FFT and I. FFT and I -- timeline. Basic MR. Basic MR Magnetic weighing machine. Based on slides from many people

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1 FFT and I -- timeline FFT and I Based on slides from many people Emigrated to Canada to perform Ph. D. studies in solid state NMR, nuclear magnetic resonance I spun samples in magnetic fields using 600,000 rpm air turbines Magnets 5000V and a few micoamps, or 300 amps plus a few volts Remember being told about the concept of subroutines greatest invention since sliced bread 1971 Came to Calgary pioneer -- which meant I designed all my own equipment PDP-11 1 K Boot Rom was on a card around 2 foot by 1 foot Learnt signal processing and biomedical physics on my own Researched in physics and pathology magnetic resonance applied to breast samples. Basic MR Place a collection of spins in a magnetic field Split into two groups aligned and nonaligned (population difference about 1 in 10^N at room temperature a fundamental MR limitation). This is a resonant system meaning that it will absorb and later emit radio signals at one frequency only Larmour Frequency Basic MR Magnetic weighing machine You excite the spins you get a signal back after you turned off the excitation pulse Amount of signal = amount of spins Free induction decay length of the decay depends on the local environment that the spins experience Later this becomes a contrast mechanism in MRI (T2) You excite the spins again you get NOTHING back You have saturated the spins, meaning as many spins aligned as unaligned You must wait for spins to recover so that you can excite them again Speed of recovery depends on the type of tissues Later this becomes a contrast mechanism in MRI (T1)

2 Diffusion Time History of MR Signal S o S = S o * (1 2 e t/t1 ) S = S o * (1 2 e t/t1 ) Normal Tissue Ischemic Tissue Dead Tissue Speed (and distance) at which water molecules move has an effect on how many different magnetic environments they experience, which in turn changes how quickly they recover -S o Each compartment that the spins experience has a different time constant Can be expressed as a convolution equation Decay signal is a series of exponentials Do deconvolution Take the experimental decay signal and multiply it by time t * sum of exponentials { exp( - t / TRECOVER) } = a sum of gamma-variate functions This allows the signal to be expressed as Series of spikes convolved with a broadening (gamma variate) signal. Position of spikes related to exponential characteristics Signal(t) = sum of spikes(t) convolved GammaVar(t) Series of spikes convolved with a broadening signal. Position of spikes related to exponential characteristics Signal(t) = sum of spikes(t) convolved GammaVar(t) DFT(Signal) = DFT(SumSpikes) x DFT(GammaVar) DFT(SumSpikes) = DFT(Signal) / DFT(GammaVar) SumSpikes( t) = IDFT( DFT(Signal) / DFT(GammaVar))

3 Limitations: Noise Filter the noise to ensure deconvolution stability Spectrum after deconvolution ENHANCED NOISE IDFT Time-domain exponential spike Function Error Ideal R (t) High frequency components are small end up dividing noise by noise Inherently unstable must apply window to remove noise re-smears the spikes Current research Application of (proper) engineering principles to the problem of magnetic resonance image reconstruction and computed aided diagnosis Signal and image processing RF issues Embedded (computer engineering) issues Biomedical engineering Software engineering (validation and verification) Collaborative work with Dr. R. Frayne in Medicine (U of C) and Dr. J. Miller in ECE (U of A) Seaman Family MR Research Centre Hardware GE 3 T Signa (40 mt m -1, 150 mt m -1 s -1 gradients)

4 Acute Stroke MR Imaging of Stroke Sag T1 DWI T2 FSE FLAIR 3D TOF PD FSE T1 SPGR E00178 Moving surgical magnet in surgical suite Basic Principle Place spins in Magnet resonant system If, and only if, you apply exactly the correct rf pulse frequency will you get resonance Off resonance no signal On resonance signal whose maximum amplitude is proportional to the total number of spins In real life you can t have exactly one frequency opf excitation as with FIR filter response, the MR response is over a range of frequencies

5 Step 2 add 1 dimension Instead of applying 1 constant magnetic field, apply a magnetic field with a field gradient in the z- direction Excite the whole system with a band of RF frequencies Demodulate the system response using a narrow band rf signal Suppose you have one group of protons at the zero point of the gradient this (when demodulated) gives a DC signal Another group is a distance of 2 cm up gradient, these spins will spin faster in the stronger field, they will produce a modulated signal of frequency constant * field gradient strength * 2 cm (distance) Another group is a distance of 4 cm down gradient, these spins will spin slower in the weaker field, they will produce a modulated signal of frequency constant * field gradient strength * -4 cm (distance) If we take these 3 sinusoidal signals (DC, f and 2f) and apply a DFT (discrete Fourier transform) we will get 3 spikes one at -4 cm, another at 0 cm and the third at 2 cm. We have started to image using nuclear magnetic resonance imaging Slice selection Spatial-varying resonance frequency during RF excitation B 1 freq band Excited location Slice profile ω ω = ω 0 + γg z z z m + = m x +im y ~ γ b 1 (t)e -iγg zzt dt = B 1 (γg z z) Magnetic Resonance Imaging Step 3 add 2 dimension and then 3 magnetic field gradients Instead of applying 1 constant magnetic field, apply a magnetic field with a field gradient in the z- direction Excite the whole system with a band of RF frequencies single frequency Suppose you have one group of protons at the zero point of the gradient this (when demodulated) gives a DC signal Another group is a distance of 2 cm up gradient, these spins will spin faster in the stronger field, they will produce a modulated signal of frequency constant * field gradient strength * 2 cm (distance) Another group is a distance of 4 cm down gradient, these spins will spin slower in the weaker field, they will produce a modulated signal of frequency constant * field gradient strength * -4 cm (distance) Only the group at 0 cm in the z-direction will be excited. Therefore the signal we receive will depend only on those spins in z-direction we have achieved slice selection While the signal is being received from these spins add another gradient, this time in the y- direction. Now this will cause some spins to speed up and some to slow down depending on their y position do a DFT on the signal we get back position information based on z and y Repeat the above steps many times, but each time add a small x-field gradient. Now the signal frequencies depends on spin position in x, y and z direction

