An Estimation Technique for Spin Echo Electron Paramagnetic Resonance

A Thesis

Presented in Partial Fulﬁllment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Frank Golub, B.S.

Graduate Program in Department of Electrical and Computer Engineering

The Ohio State University

2013

Master’s Examination Committee:

Lee C. Potter, Advisor Bradley Clymer Rizwan Ahmad © Copyright by

Frank Golub

2013 Abstract

In spin echo electron paramagnetic resonance (SE-EPR) spectroscopy, traditional methods to estimate T2 relaxation time include fitting an exponential to the peaks or the integrated areas of multiple noisy echoes. These methods are suboptimal and result in lower estimation accuracy for a given acquisition time. Here, two data processing methods to estimate T2 for SE-EPR are proposed. The first method finds the maximum likelihood estimate (MLE) of T2 via parametric modeling of the spin echo and joint least-squares fitting of the collected data. The second method exploits the underlying rank-one structure in SE-EPR data via singular-value decomposition

(SVD). The right singular vector corresponding to the largest singular value is then

fitted with an exponential to find T2. This method bears strong similarity to a non- parametric MLE-based approach that does not assume a structure of an echo. The methods are validated using simulation and experimental data. The proposed methods provide 41-fold and 3-fold acquisition time savings over the traditional methods of fitting echo peaks and areas, respectively. Interestingly, the results also indicate that the SVD-based approach generates mean squared error nearly identical to that produced by the MLE based on parametric modeling for a wide range of SNR.

ii This is dedicated to my parents, my nephews, and their sense of humor.

iii Acknowledgments

I wish to acknowledge the love and support from my friends and family, and the dedication from my advisors Lee Potter and Rizwan Ahmad. My sister Marjorie deserves much credit for inspiring me to learn math at a young age.

iv Vita

December 8, 1987 ...... Born - Canton, Ohio

2006 ...... McKinley Senior High School

2011 ...... B.S. Physics, Brandeis University 2011 ...... B.S. Mechanical Engineering, Columbia University

Fields of Study

Major Field: Electrical and Computer Engineering

Focus: Signal Processing and Electron Paramagnetic Resonance

v Table of Contents

Page

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

List of Figures ...... viii

1. Introduction ...... 1

1.1 Background ...... 1 1.2 Relevant Trends in EPR ...... 2 1.3 Approach ...... 3 1.4 Thesis Organization ...... 4

2. Spin Echoes ...... 5

2.1 Resonance ...... 5 2.2 The Initial Hahn Spin Echo ...... 5 2.3 Carr and Purcell: The Current Hahn Echo ...... 6 2.4 Signal Processing Methods ...... 8

3. Maximum Likelihood Estimation ...... 10

3.1 Maximum Likelihood Estimation ...... 10 3.2 Singular Value Decomposition ...... 12

3.3 Estimators for T2 ...... 14 3.3.1 Peak-picking and integration ...... 14 3.3.2 Parametric MLE ...... 15 3.4 The Relationship Between SVD and MLE ...... 15

vi 4. The Cram´er-RaoLower Bound ...... 18

4.1 Derivation for One Non-random Parameter ...... 18 4.2 Extension to Multiple Parameters ...... 21

5. Results ...... 24

5.1 Materials and Methods ...... 24 5.2 Results ...... 24

6. Discussion ...... 29

7. Conclusion ...... 31

Appendices 32

A. Matlab Code ...... 32

Bibliography ...... 42

vii List of Figures

Figure Page

4.1 Lower bound on the standard error of Tb2...... 23

5.1 Singular values for the data array formed from the seven acquired echoes. 25

5.2 Left: standard deviation of T2 estimation error for a simulated seven echo train at four noise levels: 17, 24, 30, and 37 dB. Right: relative

acquisition time versus SNR to achieve ﬁxed standard deviation of T2 estimation error...... 26

5.3 Left: standard deviation of T2 estimation error from simulated data versus number of acquired echoes. Right: relative acquisition time

versus number of echoes to achieve ﬁxed standard deviation of T2 estimation error...... 27

5.4 Left: standard deviation of T2 estimation error for measured data with seven echoes at four synthesized noise levels: 17, 24, 30, and 37 dB. Right: relative acquisition time versus SNR to achieve ﬁxed standard

deviation of T2 estimation error...... 28

5.5 Left: standard deviation of T2 estimation error from measured data versus number of acquired echoes. Right: relative acquisition time

versus the number of echoes to achieve ﬁxed standard deviation of T2 estimation error...... 28

viii Chapter 1: Introduction

1.1 Background

Electron paramagnetic resonance (EPR) is a spectroscopic method capable of

detecting and quantifying free radicals. The physics that governs nuclear magnetic

resonance (NMR) technology also governs electron paramagnetic resonance (EPR).

NMR focuses on the spins of protons, whereas EPR revolves around the spins of

unpaired electrons.

Felix Bloch won the 1952 Nobel Prize for discovering how nucleons move in a

magnetic ﬁeld. Using the Bloch equations, Hahn invented the concept of spin echoes,

later reﬁned by Carr and Purcell, that is prevalent in both nuclear imaging domains.

But whereas NMR can successfully map the tissue densities in the body, EPR has

failed to gain widespread clinical practice. EPR promises to measure the oxygen

concentration in tumors, but it is hindered by the need to inject a patient with a spin

label. Internal biochemistry quickly degrades signals from spin labels, highlighting the

need to assess these signals with minimal acquisition time. The proposed processing

techniques improve upon the traditional methods by oﬀering estimates with smaller

variances of the transverse decay constant T2. Because linewidth is linearly related to partial pressure of oxygen, the T2 parameter provides a sensitive indicator of oxygen

1 concentration. Thus, for a given desired estimation accuracy, the proposed methods allow for a reduction in the amount of noise averaging and hence the acquisition time.

1.2 Relevant Trends in EPR

Over the past several decades, EPR has found numerous applications in biology, chemistry, physics, and medicine [16]. Among in vivo applications of EPR, oximetry has been arguably the most investigated area of research, with emphasis on quanti- tative assessment of tumor hypoxia [13, 19].

Continuous wave (CW) EPR and pulsed EPR are competing yet often complemen- tary modes of data acquisition. At present, CW EPR remains the most widespread technique for in vivo oximetry [2] because of its simple equipment design and ability to utilize a wide variety of oxygen sensitive spin probes. However, the data acquisition in CW EPR is generally slow, resulting in long acquisition times. One way to accelerate data acquisition is to use pulsed EPR methods. With recent technologi- cal advances and the development of EPR oximetry probes with narrow linewidth, pulsed EPR oximetry has become an attractive option, especially for studying tumor hypoxia [10].

In pulsed EPR oximetry, a spin echo (SE) approach allows measuring intrinsic T2 directly, which can be readily converted to the homogeneous broadening component of the EPR lineshape. Since the homogeneous broadening is proportional to the oxygen concentration, T2 can thus be used to estimate oxygen concentration via a precom- puted calibration curve. The SE-EPR oximetry can be used in both spectroscopic and imaging modes.

