From 37284d95779aa440ca7c6da4a7a9ca38a73a5c75 Mon Sep 17 00:00:00 2001
From: "W. Trevor King" <wking@drexel.edu>
Date: Tue, 15 Mar 2011 14:11:39 -0400
Subject: [PATCH] Add pulse oxymetry post and updated master Makefile for SCons
 builds.

---
 .gitignore                                |  31 +-
 Makefile                                  |  22 +-
 posts/Pulse_oxymetry.mdwn                 |  18 +
 posts/Pulse_oxymetry/SConstruct           |  11 +
 posts/Pulse_oxymetry/fig/SPO2vR.png       | Bin 0 -> 8993 bytes
 posts/Pulse_oxymetry/fig/SfO2vR-theory.py |  42 ++
 posts/Pulse_oxymetry/fig/absorbtion.png   | Bin 0 -> 13530 bytes
 posts/Pulse_oxymetry/fig/ac-dc.png        | Bin 0 -> 11195 bytes
 posts/Pulse_oxymetry/main.bib             |  25 ++
 posts/Pulse_oxymetry/main.tex             | 499 ++++++++++++++++++++++
 10 files changed, 634 insertions(+), 14 deletions(-)
 create mode 100644 posts/Pulse_oxymetry.mdwn
 create mode 100644 posts/Pulse_oxymetry/SConstruct
 create mode 100644 posts/Pulse_oxymetry/fig/SPO2vR.png
 create mode 100755 posts/Pulse_oxymetry/fig/SfO2vR-theory.py
 create mode 100644 posts/Pulse_oxymetry/fig/absorbtion.png
 create mode 100644 posts/Pulse_oxymetry/fig/ac-dc.png
 create mode 100644 posts/Pulse_oxymetry/main.bib
 create mode 100644 posts/Pulse_oxymetry/main.tex

diff --git a/.gitignore b/.gitignore
index 365fa1e..ca58c1e 100644
--- a/.gitignore
+++ b/.gitignore
@@ -3,17 +3,28 @@
 /recentchanges
 
 # built by Makefiles inside posts/
-posts/Abax/*.pdf
-posts/Abax/*/main.aux
-posts/Abax/*/main.log
-posts/Abax/*/main.out
-posts/Abax/*/main.pdf
-posts/Abax/*/main.toc
-posts/drexel-thesis/*.pdf
-posts/drexel-thesis/drexel-thesis/
+/posts/Abax/*.pdf
+/posts/Abax/*/main.aux
+/posts/Abax/*/main.log
+/posts/Abax/*/main.out
+/posts/Abax/*/main.pdf
+/posts/Abax/*/main.toc
+/posts/drexel-thesis/*.pdf
+/posts/drexel-thesis/drexel-thesis/
 /posts/Hooke/README.rst
 /posts/Hooke/hooke
 /posts/Hooke/html
+/posts/Pulse_oxymetry/.sconsign.dblite
+/posts/Pulse_oxymetry/fig/SfO2vR-theory.png
+/posts/Pulse_oxymetry/main.aux
+/posts/Pulse_oxymetry/main.bbl
+/posts/Pulse_oxymetry/main.blg
+/posts/Pulse_oxymetry/main.fls
+/posts/Pulse_oxymetry/main.i*
+/posts/Pulse_oxymetry/main.log
+/posts/Pulse_oxymetry/main.n*
+/posts/Pulse_oxymetry/main.out
+/posts/Pulse_oxymetry/main.pdf
 /posts/problempack/problempack/
 /posts/problempack/probs.pdf
 /posts/problempack/sols.pdf
@@ -21,8 +32,8 @@ posts/drexel-thesis/drexel-thesis/
 /posts/sawsim/README.rst
 /posts/sawsim/sawsim.pdf
 /posts/sawsim/sawsim/
-posts/Thesis/thesis/
-posts/Thesis/draft.pdf
+/posts/Thesis/thesis/
+/posts/Thesis/draft.pdf
 /posts/XSLT/chapter/chapter.html
 /posts/XSLT/code/index.shtml
 /posts/Xmodmap/_Xmodmap
diff --git a/Makefile b/Makefile
index 6141a51..3acb440 100644
--- a/Makefile
+++ b/Makefile
@@ -5,9 +5,23 @@ SUBDIRS = posts/Abax posts/drexel-thesis posts/Hooke posts/problempack \
 	posts/XSLT/chapter posts/XSLT/code
 TARGETS = all clean
 
-all :
+SCONS_SUBDIRS = posts/Pulse_oxymetry
 
-$(TARGETS) :
+
+all : make-all scons-all
