Optimal and Efficient Designs for Functional Magnetic Resonance Imaging (fmri) Experiments

Transcription

1 Optimal and Efficient Designs for Functional Magnetic Resonance Imaging (fmri) Experiments Ming-Hung (Jason) Kao Arizona State University with Ching-Shui Cheng and Federick Kin Hing Phoa

2 Introduction

3 fmri experiment Functional Magnetic Resonance Imaging (fmri): a technology for studying how our brains respond to mental stimuli A subject in an MRI time 2. Present a sequence of mental stimuli (e.g. pictures) to the subject MR scanner scans the brain 4. These fmri time series are analyzed to make inference about local brain activity in response to the stimuli 3. An fmri time series is then obtained from each of the, e.g., 64x64x30 brain voxels(imaging units)

4 fmri design At the design stage, we would like to obtain the best sequence of mental stimuli to help collect the most informative data to render the most precise statistical inference. d = (d 1,..., d N ) T : an fmri design N: number of regularly spaced time points in the study d n {0, 1,..., Q}; Q = the total number of stimulus types d n = q > 0: a qth-type stimulus appears at the nth time point d n = 0: no stimulus presentation An example: Q = 2 & time between the time points: τ = 3 s,

5 fmri design At the design stage, we would like to obtain the best sequence of mental stimuli to help collect the most informative data to render the most precise statistical inference. d = (d 1,..., d N ) T : an fmri design N: number of regularly spaced time points in the study d n {0, 1,..., Q}; Q = the total number of stimulus types d n = q > 0: a qth-type stimulus appears at the nth time point d n = 0: no stimulus presentation An example: Q = 2 & time between the time points: τ = 3 s, d={ } s ISI=3s 3s 6s 9s 12s 15s 18s Times between stimulus onsets are multiples of a pre-specified ISI (e.g. 3s) 1= 2= 0= no stimulus (pesudo-event)

6 Hemodynamic Response Function (HRF) The effect of a stimulus over time is described by a function of time called the hemodynamic response function (HRF) time (seconds) Statistical inference about brain activity is normally made based on analyzing some characteristics of the HRF.

7 Model A commonly used model for studying the Q HRFs is: y = X d,1 h 1 + X d,2 h X d,q h Q + Sγ + ε y = (y 1,..., y N ) T : fmri time series of a brain voxel h q = (h 1,q,..., h K,q ) T : the HRF parameter vector of the qth-type stimulus X d,q : 0-1 design matrix for the qth-type stimulus Sγ: a nuisance term for drift/trend of y; S: a specified matrix; γ: unknown parameter vector ε = (ε 1,..., ε N ) T is noise with ε N(0, Σ). Our target: fmri designs d that are optimal in some statistically meaningful sense for estimating (some linear combinations of) h q s.

8 Estimation of an individual HRF (Q = 1)

9 Analytical results (Q = 1) We first identify optimal fmri designs with Q = 1 for N = 4t, 4t + 1, 4t + 2 and 4t + 3. Model assumptions (y = X d h + γj N + ε): h = (h 1,..., h K ) T : the HRF parameter vector of interest S = j N is the vector of N ones; γ: nuisance paramter ε N(0, σ 2 I N ). the last K 1 elements of the design d are also presented to the subject in the warm-up period (i.e. circular setting): An optimal fmri design is a d {0, 1} N minimizing some function Φ(M d [h]) of the information matrix M d [h] of h. Two common design selection criteria: A-criterion (average var. of ĥ): Φ A(M) = trace(m 1 )/K D-criterion (generalized var. of ĥ): Φ D(M) = det(m) 1/K

10 Analytical results (Q = 1) Under the previous model assumption, the information matrix for any given d {0, 1} N is M d [h] = X T d (I N N 1 J N )X d. X d is an N-by-K, 0-1 circulant matrix. A toy example: X d = In general, X d = [d, Ud, U 2 d,..., U K 1 d] [ 0 T U = N 1 1 I N 1 0 N 1 ] (cyclic shifting operator).

