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New Approaches Towards a Higher Resolution Biomedical Imaging

Magnetic Induction Tomography

04/06/2010

Nuno Brás PhD Thesis Presentation

Universidade Técnica de Lisboa Instituto Superior Técnico

PhD Dissertation Doctoral Program In Electrical and Computer

Engineering

António C. Serra and Raúl C. Martins (advisors)

Outline

04/06/2010 Nuno B. Brás - PhD Dissertation

!  Motivation !  The Problem in Hands !  State of the Art Issues !  Objectives and Developed Work

!  Experimental Work !  Forward Problem !  Inverse Problem

!  Conclusions and Original Contributions !  Further Work !  Acknowledgements

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Outline

04/06/2010 Nuno B. Brás - PhD Dissertation 3




Motivation

Imaging Systems

In Natural Sciences Astrophysics; Biophysics; Geophysics, Biology…

In Engineering

biomedical; industrial processes; oil and water prospecting; search and rescue instrumentation;…


Motivation

Magnetic Induction Tomography (as a Biomedical Imaging System)

Tomography - recreates maps or images from peripheral measurements

!  MIT is: !  Active Tomographic method (instead of Passive); !  Harmless (instead of Harmful) !  In-vivo or in-vitro; !  Functional or Steady Imaging


Motivation

What does MIT try to solve? Good things about MIT: !  (very) low price comparing with other tomographic systems

(e.g. MRI machine up to 2.5 million € + installation (25%))

!  biological passivity (low power radiofrequency)

!  excellent penetration abilities in biological phantoms, even in bone-like tissues (unlike ultrasound tomography )

!  full reconstruction is theoretically achievable


Motivation

Not so good things about MIT: 1.  The underlying image reconstruction problem is a

large, complex and non-linear problem (non-convex)

2.  Its behavior is strongly dependent on the number of measurements and their intrinsic accuracy

Actual Context There are no commercial equipments, just prototypes.

First applications are being explored


Outline


!  Motivation !  The Problem in Hands !  State of The Art !  Objectives and Developed Work



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What is MIT, conceptually

It is a distributed parameter estimation (DPE) problem under Electromagnetic Partial Differential equations

(PDE)


The Problem in Hands

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where is the kernel of the underlying physical process.

Consider an electromagnetic physical process ruled by the following relation:


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Distributed Parameter Estimation Problems in electromagnetics

where is the kernel of the underlying physical process. Electric and

magnetic fields or potentials

(state variables)

Boundary Conditions

+ Sources

Set Differential Operators

+ parameters



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where is the kernel of the underlying physical process.


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The inverse problem is the distributed parameter identification problem given by:

The forward problem (well-posed, and typically linear) is the PDE equation


The General Parameter Estimation Model (Outline of this Thesis)

Experimental Setup (PART 1)

The Forward Problem (PART 2)

The Inverse Problem (PART 3)


Problem in Hands

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The MIT Setup


Source Currents: Harmonic , tens of kHz to some MHz;

Current Amplitude: Up to 1 A;

Setup Radius: Typically 15 cm;

Conductivity Values: complex with absolute value between 0.1 S/m to 2 S/m;

Source and sensing coils size: around 5 cm and 3 cm correspondingly


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The MIT Setup

1



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The MIT Setup

2



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The MIT Setup



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The MIT Setup



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The MIT Setup Each d should be as large, independent and accurate as possible

Outline





!  Experimental Issues !  Typical layout generates ambiguities (gradiometers); !  Fixed system of acquisition (fixed and low number is allowed)

!  Forward Problem !  Developed solvers in MIT context are accurate but typically slow;

(it is not possible to use commercial solvers)

!  Inverse Problem !  Non-linear approaches are the best known reconstruction methods. !  The state of the art method is not efficient when used with a high

number of acquisitions.