+clean : make-clean scons-clean
+
+$(TARGETS:%=make-%) :
 	@for i in $(SUBDIRS); do \
-		echo "make $@ in $$i..."; \
-		(cd $$i; $(MAKE) $(MFLAGS) $@); done
+		echo "make ${@:make-%=%} in $$i..."; \
+		(cd $$i; $(MAKE) $(MFLAGS) ${@:make-%=%}); done
+
+scons-all :
+	@for i in $(SCONS_SUBDIRS); do \
+		echo "scons in $$i..."; \
+		(cd $$i; scons); done
+
+scons-clean :
+	@for i in $(SCONS_SUBDIRS); do \
+		echo "scons -c in $$i..."; \
+		(cd $$i; scons -c); done
diff --git a/posts/Pulse_oxymetry.mdwn b/posts/Pulse_oxymetry.mdwn
new file mode 100644
index 0000000..67dcc6b
--- /dev/null
+++ b/posts/Pulse_oxymetry.mdwn
@@ -0,0 +1,18 @@
+My wife was recently reviewing some [pulse oxymeter][pox] notes while
+working a round of anesthesia.  It took us a while to trace the logic
+through all the symbol changes and notation shifts, and my wife's
+math-discomfort and my bio-discomfort made us more cautios than it
+turns out we needed to be.  There are a number of nice [review
+articles][art] out there that I turned up while looking for explicit
+derivations, but by the time I'd found them, my working notes had
+gotten fairly well polished themselves.  So here's
+[[my_contribution|pdf]] to the pulse-ox noise ;).  Highlights include:
+
+* Short and sweet (with pictures)
+* Symbols table
+* Baby steps during math manipulation
+
+[pox]: http://en.wikipedia.org/wiki/Pulse_oximeter
+[art]: http://journals.lww.com/anesthesiology/Citation/1989/01000/Pulse_Oximetry.19.aspx
+
+[[!tag tags/theory]]
diff --git a/posts/Pulse_oxymetry/SConstruct b/posts/Pulse_oxymetry/SConstruct
new file mode 100644
index 0000000..306becd
--- /dev/null
+++ b/posts/Pulse_oxymetry/SConstruct
@@ -0,0 +1,11 @@
+import os
+
+env = Environment(ENV=os.environ)
+
+main = env.PDF('main.tex')
+SfO2vR_theory = env.Command(
+    'fig/SfO2vR-theory.png', 'fig/SfO2vR-theory.py', 'python $SOURCE')
+env.Depends(main, SfO2vR_theory)
+
+view = env.Alias('view', [main], 'xpdf -z page $SOURCE')
+env.AlwaysBuild(view)
diff --git a/posts/Pulse_oxymetry/fig/SPO2vR.png b/posts/Pulse_oxymetry/fig/SPO2vR.png
new file mode 100644
index 0000000000000000000000000000000000000000..b2a0c3cf749f3862c63852f8777f07dc872e6000
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diff --git a/posts/Pulse_oxymetry/fig/SfO2vR-theory.py b/posts/Pulse_oxymetry/fig/SfO2vR-theory.py
new file mode 100755
index 0000000..cdcdde5
--- /dev/null
+++ b/posts/Pulse_oxymetry/fig/SfO2vR-theory.py
@@ -0,0 +1,42 @@
+#!/usr/bin/env python
+
+import matplotlib
+matplotlib.use('Agg')
+from pylab import *
+
+# 1 = HbO2
+# 2 = HHb
+# a = 660 nm
+# b = 940 nm
+
+#author = 'Mendelson'
+author = 'King'
+
+if author == 'Mendelson':
+    e1a = 0.12
+    e2a = 0.86
+    e1b = 0.29
+    e2b = 0.20
+else:
+    e1a = 0.10
+    e2a = 0.83
+    e1b = 0.29
+    e2b = 0.17
+
+if author == 'Mendelson':
+    def SfO2(R):
+        return (-e2a+e2b*R)/(e1a-e2a + (e2b-e1b)*R)
+else:
+    def SfO2(R):
+        return 1./(1 + (R*e1b-e1a)/(e2a-R*e2b))
+
+R = linspace(0.4, 4, 100)
+figure(figsize=(4,4))
+plot(R, SfO2(R), 'k.')