11 Analytical results (Q = 1) The key idea: We made use of the following fact: M d [h] = M d[h]/4, with d = ±(2d j N ). For the previous toy example, we have X d = ; X d = Finding optimal X d {0, 1} N K is equivalent to finding optimal X d {±1}N K, which is linked to a Circulant Chemical Balance Weighing Design (CCBWD) problem with a systematic, nuisance bias in the weighings.

12 Analytical results (Q = 1) Result. (Combinatoric structures of some optimal fmri designs) k 1 elements {}}{ For any design d, let n k,p,q = #{(q p)}, p, q = 0, 1. If a d {0, 1} N satisfies the following conditions, then d is both A- and D-optimal for estimating the HRF (h) of length K. N = 4t: X T d X d = NI K & d T j N = 0; i.e., (n k,0,0, n k,1,1, n k,0,1, n k,1,0 ) = (t, t, t, t); k = 1,..., K 1. N = 4t + 1: X T d X d = (N 1)I K + J K & d T j N = ±1; i.e., (n k,0,0, n k,1,1, n k,0,1, n k,1,0 ) = (t + 1, t, t, t); k = 1,..., K 1, or (n k,0,0, n k,1,1, n k,0,1, n k,1,0 ) = (t, t + 1, t, t); k = 1,..., K 1,

13 Analytical results (Q = 1) Result. (Cont d.) N = 4t + 2 with K 2: X T d X d = (N 2)I K + 2J K & d T j N = 0; i.e., (n k,0,0, n k,1,1, n k,0,1, n k,1,0 ) = (t+1, t+1, t, t); k = 1,..., K 1. N = 4t + 3 with K 4 and N N 0 (K), where N 0 (K) is the greatest real root of the cubic function c(x) = 2x 3 + (10 7K)x 2 + 2(2K 5)(K 1)x + 4K 2 7K: X T d X d = (N + 1)I K J K & d T j N = ±1; i.e., (n k,0,0, n k,1,1, n k,0,1, n k,1,0 ) = (t + 1, t, t + 1, t + 1); k = 1,..., K 1, or (n k,0,0, n k,1,1, n k,0,1, n k,1,0 ) = (t, t + 1, t + 1, t + 1); k = 1,..., K 1,

14 Analytical results (Q = 1) How to systematically generate optimal fmri designs? We propose to make use of a normalized Hadamard matrix that contains a circulant core A toy example with N = 3: H = We may take the any column of the circulant core as d H ; e.g., d H = ( 1, 1, 1) T We then relabel d H to get a Hadamard sequence; e.g., d H = (1, 0, 1) T. d H can be viewed as an fmri design to determine the onset times of the stimulus.

15 Analytical results (Q = 1) How to systematically generate optimal fmri designs? We propose to make use of a normalized Hadamard matrix that contains a circulant core A toy example with N = 3: H = We may take the any column of the circulant core as d H ; e.g., d H = ( 1, 1, 1) T We then relabel d H to get a Hadamard sequence; e.g., d H = (1, 0, 1) T. d H can be viewed as an fmri design to determine the onset times of the stimulus.

16 Analytical results (Q = 1) Hadamard sequences are known to exist when N = 4t + 3 (1) a prime; (2) a product of twin primes; or (3) is 2 r 1. [Condition (3) gives the popularly used binary m-sequences]. Example. An d H (with N = 131 = ) generated by a Paley difference set:

17 Analytical results (Q = 1) Hadamard sequences are known to exist when N = 4t + 3 (1) a prime; (2) a product of twin primes; or (3) is 2 r 1. [Condition (3) gives the popularly used binary m-sequences]. Example. An d H (with N = 131 = ) generated by a Paley difference set:

18 Analytical results (Q = 1) Result. (Design Constructions) [N = 4t] Let d 1,g,H {0, 1} N be obtained by inserting a 0 to a run of g 0 s in a Hadamard sequence d H. Then, d 1,g,H is optimal for estimating h of length K g + 1. [N = 4t + 1] Let d 2,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 s in a Hadamard sequence d H. Then, d 2,g,H is optimal for estimating h of length K g + 1. [N = 4t + 2] Let d 3,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 0 s and inserting an 1 to a run of g 1 1 s of a Hadamard sequence d H. Then, d 3,g,H is optimal for estimating h of length 2 K g + 1 where g = min(g 0, g 1 ). [N = 4t + 3] d H {0, 1} N is optimal for estimating h of length K 4 whenever N N 0 (K).