2 In SE-EPR, the data are generally collected using conventional π/2 − τ − π−echo

pulse sequence [11]. Since the echo amplitude decays with exp(−2τ/T2), collecting

and processing multiple echoes with diﬀerent τ values enables estimation of T2. The

traditional methods of processing SE-EPR data include ﬁtting an exponential to echo

peaks (peak-picking) or echo areas (echo-integration). The peak-picking method is

ineﬃcient, because it only uses one data point, the peak, from each echo. Additionally,

echo-integration is likewise shown to be an ineﬃcient use of the acquired data.

1.3 Approach

The general approach towards selecting an estimation technique is to fully charac-

terize all samples in a data set to better ascertain T2. In this study, a measured data

set consists of a sequence of echoes. The amplitude of each echo decays exponentially

with rate constant T2. Traditional methods such as peak picking and integration fail to represent the underlying structure of each echo.

Here, we propose two methods to process SE-EPR data for estimating T2. The

ﬁrst approach parametrically models the spin echo and ﬁnds T2, along with other

nuisance parameters, by nonlinear least-squares to generate the maximum likelihood

estimate (MLE). A similar approach has been used for pH measurement using CW

EPR [1]. The second method is an SVD-based estimator for an unknown echo shape.

We note the presence of a similar MLE that also makes no prior assumption of

an echo shape. A Cram´er-Raobound investigation demonstrates the similarity in

the proposed techniques in a typical SNR region. Furthermore, the two maximum

likelihood estimators are consistent estimators for the SE-EPR signal model. All

3 techniques were compared using simulated and measured data with synthetic noise.

A small vial of activated charcoal was excited at X-band.

1.4 Thesis Organization

Chapter 2 introduces the concept of spin echoes and the early signal models.

Chapter 3 presents the two traditional and two proposed signal models in the context of maximum likelihood estimation. Chapter 4 begins with a derivation of the

Cram´er-Raolower bound and discusses the asymptotic variance qualities of maximum likelihood estimation. Chapter 5 details the experimental conditions and presents the results for the simulated and measured data set, respectively. A performance comparison of the four methods is discussed in Chapter 6. Chapter 7 concludes the

ﬁndings.

4 Chapter 2: Spin Echoes

2.1 Resonance

Resonance may be more familiar in the context of an operatic soprano shattering glassware. Glass has a tendency to vibrate at its natural frequency. By matching that frequency with her voice, an opera singer can force such large vibrations in the glass that it breaks. Similarly, a molecule with unpaired electrons will also vibrate in accordance with its distribution of resonant frequencies, or lineshape. By matching a highly resonant region of the lineshape with a radio-frequency (RF) pulse, two significant effects occur. The energy of the molecules shall increase and then decay, and the electrons shall precess in phase before their synchronicity decays. These two decays serve as intrinsic properties of the molecule. In SE-EPR, spin probes lose their energy after a significantly long period of time, but lose their phase memory too rapidly to acquire a signal. In 1950, Hahn [11] developed an RF pulse sequence to retain phase memory. He termed the resultant time-domain signal as a “spin echo.”

2.2 The Initial Hahn Spin Echo

The initial Hahn spin echo sequence reproduces nuclear phenomena in a pre- dictable fashion. Consider a collection of paramagnetic molecules in a three-dimensional

5 coordinate system with a strong magnetic ﬁeld along the positive z-axis. The unpaired

electrons shall precess about their magnetic moment vector. Due to the Zeeman eﬀect,

slightly less than half of these electrons will precess out-of-phase about the negative

z-axis, as opposed to the positive z-axis.

However, the brief presence of a new magnetic ﬁeld, perpendicular to the ﬁrst,

will cause the electrons to precess in-phase about a diﬀerent axis. These vectors

soon move out of alignment, but the reapplication of the orthogonal magnetic ﬁeld

will force the magnetic moment vectors to realign. The in-phase movement of the

electrons produces a time varying magnetic ﬁeld, which in turn causes a time-varying

electric ﬁeld, or current, in the receiver coils. These brief magnetic ﬁelds arise from

short RF pulses whose frequencies resonate with the paramagnetic molecule.

In the original paper [7], Hahn excited protons, not electrons, but the physics

remains very similar. He posited a 90◦ pulse, followed by a waiting period of duration

τ, another 90◦ pulse, followed by another waiting period τ, followed by a spin echo.

The angle depends on τ, in addition to the strength of the new magnetic field and the gyromagnetic ratio. Diffusion and field inhomogeneities complicate the analyti- cal solution. Hahn palliated those effects, producing echo amplitudes that diminish exponentially as τ increases. To identify a signal, Hahn fit the logarithm of those amplitudes to a straight line, an early form of peak picking.

2.3 Carr and Purcell: The Current Hahn Echo

Carr and Purcell [7] improved upon the earlier work by seeking a better functional understanding of the spin echoes. Instead of applying two 90◦ pulses, they suggested a 90◦ followed 180◦ pulse to remove the eﬀects of diﬀusion. This method provided a

6 better graphical understanding of how the magnetic moment vectors move. Again,

the original paper employed protons, but the physics also applies to electrons.

As before, the moment vectors of the unpaired electrons initially align towards

and against the direction of the main magnetic ﬁeld. A 90◦ RF pulse excites a narrow

frequency band particular to a spin probe, causing the vectors to rotate 90◦ and point

perpendicularly to the main magnetic ﬁeld. The protons again spin in phase with one

another before moving out of alignment. The corresponding signal decay is known as

the free induction decay (FID). Then, the moment vectors distribute outwardly about

the axis of the main magnetic ﬁeld. A 90◦ RF pulse then ﬂips the magnetic moment vector of the electrons, causing the distribution to realign before de-phasing. Carr and Purcell’s insight was that a double-sided FID, or spin echo, is the inverse Fourier transform of the excited lineshape. Consequently, if the lineshape were Gaussian, the shape of each echo would also be Gaussian.

Furthermore, time constants T1 and T2 should be distinguished [8]. In a spin echo

sequence, the time between the 90◦ and 180◦ RF pulses is same time between the

180◦ pulse and the peak of a spin echo. As the time between pulses increases, the

amplitude of the spin echo diminishes proportionally with a ﬁrst-order exponential

with rate constant T2. The T1 decay constant refers to the time required to realign the

macroscopic moment vector of unpaired electrons with the main magnetic ﬁeld. In

EPR, unpaired electrons dephase much more quickly than they realign, which means

that T2 is much smaller and more relevant to signal decay than T1.

7 2.4 Signal Processing Methods

Two recent studies [12, 18] evaluate T2 by varying the time between RF pulses and fitting the echo amplitudes to a decaying exponential, a technique known as peak picking. We note that Tseitlin [18] strove to identify the functionality of new hardware rather than demonstrate statistical robustness. An OX63 trityl radical was excited at L-band. The times between the 90◦ and 180◦ pulses varied between 1 µs and 11 µs in increments of 2 µs, which generated spin echoes of similar structure but diminishing amplitude. Over 300,000 echoes were collected and averaged. The peaks of these averages were fit to a decaying exponential. This paper demonstrates that peak picking is an acceptable, if inefficient, method of determining T2. The inefficiency stems from using only one data point in a spin echo curve. In fairness,

Tseitlin also acknowledged an integration-based technique. The areas under each echo are proportional to the peaks.

