Investigation of Multicomponent MRI Relaxation Data with Stochastic Contraction Fitting Algorithm

Transcription

1 Investigation of Multicomponent MRI Relaxation Data with Stochastic Contraction Fitting Algorithm Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master in Mathematical Science in the Graduate School of the Ohio State University By Mary Elizabeth Russell, BA Graduate Program in Mathematics The Ohio State University 2014 Thesis Committee: Dr. Gregory Baker, Advisor Dr. Petra Schmalbrock

2 c Copyright by Mary Elizabeth Russell 2014

3 ABSTRACT Brain tissue can be identified and studied with Magnetic Resonance Imaging. Instead of relying only on the image, more advanced methods obtain quantitative measurements that describe specific tissues. Differential equations govern the shape of the magnetization curve, and the six unknown parameters of these equations are fit to the curves obtained from real data. The six parameter fit is performed numerically using a stochastic contraction algorithm. The stability of the results depend on the size of the original search space as well as additional factors. Numerical simulations were performed using this method and the results are concurrent with other literature results. ii

4 ACKNOWLEDGEMENTS I would like to thank Dr. Petra Schmalbrock for all of her guidance and advice throughout the past year. I would also like to thank Dr. Baker for his help and insight on this thesis. iii

5 VITA B.A., Mathematics Canisius College Present Graduate Teaching Associate The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Computational Sciences iv

6 TABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENTS VITA LIST OF FIGURES LIST OF TABLES ii iii iv vii viii CHAPTER Page 1 INTRODUCTION BACKGROUND MATRIX VERIFICATION Longitudinal Magnetization SPGR Equations Transverse Magnetization: SSFP Equations Defining Exchange Rates FITTING ALGORITHM Description of Stochastic Contraction Fitting Potential Problems with Stochastic Contraction Fitting Numerical Results v

7 4.3.1 Restricting Search Space Reduced Parameter Search Simulating Different Data Aquisition Another Set of Parameters Simulating Modified Aquisition Strategy: Change θ Effects of Larger N Discussion of testing results CONCLUSION Future Work APPENDIX : MATLAB CODES BIBLIOGRAPHY vi

8 LIST OF FIGURES Figure Page 3.1 Signal Equations with Different Exchange Rates Compare Equations with the Same Exchange Rates Exchange Rate: Case Exchange Rate: Case Exchange Rate: Case Exchange Rate: Case Fitting algorithm with full search space Fitting algorithm with restricted search space vii

9 LIST OF TABLES TABLE Page 3.1 Exchange rate tests Parameters used to generate curves with different exchange rates Parameters used for fitting algorithm Restricted Search Space Reduced Parameter Search Restricted Search Space: Upward Slope Reduced Parameter Search: Upward Slope Restricted Search Space: New True Parameter Values Reduced Parameter Search: New True Parameter Values Different θ with Full Search Space Different θ with Restricted Search Space [5] Different α with Full Search Space viii

10 4.11 Different α with Restricted Search Space [5] Full Search Space, N1= Restricted Search Space [5], N1= ix

11 Chapter 1 INTRODUCTION Magnetic Resonance Imaging (MRI) is a noninvasive method for imaging anatomy and detecting change in diseases. New, advanced MRI methods aim to obtain tissue information beyond depicting overall anatomy. By using biophysical tissue models to quantitatively describe the observed MRI signal, one may infer information regarding tissue microstructure beyond the level that is spatially resolved in the images. This allows for a noninvasive method to measure degenerative changes in brain tissue. If relaxation parameters can be measured from the tests, information regarding changes in the tissue can indicate progression of a degenerative disease. The purpose of this thesis is to analyze and reproduce algorithms that provide quantitative information describing brain tissue. This thesis recreates the MRI signal curves to verify matrix equations that describe the biophysical phenomenon that tissues undergo during the MRI test. Additionally, this thesis implements a stochastic contraction fitting 1

12 algorithm to find parameter values that create the magnetization curve. This implementation is based on previous work [4]. Finally, conclusions about the reproducibility of the parameters as well as the accuracy of the algorithm are discussed. The second section of this thesis provides background information which includes methods to measure parameters and fitting algorithms used in previous papers. The third section verifies that the different forms of matrix equations in various papers describe the same phenomena. The fourth section describes the implementation of a stochastic contraction fitting algorithm with results. 2

13 Chapter 2 BACKGROUND The signal in MRI is generated by the magnetization response of the protons in the nuclei of water molecules in the brain tissue. When protons are placed in a magnetic field, B 0, the naturally spinning protons align either in a parallel, (lower energy state) or anti-parallel (higher energy state) direction with respect to the magnetic field. In equilibrium, more protons are aligned in the parallel direction and emit a magnetic vector M 0. When the protons are tilted away from the magnetic field, the moment precesses about the field at a specific frequency, the Larmor frequency. If an additional radio frequency (RF) pulse is applied, the magnetic moment tilts away from the direction of B 0 into the transverse plane. This magnetic moment in the transverse plane is the measured signal. When this RF pulse is removed, the protons return to equilibrium at a specific rate. These relaxation rates are defined as T 1 and T 2. The T 1 relaxation rate refers to 3

