Merge pull request #52 from jessdtate/ibbm_release

Ibbm release

Author: dcwhite

Documentation/ECGTool_math.tex

@@ -2,35 +2,36 @@ \chapter{Mathematical Background} \label{sec:math}
 
 \section{Overview}
 
-In this chapter we describe the basic mathematical formulations, and some
+In this chapter we describe the basic mathematical formulations and some
 solution approaches to the forward and inverse problems of
-electrocardiography. The goal of solutions to the forward problem is to
+electrocardiography. The goal of solutions to the forward problem, also called ECG 
+forward Simulation, is to
 predict potentials that could be measured in any accessible location
 (usually the surface of the torso) given a description of the cardiac
-electrical sources as well as the geometry and conductivities of the torso involved. The goal of
-solutions to the inverse problem is to predict cardiac sources given a set
+electrical sources as well as the geometry and conductivities of the torso involved.
+The goal of solutions to the inverse problem, also called ECG imaging or ECGI,
+is to predict cardiac sources given a set
 of measurements and the same geometry and conductivity information. It is
-important to note that a solution to the inverse problem presupposes an
-available solution to the forward problem. There exists a large literature on
-both of these problems as well as tutorial articles 
-(see \url{http://www.ecg-imaging.org/home/publications}),  and many excellent textbook
-and reference chapters. Here we briefly summarize the
-basic background to facilitate our description of the toolkit capabilities
-in subsequent chapters.
-
-Both forward and inverse solutions require a specific model formulation of
-the cardiac electrical sources. This toolkit treats two different
-equivalent source models. (These two models are arguably the two dominant
-formulations in current forward and inverse problem research.) One model,
-which we will refer to as the ``activation-based'' source model, assumes
-that the dominant feature of cardiac electrical activity is the timing of
-the arrival of the depolarization wavefront (known as ``activation times'')
+important to note that a solution to ECG Imaging presupposes an
+available solution to the forward simulation. 
+Here we briefly summarize the basic background to 
+facilitate our description of the toolkit capabilities in subsequent chapters.
+Please refer to the list of literature on the CEI webpage
+for more information (\url{http://www.ecg-imaging.org/home/publications}). 
+
+
+Solving both forward and inverse problems requires a specific model formulation of
+the cardiac electrical sources. This toolkit contains examples from the two arguably 
+most dominant equivalent source models used in current forward and inverse 
+problem research. One model, which we will refer to as the ``activation-based'' 
+source model, assumes that the dominant feature of cardiac electrical activity 
+is the timing of the arrival of the depolarization wavefront (known as ``activation times'')
 at each location in the heart. (A similar problem, in which the equivalent 
 sources called ''recovery times'' are the timing of repolarization at each 
-location, is solved in a similar fashion.) Classical
-results have shown that, under assumptions of isotropy and homogeneity of
+location, is solved in a similar fashion.) Classical results have shown that, 
+under assumptions of isotropy and homogeneity of 
 the myocardium, activation-based models can be reduced to the activation
-times on the surface of the heart. In this context, this is usually taken as
+times on the surface of the heart \cite{RSM:Oos2004}. In this context, this is usually taken as
 the epicardial (outer) and endocardial (inner) heart surfaces, connected
 across the base of the heart by an imaginary connecting surface. The source
 in this case can be modeled by a moving set of current dipoles aligned
@@ -90,9 +91,11 @@ \section{Solutions to the Forward Problem in the Forward/Inverse Toolkit}
 \noindent where $\BM{\sigma}$ contain the relevant conductivities and $\Phi$ the
 electrical potentials. The boundary conditions are given by:
 \begin{align} \Phi(x,y,z)|_{\Omega_k} &= V_k\\ \left. \frac{\partial
-\Phi}{\partial n} \right|_\Omega &= 0 \label{eq:bc}
+\Phi}{\partial \hat{n}} \right|_\Omega &= 0 \label{eq:bc}
 \end{align}
-\noindent where $\Omega_{k}$ is the surface on which the sources are located.
+\noindent where $\Omega_{k}$ is the surface on which the sources are located,
+$V_k$ are the potentials on that surface, and $\hat{n}$ are the surface normals 
+of the surface.
 