Mailer [12] collected multiple time series, each containing a set of echoes. A procedure was used to determine the common sample points between series to correspond to echo peaks. This approach prevents the noise from linearly biasing the peaks and non-linearly biasing the decay constants. For each echo in the ﬁrst time series, the peak of one Lorentzian function was positioned at approximately the maximum of a given echo. A second Lorentzian was ﬁtted to the nearby data points. The time point that maximized the second Lorentzian was chosen as a common sample point.

Sample points found in the ﬁrst time series would be applied in subsequent ones.

Raz [14] proposed a maximum likelihood model that assigned the same underlying structure to each echo in a set. His work suggested an even symmetric echo, spline

8 ﬁtting each tail with the sum of complex, exponentially damped sinusoids to amelio- rate noise artifacts. Each echo was situated in a row of data matrix µ to ease least squares ﬁtting. A gradient descent algorithm determined the unknown parameters.

9 Chapter 3: Maximum Likelihood Estimation

3.1 Maximum Likelihood Estimation

In this study, K echoes are collected, each M samples long. Intuitively, the data can be structured into a concatenated vector ~y. Consider a data set ~y = [~y1, ~y2, ...

T th ~yK ] , where each ~yi pertains to N elements in the i echo. The corresponding signal model is ~ ~y = fθ~ + ~n. (3.1)

~ ~ The model fθ~ depends on a set of parameters θ, including the transverse spin relaxation time Tb2. Each element of the noise ~n is assumed to be independent and identically distributed with variance σ2. This assumption has been veriﬁed in prior

EPR spectroscopy [15]; further for a given variance, additive white Gaussian noise

(AWGN) maximizes the entropy [9]. Solving for the likelihood function:

−(~y − f~ )T (~y − f~ ) ~ 1 θ~ θ~ pY |θ~(Y |θ) = MK exp 2 (3.2) (2πσ2) 2 2σ

2 where the notation k · kF refers to the Frobenius norm. ~ We seek a vector of maximum likelihood estimates θML. Thus, by the monotonicity of the logarithm function, the problem reduces to minimizing the argument inside the exponential in Eq. 3.2, which may be solved by the lsqnonlin function in Matlab.

Equation 3.6 demonstrates a cost or penalty for obtaining a data point that de- viates from the model. The objective is to ﬁnd the minimum of a multi-dimensional cost surface. The number of dimensions equals the number of parameters.

For a signal model dependent on only two parameters, the cost surface may be visualized as a hilly baseball ﬁeld. The baselines serve as the two coordinate axes.

A pitcher, initialized at a point on the ﬁeld, seeks the lowest cost or location in the ballpark. The function lsqnonlin employs a gradient descent algorithm to determine the minimum.

Several factors mitigate the player’s pursuit. He may stop at a local instead of the global minimum. AWGN roughens the cost surface, making local minima more al- luring and reducing the curvature of valleys. Less curvature implies greater variance.

Last, the pitcher’s eyesight depends on the characterization of the parametric model.

Failure to properly characterize data makes more data points about a minimum ap- pear as viable stopping points, which also implies a greater variance. The task could be simpliﬁed by reducing the model to only one dimension.

11 3.2 Singular Value Decomposition

Singular Value Decomposition (SVD) can reduce dimensionality, initialize the gra-

dient descent, and characterize all data. Ironically, the data must ﬁrst be restructured

T th from a vector into a two-dimensional matrix, with ~yi as the i column of a matrix

A matrix can express the relationship between two data points that are not ac-

quired consecutively. An early discovery was that each echo has the same normalized

shape. The scale factor depends on τ, the period between the two RF pulses in the

th Hahn sequence, and T2, the longitudinal relaxation time. Importantly, the m data

−2τk/T2 point in every ~yk expressed the exponential decay relationship bk(T2) = e . Let T ~ th fk be the k column of the matrix F.

T T T F = [f1 f2 ...fk ] (3.7)

~T = α~ab (T2) (3.8)

Here, α is a scalar, and ~a represents the unknown structure of the echo with unit norm. The following more suitably expresses the MLE problem.

~ 2 θML = argmin kY − F kF (3.9) θ~ ~T 2 = argmin kY − α~ab (T2)kF (3.10) θ~

Using SVD, any matrix Y can be decomposed into the unique construct

Y = UΣV T (3.11)

The columns of U and V are known as the left and right singular vectors and form

the eigenbasis of YY T and Y T Y respectively. Σ is a diagonal matrix composed of

12 singular values σi, arranged in descending order. The squares of these singular values

are the eigenvalues of matrix YY T and its transpose.

If U is an eigenbasis of YY T ,

(YY T )U = UΣΣT , (3.12)

and V is an eigenbasis of Y T Y ,

(Y T Y )V = V ΣΣT . (3.13)

Then

YY T = (UΣ)(V T V )(ΣT U T ) (3.14)

= (UΣV T )(UΣV T )T . (3.15)

Importantly, if one eigenbasis is known, then the other eigenbasis is deﬁned.

If the ratio of ﬁrst to second singular values is suﬃciently large, then SVD poses

signiﬁcant beneﬁts for our understanding of the spin echo data and the initialization

of the optimization task in Eq. 3.10. Each column of Y will then closely resemble the

ﬁrst column of U, and each row of Y will resemble the ﬁrst row of V T . In this study,

the ratio of first to second singular values was shown to be 77.1:1. This observation ~ implies ~v1 ≈ b, where ~v1 denotes the first right singular vector. For a parametric ~ MLE, ~v1 was fitted in a least squares sense to b to provide an initial guess of Tb2. This

guess was then used in the gradient descent line search.

~ 2 Tb2init. = argmin k~v1 − bk (3.16) T2 A large disparity in singular values also implies a reduction in dimensionality. Let ~ ~u1 be the ﬁrst left singular vector. If ~v1 ≈ b, then the echo shape satisﬁes α~a ≈ ~u1σ1.

13 The SVD-based modeling technique exploits this observation to obtain an estimate

of the unstructured echo decay ~b.

3.3 Estimators for T2

The purpose of data acquisition and processing is to produce an estimate, Tb2, of the T2 relaxation time. The quality, or resolution, of an unbiased estimate is judged by standard error in the estimated relaxation time. In this section, we ﬁrst review the peak picking and integration estimators. Then, we present and analyze two proposed estimators: nonlinear least-squares curve ﬁtting and an estimator based on the singular value decomposition (SVD).

3.3.1 Peak-picking and integration

In the peak picking method (e.g., [12, 18]), the estimate of T2 is determined by a least-squares ﬁt of an exponential to peaks taken from N measured echoes:

N X 2 Tb2 = argmin |yk − A exp(−2τk/T2)| , (3.17) A,T 2 k=1

th where yk is the sample from the k echo taken a ﬁxed position common to all echoes,

th A is a ﬁxed amplitude, and τk is the echo time for the k pulse. The ﬁxed position

is taken to be at approximately the peak of the echo, maximizing SNR and realizing

the name peak picking.

In the integration method each echo is summed before being ﬁt to a decaying

exponential. Thus, the estimated relaxation time is

N X 2 Tb2 = argmin |yk − A exp(−2τk/T2)| , (3.18) A,T 2 k=1

th where yk is the average of the equi-spaced sampled values for the k echo.

