Decompression theory - Calculating deco profiles

ABSTRACT
This section describes a procedure for calculating decompression profiles. Under certain circumstances a diver has to follow a decompression (depth) profile in order to ascend safely to the surface. A decompression profile comprises a number of stops for a certain period at certain depths. During the stops excess dissolved gas is removed from critical parts of the divers body by the respiratory and circulatory systems.

The procedure

At the end of the dive the diver wants to return to the surface. Each of the divers tissue compartments is loaded with inert gas (e.g. Nitrogen) to a certain level (which generally differ per compartment). The amount of inert gas may be high, inhibiting the diver to go up to the surface at once. The diver has to decompress (get rid of excess inert gas) during the ascent in order to return to the surface safely. To do so the diver includes a number of stops at certain depths during the ascent. The stop depths are chosen so that:

The partial pressure in all tissue compartments stays within the limits.
The ambient partial pressure is lower than partial pressure in the tissue (at least in the limiting tissue compartments): a gradient exists.

The diver remains at the stop depth for a certain amount of time. Due to the gradient the tissue gets rid of excess inert gas, lowering the partial inert gas pressure in the tissue compartments. When this process has proceeded long enough, the diver may ascend further to the next stop. For the ease of use, usually a number of fixed depths is used (for example 3, 6, 9, 12, etc meters).

Tissue Compartments

At the end of the dive we know:

The inert gas level (tissue tension) in each tissue compartment
The allowed excess level (supersaturation) for the inert gas for each compartment, as a function of depth. This level is the amount the tissue tension P_t is allowed to exceed the ambient pressure P_amb and expressed as P_t-P_amb. In conventional neo-Haldane decompression theory these levels are defined by the M-values. In the Varying Permeability Model (VPM) these level are defined by P_ss^min.

In general the saturation levels (at the end of the dive) as well as the allowed excess levels differ per tissue compartment. This results in a minimum allowed depth to which the diver is allowed to ascend for each compartment. When calculating decompression profiles, you always have to choose the most conservative depth. This is the largest depth. The corresponding compartment is known as the limiting compartment or leading compartment.

Calculating the decompression profile

The following steps have to be applied in order to calculate a decompression profile:

Calculate the partial pressure (tissue tension) in each tissue compartment at the end of the dive (just prior to the ascent).
For each compartment, calculate the minimum depth d_min to which the diver is allowed to ascent.
From all the values d_min obtained in step 2, choose the largest depth.
Round the largest depth calculated in step 3 to the first, deeper fixed decompression stop depth. This will be the depth for the first decompression stop.
Let the diver ascend to the decompression stop. Calculate the gas loading during this ascent.
For each compartment, calculate the period the diver has to spend at this depth before proceeding to the next, shallower fixed decompression stop depth.
From all the values obtained in step 6, choose the largest period.
Round up this period to a fixed period boundary.
Update the divers tissue tension in each tissue compartment for the stay at the decompression stop depth.
Repeat step 5-9 for each decompression stop depth until the surface is reached.

1. Calculate the gas loading of each tissue compartment at the end of the dive
Calculation of the partial pressure of the inert gas in the tissue compartment can be done by applying the exponential Haldane equation or the Schreiner equation to each stage of the dive. This result in a partial pressure value for each compartment, just prior to ascending.

2. Calculate the minimum depth for each compartment to which the diver can ascend
For each compartment the minimum depth can be calculated to which he can safely ascend. This is the depth at which the supersaturation of the tissue compartment is equal to the largest allowed supersaturation for that compartment. Using the classical decompression theory of M-values, this depth (in meter) is given by:

d _min =

P _t - M ₀

ΔM

(1)

P_t: Compartment Tissue tension at the end of the dive (bar)
M: Partial pressure limit for the tissue tension at depth d (bar)
M₀: The partial pressure limit at sea level, depending on tissue compartment (bar)
ΔM: Increase of M per meter depth, depending on tissue compartment (bar/m)
d_min: Minimum allowed depth for the diver (m)

Using VPM, this depth is given by:

d_min = (P_amb - P_{amb_sea_level}) 10 m/bar

d_min = (P_t - P_ss^min - P_{amb_sea_level}) 10 m/bar. (in meter)

(2)

P_amb: Ambient pressure (bar)
P_{amb_sea_level}: Ambient pressure at sea level (atmospheric pressure) (bar)
P_t: Compartment Tissue tension at the end of the dive (bar)
P_ss^min: Maximimum supersaturation gradient defined by VPM (bar)
d_min: Minimum allowed depth for the diver (m)

These relationships assume instantaneous ascent. In practice divers ascend slowly. This means that when the diver reaches the stop depth calculated above, its tissue may have been off gassed during the ascent that much that he may proceed to next stop depth immediately. We can take into account the ascend rate in calculating the first stop depth using the Schreiner equation. Using M-values this leads to the following equation governing the decompression:

P_t ≤ M

P_alv0 + R(t-1/k) - (P_alv0 -P_t0-R/k) e^-kt ≤ M₀ + ΔM d_min

(3)

Furthermore we know that

d_min - d_start = R_amb t

(4)

P_t: Tissue tension during the ascent (bar)
P_alv0: Alveolar pressure at the start of the ascent (bar)
P_t0: Compartment Tissue tension at the start of the ascent (bar)
d_min: Minimum allowed depth for the diver (m)
d_start: Depth at start of the ascent (m)
R: Rate of change of alveolar partial inert gas pressure, equal to R_amb Q (Q is the inert gas fraction) (bar/min)
R_amb: Rate of change of the ambient pressure. Negative on ascent (bar/min)

Equation (3) and (4) can only be solved numerically (as far as I know) for d_min, substituting ≤ by =. Another option is to calculate equation (3) for each fixed deco stop depth and verify which is the shallowest depth for which equation (3) still holds. This would be the first deco stop depth.

3. Choose the deepest value for dmin
For each tissue compartment a d_min value is obtained. These values are not equal, generally. The deepest value should be chosen as depth to which the diver safely can ascend (ceiling). At this depth, the corresponding (limiting) tissue compartment is at maximum supersaturation, whereas the other compartment are below maximum supersaturation. Ascending beyond this depth would probably cause decompression problems in the limiting tissue compartment.

4. Round this minimum depth to the closest, higher fixed decompression stop depth
For the ease of use fixed depths are chosen for decompression stops. Since the diver should not ascend above the ceiling, the closest fixed decompression stop depth which is deeper than the ceiling is chosen.

5. Let the diver ascend to the stop depth
The tissue tension in each compartment is updated for the ascent to the decompression stop. A maximum ascending speed is defined for the decompression. The Schreiner equation is used for the ascent.

6. For each compartment, calculate the time to stay at the decompression stop depth
The time period has to be calculated which the diver has to stay at the decompression stop depth. This is the time between reaching the decompression stop depth and the time the diver is allowed to go up the next, shallower decompression stop depth. The principle of this calculation for each tissue compartment is to

Calculate the maximum allowed tissue tension at the next, shallower stop. Since the depth of next stop is shallower, this limit is smaller than the tension of current limiting tissue compartment.
At current decompression stop depth, calculate the time it takes for the tissue tension to drop to this new limiting value.

When using M-values the tissue tension limit is given by the M-value:

M = M₀ + ΔM d

(5)

M: Partial pressure limit for the tissue tension at depth d (bar)
M₀: The partial pressure limit at sea level, depending on tissue compartment (bar)
ΔM: Increase of M per meter depth, depending on tissue compartment (bar/m)
d: Depth of next, shallower decompression stop depth (m)

When using VPM the tissue tension limit is given by

M = P_amb + P_ss^min

(6)

M: Partial pressure limit for the tissue tension at depth d (bar)
P_amb: The ambient pressure at the next, shallower depth (bar)
P_ss^min: Maximum supersaturation gradient defined by VPM (bar)

For the computation of the time it takes for the tissue tension to drop to the limiting value (at the next, shallower decompression stop depth), the Haldane equation can be used. The equation has to be rewritten a bit:

t_{deco_step} = -1/k ln[ (M - P_alv0) / (P_t0 - P_alv0) ]

(7)

t_{deco_step}: Time to spend at current decompression stop depth (min)
k: Tissue compartment constant: k=ln(2)/τ (bar)
M: The partial pressure limit for ascending to next, shallower depth (bar)
P_alv0: Alveolar pressure at current decompression stop depth (bar)
P_t0: Tissue tension at the start of current decompression stop (bar)

7. Choose the largest period
The ultimate time to stay at current decompression stop depth is given by the largest time calculated in previous step. If the diver stays that long at current decompression stop depth, the tissue tension in each compartment is low enough to stay within limits on ascending to next, shallower decompression stop.

8. Round up the decompression stop period
It is convenient to add the deco stop time (found at step 7) to the run time (time measured from the start of the dive) and round this run time up to the next (larger) minimum deco stop time multiple (usually one minute).

9. Update the diver's tissue tension in each compartment
At the end of the decompression stop, the tissue tension has to be updated in each tissue compartment. This is done by applying the Haldane equation to the period spent at the stop.

10. Repeat the steps 5-9 for each decompression stop
The diver proceeds to next, shallower fixed decompression stop depth. Steps 5-9 are repeated. All stops are processed this way, until the diver reaches the surface.

A deco profile example

This example is for purpose of demonstration only. It does not apply to a real situation. One shall never apply the data presented to a real diving situation!