 As an alternative to solving this problem directly, a (weighted) integral
 can be taken over the solution domain on both sides of Eq.~(\ref{eq:eq})
@@ -105,8 +108,9 @@ \section{Solutions to the Forward Problem in the Forward/Inverse Toolkit}
 In a tractable geometry, such as a set of concentric or even eccentric
 spheres, this PDE can be solved via analytical expansions. However in
 complex geometries such as realistic torso models, numerical solutions must
-be applied. Again there is a large literature on such numerical
-methods. Two of these methods have predominated in the literature for
+be applied. 
+%Again there is a large literature on such numerical methods. 
+Two of these methods have predominated in the literature for
 forward electrocardiography: the Finite Element Method (FEM) and the
 Boundary Element Method (BEM). It is these two methods which have been
 implemented in this toolkit. Thus, in the rest of this subsection, we give a
@@ -123,9 +127,9 @@ \section{Solutions to the Forward Problem in the Forward/Inverse Toolkit}
 these surfaces is then discretized into a mesh. In other words, in both FEM
 and BEM there exists a collection of points, called nodes, which define the
 respective volume or surface elements. The potential $\Phi$ (and the
-current, $\BM{\sigma} \nabla \Phi)$, is approximated by interpolation of
-potential and current across those elements, based on its value at the
-nodes, using known (usually polynomial) interpolation functions. Thus,
+current, $\BM{\sigma} \nabla \Phi)$, is approximated by interpolating the
+potential and current across those elements based on its value at the
+nodes using known (usually polynomial) interpolation functions. Thus,
 numerical integration can be applied to the weak form, and the node values
 come out of the integrals, leaving subintegrals over known functions which
 result in a set of weights. The result in either case is a system of
@@ -243,12 +247,10 @@ \subsection{FEM in the Forward/Inverse Toolkit}
 this reduces to replacing the $0$'s on the right hand side by known
 currents or fixing some values of the vector $\BM{\Phi}$ to correspond to
 known voltages. There are a variety of ways to accomplish this to preserve
-certain numerical properties of the matrix equation, and again we refer the
-reader to the vast literature on this subject for details. If the
+certain numerical properties of the matrix equation. If the
 measurement electrodes are treated as being larger than one node in size,
 this can lead to additional boundary conditions which in turn leads to
-additional modifications of the equations, and again we refer the reader to
-the literature.
+additional modifications of the equations.
 
 The result is a matrix equation whose size is the total number of nodes in
 the volume, which tends to result in a relatively large system. However, as
@@ -282,10 +284,10 @@ \subsection{FEM in the Forward/Inverse Toolkit}
 provided an example SCIRun network to implement one of them, the so-called
 ``lead field'' method, which solves the matrix equation repeatedly for
 source vectors consisting of a $1$ at each source node in turn and $0$ at
-all other source nodes. From this collection of solutions we can obtain the
-desired transfer matrix, which we will denote a $\mathbf{A}$. This network is described below in
-Ch.~\ref{ch:fwd} and once again the details are available to the interested
-reader in the literature.
+all other source nodes \cite{JDT:Gul97}. From this collection of solutions we 
+can obtain the desired transfer matrix, which we will denote as $\mathbf{A}$ 
+as used in eq.~\ref{eg:TransMat}. 
+This network is described below in Ch.~\ref{ch:fwd}.
 
 \subsection{BEM in the Forward/Inverse Toolkit}
 
@@ -298,10 +300,9 @@ \subsection{BEM in the Forward/Inverse Toolkit}
 that assumption, the surfaces of those subdomains become a sufficient
 domain upon which to solve the problem for the entire domain.
 