State of the Art Issues

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Outline






Objectives - Developed Work

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General Objective Implementation of a new Magnetic Induction Tomography prototype and numerical framework to deal with large number of acquisitions

Developed Work !  Experimental

!  A new moving prototype was implemented for large number of acquisitions while attaining the state of the art sensitivity (SCR)

!  Forward Problem !  A new 3D eddy current PDE solver was developed with a

competitive processing time and low relative error.

!  Inverse Problem

!  A new ADMM method was developed and used in 2D and 3D IP.

!  Its feasibility was proved and its advantage was clearly shown for large datasets scenarios.

Outline





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Experimental Work

!  Early Experimental Prototypes


Experimental Work


! Twin Coil Setup Source Current

Amplifier

Capacitive Shields

Source Coil

Motor systems


Experimental Work

Involved Areas While Designing This Prototype

! Electromagnetic Compatibility (EMC)

! Mechanical Setup Design and Characterization

! Current Source Design

! Acquisition System

!  Signal Processing


Experimental Work

Involved Areas While Designing This Prototype

! Electromagnetic Compatibility (EMC)

! Mechanical Setup Design and Characterization

! Current Source Design

! Acquisition System

!  Signal Processing Measuring System


Symmetry Axis

Without object: V1-V2 = Residual V ≈ 0, for any angle With object: V1-V2 = ∆V

V1

V2

Experimental Work

! Twin Coil Setup

Useful V =∆V– Residual V

!  Shielded Differential coils and cables

!  Gain (low noise) Amplifier =100; with a low-pass filter

!  ADC with 12 bits, up to 60 MSamples/s

!  The Goertzel Transform (second order IIR filter ) to calculate the frequency bin amplitude and phase (others were tested)

!  Average sliding window applied to avoid 50Hz modulation and reduce noise

!  Principal Component Analysis (PCA) and a reference coil were used to eliminate long term trends from the source current.


Experimental Work

Acquisition System

Signal Processing System


Experimental Work

!  Results a differential coil pair, during 300 sec time


Experimental Work

!  Results - Stability Applying an averaging sliding window with 50 values

A minimum measurable SCR of

was obtained which is the stated state of the art value in this case with a moving system of sensing coils

Outline





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The Forward Solver abilities:

!  Isotropic parameter model !  Constant � !  Diffusion model approach (no skin currents) !  Harmonic and stationary equations !  Multi- level grid allowing mutligrid (MG) techniques and/or

adaptive mesh refinements (AMR) in future iterations !  Analytical sources, using the Biot-Savart law

The Forward Problem


Normal Component of = 0

The Forward Problem


Formulation:

Imposed directly by (1)

(1)

in

Boundary Conditions Interface Conditions

where and

The Forward Problem

The Discretization : Finite Integration Technique


The Forward Problem


The proposed FIT formulation:

!  The resultant system is non-singular - the used gauging ensures a robust regularization of the system;

!  A Laplacian-type instead a curl-curl-type problem, where a larger set of numerical methods are available;

!  Subgriding implementation;

Moreover…

!  Several Numerical Optimization aspects were implemented;

!  Solved with iterative preconditioned methods (here, iLU factorization was used);

The Forward Problem

Results – Accuracy


The Forward Problem


Results – Performance

The Forward Problem


Results – Performance and accuracy

The Forward Problem

How this interacts with the inverse problem?


Outline



!  Experimental Work !  Forward Problem !  Inverse Problems


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Inverse Problems


!  Implemented Methods :

! Two versions of the Alternating Direction Method of Multipliers (Augmented Lagrangian Method) in an elliptic 2D inverse problem with total variation regularization and Wavelets regularization

! Gauss Newton in ellipitc 2D Inverse Problem

! ADMM version adapted to solve the MIT 3D

Inverse Problems


The ADMM – a sequence of simple problems

Closed form

Inverse Problems


The ADMM – a sequence of simple problems

Closed form

Either implementing a fixed point iteration for a continuous approximation of

(ADMM with fixed point iteration) or

Using a second Alternate Direction split Closed Solution (DIPESAL Method)