+title('Theory')
+xlabel('R')
+ylabel('SfO2')
+
+target = __file__[:-2] + 'png'
+print target
+savefig(target, dpi=300)
diff --git a/posts/Pulse_oxymetry/fig/absorbtion.png b/posts/Pulse_oxymetry/fig/absorbtion.png
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diff --git a/posts/Pulse_oxymetry/main.bib b/posts/Pulse_oxymetry/main.bib
new file mode 100644
index 0000000..3ab909e
--- /dev/null
+++ b/posts/Pulse_oxymetry/main.bib
@@ -0,0 +1,25 @@
+@article { tremper1989,
+  author = "Tremper, Kevin K. and Barker, Steven J.",
+  title = "Pulse Oximetry",
+  year = 1989,
+  month = jan,
+  journal = "Anesthesiology",
+  issn = "0003-3022",
+  volume = 70,
+  number = 1,
+  pages = "98--108",
+  url = "http://journals.lww.com/anesthesiology/Citation/1989/01000/Pulse_Oximetry.19.aspx",
+}
+
+@article { mendelson1989,
+  author = "Mendelson, Yitzhak and Kent, Joel C.",
+  title = "Variations in Optical Absorbtion Spectra of Aduly and Fetal Hemoglobins and ITs Effect on Pulse Oximetry",
+  year = 1989,
+  month = aug,
+  journal = "IEE Transactions on Biomedical Engineering",
+  issn = "0018-9294",
+  volume = 36,
+  number = 8,
+  pages = "844--848",
+  url = "http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=30810",
+}
diff --git a/posts/Pulse_oxymetry/main.tex b/posts/Pulse_oxymetry/main.tex
new file mode 100644
index 0000000..8a4a7b7
--- /dev/null
+++ b/posts/Pulse_oxymetry/main.tex
@@ -0,0 +1,499 @@
+\documentclass{article}
+
+\usepackage{amsmath}  % align and other math niceties
+\usepackage{makeidx}  % indexing
+\makeindex  % we don't actually use indexing, but having this here
+            % triggers Scons' call to makeindex
+\usepackage[intoc]{nomencl}  % nomenclature/symbol/abbreviation indexing
+\makenomenclature
+
+\usepackage[super,sort&compress,comma]{natbib} % fancy citation extensions
+% super selects citations in superscript mode
+% sort&compress automatically sorts and compresses compound citations
+%   (\citep{a,b,...})
+% comma seperates multiple citations with commas rather than the
+%   default semicolons.
+\bibliographystyle{unsrtnat} % number citations in the order referenced
+
+\usepackage{subfig}    % compound figures
+\usepackage{graphicx}  % include graphics
+
+\usepackage[final]{hyperref} % hyper-links
+\hypersetup{colorlinks}
+\hypersetup{pdfauthor={W. Trevor King}}
+\hypersetup{pdftitle={Pulse Oximetry}}
+\hypersetup{pdfsubject={Pulse Oximetry}}
+\title{Pulse Oximetry}
+\date{\today}
+\author{W. Trevor King}
+
+% parenthesis, for example (some stuff] would be \p({some stuff}]
+\newcommand{\p}[3]{\left#1 #2 \right#3}
+% concentration shortcut
+\newcommand{\C}[1]{\ensuremath{\p[{#1}]}}
+% the proper d to be used in derivatives and integrals
+\newcommand{\dd}{\ensuremath{\mathrm d}}
+% derivative of #2 with respect to #1
+\newcommand{\deriv}[2]{\ensuremath{\frac{\dd{#2}}{\dd{#1}}}}
+% average of #1
+\newcommand{\avg}[1]{\ensuremath{\left\langle {#1} \right\rangle}} % average
+% units without preceding value
+\newcommand{\bareU}[1]{\textnormal{#1}}
+% units shortcut
+\newcommand{\U}[1]{\textnormal{\ \bareU{#1}}}
+
+% chemicals and molecules
+\newcommand{\Ox}{\ensuremath{\mathrm O_2}}
+\newcommand{\Hb}{\ensuremath{\mathrm{Hb}}}
+\newcommand{\HHb}{\ensuremath{\mathrm{HHb}}}
+\newcommand{\dysHb}{\ensuremath{\mathrm{dysHb}}}
+\newcommand{\MHb}{\ensuremath{\mathrm{MHb}}}
+\newcommand{\CO}{\ensuremath{\mathrm{CO}}}
+\newcommand{\Hg}{\ensuremath{\mathrm{Hg}}}
+
+\newcommand{\xO}[1]{\ensuremath{{\mathrm{#1}}_{\Ox}}}
+
+% partial pressure PiO2
+\newcommand{\PxO}[1]{\xO{P#1}}
+\newcommand{\PO}{\PxO{}}
+\newcommand{\PiO}{\PxO{\mathit i}}
+
+% saturation SiO2.  Also O2Sat
+\newcommand{\SxO}[1]{\xO{S#1}}
+\newcommand{\SO}{\SxO{}}
+\newcommand{\SiO}{\SxO{\mathit i}}
+
+% functional saturation
+\newcommand{\SOf}{\xO{S_{\mathit f}}}
+
+% content CoO2.  Also O2CT.
