, etc). The initial citical adius coesponding to this numbe is min (equation 4). 2. The actual numbe of bubbles N actual may be highe than N safe as long as the total volume V of all fee gas always emains below a citical value V cit. This is called the citical-volume hypothesis. A initial adius new smalle than min is associated with this numbe. 3. The volume of fee gas V inflates at a ate popotional to P ss (N actual N safe ), whee P ss is the satuation P t P amb. The fist of these assumption agees with physiological studies, which state that the lungs ae able to continue functioning as a tap fo venous bubbles to a cetain degee. Fom this assumption can be deduced that the ate at which the body can dissipate fee gas by exchange in the lungs is popotional to both the supesatuation pessue P ss and N safe. The assumption defined by equation 19 is fine tuned accoding to obsevations: the adius m slowly egeneates duing satuation instead of emaining unchanged, as stated by equation 19. The egeneation is exponential, govened by a egeneation time constant τ R : ) s (t R ) = m + ( m ) (1 e t R τr (52) 12

s m t R R Nuclea adius just pio to ascent and decompession (m) Nuclea adius afte compession by P cush (m) Nuclea adius befoe descent (m) Regeneation peiod: time fom stat of dive up to stat of ascent and decompession (min) Regeneation time constant (min) If we wait long enough the cushed nucleus will end up with its initial adius pio to compession. In contast with othe decompession models, VPM takes the effect of othe gasses (wate vapo, Oxygen, Cabon Dioxide) into account in calculating the tissue tension: P t total P inet gasses P t total = P inet gasses + P othe gasses (53) Tissue tension Sum of the patial pessues of the dissolved inet gasses P othe gasses Pessue due to wate vapo, Oxygen and Cabon Dioxide. Yount specifies a nealy constant value of 12 mm Hg (coesponding to.136 ba) fo inspied patial Oxygen pessues up to 2 atm [5] The supesatuation is now defined as: The efomulated VPM now consist of the following steps: P ss = P t + P amb (54) 1. Specify the paametes defining the VPM: suface tension γ, the cumbling compession, the minimum initial adius min, the egeneation time constant τ R and a composite paamete λ. The latte is elated to the citical volume V cit. The paametes ae the same fo each compatment. 2. Calculate the initial allowed supesatuation that is just sufficient to pobe min and that esults in N safe bubbles. The equation fo this is: P min ss = 2 γ γ s t R (55) In fact, this is an enhanced equation 25, taking nuclea egeneation into account. In this equation the egeneated adius s (t R ) is given by equation 52. Since the VPM paametes ae the same fo each tissue compatments, this initial allowed supesatuation gadient will be the same fo each compatment. 3. Calculate a decompession pofile, using this Pss min. The total decompession time defined by the pofile is t D. 4. Calculate a new allowed supesatuation gadient P new ss using: Pss new = 1 [ b + ( b 2 4c ) ] 1/2 (56) 2 whee b = P min ss + λγ ( td + 1 k ) (57) ( ) 2 γ λp cush c = t D + 1 k (58) In these equation is k = ln(2)/τ, whee τ is the half-time of the tissue compatment. This esult in a lage allowed supesatuation gadient Pss new. Of couse, this step is epeated fo each tissue compatment. 5. Pefom a numbe of iteation of step 3-4, until t D and Pss new ae now substituted by P new ss. convege. Of couse, occuences of P min ss 13