14 the relaxation in the longitudinal direction, while T 2 relaxation rate refers to the transverse magnetization [3]. The T 1 relaxation rate is a measure of the exchange of energy between water protons and protons of other macromolecules. This represents an overall energy loss and is sometimes refered to as spin-lattice relaxation. The T 2 relaxation rate is indicative of the exchange of energy between different water molecules. This represents an overall energy conservation and is called a spin-spin relaxation. T 1 and T 2 are goverened by their environments, therefore their values contain information about tissue microstructure [3]. In the brain tissue, water is divided into multiple, distinct environments with their own T 1 and T 2 values. If water is able to penetrate through the boundaries of these divisions, then the protons can exchange between the areas and therefore the T 1 and T 2 values cannot be described by only one value. According to MacKay et al.[8], after analyzing the T 2 relaxation time in the central nervous system, the tissue appears to have three distinct divisions of water. These are later referred to as three exchanging water pools. From experimental data, the tissue with the shortest T 2 relaxation time of 10ms - 55ms was found in myelin membranes. The tissue with a T 2 relaxation time of 70ms - 95ms was found in intra- and extra-cellular spaces. The tissue with the longest T 2 relaxation time of over 1000ms was found in the cerebral spinal fluid [8], Although recent papers [5], perform their experiments using the three exchanging water pools, this paper 4

15 focuses on two exchanging pools. Although there are models using the three exchanging case [5], the computation time for the fitting algorithm would increase and is not studied in this paper. There are different methods to acquire these relaxation rates. Standard methods include inversion recovery and CPMG methods (e.g. GRASE). The inversion recovery sequence begins with the magnetic moment in equilibrium (measured M 0 ). Initially, an inversion pulse is applied to the magnetic moments from protons in water molecules, and they rotate 180. Between this initial time and the inversion time (TI), the magnetic moment is able to partially recover to equilibrium according to its longitudinal T 1 relaxation rate. At time TI, a 90 pulse is applied, which rotates the longitudinal magnetization into the transverse plane, where it is measured thus reflecting T 1. Measurements are taken at different times TI, and the T 1 value is then found by fitting the measured signal to a function M 0 (1 2e T I/T 1 ). Although the inverstion recovery method is an accepted method for finding T 1, it is inefficient because of long acquisition times [3]. The transverse relaxation time can be measured with a spin echo sequence. Here the equilibrium magnetization is rotated by a 90 pulse in the transverse plane. A 180 refocussing pulse applied at a time T E/2 refocuses the magnetization at time T E and is measured. The strength of the refocused signal is proportional to e T E/T 2. Acquisition of data at different T E and fitting allows to determine T 2. Discussion of specific details for acquisition methods to measure T 2 are beyond the scope of this 5

16 thesis. GRASE is one such method that has been used to measure multi-component T 2 relaxation in the brain and to determine myelin water fractions. The DESPOT 1 and DESPOT 2 methods are similar to the previously described methods, but these are much less time consuming. A train of small α pulses is used to measure the T 1 and T 2 relaxation curves by collecting data at different α. In the Spoiled Gradient imaging sequences (SPGR), signals are obtained by applying α pulses while the magnetization is in the transverse plane and not fully recovered. Since there is no transverse residual from the partially recovered magnetization, the transverse magnetization is spoiled. If the transverse magnetization is adequately spoiled, then the longitudinal magnetization is forced into a dynamic equilibrium that is determined by ρ, T 1, T 2. The signal (M SP GR ) is governed by the following equation[3]: M SP GR = ρ ( ) 1 e T R/T 1 sin α e T E/T 2 (2.1) 1 e T R/T 1 cos α DESPOT 2 is a way in which to measure the T 2 relaxation. The SSFP (Steady State Free Precession) recycles the magnetization because it is dephased, rephased, dephased, and finally rephased to produce a signal that can recover the T 2. The observed magnetization is [3]: ( ) 1 e T R/T 1 e T E/T 2 sin α S SSF P = ρ 1 e T R/T 1 e T R/T 2 + (e T R/T 1 e T R/T 2) (2.2) cos α 6

17 In more recent papers [1], DESPOT 1 and DESPOT 2 methods are combined in mcdespot. This method is advantageous because it can improve imaging speed, SNR (Signal to Noise Ratio) efficiency, high resolution and the possibility of whole brain imaging. The disadvantages of this method is the difficulty in measuring exchange rates and the error that can occur from flip angle measurements and B 0 inhomogeneity. The most prominent disadvantage is that it requires a fitting of 6 or more parameters. After performing mcdespot and obtaining signal curves, parameters including T 1 and T 2, myelin water fraction, and exchange rate of the two exchanging species can be calculated by solving an inverse problem. The equations that govern these curves are theoretically known, so if the obtained signal curve is fit to a specific equation, the parameter values can be obtained. This same paper [1] continues to fit the signal curves with equations generated by random searches of parameter values. The claim is that the results converge to the correct solution with high reproducibility [1]. This thesis attempts to reproduce these results. In order to begin the fitting algorithm, the governing matrix equations must be understood. This was not trivial since different versions of the matrix equations appear in the literature [1],[2],[10],[9], [7]. These inconsistencies are addressed and each is evaluated based on graphical analysis performed in Matlab. A few fitting algorithms have been introduced, including gradient descent search [1] and stochastic contraction [4]. The stochastic contraction method is implemented in this paper, because it is mentioned in most 7