19 Analytical results (Q = 1) Result. (Design Constructions) [N = 4t] Let d 1,g,H {0, 1} N be obtained by inserting a 0 to a run of g 0 s in a Hadamard sequence d H. Then, d 1,g,H is optimal for estimating h of length K g + 1. [N = 4t + 1] Let d 2,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 s in a Hadamard sequence d H. Then, d 2,g,H is optimal for estimating h of length K g + 1. [N = 4t + 2] Let d 3,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 0 s and inserting an 1 to a run of g 1 1 s of a Hadamard sequence d H. Then, d 3,g,H is optimal for estimating h of length 2 K g + 1 where g = min(g 0, g 1 ). [N = 4t + 3] d H {0, 1} N is optimal for estimating h of length K 4 whenever N N 0 (K).

20 Analytical results (Q = 1) Result. (Design Constructions) [N = 4t] Let d 1,g,H {0, 1} N be obtained by inserting a 0 to a run of g 0 s in a Hadamard sequence d H. Then, d 1,g,H is optimal for estimating h of length K g + 1. [N = 4t + 1] Let d 2,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 s in a Hadamard sequence d H. Then, d 2,g,H is optimal for estimating h of length K g + 1. [N = 4t + 2] Let d 3,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 0 s and inserting an 1 to a run of g 1 1 s of a Hadamard sequence d H. Then, d 3,g,H is optimal for estimating h of length 2 K g + 1 where g = min(g 0, g 1 ). [N = 4t + 3] d H {0, 1} N is optimal for estimating h of length K 4 whenever N N 0 (K).

21 Analytical results (Q = 1) Result. (Design Constructions) [N = 4t] Let d 1,g,H {0, 1} N be obtained by inserting a 0 to a run of g 0 s in a Hadamard sequence d H. Then, d 1,g,H is optimal for estimating h of length K g + 1. [N = 4t + 1] Let d 2,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 s in a Hadamard sequence d H. Then, d 2,g,H is optimal for estimating h of length K g + 1. [N = 4t + 2] Let d 3,g,H {0, 1} N be obtained by inserting two 0 s to a run of g 0 0 s and inserting an 1 to a run of g 1 1 s of a Hadamard sequence d H. Then, d 3,g,H is optimal for estimating h of length 2 K g + 1 where g = min(g 0, g 1 ). [N = 4t + 3] d H {0, 1} N is optimal for estimating h of length K 4 whenever N N 0 (K).

22 Comparison of two HRFs (Q = 2)

23 Analytical results (Q = 2) For Q = 2, there are two HRFs, h 1 and h 2, evoked by the two types of stimuli, respectively. Model: y = X d,1 h 1 + X d,2 h 2 + γj N + ε d {0, 1, 2} N is an fmri design for Q = 2. X d,q is the N-by-K, circulant, 0-1 design matrix for the qth-type stimuli; q = 1, 2. The first column of X d,q is δ q whose nth element is: { 1, dn = q; ((δ q )) n = 0, otherwise; h q = (h 1,q,..., h K,q ) T is the HRF parameter vector of the qth-type stimulus; q = 1, 2. The remaining terms are as in the previous model for Q = 1. Our focus is on finding an optimal fmri design d {0, 1, 2} N to yield the most precise estimate of θ = h 1 h 2.