14 3.3.2 Parametric MLE

Parametric modeling relies upon prior knowledge of the noiseless echo shape, as expressed by a function φ(t, θ~) parametrized by one or more variables in θ~. An

~ 2 example is a Gaussian model, φ(t, θ) = exp {−(t − θ1) /θ2}, where θ1 is a time shift of the Gaussian shape, and θ2 is twice the variance; Lorentzian and Voigt functions are other examples [14]. As an extreme case, φ(θ~) = ~a

Thus, under the assumption of known parametric model of echo shape and additive white Gaussian noise, the maximum likelihood estimate (MLE) of relaxation time is

N 2 X ~ Tb2 = argmin Yk − Aφ(θ) exp(−2τk/T2) (3.19) ~ A,T2,θ k=1

2 where k·k denotes the sum of squares of a vector, Yk is the vector of samples from the kth echo, and φ(θ~) is the vector of samples, at the same sampling instants, from the parametric model of the echo shape. For the signal model in Eq. 3.19, a theoretical lower bound on the standard error of any unbiased estimator is given by the Cram´er-

Rao lower bound [17]. For the Gaussian noise model, the bound predicts that standard error versus signal-to-noise ratio (SNR) is linear in a log-log plot. The MLE achieves the theoretical bound at high SNR and is hence asymptotically a statistically eﬃcient estimator of T2.

3.4 The Relationship Between SVD and MLE

If we assume an arbitrary shape of the echo, then we arrive at a non-parametric

MLE, which, surprisingly, can be easily computed.

15 The exact MLE can be derived; following from Eq. 3.10 [3],

~ ~T 2 θ = argmin kY − α~ab (Tb2)kF (3.20) θ~ ~ ~T Y b(Tb2)b (Tb2) = argmin kY − k2 (3.21) ~ 2 F θ~ kb(Tb2)kF Now consider X = (bbT )/(bT b).

~ 2 θ = argmin kY (I − X)kF (3.22) θ~ = argmin tr(Y (I − X)(I − X)Y T ) (3.23) θ~ = argmin tr(Y (I − X − X + X2)Y T ) (3.24) θ~ = argmin tr(YY T − YXY T ) (3.25) θ~ = argmax tr(YXY T ) (3.26) θ~ ~ ~T T ! Y b(Tb2)b (Tb2)Y Tb2 = argmax tr (3.27) ~T ~ Tb2 b (Tb2)b(Tb2) ~T T ~ ! b (Tb2)Y Y b(Tb2) Tb2 = argmax tr (3.28) ~T ~ Tb2 b (Tb2)b(Tb2) For an unstructured ~b, the vector that maximizes Eq. 3.28 is the ﬁrst right singular vector. However, for the exponential decay model, Eq. 3.28 can be directly optimized via a one-dimensional line search. In the results section below, we observe that the

SVD approximation to the MLE gives nearly identical estimation performance for signal-to-noise ratios of interest.

In the proposed SVD technique, sampled data from each echo forms a single column in a rectangular data array Y . The SVD [5] provides the matrix decomposition,

N X T Y = σrurvr , σ1 ≥ σ2 ≥ ... ≥ σN (3.29) r=1

where, for M samples per echo, ur is an M-by-1 vector of norm 1 and vr is an N-by-1 vector of norm 1. Under the assumption that the noiseless echo has the same shape

16 for each delay time τk, then Y must be a rank 1 matrix. Accordingly, in the noiseless case, u1 is the normalized echo shape, and the N values in v1 are samples of the decay curve. Then, in noise, the estimated relaxation time for an exponential decay is given by

N X 2 Tb2 = argmin kYk − Au1 exp(−2τk/T2)k (3.30) A,T 2 k=1 N X 2 = argmin |v1[k] − A exp(−2τk/T2)| , (3.31) A,T 2 k=1

th where v1[k] is the k entry of the principal right singular vector, v1. Thus, using the

SVD of the data array, the relaxation time can be estimated without prior knowledge of the echo shape.

The SVD-based estimator is defined assuming a single echo shape and one relaxation time. The same estimator results if multiple paramagnetic species are present with different echo shapes, but share the same exponential decay. On the other hand, for R > 1 species with different relaxation times, T2,r, the estimator may be modified from Eq. 3.30 to obtain

N R 2 X X Tb2,r = argmin Yk − A σrur exp(−2τk/T2,r) (3.32) A,T 2,r k=1 r=1

17 Chapter 4: The Cram´er-RaoLower Bound

The Cram´er-Raolower bound (CRLB) is the lowest achievable variance for an unbiased estimator. An MLE attains this lower bound in a suﬃciently high signal-to- noise (SNR) region. The two proposed signal models possess a smaller CRLB than the two traditional ones.

4.1 Derivation for One Non-random Parameter

This section reviews the derivation of the CRLB for a single parameter [17]. The

CRLB for one non-random parameter relies on the Cauchy-Schwartz or triangle inequality. This inequality states that the longest side of a triangle cannot be greater in length than the sum of the other two sides. Equality holds only when the two smaller sides are linearly dependent. This inequality extends to the concept of correlation.

Let the covariance of two random variables serve as the hypotenuse of a triangle, and let the square root of the individual variances serve as the smaller two sides. The two considered variables are the unbiased and the maximum likelihood estimates. Here,

Tb2 represents the parameter to be estimated. Y represents the data.

d ln p Y |T2 E[(Tb2 − T2)( )] dT2 ≤ 1 (4.1) q q d ln p Y |T2 V ar[T2 − T2] V ar[ ] b dT2

18 Equality holds only when both estimates yield the same information, whereas a large noise variance hinders linear dependence. The following derivation closely follows the literature. Consider the following deﬁnitions:

√ f(x) = (Tb2 − T2) pT2|Y (4.2)

d ln pY |T2 √ g(x) = pY |T2 (4.3) dT2

~a = [f(x1), f(x2), ...f(xn)] (4.4)

~ b = [g(x1), g(x2), ...g(xn)] (4.5)

4x = xi+1 − xi where i = 1, 2, ...N . (4.6)

In vector notation, the triangle inequality reads:

~a ·~b ≤ |~a||~b| (4.7) v v N u N u N X uX 2uX 2 f(xi)g(xi) ≤ t f(xi) t g(xi) (4.8) k=1 k=1 k=1 v v N u N u N X uX 2 uX 2 f(xi)g(xi) 4 x ≤ t f(xi) 4 xt g(xi) 4 x . (4.9) k=1 k=1 k=1

Integrals are utilized in this tutorial for intuition. Taking the limit as 4x approaches zero: Z sZ sZ f(x)g(x)dx ≤ f(x)2dx g(x)2dx . (4.10)

Last, rearranging the inequality satisﬁes Eq. 4.1:

19 Moreover, the numerator in Eq. 4.1 equals one. Below, Eq. 4.14 employs the chain

rule, and Eq. 4.15 takes the derivative of a constant:

Inserting this information into Eq. 4.1.