18 recent papers [4][5]. The fitting algorithm is difficult to implement because it has high computational cost and larger than previously recorded error [1]. Several papers have addressed the apparent irreproducibility of the stochastic contraction fitting algorithm. A recent paper [6], reports the results of a statistical analysis of mcdespot and the calculation of the Cramér-Rao lower bounds for variance. The conclusion is that because the fitting method involves 6 parameters, the estimates cannot be produced with useful levels of precision. However, the precision can be improved if the original search spaces are constricted based on apriori information. Unfortunately, this can lead to biased solutions, which may not be useful. This thesis tests precision of the results when the original search space is influenced by parameter values from the literature. Another critical paper [7] questions the accuracy of a fitting algorithm that evaluates 6 parameters. In order to simplify the problem, only one parameter, myelin water fraction, is fit while the other 5 parameters are held constant. After the one parameter and measured magnetization are fit, the results proved accurate. Further justification for this 1 parameter fit is that in experiments, variation to T 1, T 2 values cause only ±10% variations to myelin water fraction. However, small variations to myelin water fraction cause greater changes in T 1, T 2 values. Additionally, the standard devation of fitted myelin water fraction values are smaller than T 1, T 2 fitted values[7]. This experiment is repeated as part of the research for this thesis and 8

19 similar results are found. In summary, the purpose of this thesis was to implement and test the mcdespot signal equation (Chapter 3) and to implement and test the SRC fitting (Chapter 4). 9

20 Chapter 3 MATRIX VERIFICATION MRI is used to determine tissue structure within the brain by measuring a magnetization curve that is dependent on the relaxation times, which characterize tissues. There is relaxation in both the longitudinal and transverse directions. Using methods, DESPOT 1 and DESPOT 2, Spoiled Gradient recalled echo (SPGR) and fully-balanced steady state free precision (SSFP) data is acquired. From this data, T 1 and T 2 information can be obtained. These T 1, T 2 relaxation times are tissue-specific, and help identify the structure [1]. A standard method of acquiring a magnetization signal is to apply a long train of pulses with a flip angle α which are delayed by a time of T R. T R is referred to as the relaxation time. The signal is measured after each flip angle and eventually the signal reaches a steady state, M ss [9]. 10

21 3.1 Longitudinal Magnetization The T 1 relaxation time governs the regrowth of the longitudinal magnetization. This magnetization in the z-direction (longitudinal magnetization) is described by the Bloch equation: dm dt = M 0 M T 1. (3.1) M 0 is the equilibrium magnetization and M is the magnetization at time t. T 1 is the relaxation parameter that is unknown. The solution to the above equation is: M(t) = M 0 (M 0 M(0))e t/t 1. (3.2) Let M(0) = M ss cos α, where M ss is the steady-state magnetization along the z-axis and α is the flip angle. At time T R, M(T R) = M ss, so equation (3.2) becomes [9]: M ss = M 0 (M 0 M ss cos α)e T R/T 1. (3.3) Rewriting equation (3.3): M ss M 0 = 1 e T R/T1 1 e T R/T 1 cos α. (3.4) 11

22 The observed magnetization, or the signal obtained from acquisition methods is given by: S = M ss sin α S = M 0(1 e T R/T 1 ) sin α 1 e T R/T 1 cos α. (3.5) SPGR Equations The signal S is the magnetization of a single species as a function of the flip angle α. However, literature suggests magnetization without exchange is a valid model only with specific values of T R. Furthermore, there is evidence [8] that suggests up to three species in exchange are present to characterize the magnetization. This thesis only considers 2-species exchange for simplicity and this may be enough to describe the magnetization [1]. Consider two species in exchange, a slow species S and a fast species F. Species S represents the intra-, extracellular tissue and species F represnts myelin. When considering two species in exchange, the original Bloch equation expands to become [9]: dm S dt dm F dt = M 0,S M S T 1S k SF M S + k F S M F = M 0,F M F T 1F k F S M F + k SF M S, (3.6) 12

23 where M 0,S (M 0,F ) is the equilibrium magnetization of species S (F) and k F S (k SF, respectively) is the exchange rate from species F to species S (S to F). In steady-state, the following must hold: k SF M 0,S = k F S M 0,F. (3.7) To simplify notation, the differential equations can be written using matrix notation: dm dt = AM + C where M = [M S, M F ] T [ M0S C =, M 0F T 1S T 1F A = ] T, 1 T 1F k F S k SF k F S 1 T 1S k SF (3.8) Then solving this differential equation: M = e (t 0)A M(0) + = e ta M(0) + t 0 t 0 e (t s)a Cds e ta e sa Cds = e ta M(0) + e ta ( A 1) e sa C t 0 (3.9) = e ta M(0) + e ta ( A 1) e ta C e ta ( A 1) e 0A C = e ta M(0) A 1 C + e ta A 1 C 13

24 The solution is: M = e ta ( M(0) + A 1 C ) A 1 C (3.10) Assuming: M(0) = M ss cos α, and M ss = M(T R) M ss = e T RA M ss cos α + e T RA A 1 C A 1 C ( I e T RA cos α ) M ss = e T RA A 1 C A 1 C (3.11) M ss = ( I e T RA cos α ) 1 ( e T RA I ) A 1 C To obtain the observed magnetization of the data, S, assume: A 1 C = M(0), and S = M ss sin α (3.12) The observed magnetization is [9]: S = ( I e T RA cos α ) 1 ( I e T RA ) M 0 sin α (3.13) The magnetization signal, acquired through the DESPOT 1 method, is labeled S SP GR. The notation S SP GR will be used for the remainder of this thesis. S T SP GR = M T SP GR ( I e T RA SP GR ) sin α ( I e T RA SP GR cos α ) 1 [1] (3.14) 14

