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for another 5 minutes and finally surfaces. As you can see in figure \ref{fig_tissue4} the pressure 
increases with the predicted exponential form, untile we decrease the ambient pressure. A distinct
 ``knack'' in de the graph marks the start of the off-gassing phase.

\section{Supersaturation limits and M-Values}

So we are now able to calculate inert gas levels and the amount of supersaturation in all tissue
 compartments of the diver. As we stated a certain amount of supersaturation is allowed, without
  developing DCS symptoms. In this section we will define limits applying to supersaturation levels. 
  As we will see these limits depend on:
\begin{itemize}
\item Type (half-time) of the tissue
\item Ambient pressure, i.e. the pressure of the breathing gas (depending on depth and atmospheric pressure)
\end{itemize}

\subsection{Limits according to Haldane}

In 1908 Haldane presented the first model for decompression. He noticed that divers could surface from a depth of 10 meter, 
without developing DCS. He concluded that the pressure in the tissue can exceed the ambient pressure by a factor of 2. 
(Actually the factor the partial pressure of the Nitrogen in the body exceeds the ambient pressure is 0.78*2=1.56, as Workman concluded)

Haldane used this ratio to construct the first decompression tables. Up to 1960 ratio's were used. Different ratio's were defined by various scientists. 
In that period most of the US Navy decompression tables were calculated using this method.  


\subsection{Workman M-values}

At longer and deeper dives, the ratio limits did not provide enough safety. Further research into supersaturation limits was performed by 
Robert D. Workman around 1965. Workman performed research for the U.S. Navy Experimental Diving Unit (NEDU). He found that each tissue compartment had a 
different partial pressure limit, above which DCS symptoms develop. He called this limiting pressure M. He found a linear relationship between this M-value and depth. 
Hence he defined this relationship as: 
\begin{equation}
M  =  M_0 + \Delta M d
\label{workman}
\end{equation}

where

\begin{itemize}
\item $M$    Partial pressure limit, for each tissue compartment (bar)
\item $M_0$    The partial pressure limit at sea level (zero depth), defined for each tissue compartment (bar)
\item $\Delta M$   Increase of M per meter depth, defined for each compartment (bar/m)
\item $d$    Depth (m) 
\end{itemize}

\section{No-decompression times}

A number of tables, like the PADI RDP express no-decompression times. These are maximum times a diver can stay at a certain depth being able to go to the surface 
without the need for decompression stops. Based on equation (\ref{haldane}) we can calculate this time for a particular tissue compartment:

\begin{equation}  
t_{no\_deco}  =   - \frac{1}{k} \ln \left ( \frac{ P_{no\_deco} - P_{alv0} } { P_{t0} - P_{alv0} } \right )
\label{no_deco_time}
\end{equation}

Ofcourse we have to calculate this time for every tissue and take the minimum value as limit for 
the diver to remain at the depth. 

\chapter{Oxygen}

Oxygen is the gas that keeps you alive. The human body needs oxygen to generate energy. We can't do very long 
without it. Oxygen however can also kill you. In certain cases, it becomes poisenous. In recreational diving with
air you will hardly run into the oxygen limits. If you stick to not going deeper than the advised maximum depth
of 40 meters and keeping within the no-decompression limits.

\section{CNS}
There are 2 ways in which oxygen becomes toxic. The first one is directly related to the depth and gas you are breathing.
When the partial pressure of oxgyen (the $pO_2$) exceeds 1.6 bar, you are at direct risk of convulsing.
This would be pretty harmless if you weren't submerged. A diver convulsing is bound to lose his regulator and drown.
So to avoid problems it is advised to keep the $pO_2$ below 1.4 and only use 1.6 when in rest during a decompression stop.


\section{Pulmonary Oxygen Toxicity}

When the lungs are exposed to partial oxygen pressures of 0.5 bar or more, they become irritated. This version of
oxygen toxicity is not life threatening, but a temporary drop in the functioning of your breathing apparatus.
To track the exposure to high oxygen levels, the \bf{Oxygen Tolerance Units} have been invented. These are defined as:
\begin{equation}  
OTU  =   t * \left ( \frac{pO_2 - 0.5} { 0.5 } \right ) ^ { - \frac{5}{6} }     , when pO_2 > 0.5
\label{otu}
\end{equation}

where

\begin{itemize}
\item $pO_2$ partial Oxygen pressure in bar
\item $t$    time in minutes exposed to the $pO_2$
\end{itemize}

Depending on how many days in a row you are diving there are limits on the number of OTU's you can acquire.

\section{CNS}

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