1 V ar[Tb2 − T2] ≥ d ln p (4.17) E[( Y |T2 )2] dT2 1 CRLB = d ln p (4.18) E[( Y |T2 )2] dT2 1 = . (4.19) d2 ln p Y |T2 E[( 2 )] dT2 The two formulations of the CRLB lead to geometrically intuitive results. The

minimum variance exhibits an inversely proportional relationship with the expected

slope about the estimate squared (Eq. 4.18) or the expected curvature about the

estimate (Eq. 4.19). Maximizing the log-likelihood function reduces to minimizing

a parabolic cost. A large measure of curvature about the minimum cost implies a

large penalty for obtaining an incorrect result. To avoid this penalty, an estimation

procedure would seek a value within a narrow range.

20 4.2 Extension to Multiple Parameters

Just as the ballplayer in our earlier example relies on gradients to determine

the smallest cost, the Fisher information matrix J relies on expected gradients of

a functional model. This metric is used to determine the smallest variance when

multiple parameters are unknown. J is symmetric, and the inverse of its diagonal

elements represent the CRLBs of the diﬀerent estimates. The elements of J are

deﬁned and simpliﬁed for the general model.

1 V ar[θbi − θi] ≥ , where (4.20) Jii

T d ln pY |θi d ln pY |θj Jij = E[ ] (4.21) dθi dθj T ∂ 1 ~ T ~ ∂ 1 ~ T ~ = E[ − 2 (~y − f) (~y − f) − 2 (~y − f) (~y − f) ] (4.22) ∂θi 2σ ∂θj 2σ ~T ~ 1 1 ∂f ~ ~ T ∂f = 2 2 E[(~y − f)(~y − f) ] (4.23) σ σ ∂θi ∂θj ~T ~ 1 1 ∂f ~ ~ T ∂f JT2 = 2 2 E[(~y − f)(~y − f) ] . (4.24) σ σ ∂T2 ∂T2

This general construct applies to the four speciﬁc estimators diﬀerently. The covariance matrix depends on the estimation technique. For the peak picking, para-

T metric, and SVD estimators, E[(~y − f~)(~y − f~) ] = Iσ2. However, for the integration

model, the addition of M random noise variables from each echo K reduces the vari-

~ ~ T σ2 ance by a factor of M. E[(~y − f)(~y − f) ] = I M . This diagonal matrix may be replaced with a constant. These covariance relationships imply that, as the SNR

approaches inﬁnity, the variances of all four estimators decrease linearly and with the

same slope. The other discriminating factor in determining JT2 is contained in the

21 ~T ~ quantity ∂f ∂f . ∂T2 ∂T2 ~T ~ N ~T N ~ ∂f ∂f X ∂f X ∂fk = k (4.25) ∂T2 ∂T2 ∂T2 ∂T2 k=1 k=1

N T N X ∂(α~abk(T2)) X ∂α~abk(T2) = (4.26) ∂T2 ∂T2 k=1 k=1

N T N X ∂(αbk(T2)) X ∂αbk(T2) = (~aT~a) . (4.27) ∂T2 ∂T2 k=1 k=1 The two summation terms remain invariant of the signal model, but not ~a. Here, the

subscripts pp, int, par, and np, refer to the peak picking, integration, parametric, and

non-parametric based methods. For the parametric and non-parametric estimators,

2 k~apar,npk = 1 by deﬁnition. For the integration model, aint contains one element, PM because the proposed data points are added together ﬁrst, aint = i=1 ~apar,i. The

2 2 triangle inequality demonstrates that kaintk ≤ k~apar,npk . Last, in the picking technique, only the approximate peak from each echo is selected, which also implies that

2 2 kappk ≤ |~apar,np| .

These observations imply the following: T 2 ~ ~ JT2par,np k~apar,npk Epar,np[(~y − f)(~y − f) ] = T (4.28) JT 2 ~ ~ 2int kaintk Eint[(~y − f)(~y − f) ] k~a k2σ2 = par,np (4.29) 2 σ2 kaintk M ≥ 1 . (4.30)

Similarly, T 2 ~ ~ JT2par,np k~apar,npk Epar,np[(~y − f)(~y − f) ] = T (4.31) JT 2 ~ ~ 2pp k~appk Epp[(~y − f)(~y − f) ] 2 2 k~apar,npk σ = 2 2 (4.32) k~appk σ ≥ 1 . (4.33)

22 More succinctly, the two proposed models enjoy smaller lower bounds.

CRLBpar,np ≤ CRLBpp (4.34)

CRLBpar,np ≤ CRLBint . (4.35)

The CRLB of the two proposed estimators is 2.9 and 44.2 times smaller than the integration and peak picking methods. The following ﬁgure illustrates the relative diﬀerence in standard error.

3 10 A) Peak Picking B) Integration C) Parametric 2 10 D) Non−parametric

1 10

0 10 Standard Error Bound (a.u.)

−1 10 −10 −5 0 5 10 15 20 25 30 35 40 SNR (dB)

Figure 4.1: Lower bound on the standard error of Tb2.

23 Chapter 5: Results

5.1 Materials and Methods

EPR spin echo data were collected on a pulsed X-band system (ELEXSYS, Bruker,

MA) using a volume resonator. Activated charcoal, under room air conditions, was used as the paramagnetic probe. A total of seven diﬀerent echos were recorded using

π/2 − τ − π − τ sequence, with τ varying from 200 ns to 440 ns in uniform increments of 40 ns. Other imaging parameters were: 9.67 GHz frequency, 256 samples to record each echo across a 256 ns wide acquisition window, 1024 averages per echo, and

1 ms shot repetition time. After acquisition, the data were transferred to Matlab

(Mathworks, Natick, MA) for further processing.

5.2 Results

The seven acquired echoes were used to form the data array Y and compute singular values, shown in Fig. 5.1. A single dominant singular value was 77.1 times

0 larger than any others, and the rank 1 approximation σ1u1v1 to the data array Y captured 96.1% of the observed energy in Y . For the charcoal probe, the principal left singular vector, u1, was observed to be Gaussian. The residual after ﬁtting was minimal and lacked structure.

24 6 10

5 10 Magnitude 4 10

3 10 0 2 4 6 8 Singular Values

Figure 5.1: Singular values for the data array formed from the seven acquired echoes.

To evaluate the noise sensitivity of each estimator, ﬁrst consider simulated data generated according to

2 Yk = exp −(t − θ1) /θ2 exp(−2τk/T2) + noise

2 for θ1 = 100 ns, θ2 = 1245.7 ns , T2 = 417.8 ns, with τk varying from 200 ns to 560 ns in uniform increments of 40 ns, and with Gaussian noise yielding signal-to-noise ratios of 17 to 37 dB. SNR is deﬁned as the ratio of the peak of the ﬁrst noiseless echo to the noise variance. From 10, 000 random trials, the resulting estimation performance is shown in Fig. 5.2 for each of the four estimation procedures. The left panel shows standard error, while the right panel shows relative acquisition time to achieve a given precision. Estimation standard error increases proportionally with the standard deviation of the additive noise; however, additive noise standard deviation decreases

25 proportionally with the square of the acquisition time. Hence, the curves in the right panel have twice the slope as the curves on the left.

4 4 10 10 A) Peak Picking B) Integration 3 C) SVD 3 10 10 D) Parametric

2 2 10 10 std. (ns) 2 T 1 1 10 10 Experimental Time (a. u.)