25 Notice, equation (3.14) has reversed the order of multiplication of equation (3.13). These equations represent the observed signal, so should be the same. The question arises under what conditions are these two expressions equivalent. Since the equations are matrix equations, multiplication in not always commutative. The next step is to show when the matrices commute, and therefore show when the equations are equivalent. Consider two matrices, A, B. If A and B are both invertible and commute: AB = BA Pre and post multiply each side of the equation by A 1 : A 1 ABA 1 = A 1 BAA 1 BA 1 = A 1 B (3.15) This shows that if A and B commute, then A 1 and B also commute. Consider: Ā = ( I e T RA cos α ) B = ( I e T RA) Ā B = ( I e T RA cos α ) ( I e T RA) = I (1 + cos α)e T RA 2T RA + cos αe BĀ = ( I e T RA) ( AT I cos αe R) (3.16) = I (1 + cos α)e T RA 2T RA + cos αe 15

26 This shows that Ā, B commute, so Ā 1, B also commute. What is left to show is Ā 1 BM = M T BĀ 1. Consider the following: (Ā 1 BM T ) T = M B T (Ā 1 ) T = M BĀ 1 (3.17) if Ā is symmetric. A SP GR is symmetric when k SF = k F S. Therefore, equations (3.13) is equivalent to (3.14) only when k F S = k SF. This result can be shown numerically. Both expressions require four parameters, T 1S, T 1F, f F, τ F. The parameters T 1S, T 1F are the spin-lattice relaxation times for the S and F species. The parameter τ F is the myelin residence time and is the reciprocal of the exchange rate from species F to S. τ F = 1 k F S. The parameter f F is the myelin fraction of the F species. The fraction of species S is from the given relationship f S = 1 f F. In equilibrium, the following constraint must hold. k F S f F = k SF f S. (3.18) Given the example parameter values, T 1F = 465, T 1S = 1070, f F = 0.2, and τ F = 100, the SPGR curves (3.13) and (3.14) are generated in Figure 3.1. The signals, with TR = 6.5, are plotted against flip angle α = 2,4,6,8,10,12,14,16,18. 16

27 Figure 3.1: Signal Equations with Different Exchange Rates This plot compares equation (3.13) (black) and equation (3.14) (blue). As the results show, the curves are different. Because k F S k SF, the matrix A SP GR is not symmetric, so the equations are not equivalent. 17

28 Figure 3.2: Compare Equations with the Same Exchange Rates This plot compares equation (3.13) and equation (3.14) with two cases. The first case (black) has no exchange rate (k SF = k F S = 0). The second case (blue) sets k SF = k F S = k m = 1 τ F. In both cases equation (3.13) is represented by a solid line and equation (3.14) is represented by circles. In Figure 3.2, the curves generated by equation (3.13) and equation (3.14) are compared in two cases when k SF = k F S. In both cases, the equations are equivalent. The curves represent this because in case 1 (k SF = k F S = 0) the black solid curve (3.13) is the same as the black circles (3.14). The result is also seen in case 2 (blue). Although (3.14) is has been used in multiple papers [4],[1], the remainder of this thesis uses equation (3.13). Another interpretation of equation (3.14) is that it is the solution to the 18

29 transpose of (3.8). Consider the transpose of (3.8): dm T dt = (AM + C) T dm T dt = M T A T + C T The solution is: (3.19) M T = C T (A 1 ) T + ( M T (0) C T (A 1 ) T) e tat The observed magnetization becomes: S T SP GR = M T SP GR ( ) ( 1 I e T RAT SPGR I cos αe SPGR) T RAT (3.20) Notice that equation (3.20) is the same equation as (3.14), except matrix A is transposed within the equation. Therefore, equation (3.20) is equivalent to equation (3.13), except the magnetization is represented as a column vector in (3.13) and a row vector in (3.20). 3.2 Transverse Magnetization: SSFP Equations The previous subsection derived the equation governing the longitudinal magnetization. This data is acquired through T 1 - weighted methods and therefore information about T 1 can be deduced from the SPGR data. An extension of the above analysis can then include the transverse magnetization, or the x and y components of the magnetization. The new 19

30 equation will describe both the longitudinal and transverse magnetizations, which will give information about the T 2 parameter. This information is obtained from SSFP data after performing the DESPOT 2 method. To include the x and y components, the extended Bloch-McConnell equations are [2]: δm z,s δt δm x,s δt δm y,s δt δm z,f δt δm x,f δt δm y,f δt = M z,s( ) M z,s T 1S = M z,s T 2S = M z,s T 2S k SF M z,s + k F S M z,f + ω S M y,s k SF M x,s + k F S M x,f + ω S M x,s k SF M y,s + k F S M y,f = M z,f ( ) M z,f T 1F = M z,f T 2S = M z,f T 2S k F S M z,f + k SF M z,s + ω F M y,f k F S M x,f + k SF M x,s + ω F M x,f k F S M y,f + k SF M y,s (3.21) M i,s is the i th directional component of the magnetization of species S. k SF, (k F S ) is the exchange rate from species S to species F (F to S). T 2S is the T 2 relaxation time for species S. ω S is the off-resonance factor. Define ω S = ω F = θ/t R [2]. The solution to this system of equations is denoted M SS or steady-state magnetization. The following is the solution written in matrix notation. 20

31 M SS = [ I e T RA R(α) ] 1 ( e T RA I ) A 1 C (3.22) where, 1 T 2F k F S k SF ω F k F S T 2S k SF 0 ω S ω A = F 0 T 2F k F S k SF ω S k F S T 2S k SF T 1F k F S k SF k F S T 1S k SF (3.23) C = [ M z,f ( ) T 1F ] T M z,s ( ) (3.24) T 1S cos α 0 sin α 0 R(α) = cos α 0 sin α 0 0 sin α 0 cos α sin α 0 cos α (3.25) The matrix R(α) is a rotation matrix of flip angle α and T R is the time between the RF pulses of α degrees. The magnetization is measured just 21