+\newcommand{\CoO}{\xO{Co}}
+
+% Beer-Lambert law coefficients
+\newcommand{\Iol}{\ensuremath{I_{0 \lambda}}}
+\newcommand{\Il}{\ensuremath{I_\lambda}}
+\newcommand{\Iacl}{\ensuremath{I_{AC\lambda}}}
+\newcommand{\Idcl}{\ensuremath{I_{DC\lambda}}}
+\newcommand{\exl}[1]{\ensuremath{\epsilon_{#1 \lambda}}}
+\newcommand{\eil}{\exl{i}}
+\newcommand{\cdcx}[1]{\ensuremath{c_{DC#1 \lambda}}}
+\newcommand{\cdci}{\cdcx{i}}
+\newcommand{\cacx}[1]{\ensuremath{c_{AC#1}}}
+\newcommand{\caci}{\cacx{i}}
+\newcommand{\edcxl}[1]{\ensuremath{\epsilon_{DC#1 \lambda}}}
+\newcommand{\edcil}{\edcxl{i}}
+\newcommand{\eacxl}[1]{\ensuremath{\epsilon_{AC#1 \lambda}}}
+\newcommand{\eacil}{\eacxl{i}}
+\newcommand{\eiA}[1]{\ensuremath{\epsilon_{AC#1,660}}}
+\newcommand{\eiB}[1]{\ensuremath{\epsilon_{AC#1,940}}}
+\newcommand{\Ldc}{\ensuremath{L_{\mathrm{DC}}}}
+
+% use most of the page
+\topmargin -0.5in
+\headheight 0.0in
+\headsep 0.0in
+\textheight 9.5in % leave a bit of extra room for page numbers                  
+\oddsidemargin -0.5in
+\textwidth 7.5in
+
+\begin{document}
+
+\maketitle
+
+\section{Oxygen content}
+
+The circulatory system distributes oxygen (\Ox) throughout the body.
+The amount of \Ox\ at any given point is measured by the \Ox\ content
+(\CoO), usually given in $\frac{\bareU{mL \Ox at BTP}}{\bareU{dL blood}}$
+(BTP is my acronym for body temperature and pressure).  Most
+transported \Ox\ is bound to
+\href{http://en.wikipedia.org/wiki/Hemoglobin}{hemoglobin} (\Hb), but
+there is also free \Ox\ disolved directly in the plasma and cytoplasm
+of suspended cells.
+\nomenclature{\Ox}{Molecular oxygen}
+\nomenclature{\CoO}{Oxygen content of blood}
+\nomenclature{BTP}{Body temperature and pressure}
+\nomenclature{\Hb}{Hemoglobin monomer}
+\nomenclature{\SO}{Fractional \Ox\ saturation}
+\nomenclature{\PO}{\Ox\ partial pressure}
+\begin{equation}
+  \CoO = a\C{\Hb}\SO + b\PO  \;,  \label{eq:CoO-symbolic}
+\end{equation}
+where \SO is the \Hb's \Ox\ saturation and \PO is the \Ox\ partial
+pressure.  Don't worry about the coefficients $a$ and $b$ yet, we'll
+get back to them in a second.
+
+The amound of dissolved \Ox is given by its
+\href{http://en.wikipedia.org/wiki/Partial_pressure}{partial pressure}
+(\PO).  Partial pressures are generally given in \bareU{mm} of mercury
+(\Hg) at standard temperature and pressure
+(\href{http://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressure}{STP}).
+Because the partial pressure changes as blood flows through the body,
+an additional specifier $i$ may be added (\PiO) to clarify the
+measurement location.  \nomenclature{\Hg}{Mercury}
+\nomenclature{STP}{Standard temperature and pressure}
+\nomenclature{\PiO}{\Ox\ partial pressure at location $i$}
+\begin{center}
+\begin{tabular}{ccl}
+  $i$ & Full symbol & Location descriptor \\
+  a & \PxO{a} & arterial \\
+  p & \PxO{p} & peripheral or pulsatile \\
+  t & \PxO{t} & tissue \\
+  v & \PxO{t} & venous \\
+\end{tabular}
+\end{center}
+
+\Ox\ is carried in the blood primarily through binding to hemoglobin
+monomers (\Hb), with each monomer potentially binding a single \Ox.
+Oxygen saturation (\SO) is the fraction of hemoglobin monomers (\Hb)
+that have bound an oxygen molecule (\Ox).
+\begin{equation}
+  \SO = \frac{\C{\Hb\Ox}}{\C{\Hb}}  \;.  \label{eq:SO}
+\end{equation}
+\SO, as a ratio of concentrations, is unitless.  It is often expressed
+as a percentage.  \C{\Hb} is often given in \bareU{g/dL}.  As with
+partial pressures, an additional specifier $i$ may be added (\SiO) to
+clarify the measurement location (\SxO{a}, \SxO{p}, \ldots).
+\nomenclature{\SiO}{\Ox\ saturation at location $i$}
+
+Now we can take a second look at our \Ox\ content formula
+(Eq.~\ref{eq:CoO-symbolic}).  The coefficient $a$ must convert
+\bareU{g/dL} to $\frac{\bareU{mL \Ox\ at BTP}}{\bareU{dL blood}}$.
+Using the molecular weight of \Hb\ and the
+\href{http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html}{volume
+  of a mole of ideal gas at STP}.