Inverse Problems

2D Elliptical IP - The same model, a simpler problem


Inverse Problems


Gauss Newton using an unconstrained version of

the problem

ADMM with a fixed point iteration and TV Regularization

ADMM with a fixed point iteration and Wavelet

Regularization

Inverse Problems


GN ADMM

Inverse Problems


GN

ADMM

Inverse Problems


DIPESAL

ADMM with fixed point iteration

2.2% better reconstruction and 7% faster, with better image reconstruction quality

5% noise and LARGER problem

Inverse Problems


!  Some remarks for 3D MIT inverse problem

! Two things changes for MIT problem:

!  Size: Each Iterations of the ADMM problem have to be solved iteratively: The Second Order Stationary method was applied.

!  Different discretization and equations: The new equations were referred before;

Inverse Problems


!  3D MIT inverse problem

Inversion scenarios

Inverse Problems


Inverse Problems

!  Relative Error Evolution and Constrain Imposition


Inverse Problems

!  Performance Assessment


Outline





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Conclusions and Original Contributions

!  Experimental Work

!  A new prototype was implemented for moving sensing and source coils, allowing acquire large sets of accurate data

!  System sensitivity, a state of the art value, is stable during a large amount of time (>300 sec), allowing to implement a moving setup with high sensitivity.


This work resulted in 7 congress papers during the PhD period


!  Forward Problem

!  A new hybrid formulation of Finite Integration Technique

!  The processing time was optimized to be included in an inverse problem attaining a low relative error

!  Total relative error ~ 1.5 %

!  Processing time ~19 sec. per eddy current problem


This work was published on IEEE Transactions on Magnetics (May 2010)



!  2D Elliptic Inverse Problem

!  A new method ADMM algorithm (DIPESAL) was implemented with closed solution for Total Variation regularization

!  Lower relative error (2,2%);

!  7% faster.

!  Better qualitative images in general

This work was submitted to IEEE Transactions on Image Processing (May 2010)



!  3D MIT Inverse Problem Solution

! The ADMM algorithm feasibility in 3D MIT problems was achieved

!  A complete new approach with clear advantages in large number of measurements conditions.

!  Its application to the MIT originates state of the art simulated reconstructed maps.

This work was submitted to IEEE Transactions on Medical Imaging (May 2010)

Outline





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Further Work


!  Experimental !  Use large arrays of Giant Magnetic Resistors as sensors – This can

dramatically change MIT since it can increase dramatically the number of acquisitions and their accuracy.

!  Use strong permeability material cores to increase magnetic field. !  Use of transient signals to increase SNR

!  Numerical !  Multigrid scheme for preconditioning and code parallelization to

accelerate even more the forward and inverse problem. !  Field Regularization – there is very experiments clearly shows

advantages in using this !  Improve ADMM to MIT – There is space to improve, namely in the

regularization process

Outline





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Acknowledgements

!  Supervisors: A. C. Serra And Raúl C. Martins

!  Other professor and researchers !  Professor José Bioucas Dias (Elect. and Comp. Dep.) !  Professor Artur Lopes Ribeiro (Elect. and Comp. Dep.) !  Professor Paulo Martins (Mechanical Dep.) !  Professor Helena Ramos (Elect. and Comp. Dep.) !  Dr. Tomas Radil (Elect. and Comp. Dep.) !  Professor Pedro Santos (Mathematics Dep.) !  Professor Carlos Alves (Mathematics Dep.) !  Eng. Luis Soares !  Eng. Alexandre Pestana !  Eng. José Gouveia


New Approaches Towards a Higher Resolution Biomedical Imaging

Magnetic Induction Tomography

04/06/2010

Nuno Brás PhD Thesis Presentation

Universidade Técnica de Lisboa Instituto Superior Técnico

PhD Dissertation Doctoral Program In Electrical and Computer

Engineering

António C. Serra and Raúl C. Martins (advisors)

phd presentation_web

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brs phd dissertation

forward problem

inverse problem

developed work

experimental work

motivation imaging systems

nonlinear problem nonconvex

underlying physical