32 before the RF pulse. Because this magnetization is given by the SSFP curve, the steady state magnetization notation becomes M SSF P. Furthermore, the observed SSFP signal is obtained from the following: (M 2 S SSF P = x,f + M ) ( ) y,f 2 + M 2 x,s + My,F 2 (3.26) Note that a version similar to equation (3.22) has been published, but in an incorrect form[1],[4]: M SS = ( e T RA I ) A 1 C [ I e T RA R(α) ] 1 (3.27) This equation is impossible to evaluate because the matrix dimensions do not match correctly for matrix multiplication. Thus for the remainder of the work in this thesis, the correct form (3.22) of the equation is used, as in [4]. 3.3 Defining Exchange Rates Before a fitting algorithm can be implemented, the matrix equations that govern the curves must be correct. This is not a trivial feat because there is not a unanimous definition for the exchange rates of the fast and slow species, and there is not a clear definition of the τ or mean residence 22

33 times. One paper [1], gives the relationships f F = 1 f S k F S = 1 τ F (3.28) k SF = f F k F S f S. The mean residence time τ F is given, so the remaining exchange rates can be determined. A recent paper [7] defines an exchange rate as the sum of the exchange rates of the two species. k = 1 τ = k F S + k SF. (3.29) Additionally, k F S = (1 f F )k k SF = f F k. (3.30) These two definitions of exchange rates are equivalent as will be shown below. A last definition of exchange rates introduces a mean exchange rate k m = f F k F S + f S k SF. When using k m in conjunction with the relationships f F = 1 f S, k F S = 1 τ F, and k SF = f F k F S, the exchange rates between f S 23

34 species can be defined in terms of k m. k m = f F k F S + k SF f S = f f k F S + f F k F S f S f S = 2f F k F S (3.31) k F S = k m 2f F A similar argument is used to obtain the relationship, k SF = km 2f S. One definition of exchange rate is[7]: 1 τ = k = k F S + k SF where k F S = (1 f F )k k SF = f F k ( ) 1 also k F S = 1 f F k SF (3.32) f F k F S = f S k SF The specific values of mean residence times τ F or τ S are never given. Instead everything is done using k = 1 τ where τ is the timescale of exchange. Another definition of specific exchange rates [1]: f F k F S = f S k SF 24

35 Where: k F S = 1 τ F, k SF = 1 τ S The same relationship is seen in both papers: f F k F S = f S k SF. Assuming the exchange rates k F S, k SF have a constant value, then the relationship between τ F, τ S [1] and τ [7] must be determined. k SF = f F τ = 1 τ S τ S = τ f F k F S = f S τ = 1 τ F (3.33) τ F = τ f S Consider definitions of k F S, k SF and sum these two values together: k F S + k SF = 1 τ F + 1 τ S = f S τ + f F τ = 1 f F τ (from above) + f F τ (3.34) = 1 τ This is the exact relationship in both papers: k = 1 τ = k F S + k SF. What remains to find is the relationship between mean exchange rate k m and exchange rate k. 25

36 From [7]: k F S = (1 f F )k k SF = f F k (3.35) Additionally, from case 4 : k F S = k m 2f F k SF = k m 2f S (3.36) Combining equation (3.35) and (3.36): k F S = (1 f F )k = k m 2f F (3.37) k m = (2f F f S )k Now that definitions of exchange rates from [1] and [7] are equivalent, and that k m is not always satisfied, graphical tests can be implemented. 26

37 Table 3.1: Exchange rate tests Case k / k m k F S k SF Comment 1-1 τ F k F S Set k SF = k F S, not done in literature, but 2-1 τ F produces accurate curves f F k F S f S From [1] 3 k = 1 τ (1 f F )k f F k Assuming τ = τ F, if τ = τ F (1 f F ) then same as case 2. [7] 4 k m = 1 τ m k m 2fS k m 2fF k m = 2f S f F k τ m = 1 k m = 1 2f S f F k = 1 2f S f F τ where, 1 = k τ The following plots compare the curves generated in [1] figure 8, region 1. In addition, curves generated with equation (3.14) (red), equation (3.13) (blue), and equation (3.20) (green stars) are compared. The parameter values used were T 1F = 373, T 1S = 919, T 2F = 11.5, T 2S = 116.5, f F = 27.4, τ F = 126. Additionally, when the curves were generated, a scaling factor (s.f.) was used to raise the amplitude of the curve, while retaining the shape. The following table contains the results for the above 5 cases. 27

38 Table 3.2: Parameters used to generate curves with different exchange rates SPGR SSFP Values α = 2, 4, 6, 8, 10, 12, 14, 16, 18 α = 6, 14, 22, 30, 38, 46, 54, 62, 70 TR = 6.5 TR = 5, θ = π S = [0.52, 0.94, 1.13, 1.19, S = [0.53, 0.94, 1.14, 1.17, 1.17, 1.10, 1.04, 0.95, 0.87] 1.13, 1.07, 0.96, 0.88, 0.82] (a) SPGR: s.f. = (b) SSFP: s.f. = Figure 3.3: Exchange Rate: Case 1 28