0 0 10 10

17 24 30 37 17 24 30 37 SNR (dB) SNR (dB)

Figure 5.2: Left: standard deviation of T2 estimation error for a simulated seven echo train at four noise levels: 17, 24, 30, and 37 dB. Right: relative acquisition time versus SNR to achieve ﬁxed standard deviation of T2 estimation error.

Estimation performance on the simulated data is evaluated in Fig. 5.3 versus the number of acquired echoes. The additive noise variance is held ﬁxed, resulting in an SNR of 37 dB for the ten echo case. For the given experimental conditions, the acquisition time is minimized with the acquisition of eight echoes, but grows rapidly after the acquisition of twelve (data not shown).

For measured spin echo data, simulated additive white Gaussian noise was added for eﬀective signal-to-noise ratios of 17 to 37 dB. From 10, 000 random noise trials, the resulting estimation performance is shown in Fig. 5.4 for each of the four estimation procedures. Estimation performance on the measured echoes is shown versus number

26 of acquired echoes in Fig. 5.5; the additive noise has a ﬁxed variance resulting in

37 dB SNR for the seven echo case.

3 3 10 10

2 2 10 10

1 1 10 10 std. (ns) 2 T

0 0 10 10 Experimental Time (a. u.)

−1 −1 10 10 3 5 7 10 3 5 7 10 Number of Echoes Number of Echoes

Figure 5.3: Left: standard deviation of T2 estimation error from simulated data versus number of acquired echoes. Right: relative acquisition time versus number of echoes to achieve ﬁxed standard deviation of T2 estimation error.

27 4 4 10 10

3 3 10 10

2 2 10 10 std. (ns) 2 T 1 1 10 10 Experimental Time (a. u.)

0 0 10 10 17 24 30 37 17 24 30 37 SNR (dB) SNR (dB)

Figure 5.4: Left: standard deviation of T2 estimation error for measured data with seven echoes at four synthesized noise levels: 17, 24, 30, and 37 dB. Right: relative acquisition time versus SNR to achieve ﬁxed standard deviation of T2 estimation error.

3 3 10 10

2 2 10 10

1 1 10 10 std. (ns) 2 T

0 0 10 10 Experimental Time (a. u.)

−1 −1 10 10 3 5 7 3 5 7 Number of Echoes Number of Echoes

Figure 5.5: Left: standard deviation of T2 estimation error from measured data versus number of acquired echoes. Right: relative acquisition time versus the number of echoes to achieve ﬁxed standard deviation of T2 estimation error.

28 Chapter 6: Discussion

Over the range of SNRs and numbers of echoes considered, the simulated data predicted that SVD-based estimation of T2 oﬀers a 3:1 acceleration of acquisition time versus the integration approach, to achieve a given estimation accuracy. The measured data results showed the same. Much larger performance gains were observed in comparison to the peak-peaking approach, which does not utilize all data points in the estimation procedure.

The estimation error curves for the SVD-based estimator and the ML curve ﬁt nearly coincide in Figs. 5.2 - 5.4. Thus, for the range of cases considered the SVD- based estimator yields approximately the same performance a nonlinear least-squares curve ﬁtting computed with prior knowledge of a parametric model for the echo shape.

In Figs. 5.2 and 5.4, the log of the standard error in T2 estimates is seen to decrease linearly with the SNR in decibels, thus conﬁrming the Cram´er-Raobound prediction.

At low SNR, the error curve for the peak-peaking technique increases above this linear trend.

The presence of only one dominant singular value in Fig. 5.1 conﬁrms, for the charcoal probe, the supposed model of single spin echo behavior with decaying amplitude. Equation 3.32 provides a estimator when multiple probes are present, each with a potentially diﬀerent relaxation time.

29 Although the proposed SVD approach is limited to SE spectroscopy, it can be extended to imaging applications. In the case of SE imaging, a series of two- or three-dimensional images are collected with diﬀerent, monotonically increasing values of τ. The overall intensity of each image is proportional to the peak value of the corresponding echo. The resulting matrix Y has pixel intensities along the column- direction and pixel-wise relaxation rates along the row-direction. Since the relaxation rates vary spatially, the matrix Y has rank larger than 1. In this case, SVD can be applied to denoise images, with some appropriate choice of truncation of singular values in Eq. 3.29. Similar denoising approaches has been previously used [4, 5, 6].

Finally, the local T2 values can be determined by pixel-wise ﬁtting of the denoised image with an exponential function. This approach can potentially overcome the

“missing voxel” issue encountered in SE-EPR imaging where extremely poor SNR at a given pixel or voxel results in a meaningless exponential ﬁt [10].

Finally, for relaxation times and echo times considered, Cram´er-Raobound analysis (not shown) reveals that uniformly spaced echo times provide slightly better estimation results than logarithmically spaced samples; however, the opposite is true when the late echo times are large.

30 Chapter 7: Conclusion

The proposed SVD-based estimate of the T2 relaxation time provides signiﬁcant reduction in standard error – and hence acquisition time – versus the peak-picking and integration techniques commonly used in the literature. Further, the SVD technique, although agnostic to the observed echo shape, was proven to provide the same estimation accuracy as the maximum likelihood estimator that is endowed with per- fect prior knowledge of the echo shape. For the experimental parameters considered, a time savings of 3:1 was observed versus integration, with much larger savings versus peak-picking. Thus, simple SVD post-processing can provide low-variance estimates of T2 relaxation times using accelerated acquisition times and without prior knowledge of a functional form for the spin echo.

31 Appendix A: Matlab Code

%% Initialization % Author: Frank Golub % Date: October 9, 2012 % Purpose: To rearrange the original data intoa usable format and % generate the initial guesses. function [x, t, Y, U]= Initialization(hypothesis, percent, synthetic, num echoes) % close all; clear all; load('Original Spin Echo');% Load the reformatted data

% Define the hypothesis % hypothesis= 2;%1= simple exponential,2= SVD,3= Lorentzian

%% Rearrange the data into one matrix raw data = zeros(length(data200(:, 2)), 7); raw data(:, 1) = data200(:, 3); raw data(:, 2) = data240(:, 3); raw data(:, 3) = data280(:, 3); raw data(:, 4) = data320(:, 3); raw data(:, 5) = data360(:, 3); raw data(:, 6) = data400(:, 3); raw data(:, 7) = data440(:, 3); if (synthetic == 1) raw data = synthetic gaussian(num echoes); else raw data = raw data(:, 1:num echoes); end

%% Find the index of the maxima before noise peakIndices = zeros(1, num echoes);% The indices where the peaks occur peaks = zeros(1, num echoes);% The values at those indices for i = 1:num echoes peakIndices(i) = find(raw data(:, i) == max(raw data(:, i))); % peaks(i)= max(raw data(:,i));

32 end raw data = raw data + percent; for i = 1:num echoes peaks(i) = raw data(peakIndices(i), i); end

%% Initial Guesses switch hypothesis case1% Simple Exponential %% Log Least Squares − Using already written code t n = linspace(0, 40*(num echoes − 1), num echoes); X = [t n; ones(1, length(t n))]'; B = inv(transpose(X)*X)*transpose(X)*(log(peaks)'); T2 = −1/(B(1));% T2= −1/m A = exp(B(2));%A= exp(b)