In this section we will calculate a decompression profile. For the ease of calculation we will only consider two tissue compartments, a 8 minutes compartment (k=ln 2/8=0.0866) and a 38.3 minutes compartment (k=ln 2/38.3=0.0181). This will show the principles and can easily be extended to real compartment models using more compartments.

We consider a dive from sea level to a depth of 48 meter. The dive consists of two sections: a 6 minutes descent at a constant rate of 8 meter per minute to 48 meters depth and a stay at 48 meter for 38 minutes. So after 44 minutes the diver decides to ascend and head for the surface. The diver breathes normal compressed air (fraction Nitrogen Q_N2=78%). This dive is not a repetitive dive. For the two tissues we use the Bühlmann ZH16-L model:

Compartment half time (min)	k = ln(2)/τ	M₀ (msw)	ΔM (msw/msw)
8	0.0866	25.4	1.5352
38.3	0.0181	17.8	1.1857

For the decompression we define a maximimum ascending speed of -9 meter/minute (negative, since the depth and hence the ambient pressure, decreases). Furthermore, we like to define deco stops at depths which are a multiple of 3 meters.

1. Calculate the gas loading of each tissue compartment at the end of the dive
At the end of the 6 minutes descent the tissue compartment tension is given by the Schreiner equation. First we specify a few components in the Schreiner equation. The alveolar N₂ pressure is given by the alveolar ventilation equation:

P_alv = [ P_amb -P_H2O + (1-RQ)/RQ P_CO2 ] Q_N2

(8)

Using RQ=0.9, P_CO2=0.0534 bar and P_H2O=0.0627 bar this equation result in P_alv0=0.736 bar at sea level. The alveolar partial pressure rate of change R is given by R = Q_N2 R_amb = 78% 8 m/min 0.1 bar/m = 0.624 bar/min. R_amb is the ambient pressure rate of change. The tissue tension in each compartment P_t0 before the dive is equal to the alveolar partial pressure. The Schreiner equation can be applied now:

P_t=P_alv0 + R(t-1/k) - (P_alv0 -P_t0-R/k) e^-kt
P_{t_8} = 0.736 + 0.624 (6-1/0.0866) - (0.736 - 0.736 - 0.624/0.0866) e^{- 0.0866 6}
P_{t_8} = 1.560 bar
P_{t_38} = 0.736 + 0.624 (6-1/0.0181) - (0.736 - 0.736 - 0.624/0.0181) e^{- 0.0181 6}
P_{t_38} = 0.932 bar

(9)

The gas loading during the stay at 48 meters is given by the Haldane equation. The alveolar pressure at a depth of 48 meters is given by the alveolar ventilation equation (8), using P_amb = 48 meter times 0.1 bar/meter + 1 bar = 5.8 bar. The alveolar pressure given by equation (8) is P_alv0 = 4.480 bar:

P_t=P_alv0 + (P_t0 - P_alv0) e^-kt
P_{t_8} = 4.480 + (1.560 -4.480) e^{- 0.0866 38} = 4.371 bar
P_{t_38} = 4.480 + (0.932 -4.480) e^{- 0.0181 38} = 2.696 bar

(10)

The 1st step is finished: we know the partial N₂ pressures in all tissue compartments at the end of the dive. The dive is shown in the figure below, which displays the entire decompression profile. The yellow line shows the ambient pressure. The purple line corresponds to the partial N₂ pressure in the alveoli (equation (8)). The thick lines represent the partial N₂ pressure in the 8 minutes (green) and 38.4 minutes (blue) tissue compartments.

Figure 1: The example decompression profile

2. Calculate the minimum depth for each compartment to which the diver can ascend
We use equation (1) to calculate for both compartments the minimum depth to which the diver is allowed to ascend so that the supersaturation stays within the limits defined by the M-values.

d_min=(P_t-M₀)/ΔM (in meter)
d_{min_8}=(4.371 10 meter/bar - 25.4) / 1.5352= 11.93 m
d_{min_38}=(2.696 10 meter/bar - 17.8) / 1.1857= 7.73 m

(11)

As we can see the 8 minutes tissue compartment is the limiting compartment, since it according to this compartment one is allowed only to ascend to 11.93 meter, whereas the 38.3 minutes compartment allowes ascending to only 7.73 meter. As you can see in the graph this is to be expected, since the faster 8 minutes compartment has loaded much more gas during the dive than the 38 minutes compartment.

3. Choose the deepest value for d_min
As stated, the deepest d_min value is the one corresponding to the 8 minutes tissue. This value d_{min_8} = 11.93 m will be the base for the first stop.

4. Round this minimum depth to the closest, higher fixed decompression stop depth
For the deco stops we use depths which are a multiple of 3. The first deco stop depth is the rounded up value of 11.93 meters, which is 12 meters.