39 (a) SPGR: s.f. (3.14) = s.f.(3.13) = (b) SSFP: s.f. = Figure 3.4: Exchange Rate: Case 2 (a) SPGR: s.f. (3.14) = s.f.(3.13) = (b) SSFP: s.f. = Figure 3.5: Exchange Rate: Case 3 29

40 (a) SPGR: s.f. (3.14) = s.f.(3.13) = (b) SSFP: s.f. = Figure 3.6: Exchange Rate: Case 4 From the previous tests, the way exchange rate is defined changes the shape of the magnetization curve. Although the implementation of case 1 match the curve in the literature, the definition of exchange rate does not reflect reality. In case 1, k SF = k F S, but this is not always true. Case 4 combines the definitions from different papers [2],[1]. Case 2 is used in many papers [4],[1],[9], and used throughout the remainder of this thesis. 30

41 Chapter 4 FITTING ALGORITHM 4.1 Description of Stochastic Contraction Fitting The magnetization curve can be described as a matrix equation defined in the previous section. The magnetization values for given flip angles are obtained from patient MRI s. However, the parameters T 1F, T 1S, T 2S, T 2F, f F, τ F, that define this curve are unknown. To estimate these values, a fitting algorithm is employed. The fitting algorithm uses stochastic contraction as implemented in the literature [4]. The algorithm first creates many magnetization curves that are generated from randomly selected parameter values from a search space. The curves that are closest to the actual curve, in least squares sense, are selected and their parameter values further restrict the search space. The process repeats until the search space is contracted to ±5%. 31

42 The first step is to create a search space. The search space for each parameter value is a range of possible values based on a priori knowledge. The second step is to randomly pick N1 parameter values from the uniformly distribued search space. N1 many magnetization curves are created. Then the sum of least squares is calculated between these generated curves and the true magnetiztion curve. The N2 closest curves are chosen as the best approximations. From all of the parameter values that generated these best curves, the smallest and largest values for each parameter are used to contract the original search space. Once the search space contracts to a space of width ±5% of the original search space, the algorithm ends and the mean and standard deviation of the parameters from the last test are calculated. These parameter values should match the parameters that are used to produce the true curve. 4.2 Potential Problems with Stochastic Contraction Fitting This algorithm, when impeimented in Matlab, proves to be inefficient. The algorithm creates large matrices that are inverted, which is computationally expensive in Matlab. In the literature, N1 = 5000 and N2 = 50 [4]. In interest of time, many of the tests were done using N1 = 1000 and N2 = 50. When testing the efficiency of the algorithm, published parameters were used to produce the true magnetization curve. The algorithm was then implemented to see if the known parameters could be 32

43 reproduced. After preliminary tests, the resulting curve fit the true curve perfectly, however, the parameters were often not correct. Although the f F parameter was close to true values in many tests, the other parameters were not. After observing these results, modifications to the agorithm were made, which include restricting the search space. Various papers, [11],[7],[6] question the validity of this method including the results that are published. Some of the concerns raised include the probablity of performing the 6 parameter fit with a certain level of precision. According to recent literature [6], this is impossible without restricting the original search space. This restriction would be based on a priori knowledge, which may lead the results to a biased value. Another concern is with the difficulty of fitting 6 parameters. One published approach was to fix five of the six of the parameters to literature values and performing a 1 parameter fit (for example) [7]. The last concern is when comparing the mcdespot and GRASE method, the values for myelin water fraction are very different. f F is about.2 to.25 for mcdespot, and 0.05 for GRASE. This may be a result of including exchange in mcdespot, or because any restricted search space causes biased answers [11]. This recent paper showed that the effect of exchange is not large enough to cause the large discrepancy between the methods. It is of interest, however, to test the results of restricting the search space for particular parameters. Before the results of the tests are stated, a graphical result of a fitting 33

44 algorithm is given. Figure 4.1 shows the results of fitting the 6 parameters with the full search space and with a restricted search space [5]. Table 4.1 gives the conditions and parameter values that are used to create the true curve. Table 4.1 also includes the fitted values as a result of the algorithm. Table 4.1: Parameters used for fitting algorithm SPGR SSFP Values α = 2, 4, 6, 8, 10, 12, 14, 16, 18 α = 6, 14, 22, 30, 38, 46, 54, 62, 70 TR = 6.5 TR = 5, θ = 180 N1=1000, N2=50 N1=1000,N2=50 True Values T 1F = 465, T 1S = 1070, T 2F = 26, T 2S = 117, f F = 0.2, τ F = 100 Fitted Values: Full Search Space T 1F = 1109, T 1S = 814, T 2F = 56, T 2S = 81, f F = 0.203, τ F = 241 Restricted Search Space [5] T 1F = 432, T 1S =1085, T 2F = 23, T 2S = 108, f F = 0.196, τ F = 263 (a) SPGR curve with full parameter fit (b) SSFP curve with full parameter fit Figure 4.1: Fitting algorithm with full search space. 34

45 (a) SPGR curve with restricted search space[5] (b) SSFP curve with restricted search space[5] Figure 4.2: Fitting algorithm with restricted search space. Using the full search space, the fitted parameter values are not matching to the true values even though the fit result perfectly matches the data. However, if the search space is restricted, the fitting algorithm fits the parameter values well. It is interesting that although the parameter values do not match in the two cases, the curves are identical. The fitting algorithm fits the curve well, but the parameter search does not always converge to the correct value. These concerns illustrate that this problem is ill-posed. The algorithm is attempting to fit 6 parameters, or trying to find an optimal solution in a six dimensional space. The full original search space leaves room for the algorithm to choose incorrect parameter values, but the combination of incorrect values can produce a correct curve. By restricting the search space, the algorithm is not able to fit the curves with incorrect values, so converges to the true values. 35