%% Initial Guesses and Data x = [T2, A];% Initial guess vector − x(1)= T2,x(2)=A t = linspace(0, 40*(num echoes − 1), num echoes); Y = peaks; U = 1; case2% SVD modulated exponential %% First Rank SVD [U, S, V] = svd(raw data); U = U(:, 1);% Parametric Form S = S(1, 1);% Rank1 singular values V = V(:, 1);% Amplitudes

if (sum(V) < 0) U = −U; V = −V; end

%% Log Least Squares t n = linspace(0, 40*(num echoes − 1), num echoes); X = [t n; ones(1, length(t n))]'; B = inv(transpose(X)*X)*transpose(X)*(log(V));% UsingV instead T2 = −1/(B(1));% T2= −1/m %A= exp(B(2));%A= exp(b) %A=A *S;% To scale the problem to the data A = S*max(V);

%% Initial Guesses and Data x0 = mean(peakIndices);

x = [T2, A, x0];% Initial guess vector t = linspace(0, 40*(num echoes − 1), num echoes); Y = raw data;

33 case3% Gaussian modulated exponential %% First Rank SVD [U, S, V] = svd(raw data); U = U(:, 1);% Parametric Form S = S(1, 1);% Rank1 singular values V = V(:, 1);% Amplitudes

if (sum(V) < 0) U = −U; V = −V; end

%% Log Least Squares t n = linspace(0, 40*(num echoes − 1), num echoes); X = [t n; ones(1, length(t n))]'; B = inv(transpose(X)*X)*transpose(X)*(log(V));% UsingV instead %A= exp(B(2));%A= exp(b) %A=A *S;% To scale the problem to the data A = S*max(V);

%% Initial Guesses and Data x0 = peakIndices(1); G1 = min(find(U >= max(U)/2)); G2 = max(find(U >= max(U)/2)); G = G2 − G1; x = [T2, A, x0, G];% Initial guess vector t = linspace(0, 40*(num echoes − 1), num echoes); Y = raw data; case4 %% Using the same initialization procedure as in Step3 [U, S, V] = svd(raw data); U = U(:, 1);% Parametric Form S = S(1, 1);% Rank1 singular values V = V(:, 1);% Amplitudes

if (sum(V) < 0) U = −U; V = −V; end

%% Log Least Squares t n = linspace(0, 40*(num echoes − 1), num echoes); X = [t n; ones(1, length(t n))]'; B = inv(transpose(X)*X)*transpose(X)*(log(V));% UsingV not peaks T2 = −1/(B(1));% T2= −1/m %A= exp(B(2));%A= exp(b) %A=A *S;% To scale the problem to the data A = S*max(V);

%% Initial Guesses and Data

34 x0 = peakIndices(1); G1 = min(find(U >= max(U)/2)); G2 = max(find(U >= max(U)/2)); G = G2 − G1; x = [T2, A];% Initial guess vector t = linspace(0, 40*(num echoes − 1), num echoes); Y = zeros(1, length(num echoes));

for i = 1:num echoes Y(i) = sum(raw data(:, i)); end end end

%% Costfun % Author: Frank Golub % Date: October 9, 2012 % Purpose: To create the error function for all four hypotheses. function [err] = costfun(x, Y, t, U, hypothesis)

% lsqnonlin minimizes | | err||ˆ2 , where | | . | | is anl −2 norm

%x= parameters {T2,A } %Y= data from EPR %t= time vector switch hypothesis case1 err = x(2)*exp(−t./(x(1))) − Y;% Simple Exponential case2 err = x(2).*U*exp(−t./x(1)) − Y;% SVD Modulated Exponential err = reshape(err, 1, length(err(1, :))*length(err(:, 1))); % Rearrange into one long vector case3 r = [1:length(Y(:, 1))]'; err = x(2).*exp(−((r − x(3)).ˆ2)./(2*x(4).ˆ2))*exp(−t./x(1)) − Y; % Gaussian Exponential err = reshape(err, 1, length(err(1, :))*length(err(:, 1))); % Rearrange into one long vector case4 err = x(2)*exp(−t./(x(1))) − Y;% Simple Exponential end end

35 % Author: Frank Golub % Date: October 9, 2012 % Purpose: To employ lsqnonlin to find the appropriate parameters T2 andA % fora peak picking exponential % Notes: Relies heavily on example from Rizwan Ahmad.

% close; clear; clc; function [T2] = main(hypothesis, graph on, percent, synthetic, ... percent nv, num echoes) [x, t, Y, U] = Initialization(hypothesis, percent, synthetic, num echoes); %% Known Quantities % Related Function: case one costfun(initial guess, EPR data, time); % Initial Guess Vector:x=[T2,A]; %Y= EPR data; %t= time;

%% Fitting X0 = x; if (hypothesis == 1) Xlb = [0, 0];%I am very inclusive here Xub = [2*x(1), inf]; elseif (hypothesis == 2) Xlb = [0, 0, 0]; Xub = [2*x(1), inf, inf]; elseif (hypothesis == 3) Xlb = [0, 0, 0, 0]; Xub = [2*x(1), inf, inf, inf]; elseif (hypothesis == 4) Xlb = [0, 0]; Xub = [2*x(1), inf]; end options=optimset('MaxFunEvals',1000,'MaxIter',500,'TolFun',1e−8,'TolX',1e−8, ... 'display','off'); X=lsqnonlin(@costfun,X0,Xlb,Xub,options,Y,t,U,hypothesis); ... % Estimated parameters Y hat=costfun(X,zeros(size(Y)),t, U, hypothesis); ... % Data fitted using estimated parameters if ((hypothesis == 2) | | (hypothesis == 3))

r = [1:256]; t altered = zeros(1, length(r)*num echoes); t altered = [r + t(1)]; for i = 2:num echoes t altered = [t altered, r + t(i)]; end t = t altered; Y = reshape(Y, 1, length(Y(:, 1))*length(Y(1, :))); Y hat = reshape(Y hat, 1, length(Y hat(:, 1))*length(Y hat(1, :)));

36 end

%% Plotting if (graph on == 1) % figure; % subplot(211); plot(t,Y,'b',t,Y hat,'−r'); legend('Y','Y {hat}'); % subplot(212); plot(t,Y −Y hat); legend('residual'); % switch hypothesis case1 figure; plot(t,Y,'b+',t,Y hat,'−−r');% legend('Y','Y {hat}'); hold on; plot(t, (Y−Y hat),'kx'); legend('Measured','Fitted','Residual');

string T2 = num2str(X(1)); string Noise = num2str(percent nv); xlabel('Time(ns)'); ylabel('Amplitude(A.U.)'); title(['Peak Picking T2 decay, T2=', string T2(1:5), ... ' Noise=', string Noise]); case2 figure; for i = 1:num echoes plot(t([1:256] + 256*(i − 1)),Y([1:256] + 256*(i − 1)), ... 'b',t([1:256] + 256*(i − 1)),Y hat([1:256] + ... 256*(i − 1)),'−−r');% legend('Y','Y {hat}'); hold on;

if (i == 1) t abbr = [0:max(t)]; plot(t abbr, X(2)*max(U)*exp(−(t abbr − x(3))/(X(1))), ... 'g−−'); hold on; end

plot(t([1:256] + 256*(i − 1)), (Y([1:256] + ... 256*(i − 1))−Y hat([1:256] + 256*(i − 1))),'k');

if (i == 7) legend('Measured','Fitted','Exponential','Residual'); end hold on; end;

xlabel('Time(ns)'); ylabel('Amplitude(A.U.)'); string T2 = num2str(X(1)); string Noise = num2str(percent nv);

