The current section covers the mathematical aspects on how the aforementionedndimpe-
dance measurements obtained for each EIT frame are transformed into tomographic images. To this end, thendmeasurements of one EIT frame are stored in the vectorv∈Rnd.
We haveΩas the body under measurement (i.e. the thorax in 3D) with boundary∂Ωand internal conductivity distributionσ(x) as a function of the spatial variable x. The scalar potentialφ(x) inside the body is related toσ(x) via the continuum version of Kirchhoff’s law3 [80, 7]:
∇ ·¡σ(x)∇φ(x)¢=0 forx∈Ω\∂Ω. (3.1)
3This is by using the quasi-static approximation assuming that the magnetic field can be neglected due to the use of low current frequencies [80]. Furthermore, it is only valid for isotropic media. More general definitions can be found in [7] or [80].
3.1. Principle of Thoracic EIT
The boundary conditions are given by the current densityJ(x), the outward unit normal to∂Ω denoted asnandxe∈∂Ωas the set including all positions of current injecting electrodes:
σ(x)∂φ(x) ∂n = J(x)6=0 forx∈xe, J(x)=0 forx∈∂Ω\xe. (3.2)
Using Equations (3.1) and (3.2) to calculate the scalar potentialφ(x) inside a bodyΩwith conductivity distributionσ(x) is known as theforward problemdescribed hereafter.
Forward Problem
The aforementioned equations can only be solved analytically for very simple geometries and conductivity distributions [80]. However, for more practical cases numerical methods are used which requires both the geometry and the conductivity to be discretized. To this end, the body is typically represented as a finite element model (FEM) of a mesh withnetetrahedral
elements and the element-wise conductivity vectorσ∈Rne[7, 80]. The forward model can
be expressed by the operatorF(·) which provides the vectorvcontainingndsurface voltage4
measurements for a given conductivity distributionσ.
However, in EIT the problem is the opposite: it is aninverse problemwhere we seek the internal conductivity distributionσfor a given set of surface measurementsv. As the number of surface measurementsndis (much) lower than the number of internal conductivitiesne
it is anill-posedproblem. In addition, in EIT the physics of the probing energy is diffusive which results in wide variations of sensitivity across the body, i.e. EIT is much more sensitive to conductivity changes near its electrodes (e.g. variations in skin-electrode impedance) than to changes deeper in the body (e.g. variations in heart or aorta volume). Solving this inverse problem requires regularization which is addressed in the next section.
In the following, we first consider an important principle typically applied for the recon- struction of clinical data: differenceEIT. In time differenceEIT difference data is used to provide more robust images. That is, both the conductivity and the surface measurements are expressed as changes with respect to a working point (called reference or baseline), i.e.
∆σ=σ−σr andd=∆v=v−vr. In doing so the reconstructed images are less affected by
imperfections which remain stable during the measurement, such as electrode errors, differen- ces in channel gains or mismatch between model and real body shape [1, 29, 2]. The reference measurementvr is usually defined as the temporal mean over a section of the recording and
is implicitly related toσr, i.e.vr=F(σr).
As we only seek to reconstruct changes∆σwith respect toσr, we can further simplify the
4Note that the EIT raw datavare usually available as voltages and not necessarily as impedances. As they are related by a simple scaling factor – the amplitude of the injected current – we use the following three terms interchangeably: voltage measurements, impedance measurements or raw measurements.
problem by linearizing the functionF(σ) in its current operating pointσr [80, 93, 7]: F(σ)≈F(σr)+J∆σ, with [J]i,j= ∂ [d]i ∂[∆σ]j ¯ ¯ ¯ ¯σr , (3.3)
whereJ∈Rnd×neis the Jacobian or sensitivity matrix, with its elements [J]
i,jrepresenting the
sensitivity of theithsurface voltage measurement to a conductivity change in thejthmodel element. We can now attempt to estimate the change in intra-thoracic conductivity∆σˆ:
∆σˆ=arg min σ ||(v−vr)−(F(σ)−F(σr))|| 2 −σr≈arg min ∆σ ||d−J∆σ|| 2. (3.4)
However, this will not lead to a satisfying solution as we are dealing with an ill-posed and ill-conditioned problem which requires regularization.
Regularized Inverse Problem
The reconstruction is regularized by introducing additional constraints which allow for nume- rically stable results and make them more robust to interference and noise [80, 93]. A usual approach is Tikhonov regularization which seeks the estimate∆σˆby using the matrixPwhich penalizes noisy images:
∆σˆ=arg min ∆σ © ||d−J∆σ||W2 +λ2||∆σ||2P ª , (3.5) where:
• W ∈Rnd×ndis theweighting matrixused to attenuate measurements which are classi-
fied as unreliable, e.g. too noisy.W can be adapted to either completely remove failing electrodes [97] or attenuate measurements classified as noisy [71, 98].
• Thehyperparameterλ∈Rcontrols the amount of regularization, and can be seen as a trade-off parameter between image robustness and accuracy [2]. In many cases, this choice can be described as a “resolution-noise performance trade-off”.
• Theregularization matrixP∈Rne×necan be chosen in various ways [80, 2], such as:
i) When set equal to the identity matrix (P=I), zeroth-order Tikhonov regularization is used. This simply penalizes for too high amplitudes of∆σˆ [5];
ii) By settingP based on edge-sensitive spatial filters (e.g. Laplacian) the recon- structed image is penalized for sharp edges and thus forced to smoothness [5]; iii) The NOSER prior [33] calculated fromJpenalizes elements with higher sensitivity. Equation (3.5) can be further simplified to matrix form:
∆σˆ=¡JTW J+λ2P¢−1
JTW d=Rd, (3.6)
3.1. Principle of Thoracic EIT
single matrix multiplication with the difference datad. This is known as one-step Gauss- Newton (GN) reconstruction, a typical algorithm used for reconstructing clinical EIT in real- time. Another typical approach is the GREIT algorithm which is extensively described in [6, 66]. Various other algorithms (non-linear, recursive, etc.) exist, which would exceed the scope of this work. The interested reader is referred to [10, 80, 7].
Issues Concerning Hyperparamter Selection
When creating and configuring an EIT reconstruction algorithm, a key decision is the selection of an appropriate value for the hyperparameterλ. This value is often chosen empirically, which is, however, not the solution of choice when one wants to compare two different types of algo- rithms. To do this more systematically, different approaches exist which automatically choose
λ. Yet, none of these work well when one wants to compare measurement configurations which differ in electrode position, electrode number or skip pattern.
This particular issue has been investigated in more detail in the present thesis and is presented later on in Chapter 11. There, we also review the existing methods to selectλalong with the related problems and then present our novel approach able to overcome some of these problems.