46 The following section explores these concerns. The first tests systematically restrict the original search space of specific parameter values. This shows how the original search space affects the outcome, including if a priori knowledge produces a biased result. More tests fix certain parameter values to implement the fitting algorithm on fewer parameters. Finally, additional tests are performed to test the effect of certain flip angles or α on the fitting. 4.3 Numerical Results The next set of tests were performed to evaluate the effects of restricting the original search space on the fitted parameter values. Literature suggests that the only way to achieve desired precision in fitted parameters is to restrict the search space with a priori knowledge [6]. However, other papers suggest that the restricted search space could give biased results or results that tend toward the endpoints of the search space range [7] Restricting Search Space All of the following tests used true parameter values T 1F = 465, T 1S = 1070, T 2S = 26, T 2F = 117, f F = 0.2, τ F = 100 [1]. Test 1 used the full search space which is defined as: 100 T 1F, T 1S 2500, 1 T 2F, T 2S 150, 0 f f 0.45, 25 τ f 500. Test 2 restricted both of the T 2 values, T 2F = 26 ± 20%, T 2S = 117 ± 20%. Test 3 had a smaller 36

47 restricted search space for T 2 values, T 2F = 26 ± 10%, T 2S = 117 ± 10%. Test 4 restricted both of the T 1 values, T 1F = 465 ± 10%, T 1S = 1070 ± 10%. Test 5 restricted the search space for each parameter to ±10% with the exception of f F, which had the original search space. Test 6 had the original search space for all parameters except for f F, where the new search space was 0 f F Test 7 uses the restricted search space from [5]. These restrictions are based on a priori knowledge. 300 T 1F 650, 900 T 1S 2500, 1 T 2F 30, 50 T 2S 165, 0 f f 0.35, and 25 τ F 600. This search space restricts all 6 parameter values to around ±40% of the true value. This may not be useful in real tests, because the true parameter values are not known. For each test, N1=1000, N2 = 50, and 5 independent iterations were performed, where each iteration ran until an error < 0.1 was calculated. The tests were performed with SPGR flip angles α spgr = 2, 4, 6, 8, 10, 12, 14, 16, 18 and TR = 6.5, and SSFP flip angles α ssfp = 6, 14, 22, 30, 38, 46, 54, 62, 70 and TR = 5. The results of these tests are given in Table 4.2. Table 4.2: Restricted Search Space T 1F T 1S T 2F T 2S f F τ F True Values Test Time mean std mean std mean std mean std mean std mean std

48 From the results in Table 4.2, the fitted parameter values are always closer to the true values when the original search space is smaller. Tests 3 and 7 have the best results, however, because the search space was restricted so much for each parameter, these results may be biased. Tests 2 and 3 result in parameter values close to the true values. These tests are performed with a restricted search space for T 2. This may indicate that if there is some a priori knowledge of the T 2 parameters, the fitting may perform much better. Test 6 is also interesting, because with the larger search space for f F, the algorithm took much longer and the figures (not given) show the curve with true values does not match the curve with the fitted values. Furthermore, the fitted value for f F is much larger than the true value. This may indicate that a search space obtained from a priori knowledge results in a fitted value closer to the true value. Since the results of Test 7 are close to the true parameter values, this restricted search space is used in later tests. It serves as the optimal search space in which perform the fitting algorithm and so other conditions of the algorithm can be tested Reduced Parameter Search The next set of tests performed a reduced number of parameter search. By fixing one or more parameter values, the fitting is done for only 4 or less parameters [7]. Test 1 is a full 6 parameter search, results dupicated from 38

49 above table for easier comparison. Test 2 fixed T 2F = 26, and T 2S = 117. Test 3 fixed T 1F = 465 and T 1S = Test 4 fixed τ F = 100. Test 5 fixed each parameter value to the true value except f F, which had the original search space. The other testing conditions remained the same as the previous set of tests. Table 4.3: Reduced Parameter Search T 1F T 1S T 2F T 2S f F τ F True Values Test Time mean std mean std mean std mean std mean std mean std From the results in Table 4.3, fixing τ F has no beneficial effect on the fitted parameter values. Test 2, fixing T 2 values gives a very close estimate of f F, with very small standard deviation while keeping the other fitted parameter values close to the true values. A very interesting result is test 5. When only one parameter, f F, is fit, the fitted value is farther away from the true value than in any other test. This test was motivated by [7], which claimed that a 6 parameter fit was too large to perform with desired precision. This resulted in a 1 parameter fit with published results very close to the true value. The results published in this paper differ from those in [7]. 39

50 4.3.3 Simulating Different Data Aquisition The next set of tests were performed on the upward slope of the magnetization curve. This translates to SPGR flip angles of α spgr = 2, 3, 4, 5, 6, 7, 8, 9, 10 and SSFP flip angles of α ssfp = 6, 10, 14, 18, 22, 26, 30, 34, 38. These tests were done to see if the parameters could be fit better or worse using at a different section of the MR signal curve. For the restricted search space tests, the parameters were restricted to ±10% the true value. Test 1 was a full search space, Test 2 restricted T 2, Test 3 restricted T 1, Test 4 restricted τ F, and Test 5 used the most recent search space based on a priori knowledge [5]. Table 4.4: Restricted Search Space: Upward Slope T 1F T 1S T 2F T 2S f F τ F True Values Test Time mean std mean std mean std mean std mean std mean std Table 4.5 shows the results of reduced parameter search fit. Test 1 is unrestricted, rewritten for comparison, Test 2 fixes T 2, Test 3 fixes T 1, Test 4 fixes τ F, and Test 5 fixes all parameters except f F. 40