+\begin{align}
+  \C{\Hb} &= \chi\frac{\bareU{g \Hb}}{\bareU{dL}}
+    \cdot\frac{1\U{mol \Hb}}{17\U{kg \Hb}}
+    \cdot\frac{1\U{mol \Ox}}{1\U{mol \Hb}}
+    \cdot\frac{22.4\U{L ideal gas}}{1\U{mol ideal gas}} \\
+    &= 1.32 \frac{\bareU{mL \Ox}}{\bareU{g \Hb}} \cdot \chi
+\end{align}
+where $\chi$ is a pure number.  Therefore, $a=1.32 \frac{\bareU{mL
+    \Ox}}{\bareU{g \Hb}}$.  The powers that be seem to have used a
+slightly different STP, since the more commonly used value is 5\%
+higher at $1.39$.
+
+The coefficient $b$ must convert \bareU{mm \Hg\ at STP} to
+$\frac{\bareU{mL \Ox\ at BTP}}{\bareU{dL blood}}$.  Empirical
+experiments (?) give a value of
+$b=0.003\frac{\bareU{mL \Ox\ at BTP}}
+             {\bareU{dL blood $\cdot$ mm \Hg\ at STP}}$.
+Now we can write out the familiar form
+\begin{equation}
+  \CoO =
+      1.39 \frac{\bareU{mL \Ox}}{\bareU{g \Hb}} \C{\Hb}\SO
+    + 0.003\frac{\bareU{mL \Ox\ at BTP}}
+                {\bareU{dL blood $\cdot$ mm \Hg\ at STP}}
+      \PO  \;.  \label{eq:CoO}
+\end{equation}
+Reasonable levels are
+\begin{center}
+\begin{tabular}{rl}
+  $\C{\Hb}$ & $14\frac{\bareU{g \Hb}}{\bareU{dL blood}}$ \\
+  $\SO$ & 98\% \\
+  $\PO$ & $100\U{mm \Hg\ at STP}$ \\
+  $1.39 \frac{\bareU{mL \Ox}}{\bareU{g \Hb}} \C{\Hb}\SO$ &
+    $19.1\frac{\bareU{mL \Ox\ at BTP}}{\bareU{dL blood}}$ \\
+  $0.003\frac{\bareU{mL \Ox\ at BTP}}
+                {\bareU{dL blood $\cdot$ mm \Hg\ at STP}}
+      \PO$ &
+    $0.299\frac{\bareU{mL \Ox\ at BTP}}{\bareU{dL blood}}$
+\end{tabular}
+\end{center}
+Because the dissolved \Ox\ has such a tiny contribution (1.5\% of the
+total in my example), it is often measured at BTP rather than STP.
+Sometimes it is dropped from the calculation entirely.  We focus on
+the more imporant $\C{\Hb}\SO$ in the next section.
+
+\section{Oxygen saturation}
+
+The preceding discussion used $\C{\Hb}\SO$ to represent the
+concentration of \Hb\Ox\ complexes (Eqs.~\ref{eq:SO} and
+~\ref{eq:CoO}).  This was useful while we were getting our bearings,
+but now we will replace that term with a more detailed model.  Let us
+sort the \Hb\ monomers into species
+\begin{center}
+\begin{tabular}{ll}
+  \Hb & all hemoglobin monomers \\
+  \Hb\Ox & monomers complexed with \Ox \\
+  \HHb & reduced \Hb\ (not complexed with \Ox) \\
+  \dysHb & dys-hemoglobin (cannot complex with \Ox) \\
+  \MHb & \href{http://en.wikipedia.org/wiki/Methemoglobin}{methemoglobin} \\
+  \Hb\CO & \href{http://en.wikipedia.org/wiki/Carboxyhemoglobin}
+                {carboxyhemoglobin}
+\end{tabular}
+\end{center}
+\nomenclature{\Hb\Ox}{\Hb\ monomers complexed with \Ox}
+\nomenclature{\HHb}{Reduced \Hb\ (not complexed with \Ox)}
+\nomenclature{\dysHb}{Dys-hemoglobin (cannot complex with \Ox)}
+\nomenclature{\MHb}{Methemoglobin}
+\nomenclature{\Hb\CO}{Carboxyhemoglobin}
+These species are related as follows
+\begin{align}
+  \C{\Hb} &= \C{\Hb\Ox} + \C{\HHb} + \C{\dysHb} \\
+  \C{\dysHb} &= \C{\MHb} + \C{\Hb\CO} + \text{other broken forms}
+\end{align}
+
+Because modern two-color pulse-oximeters don't measure \SO exactly,
+the related quantity that they \emph{do} measure has been given a name
+of its own: the \emph{functional} saturation (\SOf).
+\nomenclature{\SOf}{Functional \Ox\ saturation}
+\begin{equation}
+  \SOf = \frac{\C{\Hb\Ox}}{\C{\Hb\Ox} + \C{\HHb}}  \;.  \label{eq:funcSO}
+\end{equation}
+Rephrasing our earier saturation in Eq.~\ref{eq:SO}, we see
+\begin{equation}
+  \SO = \frac{\C{\Hb\Ox}}{\C{\Hb}}
+    = \frac{\C{\Hb\Ox}}
+           {\C{\Hb\Ox} + \C{\HHb} + \C{\dysHb}}  \;.  \label{eq:fracSO}
+\end{equation}
+To avoid confusion with \SOf, our original \SO\ is sometimes referred
+to as the\emph{fractional} saturation.