Afte the iteations we end up with a moe sevee decompession pofile and a Pss new coesponding to a new initial citical adius new, which is smalle than min. This new adius esults in a lage numbe of bubbles N actual and a maximum volume of fee gas appoaching V cit. Moe inet gasses In some (tech) diving situations, othe gas mixtues ae used consisting of moe than one inet gas (fo example Timix, containing Oxygen, Nitogen and Helium). In next emaks we assume Helium and Nitogen to be the inet gases. 1. Fo each gas, the VPM paametes should be specified. Fo each tissue compatment a half-time fo each gas should be specified. 2. Fo each gas, the allowed supesatuation gadient should be calculated using the method in pevious section. In this case the supesatuation gadient fo Helium is P ss He and fo Nitogen is P ss N 3. If P t He and P t N ae the Helium and Nitogen tissue tensions, the total tissue tension is given by: P t total = P t He + P t N + P othe gasses (59) 4. The allowed supesatuation gadient is given by the weighted aveage: P ss total = P t total P amb = P t HeP ss He+Pt N P ss N P t He + P t N (6) Deivation In this section we will deive the new VPM equations 56-58. The allowed supesatuation gadient Pss min as given by equation 25, 26 and 55 can be applied to diving as a safe-ascent citeion. Wheeas they can be deived diectly fom VPM, the deivation of Pss new in equation 56-58 involves a numbe of ad hoc assumptions. Assumption 1: The total volume of fee gas in the dives body should neve exceed a citical volume value V cit at any time t (not duing the dive, no theeafte). Assumption 2: The ate at which the fee gas inflates is popotional to P ss (t)(n actual N safe ). In this equation P ss(t) = P t (t) P amb (t). Assumption 1 and 2 esult in the decompession citeion: t P ss (t) (N actual N safe ) dt αv cit (61) In this equation is α a popotionality constant. This citeion should hold fo any t. To minimise the decompession time t D, the sign is eplaced by the = sign. Assumption 3: The actual numbe of bubbles N actual and the numbe of bubbles always allowed N safe ae detemined by the initial decompession stop and emain constant theeafte. The decompession citeion now eads: tmax αv cit = (N actual N safe ) P ss (t)dt (62) In this equation t max is the value of t at which the integal eaches the maximum value. Assumption 4: The decompession pofile is chosen so that P ss (t) emains at constant value Pss new duing the ascent peiod t D and decays exponentially to zeo theeafte (at the suface). This is in ageement with Assumption 3: P ss (t) is always positive and neve exceeds its initial value Pss new. This initial value is the maximum value defining N actual. The latte emains constant theeafte. The exponential decay to zeo is a consevative appoximation: accoding to Yount&Lally [5] humans ae inheently unsatuated when equilibated at atmospheic pessue by about 54 mm Hg (.72 ba). Eventually, P ss (t) will become negative by this amount. Due to Assumption 4 and the exponential decay, the integal of equation 62 eaches it maximum value in the limit as t max appoaches. The citeion fo decompession now becomes: 14

td αv cit = (N actual N safe ) Pss new (t)dt + t D P new ss e k(t t D) (63) In this equation is k = ln(2)/τ whee τ is the tissue compatment halftime. Assumption 5: The distibution of nuclei in humans is not known. assumed, obseved in vito: An decaying exponential elation is N actual = N e β S new 2kT (64) β N S k T VPN Constant Nomalisation constant Constant aea, occupied by one sufactant molecule in situ Bolzmann contant Absolute tempeatue in K The decompession citeion can be ewitten: N safe = N e β S min 2kT (65) whee P new ss = αv cit (N actual N safe ) ( t D + 1 k ) (66) (N actual N safe ) = N [ e β S new 2kT ] e β Smin 2kT (67) Assumption 6: The exponential aguments in equation ae small enough so that they can be expanded. Accoding to [3] this appoximation is in some question, since the model paametes ae not fixed no well known. The tue distibution is unknown. Accoding to this assumption equation 67 becomes: (N actual N safe ) N β S min 2kT ) (1 new min (68) Substituting 68 in 66 and ewiting this a bit esults in: ( N β S 1 2kT min new 1 ) ( min t D + 1 ) P new k ss αv cit 1 new = (69) The adii new and min can now be eplaced using the VPM equations (ewiting 38-42 and 6): min = 2 γ β (7) 1 (Pss min min = 1 new = (P new ss P cush γ ) 2γ ( γ) P cush γ ) 2γ ( γ) P cush is hee P m P. The equations 7-72 apply to the pemeable egion. Applying them to the impemeable egion esults in an acceptable eo of only 3% fo values of P cush below 1 ba. Substituting min, min 1 and new 1 in the elation 69 by the equations given in 7-72 esults in: N ( γ) S kt ( P new ss P min ss ) P new ss (71) (72) ( t D + 1 ) ( ) αv cit Pss new γ P cush (73) k 15

Rewiting this leads to the quadatic equation: apss new2 bpss new + c = (74) a = 1 (75) b = P min ss + λγ ( td + 1 k ) (76) ( ) 2 γ λp cush c = t D + 1 k (77) Equation 56-58 is the solution of equation 74-77, whee: λ = αv cit kt γn ( γ) S (78) Paamete values The Yount aticle [3] epots the following paamete values: Paamete Value γ 17.9 dyn/cm =.179 N/m 257 dyn/cm = 2.57 N/m τ R 216 min min 8 µm λ 75 fsw min = 25 ba min Some adaptation to VPM In the Yount/Maiken/Bake aticle [4] VPM is applied to evese dive pofiles (P ss < P cush ). Some adaptations ae made to the VPM. Fist, the descent is not assumed to be instantaneous but takes place at a cetain ate. Duing descent gas is loaded into the tissue compatments, leading to a smalle P cush value than on instantaneous descent (no gas loading). This effects the faste tissues moe than the slowe ones. Using the Scheine equation one can deive a new, moe geneal vesion of equation 55 fo compatment j: Pss new j = 2γ ( γ) min + γ j (79) j The effects of nuclea egeneation have not been taken into account in this equation. In this equation the set of effective cushing pessues j is given by j = P cush (1 Q N2 ) + Q N2R c k j ( 1 e k jt c ) (8) In this equation is Q N2 the Nitogen faction in the beathing gas mixtue and R c is the cushing change ate of the patial Nitogen pessue. In the case of apid descent, whee in the limit t c appoaches, R c t c appoaches P cush and j goes to P cush. This esults in the oiginal equation 25. In the Yount/Maiken/Bake aticle [4] equations 56-58 has been eplaced by: whee Pss new = 1 [ b + ( b 2 4c ) ] 1/2 (81) 2 16