24 Analytical results (Q = 2) To that end, we work on an equivalent model: y = A d η + B d θ + γj N + ε A d = (X d,1 + X 2,d )/2; B d = (X d,1 X d,2 )/2 η = h 1 + h 2 ; θ = h 1 h 2 The information matrix of interest is then: M d [θ] = B T d [I N w{[j N, A d ]}]B d, w( ) is the orthogonal projection matrix A d {0, 0.5} N K is a circulant matrix; A d = J N,K /2 when d contains no zero B d {0, ±0.5} N K is a circulant matrix

25 Analytical results (Q = 2) Finding a d {0, 1, 2} N minimizing Φ(M d [θ]) is challenging Our strategy: 1 Find a manageable upper bound of M d [θ]; in general, M d [θ] = B T d [I N w{[j N, A d ]}]B d L B T d [I N N 1 J N ]B d L represents the Loewner ordering. the equality holds when d contains no zero; i.e., A d = J N,K /2 Note. For some cases, we find a sharper upper bound. 2 Find a d so that Φ(B T d [I N N 1 J N ]B d ) is minimized 3 If d contains no zero, the corresponding d is optimal. This is because, for any other d {0, 1, 2} N, Φ(M d [θ]) = Φ(B T d [I N N 1 J N ]B d ) Φ(B T d [I N N 1 J N ]B d ) Φ(M d [θ]) The last inequality follows from the fact that: M d [θ] L B T d [I N N 1 J N ]B d

26 Analytical results (Q = 2) Finding a d {0, 1, 2} N minimizing Φ(M d [θ]) is challenging Our strategy: 1 Find a manageable upper bound of M d [θ]; in general, M d [θ] = B T d [I N w{[j N, A d ]}]B d L B T d [I N N 1 J N ]B d L represents the Loewner ordering. the equality holds when d contains no zero; i.e., A d = J N,K /2 Note. For some cases, we find a sharper upper bound. 2 Find a d so that Φ(B T d [I N N 1 J N ]B d ) is minimized 3 If d contains no zero, the corresponding d is optimal. This is because, for any other d {0, 1, 2} N, Φ(M d [θ]) = Φ(B T d [I N N 1 J N ]B d ) Φ(B T d [I N N 1 J N ]B d ) Φ(M d [θ]) The last inequality follows from the fact that: M d [θ] L B T d [I N N 1 J N ]B d

27 Analytical results (Q = 2) Finding a d {0, 1, 2} N minimizing Φ(M d [θ]) is challenging Our strategy: 1 Find a manageable upper bound of M d [θ]; in general, M d [θ] = B T d [I N w{[j N, A d ]}]B d L B T d [I N N 1 J N ]B d L represents the Loewner ordering. the equality holds when d contains no zero; i.e., A d = J N,K /2 Note. For some cases, we find a sharper upper bound. 2 Find a d so that Φ(B T d [I N N 1 J N ]B d ) is minimized 3 If d contains no zero, the corresponding d is optimal. This is because, for any other d {0, 1, 2} N, Φ(M d [θ]) = Φ(B T d [I N N 1 J N ]B d ) Φ(B T d [I N N 1 J N ]B d ) Φ(M d [θ]) The last inequality follows from the fact that: M d [θ] L B T d [I N N 1 J N ]B d

28 Analytical results (Q = 2) With this strategy, we can show that optimal designs for studying θ with Q = 2 can be obtained by relabeling the previously mentioned optimal designs for estimating the HRF h with Q = 1. Result. Let u H, u 1,g,H, u 2,g,H, and u 3,g,H be respectively obtained by replacing the two symbols of d H, d 1,g,H, d 2,g,H, and d 3,g,H with 1 and 2. (a) For N = 4t, u 1,g,H is A- and D-optimal over {0, 1, 2} N whenever K g + 1 (b) For N = 4t + 1, u 2,g,H is A- and D-optimal whenever K g + 1 (c) For N = 4t + 2, u 3,g,H is A- and D-optimal whenever 2 K g + 1 (d) For N = 4t + 3, u H is is A- and D-optimal whenever K 4 and N N 0 (K).

29 Contrasts of three or more HRFs (Q 3)

30 Analytical results (Q 3) Model: y = X d,1 h 1 + X d,2 h X d,q h Q + γj N + ε Parametric function of interest: θ = [(I Q Q 1 J Q ) I K ] h 1. h Q E T h is the Kronecker product θ = (θ T 1,..., θ T Q) T is the standardized contrast of the Q HRFs θ q = h q h, and h = h1+h2+ +h Q Q. We would like a design d {0, 1,..., Q} N that minimizes Φ(M d [θ]) for some Φ.