37 title(['SVD T2 decay, T2=', string T2(1:5),' Noise=', ... string Noise]);

%% Individual Echos figure; i = 1; plot(t([1:256] + 256*(i − 1)),Y([1:256] + 256*(i − 1)), ... 'b',t([1:256] + 256*(i − 1)),Y hat([1:256] + ... 256*(i − 1) ),'−−r');% legend('Y','Y {hat}'); hold on; plot(t([1:256] + 256*(i − 1)), ((−1e4) + (Y([1:256] + ... 256*(i − 1))−Y hat([1:256] + 256*(i − 1)))),'k'); xlabel('Time(ns)'); ylabel('Amplitude(A.U.)'); string T2 = num2str(X(1)); string Noise = num2str(percent nv); title(['SVD T2 decay, Echo#', num2str(i),', Noise=',... string Noise]); set(gca,'XLim',[1 256]) legend('Measured','Fitted','Residual'); case3 figure; for i = 1:num echoes plot(t([1:256] + 256*(i − 1)),Y([1:256] + 256*(i − 1)), ... 'b',t([1:256] + 256*(i − 1)),Y hat([1:256] ... + 256*(i − 1) ),'−−r');% legend('Y','Y {hat}'); hold on;

if (i == 1) t abbr = [0:max(t)]; plot(t abbr, X(2)*exp(−(t abbr − X(3))/(X(1))),'k −−'); hold on; end

plot(t([1:256] + 256*(i − 1)), (Y([1:256] + ... 256*(i − 1))−Y hat([1:256] + 256*(i − 1))),'k');

if (i == 7) legend('Measured','Fitted','Exponential','Residual'); end hold on; end;

xlabel('Time(ns)'); ylabel('Amplitude(A.U.)'); string T2 = num2str(X(1)); string Noise = num2str(percent nv); title(['Gaussian T2 decay, T2=', string T2(1:5), ... ', Noise=', string Noise]);

38 %% Individual Echos figure; i = 1; plot(t([1:256] + 256*(i − 1)),Y([1:256] + 256*(i − 1)),'b', ... t([1:256] + 256*(i − 1)),Y hat([1:256] + 256*(i − 1) ) ... ,'−−r');% legend('Y','Y {hat}'); hold on; plot(t([1:256] + 256*(i − 1)), ((−3e4) + (Y([1:256] + ... 256*(i − 1))−Y hat([1:256] + 256*(i − 1)))),'k'); xlabel('Time(ns)'); ylabel('Amplitude(A.U.)'); string T2 = num2str(X(1)); string Noise = num2str(percent nv); title(['Gaussian T2 decay, Echo#', num2str(i),', Noise=', ... string Noise]); set(gca,'XLim',[1 256]) legend('Measured','Fitted','Residual'); case4 figure; plot(t,Y,'b+',t,Y hat,'−−r');% legend('Y','Y {hat}'); hold on; plot(t, (Y−Y hat),'kx'); legend('Measured','Fitted', .... 'Residual');

string T2 = num2str(X(1)); string Noise = num2str(percent nv); xlabel('Time(ns)'); ylabel('Amplitude(A.U.)'); title(['Integration T2 decay, T2=', string T2(1:5),... ' Noise=', string Noise]); end end

%% Answer T2 = X(1);

%% Noisy Distribution % Author: Frank Golub % Date: October 10, 2012 % Purpose: To add noise to the hypotheses and determine how much T2 varies % from the true value. function [] = noisy distribution(percent nv, N, synthetic, num echoes) %% Parameters to Play With % percent nv= [0.00]; % percent nv=0.10;% See Line 18. % percent nv= percent nv entered;

39 % synthetic= 0;%0 −> Measured with EPR,1 −> Synthetic Data %N= 1000;% Try settingN=1 %N= 10;% Number of Samples, Try settingN= 1;

%% Section Not to Play With for j = 1:length(percent nv) noise variance = percent nv(j)* ... (max(max(synthetic gaussian(num echoes))))ˆ2; % max(max(synthetic gaussian)) ˜˜ first peak of measured data.

T2 array = zeros(4, N); mean T2 array = zeros(4, N); std T2 array = zeros(4, N); true T2 = main(1, 0, 0, synthetic, percent nv(j), num echoes);

for i = 1:N percent = random('Normal', 0, sqrt(noise variance), 256, num echoes);

if (N == 1) T2 array(1, i) = main(1, 1, percent, synthetic, ... percent nv(j), num echoes); T2 array(2, i) = main(2, 1, percent, synthetic, ... percent nv(j), num echoes); T2 array(3, i) = main(3, 1, percent, synthetic, ... percent nv(j), num echoes); T2 array(4, i) = main(4, 1, percent, synthetic, ... percent nv(j), num echoes); else T2 array(1, i) = main(1, 0, percent, synthetic, ... percent nv(j), num echoes); T2 array(2, i) = main(2, 0, percent, synthetic, ... percent nv(j), num echoes); T2 array(3, i) = main(3, 0, percent, synthetic, ... percent nv(j), num echoes); T2 array(4, i) = main(4, 0, percent, synthetic, ... percent nv(j), num echoes); end

mean T2 hyp1(i) = mean(T2 array(1, 1:i)); mean T2 hyp2(i) = mean(T2 array(2, 1:i)); mean T2 hyp3(i) = mean(T2 array(3, 1:i)); mean T2 hyp4(i) = mean(T2 array(4, 1:i));

std T2 hyp1(i) = std(T2 array(1, 1:i)); std T2 hyp2(i) = std(T2 array(2, 1:i)); std T2 hyp3(i) = std(T2 array(3, 1:i)); std T2 hyp4(i) = std(T2 array(4, 1:i)); % % if((i == 10) | | (i == 100) | | (i == 1000) | | (i == 1500)) % toc; % end

40 end

if (N ˜= 1) figure(j); subplot(2,1,1); sample number = [1:N]; plot(sample number, T2 array(1, :),'b', sample number, ... T2 array(2, :),'r', sample number, T2 array(3, :), ... 'g', sample number, (zeros(size([1:N])) + true T2),'k'); legend('Peak Picking','SVD','Gaussian','True', ... 'Location','NorthEastOutside'); % axis([0N 200 250]) xlabel('Sample') ylabel('Mean of T2'); title(['Mean, Noise=', num2str(percent nv(j))]); set(gca,'XLim',[1 N])

subplot(2, 1, 2); y = mean(T2 array'); e = 3*std(T2 array'); errorbar(y,e,'xb'); set(gca,'XTick',[1:4]) hypothesis label = ['Peak Picking';' SVD'; ... ' Gaussian';'Integration']; set(gca,'XTickLabel',hypothesis label) ylabel('T2'); title(['T2 +/− 3',' \sigma' ]);

hold on; plot([0.5:4.5] ,(zeros(size([0.5:4.5])) + true T2),'k'); end end end

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