-Briefly (and once again we refer the reader to the literature for the
-details and complications), one of the Green's Theorems from vector
+Briefly, one of the Green's Theorems from vector
 calculus is applied to an integrated form of Laplace's equation to transform the
-differential problem into a Fredholm integral problem. The surfaces are
+differential problem into a Fredholm integral problem \cite{RSM:Bar77}. The surfaces are
 each subdivided (tessellated) into a collection of small surface (or
 boundary) elements. Then (two-dimensional) basis functions (again usually
 low-order polynomials) are used to
@@ -314,7 +315,8 @@ \subsection{BEM in the Forward/Inverse Toolkit}
 In the BEM method, these integrals involve as unknowns the
 potential and its gradient. The integration involves the computation of the
 distance between each node within the surface and to all others surfaces.
-In complicated geometries, and in all cases when the node is integrated against the points on its ``own''
+In complicated geometries, and in all cases when the node is integrated 
+against the points on its ``own''
 surface, there are numerical difficulties computing these integrals. In those cases
 there are a number of sophisticated solutions which have been proposed in the literature (and
 some of them are adopted in the SCIRun implementation).
@@ -388,8 +390,8 @@ \section{Solutions to the Inverse Problem in the Forward/Inverse Toolkit}
 
 \subsection{Activation-based Inverse Solutions in the Forward/Inverse Toolkit}
 
-Since this problem is non-linear, iterative solutions are employed. An
-``initial guess'', or starting point, is required. Then
+Since the acitvation-based inverse problem is non-linear, iterative solutions 
+are employed. An ``initial guess'', or starting point, is required. Then
 solutions are iteratively re-computed until a desired convergence criterion
 is met. Currently in the toolkit, we have a Matlab version of a Gauss-Newton
 iterative solver which can be called from within SCIRun. The starting
@@ -409,8 +411,17 @@ \subsection{Potential-based Inverse Solutions in the Forward/Inverse Toolkit}
 
 \subsubsection{Standard Tikhonov regularization}
 
-The Tikhonov regularization minimizes $ \| Ax - y \|$ in order to solve $Ax=y$, where $x$ is the unknown solution and $y$ the given measurement. The forward solution matrix $\mathbf{A}$ is of size $m\times n$, where $m$ is the number of measurement channels and $n$ is the number of source reconstruction points).
-The system $Ax$ can be overdetermined ($m > n$), underdetermined ($m <n$) or $m=n$. $A$ is often ill-conditioned or singular, so it needs to be regularized. The Tikhonov regularization is often used to overcome those problems by introducing a minimum solution norm constraint, such as $\lambda \|Lx\|_2^2$. The solution of the problem can be found by minimizing the following function:
+The Tikhonov regularization minimizes $ \| Ax - y \|$ in order to solve $Ax=y$, 
+which is the same as eq.~\ref{eq:TransMat} with $x$ replacing $x(t)$ for the 
+unknown solution and $y$ replacing $y(t)$ for the given measurement. The 
+forward solution matrix $\mathbf{A}$ is of size $m\times n$, where $m$ is the 
+number of measurement channels and $n$ is the number of source reconstruction 
+points). The system $Ax$ can be overdetermined ($m > n$), underdetermined 
+($m <n$) or $m=n$. $A$ is often ill-conditioned or singular, so it needs to be 
+regularized. The Tikhonov regularization is often used to overcome those 
+problems by introducing a minimum solution norm constraint, such as 
+$\lambda \|Lx\|_2^2$. The solution of the problem can be found by 
+minimizing the following function:
 \begin{center}
 \begin{eqnarray}
     f (x) = \| P (y - A x) \|^{2}_{2} + \lambda^{2} \| Lx \|^{2}_{2},
@@ -419,8 +430,6 @@ \subsubsection{Standard Tikhonov regularization}
 \end{center}
 