51 Table 4.5: Reduced Parameter Search: Upward Slope T 1F T 1S T 2F T 2S f F τ F True Values Test Time mean std mean std mean std mean std mean std mean std The results in Tables 4.4 and 4.5 show that if flip angles selected such that they are approximately distributed around the signal curve peak, the parameter fits are worse than using a larger range of flip angles. Therefore, for the remainder of this thesis, the tests are performed over the entire curve Another Set of Parameters The following tests repeat the tests with restricted search space and reduced parameter search with different true parameter values. These tests were performed to confirm that the problems with the fitting algorithm are not sensitive to a specific set of parameter values. Parameters used for the next tests were T 1F = 350, T 1S = 1400, T 2F = 30, T 2S = 120,f F = 0.2, τ F = 500. The original search space is the same, except 25 τ F 750. Additionally, when the search space is restricted for a specific parameter, the search space is equal to the true value ±10%. Test 1 is the full search space, test 2 restricts T 2, test 3 restricts T 1, test 4 restricts 41

52 τ, and test 5 uses [5]. The results are in the following tables. Table 4.6: Restricted Search Space: New True Parameter Values T 1F T 1S T 2F T 2S f F τ F True Values Test Time mean std mean std mean std mean std mean std mean std Table 4.7: Reduced Parameter Search: New True Parameter Values T 1F T 1S T 2F T 2S f F τ F True Values Test Time mean std mean std mean std mean std mean std mean std Tables 4.6 and 4.7 reveal similar patterns to the previous tests. The most accurate parameter values are produced when the T 2 search space is restricted. Additionally the search space from [5] produces parameter values close to true values Simulating Modified Aquisition Strategy: Change θ The following tests are to check if the fit performs better given different values of θ. The parameters used for the fitting were T 1S = 465, T 1F = 1070, T 2S = 26, T 2F = 117, f F = 0.2, τ F = 100. The algorithm used 42

53 N1 = 1000, N2 = 50, error < 0.1 and 5 iterations (in interest of time). The tests perform a 6 parameter fit using a full search space and with θ= 10,30,50,90,130,150,180. Results of the 6 parameter fit using the full search space are in Table 4.8. Table 4.8: Different θ with Full Search Space T 1F T 1S T 2F T 2S f F τ F True Values θ Time mean std mean std mean std mean std mean std mean std Unfortunately, there is no evidence in Table 4.8 of a preferred choice for θ that would produce better fits. In conclusion, each test gives poor parameter fits, owing to the open search space. Results of the tests with restricted search space from [5] are given in Table 4.9. Table 4.9: Different θ with Restricted Search Space [5] T 1F T 1S T 2F T 2S f F τ F True Values θ Time mean std mean std mean std mean std mean std mean std

54 As expected, the fits with restricted search space are much more accurate, when compared to the full search space. However, these tests give no evidence for a particular θ producing the best parameter fit. The tests with θ = 10, 50, 180 are marginally better than other tests, but the results are not dramatically better and may represent the randomness of the testing rather than improved accuracy. The next tests were to check if the fit performs better given different values of α. The SSFP curve was plotted as a function θ = 10,20,30,40,50,60,70,80,90. Since only the SSFP curve is dependent on θ, the SPGR curve is fit with the usual range of flip angles. The same paramerters and conditions of the previous tests were used. The results are in Table Table 4.10: Different α with Full Search Space T 1F T 1S T 2F T 2S f F τ F True Values α Time mean std mean std mean std mean std mean std mean std There is little evidence for an optimal α flip angle given a range of θ angles. However, if only the fitted values for f F are considered, α = 10, gives the best result. The results of the tests with a restricted search space are in Table

55 Table 4.11: Different α with Restricted Search Space [5] T 1F T 1S T 2F T 2S f F τ F True Values α Time mean std mean std mean std mean std mean std mean std Once again, the results with restricted search space are closer to the true values, but these tests do not reveal an optimal flip angle. However, if only the fitted values of f F are considered, α= 10, 60,and 90 give the closest value to the true value Effects of Larger N1 The following tests were performed to evaluate the effect of using a larger N1, i.e. are the fitted parameter values better if a larger number of randomly generated curves are evaluated each iteration. The following tests use N1 = 5000, N2 = 50, error < 0.1 with 5 iterations. The following table shows the results of thses tests using a full search space. 45

56 Table 4.12: Full Search Space, N1=5000 T 1F T 1S T 2F T 2S f F τ F True Values θ Time mean std mean std mean std mean std mean std mean std The results are very interesting. Each test converges to incorrect T 1 values. The T 1F values are over twice the size as the actual values. The T 2 values are the same estimates as the choice N1 = The estimates for f F and τ F are more accurate. Table 4.13 shows the results of the 6 parameter fit with a restricted search space,[5]. Table 4.13: Restricted Search Space [5], N1=5000 T 1F T 1S T 2F T 2S f F τ F True Values θ Time mean std mean std mean std mean std mean std mean std The results reveal better accuracy when N1 = 5000 for a restricted search space. Most importantly, the f F values are close to the acutal results with small standard deviations. Note that in both Table 4.12 and Table 4.13, the best fitted value of f F occured at θ =