+
+\section{The Beer-Labmert law}
+
+So far we've been labeling and defining attributes of the blood.  The
+point of this excercise is to understand how a pulse oximeter measures
+them.  People have known for a while that different hemoglobin
+complexes (\Hb\O, \HHb, \MHb, \Hb\CO, \ldots) have differnt absorbtion
+spectra (Fig.~\ref{fig:absorbtion}), and they have been using this
+difference since the 1930's to make pulse-oximeters based on two-color
+transmittance measurements\citep{tremper1989}.
+
+\begin{figure}
+  \begin{center}
+    \includegraphics[width=0.6\textwidth]{fig/absorbtion}
+  \end{center}
+  \caption{Absorbance spectra for assorted hemoglobin
+    species\citep{tremper1989}.\label{fig:absorbtion}}
+\end{figure}
+
+By passing different wavelengths of light through perfused tissue, we
+can measure the relative quantities of the different \Hb\ species.
+The basis for this analysis comes from the
+\href{http://en.wikipedia.org/wiki/Beer\%E2\%80\%93Lambert_law}{Beer-Lambert
+  law}
+\begin{equation}
+  I = I_0 e^{-c \epsilon L}  \;,  \label{eq:BL}
+\end{equation}
+where $I_0$ is the incident intensity (entering the tissue), $I$ is
+the tranmitted intensity (leaving the tissue), $c$ is the tissue
+density (concentration), $\epsilon$ is the extinction coefficient
+(molar absorbtivity), and $L$ is the tissue thickness.  Rephrasing the
+math as English, this means that the intensity drops off exponentially
+as you pass through the tissue, and more tissue (higher $c$ or $L$) or
+more opaque tissue (higher $\epsilon$) mean you'll get less light out
+the far side.  This is a very simple law, and the price of the
+simplicity is that it brushes all sorts of things under the rug.
+Still, it will help give us a basic idea of what is going on in a
+pulse-oximeter.
+
+Rather than treat the the tissue as a single substance, lets use the
+Beer-Labmert law on a mixture of substances with concentrations $c_1$,
+$c_2$, \ldots and extinction coefficients $\epsilon_1$, $\epsilon_2$,
+\ldots.
+\begin{equation}
+  I = I_0 e^{-(c_1 \epsilon_1 + c_2 \epsilon_2 + \ldots) L} \;.
+\end{equation}
+
+We also notice that the intensities and extinction coefficients may
+all depend on the wavelength of light $\lambda$, so we should really
+write
+\begin{equation}
+  \Il = \Iol e^{-(c_1 \exl{1} + c_2 \exl{2} + \ldots) L}
+    \;.  \label{eq:BL-lambda}
+\end{equation}
+\nomenclature{\Iol}{Intensity of incident light at wavelength $\lambda$}
+\nomenclature{\Il}{Intensity of transmitted light at wavelength $\lambda$}
+\nomenclature{$c_i$}{Concentration of species $i$}
+\nomenclature{\eil}{Extinction coefficient of species $i$ at
+  wavelength $\lambda$}
+\nomenclature{$L$}{Length of tissue through which light must pass}
+
+Once isolated, a simple spectroscopy experiment can measure the
+extinction coefficient $\eil$ of a given species across a range of
+$\lambda$, and this has been done for all of our common \Hb\ flavors.
+We need to play with Eq.~\ref{eq:BL-lambda} to find a way to extract
+the unknown concentrations, which we can then use to calculate the
+\SO\ (Eqs.~\ref{eq:funcSO} and \ref{eq:fracSO}) which we can use in
+turn to calculate \CoO\ (Eq.~\ref{eq:CoO}).
+
+Note that by increasing the number of LEDs (adding new $\lambda$) we
+increase the number of constraints on the unknown $c_i$.  A
+traditional pulse-oximeter uses two LEDs, at $660\U{nm}$ and
+$940\U{nm}$, to measure \SOf (related to $\C{\Hb\Ox}$ and $\C{\HHb}$).
+More recent designs called
+\href{http://en.wikipedia.org/wiki/Co-oximeter}{pulse CO-oximeters}
+use more wavelengths to allow measurement of quanties related to
+additional species (approaching the end goal of measuring \SO).
+\nomenclature{LED}{Light emitting diode}
+
+Let us deal with the fact that there is a lot of stuff absorbing light
+that is not arterial blood (e.g. venous blood, other tissue, bone,
+etc).  The good thing about this stuff is that it's just sitting there
+or moving through in a smooth fasion.  Arterial blood is the only
+thing that's pulsing (Fig.~\ref{fig:ac-dc}).