 \noindent where $\lambda$ is the regularization parameter, which is a user defined scalar value. The matrix $\mathbf{P}$ represents the \textit{a priori} knowledge of the measurements. The matrix $\mathbf{L}$ describes the property of the solution $x$ to be constrained.
-%The following sentence doesn't fit here and is removed. (Dafang)
-%as well as the posed \textit{a prior knowledge} expressed as a weighting of the measurements $P$ (also $C$ see below, e.g. sensor covariance matrix) and $L$ (also $W$ see below) which constraints the solution $x$.
 Conceptually, $\lambda$ trades off between the misfit between predicted and measured data (the first term in the equation) and the \textit{a priori} constraint.
 An approximate solution $\hat{x}$ of \ref{tik_problem} is given for the
 overdetermined case ($n > m$) as follows:

Documentation/ECGToolkitGuide.tex

@@ -42,7 +42,7 @@
 % Inverse ---------------------------------------------------------------
 \input{ECGTool_inverse}
 
-\bibliography{strings,biglit,sci,jdt}
+\bibliography{toolkit}
 \bibliographystyle{unsrt}
 
 \end{document}

Documentation/ECGToolkitGuide_figures/fwd_inv_diag.pdf

@@ -0,0 +1,75 @@
+@INPROCEEDINGS{JDT:Gul97, 
+    author="R. M. Gulrajani", 
+    booktitle="EMBS, 1997. Proceedings of the 19th Annual International Conference of the IEEE", 
+    title="The forward problem of electrocardiography: from heart models to body surface potentials", 
+    year="1997", 
+    volume="6", 
+    pages="2604-2609 vol.6", 
+    doi="10.1109/IEMBS.1997.756866", 
+    ISSN="1094-687X", 
+    month="Oct"
+}
+
+@Article{RSM:Gul98,
+  author =       "R.M. Gulrajani",
+  title =        "The forward and inverse problems of
+                 electrocardiography",
+  journal =      "EMBS Magazine",
+  year =         "1998",
+  volume =       "17",
+  number =       "5",
+  pages =        "84--101",
+  robnote =      "New review by Ramesh from his book",
+  bibdate =      "Wed Jun 10 18:57:00 1998",
+}
+
+
+@Article{RSM:Bar77,
+  author =       "R.C. Barr and M. Ramsey and M.S. Spach",
+  title =        "Relating Epicardial to Body Surface Potential
+                 Distributions by Means of Transfer Coefficients Based
+                 on Geometry Measurements",
+  journal =       "{IEEE} Trans. Biomed. Eng.",
+  year =         "1977",
+  volume =       "24",
+  pages =        "1--11",
+  robnote =      "The basis of the forward solution method is given in
+                 this paper. This is an important one for the thesis as
+                 the intrgral method based on Green's theorum is used to
+                 calculate torso potentials from epicardial ones. The
+                 data is from the dog. We start from this point in our
+                 development of the problem., ECG055",
+}
+
+
+@Article{RSM:Oos2004,
+  author =       "A. van Oosterom and T.F. Oostendorp",
+  title =        "{ECGSIM}: an interactive tool for studying the genesis
+                 of {QRST} waveforms.",
+  journal =      "Heart",
+  year =         "2004",
+  month =        feb,
+  volume =       "90",
+  number =       "2",
+  pages =        "165--168",
+  robnote =      "DESIGN: A numerical method was developed for computing
+                 ECG signals on the thorax, as well as electrograms on
+                 both endocardium and epicardium. The source
+                 representation of the myocardial electric activity is
+                 the equivalent double layer. The transfer factors
+                 between the electric sources and the resulting
+                 potentials on the heart surface as well as on the body
+                 surface were computed using a realistic thorax model.
+                 RESULTS AND CONCLUSION: The resulting transfer factors
+                 were implemented in a simulation program. The program
+                 allows the user to make interactive changes in the
+                 timing of depolarisation and repolarisation on the
+                 ventricular surface, as well as changing the local
+                 source strength, and to inspect or document the effect
+                 of such changes instantaneously on electrograms and
+                 body surface potentials, visualised by waveforms as
+                 well as by potential maps and movies. The entire
+                 simulation package can be installed free of charge from
+                 www.ecgsim.org.",
+  bibdate =      "Wed Aug 11 06:37:42 2004",
+}