( ) b = Pss min j + λγ P dive tj P m td ( ) ) (82) t D + 1 k j 2 (t D + 1kj ( ) 2 γ λp cush c = t D + 1 k j P ( ss min j P dive tj 2 ) P m td ) (83) (t D + 1kj In this equation Ptj dive denotes the set of compatment tissue tensions. The last tems have been added to b and c, compaed to equations 56-58. These tems become zeo fo satuated, not-metabolizing systems, whee Ptj dive is P t P m. The Reduced Gadient Bubble Model Buce Wienke extended the VPM by incopoating epetitive diving, including multi day diving [6]. This esulted in the Reduced Gadient Bubble Model (RGBM). In this section we follow the deivation given in [6]. Following equation 63, the citical volume hypothesis states fo J epetitive dives becomes: J j=1 [ td tsj (N actual N safe ) Pss new dt + J j=1 [ tsj (N actual N safe ) Pss new t D + ] Pss new e kt dt < αv cit (84) ] Pss new e kt dt < αv cit (85) In this equation t sj is the time of the suface inteval afte the j th dive. In [6] Wienke uses G (of Gadient) instead of Pss new. Hence, we followed this Yount notation so fa, we keep up doing so. Afte the last dive no moe dives ae made. Hence t sj goes to infinity. Rewiting equation 84 (intoducing N = N actual N safe ) esults in: We now define G j : Fo j = 1: J j=1 [ NPss (t new D + 1k 1k )] ( + NP new e ktsj ss t DJ + 1 ) < αv cit (86) k Fo j = 2..J: ( NG j t Dj + 1 ) ( = NPss new t Dj + 1 ) k k Using equations 87 and 88, equation 86 now eads: J j=1 NG 1 = NP new ss (87) NPss new 1 k e kt D(j 1) (88) ( [ G j t Dj + 1 )] < αv cit (89) k One impotant popety of G is: G j P new ss (9) 17

Compaing this esult to equation 63, equation 9 teats the dives as if they wee independent dives. Howeve, to account fo the influence of the pevious dives, educed gadients ae used fo subsequent dives. The educed gadients can be witten as: with G j = ξ j P new ss (91) ξ j 1 (92) We will look at the factos that influence ξ. Howeve, [6] only shows the esulting equations, not the deivation. Regeneation Duing the suface intevals bubble sizes egeneate. η eg j = N ( ) t cum j 1 = e τ Rt cum j 1 (93) N Revese diving pofiles η exc j = (94) Repetitive dives ξ j = η eg j ηj exc η ep j = ( N) [ max 1 ( N) j η ep j = (95) (1 Gmin P new ss ) e τ M t j 1 ] e τ Rt cum j 1 (96) Refeences [1] David E. Yount, Richad H. Stauss, Bubble fomation in gelatin: A model fo decompession sickness, Jounal of Applied Physics, Vol. 47, No. 11, Novembe 1976, p581-589 [2] David E. Yount, Skins of vaying pemeability: a stabilization mechanism fo gas cavitation nuclei, Jounal of the Acoustical Society of Ameica 65 (6), june 1979, p1429-1439 [3] David E. Yount, D.C. Hoffman, On the use of a bubble fomation model to calculate diving tables, Aviation, Space and envionmental medicine, feb. 1986, p149-156 [4] David E. Yount, Eic B. Maiken, Eik C. Bake, Implications of the Vaying Pemeability Model fo Revese Dive Pofiles, pesented at the Revese Dive Pofiles Wokshop Octobe 29 and 3, 1999 Smithsonian Institution, Washington DC. See http:www.phys.hawaii.edu/ dey. [5] David E. Yount, D. A. Lally, On the use of oxygen to facilitate decompession, Aviation, Space and envionmental medicine, 198, 51 page 544-55. [6] Buce R. Wienke, Abyss/educed gadient bubble model: algoithm, bases, eductions, and coupling to ZHL citical paametes. 18