+
+\begin{figure}
+  \begin{center}
+    \includegraphics[width=0.6\textwidth]{fig/ac-dc}
+  \end{center}
+  \caption{AC and DC transmission
+    components\citep{tremper1989}.\label{fig:ac-dc}}
+\end{figure}
+
+During a pulse, the pressure in the finger increases and non-arterial
+tissue is compressed, changing $L$ and $c_i$ from their trough values
+to peak values $L'$ and $c_i'$.  Since the finger is big, the
+fractional change in width $\dd L/L=(L'-L)/L$ is very small.  Assuming
+the change in concentration is even smaller (since most liquids are
+fairly incompressible), we have
+\begin{align}
+  \deriv{L}{\Il}
+    &= \deriv{L}{}\p({\Iol e^{-(c_1 \exl{1} + c_2 \exl{2} + \ldots) L}})
+    = \deriv{L}{}\p({\Iol e^{-X L}})
+    = -X \Iol e^{-X L}
+    = -X\Il \\
+  \frac{\dd \Il}{\Il} &= -X \dd L  \;,  \label{eq:dI-I-naive}
+\end{align}
+where $X=c_1 \exl{1} + c_2 \exl{2} + \ldots$ is just a placeholder to
+reduce clutter.  \dd \Il is the AC amplitude (height of wiggle top of
+the detected light intensity due to pulsatile arterial blood), while
+\Il is the DC ampltude (height of the static base of the detected
+light intensity due to everything else).  This is actually a fairly
+sneaky step, because if we can also use it to drop the DC compents.
+Because we've assumed fixed concentrations (incompressible fluids),
+and there is no more DC material comming in during a pulse (by
+definition), we can pull out the effective $L$ for the DC components
+does not change.  Separating the DC and AC components and running
+through the derivative again, we have
+\nomenclature{\cdci}{Concentration of the $i$th DC species at
+  wavelength $\lambda$} \nomenclature{\caci}{Concentration of the
+  $i$th AC species at wavelength $\lambda$}
+\nomenclature{\edcil}{Extinction coefficient of the $i$th DC species
+  at waveleng th $\lambda$} \nomenclature{\eacil}{Extinction
+  coefficient of the $i$th AC species at waveleng th $\lambda$}
+\nomenclature{\Ldc}{DC finger width}
+\begin{align}
+  \deriv{L}{\Il}
+    &= \deriv{L}{}\p({
+         \Iol e^{-(\cdcx{1}\edcxl{1} + \cdcx{2}\edcxl{2} + \ldots) \Ldc
+                 -(\cacx{1}\eacxl{1} + \cacx{2}\eacxl{2} + \ldots) L}}) \\
+    &= \Iol e^{-(\cdcx{1}\edcxl{1} + \cdcx{2}\edcxl{2} + \ldots) \Ldc}
+       \deriv{L}{}\p({e^{-(\cacx{1}\eacxl{1} + \cacx{2}\eacxl{2} + \ldots) L}
+                      }) \\
+    &= \Iol Y \deriv{L}{}\p({e^{-Z L}})
+    = -Z \Iol Y e^{-Z L}
+    = -Z \Il  \\
+  \frac{\dd \Il}{\Il} &= -Z \dd L \;,  \label{eq:dI-I}
+\end{align}
+where $Y=e^{-(\cdcx{1}\edcxl{1} + \cdcx{2}\edcxl{2} + \ldots) \Ldc}$
+and $Z=\cacx{1}\eacxl{1} + \cacx{2}\eacxl{2} + \ldots$ are just
+placeholders to reduce clutter.  Note that Eq.~\ref{eq:dI-I} looks
+just like Eq.~\ref{eq:dI-I-naive} with the translation $X\rightarrow
+Z$.  This means that if we stick to using the AC-DC intensity ratio
+($\frac{\dd \Il}{\Il}$) we can forget about the DC contribution
+completely\footnote{
+If the changing-$L$-but-static-$\Ldc$ thing bothers you, you can
+imagine insteadthat $\Ldc$ grows with $L$, but \cdci shrinks
+proportially (to conserve mass).  With this proportionate stretching,
+there is still no change in absorbtion for that component so
+$\deriv{}{L}\exp(-\cdci\edcil L)=0$ and we can still pull the DC
+terms out of the integral as we did for Eq.~\ref{eq:dI-I}.}.
+
+Taking a ratio of these amplitudes at two different wavelengths, we
+get optical density ratio
+\nomenclature{$R$}{Optical density ratio}
+\begin{equation}
+  R = \frac{\frac{AC_{660}}{DC_{660}}}{\frac{AC_{940}}{DC_{940}}}
+    = \frac{\frac{\dd I_{660}}{I_{660}}}{\frac{\dd I_{940}}{I_{940}}}
+    = \frac{-Z_{660} \dd L}{-Z_{940} \dd L}
+    = \frac{Z_{660}}{Z_{940}}  \;,
+\end{equation}
+because $\dd L$ (the amount of finger expansion during a pulse)
+obviously doesn't depend on the color light you are using ;).