6 Frequency encoding - 1D imaging Image Acquisition Spatial-varying resonance frequency during RF detection Image Space k-space B= B 0 + G x x S(t) ~ e iγβt S(t) ~ m(x)e iγg xxt dx m(x) k x = γg x t x S(t) = m(x)e ik xx dx = S(k x ), m(x) = FT{S(k x )} Fourier Transform Hardware Overview Pulse Sequence: Spin-Echo Diffusion Weighting 90 o RF Excitation G 180 o G Gx Image Gy Acquisition Gz

7 Pulse Sequence: Time-of-Flight Contrast The excited spins exit the system, then you look MR Angiography RF Gx 90 o Excitation Time to allow fresh flow enter the slice 90 o Gy Saturation Image Acquisition Gz 3D TOF 3D TOF + contrast Pulse Sequence: Gradient-Echo Diffusion Weighting DSC RF Gx Gy Gz Excitation 90 o G -G Image Acquisition Dynamic susceptibility contrast magnetic resonance imaging Arrange to take 19 different slices of the brain Inject a paramagnetic bolus into the arm this bolus destroys the signal in the slice Every second for 90 seconds, take a picture of the brain Assume that the rate at which the bolus moves through the picture is the same as the way the blood flows through the brain

8 Perfusion Imaging 1 Perfusion Imaging 2 0 s 6 s 12 s 18 s 24 s 30 s 36 s 42 s 48 s 54 s 60 s 66 s 72 s 78 s 84 s 2 3 E00178 E00178 Dynamic Susceptibility Contrast Studies Background: What is a Stroke? Bolus concentration c (t) VOI Time [s] ( ) ( ) '( ) a 1. c a (t) - Arterial input function 2. c VOI (t) - Tissue volume-of-interest (VOI) concentration ca ( t ) R'( t) cvoi ( t ) Scaled residue function c t = c t R t CBF = max( R '( t )) Cerebral vascular accident characterized by an insufficient supply of blood to the brain tissue TISSUE DIES Incidence: On average, every 50 seconds someone in North America has a stroke*. Mortality: Stroke ranks 3 rd among all causes of death, behind heart diseases and cancer. On average, every 3 minutes someone dies of a stroke in North America*. IDENTIFICATION Time Domain Deconvolution using Singular Value Decomposition Frequency Domain Deconvolution using Fourier Transform Cost: In 2004 the estimated cost of stroke is $53.6 billion*. (*) Heart Disease and Stroke Statistics 2004 Update, American Heart Association

9 Some of the results People measured the arterial signal at one time, and the tissue signal at another time They expect to find dispersion of the blood signal because of the time of travel between the measurement sites They found enormous dispersion to which they attributed many medical issues. They consider dispersion a major issue in obtaining accurate diagnosis of whether tissue will or will not die after stroke. They were using time-domain deconvolution approach singular valued decomposition (SVD) which was successfully used in other fields for 50+ years They said that the noise on the signals had certain characteristics and had to be removed a specific way because of these characteristics. We started Ignorance is bliss? The noise did not have the behaviour that was expected The difference in noise characteristics meant that the optimization techniques being applied where not ideal Has been proven experimentally to be correct However, even though the noise behaviour is wrong, there are many other issues, and in the end people chose a certain way because it worked. We are proposing new techniques that use our optimization approaches to provide improved responses We started Ignorance is bliss! I was not familiar with SVD deconvolution, and as FFT-King, used FFT deconvolution approach Expect minor differences but I was unable to reproduce the issues expressed in 10 years of literature. Assumed (as new boy on the block) we did not understand the system correctly, and therefore getting wrong results After much experimentation and theoretical analysis have shown that everybody else is wrong this has not gone down well The wrongness made certain correct, but underestimated, values look larger under the standard simulation conditions (no delay). The wrongness caused the appearance of phantom dispersion effects, in exactly the way that the true dispersion effects were expected to behave. The only problem was the simulation models were showing dispersion effects when dispersion was not part of the model. We are stating that dispersion is no big deal and that certain techniques, discarded because they could not account for dispersion, should be re-examined