5. Let the diver ascend to the stop depth
The diver will ascend to the first deco stop depth (12 meter as calculated in previous step) at a speed of -9 meter/minute. It wil take 48-12/9=4 minutes to reach the first deco stop depth. The run time at which the diver reaches the first stop depth is 44+4=48 minutes. The alveolar pressure change rate R = Q_N2 R_amb = 78% -9 m/min 0.1 bar/m = -0.702 bar/minute. We first calculate the tissue tensions at the moment of arrival at the stop depth by using the Schreiner equation:

P_t=P_alv0 + R(t-1/k) - (P_alv0 -P_t0-R/k) e^-kt
P_{t_8} = 4.480 - 0.702 (4-1/0.0866) - (4.480 - 4.371 + 0.702/0.0866) e^{- 0.0866 4}
P_{t_8} = 3.968 bar
P_{t_38} = 4.480 - 0.702 (4-1/0.0181) - (4.480 - 2.696 + 0.702/0.0181) e^{- 0.0181 4}
P_{t_38} = 2.721 bar

(12)

6. For each compartment, calculate the time to stay at the decompression stop depth
At the next shallower stop depth (12-3=9 meter) the maximum allowed tissue tension for a compartment is given by the M-Value equation (5):

M=M₀+ ΔM d
M₈ = 25.4 0.1 bar/m + 1.5352 0.1 bar/m 9 = 3.922 bar
M₃₈ = 17.8 0.1 bar/m + 1.1857 0.1 bar/m 9 = 2.847 bar

(13)

We now know the tissue tension at the start of the stop (step 5) and at the end of the stop. We use equation (7) to calculate the time at the deco stop.

t_{deco_step} = -1/k ln[ (M - P_alv0) / (P_t0 - P_alv0) ]
t_{deco_step_8} = -1/0.0866 ln[ (3.922 - 1.672) / (3.968 - 1.672) ] = 0.24 min
t_{deco_step_38} = -1/0.0181 ln[ (2.847 - 1.672) / (2.721 - 1.672) ] = -6.25 min

(14)

The stop depth defined by the 38.3 minutes compartment is negative. This indicates that this tissue is not a limiting factor for ascending to the next deco stop depth. This is consistent with the minimimum depth of 7.73 meters for this tissue calculated in step 2: the next deco stop depth of 9 meters is well below this minimum depth.

7. Choose the largest period
We've seen the largest period is for the 8 minutes compartment. It is 0.24 minutes.

8. Round up the decompression stop period
We add the deco stop time (step 7) to the run time and round this to the nearest larger minute value to get the run time to head for the next decompression stop depth: t=round_up(48+0.24)=49 minutes. So the first deco stop is dominated by the 8 minutes tissue compartment. It is however a fairly short stop of 1 minute.

9. Update the diver's tissue tension in each compartment
During the stay at the deco stop depth the tissue compartment partial pressures change according to the Haldane equation:

P_t=P_alv0 + (P_t0 - P_alv0) e^-kt
P_{t_8} = 1.672 + (3.968 - 1.672) e^{- 0.0866 1} = 3.777 bar
P_{t_38} = 1.672 + (2.721 - 1.672) e^{- 0.0181 1} = 2.703 bar

(15)

8. Repeat the steps for each decompression stop
The next stop is at 3 meter shallower than current deco stop depth. From this point on we follow the steps from point 5 until we reach or surface. In this way we find the following profile parameters:

run time	depth (m)	action	P_{t_8} (bar)	P_{t_38} (bar)	M₈ (bar)	M₃₈ (bar)	t_{deco_8} (min)	t_{deco_38} (min)
0'	0	descend	0.7357	0.7357
6'	48	dive	1.560	0.932
44'	48	ascend	4.371	2.696
48'	12	1st deco	3.968	2.721	4.38	3.20	0.29	-6.25
49'	12	ascend	3.777	2.703
49'20"	9	2nd deco	3.714	2.696	3.92	2.85	1.36	9.79
60'	9	ascend	2.341	2.475
60'20"	6	3rd deco	2.312	2.468	3.46	2.49	-5.58	16.85
78'	6	ascend	1.444	2.122
78'20"	3	4th deco	1.433	2.116	3.00	2.14	-14.08	19.16
98'	3	ascend	1.020	1.690
98'20"	0	surface	1.015	1.685	2.54	1.78

This data is shown in the graph of Figure 1. Whereas the 12 meter deco stop is dominated by the 8 minutes tissue, from the 9 meter deco stop depth on the 38.3 minutes tissue is the limiting tissue, dominating the deco profile.

Notes

Most efficient decompression profile

Applying deco stops is not the most efficient way of decompressing, though it might approximate the most effienct deco profile quite well.

Last modified on March 13 2005 22:51:32.
Copyright 1999-2008 Deep Ocean
Please read the disclaimer