+Plugging back in for $Z$,
+\begin{equation}
+  R = \frac{\cacx{1}\eiA{1} + \cacx{2}\eiB{2} + \ldots}
+           {\cacx{1}\eiB{1} + \cacx{2}\eiB{2} + \ldots}  \;.
+\end{equation}
+Assuming, for now, that there are only two species of \Hb, \Hb\Ox and
+\HHb, we can solve for $\cacx{1}/\cacx{2}$.
+\begin{align}
+  R &= \frac{\cacx{1}\eiA{1} + \cacx{2}\eiA{2}}
+            {\cacx{1}\eiB{1} + \cacx{2}\eiB{2}} \\
+  R(\cacx{1}\eiB{1} + \cacx{2}\eiB{2})
+    &= \cacx{1}\eiA{1} + \cacx{2}\eiA{2} \\
+  \cacx{1} (R\eiB{1} - \eiA{1}) &= \cacx{2} (\eiA{2} - R\eiB{2}) \\
+  \frac{\cacx{1}}{\cacx{2}} &= \frac{\eiA{2} - R\eiB{2}}{R\eiB{1} - \eiA{1}}
+    \;.
+\end{align}
+So now we know $\C{\Hb\Ox}/\C{\HHb}$ in terms of the measured quantity
+$R$.
+
+Plugging in to Eq.~\ref{eq:funcSO} to find the functional saturation
+\begin{align}
+  \SOf &= \frac{\C{\Hb\Ox}}{\C{\Hb\Ox} + \C{\HHb}}
+    = \frac{1}{1 + \frac{\C{\HHb}}{\C{\Hb\Ox}}}
+    = \frac{1}{1 + \frac{\cacx{2}}{\cacx{1}}}
+    = \frac{1}{1 + \frac{R\eiB{1} - \eiA{1}}{\eiA{2}-R\eiB{2}}}
+    \;.  \label{eq:SOfvR}
+\end{align}
+
+As a check, we can rephrase this as
+\begin{align}
+  \SOf &= \frac{1}{1 + \frac{R\eiB{1} - \eiA{1}}{\eiA{2} - R\eiB{2}}}
+    = \frac{\eiA{2} - R\eiB{2}}{\eiA{2} - R\eiB{2} + R\eiB{1} - \eiA{1}} \\
+    &= \frac{\eiA{2} - \eiB{2}R}{\eiA{2} - \eiA{1} + (\eiB{1} - \eiB{2})R}
+    = \frac{-\eiA{2} + \eiB{2}R}{\eiA{1} - \eiA{2} + (\eiB{2} - \eiB{1})R}
+    \;,
+\end{align}
+which matches \citet{mendelson1989}, Eq.~8 with the translations
+$\SOf\rightarrow\SxO{p}$, $R\rightarrow R/IR$,
+$\eiA{2}\rightarrow\epsilon_R(\HHb)$,
+$\eiB{2}\rightarrow\epsilon_IR(\HHb)$,
+$\eiA{1}\rightarrow\epsilon_R(\Hb\Ox)$, and
+$\eiB{1}\rightarrow\epsilon_IR(\Hb\Ox)$.
+
+And that is the first-order explaination of how a pulse-oximeter
+measures the functional saturation!
+
+Reading extinction coefficients off Fig.~\ref{fig:absorbtion}, I get
+\begin{align}
+  \epsilon_{\Hb\Ox,660} &= \eiA{1} = 0.10 \\
+  \epsilon_{\HHb,660} &= \eiA{2} = 0.83 \\
+  \epsilon_{\Hb\Ox,940} &= \eiB{1} = 0.29 \\
+  \epsilon_{\HHb,940} &= \eiB{2} = 0.17
+\end{align}
+which are comfortingly close to those given by \citet{mendelson1989}
+in their Table~1.  The corresponding $\SOf(R)$ plot can be seen in
+Fig.~\ref{fig:SfOvR-theory}.
+
+\begin{figure}
+  \begin{center}
+    \subfloat[]{\label{fig:SfOvR-experiment}
+      \includegraphics[width=0.48\textwidth]{fig/SPO2vR}}
+    \subfloat[]{\label{fig:SfOvR-theory}
+      \includegraphics[width=0.40\textwidth]{fig/SfO2vR-theory}}
+  \end{center}
+  \caption{Comparison of (a) an experimental calibration
+    curve\citep{tremper1989} with (b) a theoretical calibration curve
+    calculated using Eq.~\ref{eq:SOfvR}.  This is why it's a good idea
+    to use an empirical calibration curve ;).\label{fig:SfOvR}}
+  % concave theoretical SfO2vR theory curve supported by
+  % mendelson1989, Fig. 4.
+\end{figure}
+
+%, because LEDs are finicky, it's hard to know how bright they are
+%without looking ;).  Therefore, we'd like to work $\Iol$ out of the
+%picture.  As blood pulses through the finger, imagine that the
+%concentrations
+
+\printnomenclature
+
+\phantomsection
+\addcontentsline{toc}{section}{References}
+\bibliography{main}
+
+\end{document}
-- 
2.26.2