31 Analytical results (Q 3) We again identify optimal designs by finding manageable upper bounds of M d [θ]. This can be done by using the following two Lemmas. Lemma 1. (Bailey & Druilhet,2004) Let θ = E T h. We have M d [θ] L F d [E T E] + E T M d [h]e[e T E] + The equality holds if and only if M d [h] commutes with w(e). [E T E] + is the Moore-Penrose inverse of E T E M d [h] is the information matrix for the HRF parameter vector h = (h T 1,..., h T Q) T.

32 Analytical results (Q 3) Lemma 2. Let G be the symmetric group of all the Q! permutations on {1, 2,..., Q}, and d {0, 1,..., Q} N be an fmri design. Suppose d g is obtained from d by permuting the labels 1,..., Q based on a g G, and keeping 0 s intact. Then, F d S g G F d g /Q! F d, where F d is the upper bound of M d [θ] defined in Lemma 1 S represents the Schur ordering: Note 1. For any two non-negative definite matrices G 1 and G 2 of order m, G 1 S G 2 if m i=l λ i(g 1) m i=l λ i(g 2) for all l = 1,..., m, where λ i (G) is the ith greatest eigenvalue of G. Note 2. A useful result states that G 1 S G 2 iff there exists a G conv{w G 1W T : W O(m)} such that G L G 2, where conv{ } denotes the convex hull, and O(m) is the group of orthogonal matrices of order m (Harman, 2008). Note. 3. G 1 L G 2 G 1 S G 2 Φ(G 1) Φ(G 2) for most commonly used criteria Φ, including Φ A and Φ D.

33 Analytical results (Q 3) With some algebra, we have for any d {0, 1,..., Q} N, F d = (I Q Q 1 J Q ) Λ d ; α 0 α 1 α K 1 Λ d = α 1 α α1 βj K, α K 1 α 1 α 0 α 0 = {N n 0 (d)}/q; n 0 (d) =number of 0 s in d α k = {Q Q q=1 n k,q,q(d) N +2n 0 (d) n k,0,0 (d)}/{q(q 1)}, k = 1,..., K 1, β = [Q Q q=1 n q(d) 2 {N n 0 (d)} 2 ]/{NQ(Q 1)}. Note. For any d allowing estimable θ, the average of the orthogonal transformations of Λ d is α 0 I K L (N/Q)I K

34 Analytical results (Q 3) Result. Let d {0, 1,..., Q} N be such that M d [θ] = (I Q Q 1 J Q ) (N/Q)I K. Then, M d [θ] S M d [θ] for any d allowing estimable θ. This is because: M d [θ] S F d S F d S (I Q Q 1 J Q ) (N/Q)I K Result. (Combinatoric structure of optimal designs) If d {0, 1,..., Q} N is such that n k,p,q (d ) = t, p, q = 1,..., Q, k = 1,..., K 1, then d is A- and D-optimal for estimating θ of length K.

35 Analytical results (Q 3) Result. Let d {0, 1,..., Q} N be such that M d [θ] = (I Q Q 1 J Q ) (N/Q)I K. Then, M d [θ] S M d [θ] for any d allowing estimable θ. This is because: M d [θ] S F d S F d S (I Q Q 1 J Q ) (N/Q)I K Result. (Combinatoric structure of optimal designs) If d {0, 1,..., Q} N is such that n k,p,q (d ) = t, p, q = 1,..., Q, k = 1,..., K 1, then d is A- and D-optimal for estimating θ of length K.

36 Analytical results (Q 3) To construct an optimal designs, we consider to use the popularly used m-sequences: exist when N = s r 1 where s,the number of different symbols, is a prime power can be generated by a primitive polynomial of a finite field toy example (N = 3 3 1): Buračas & Boynton (2002) proposed to use m-sequences as fmri designs for estimating Q = s 1 individual HRFs Result (Design construction) Let N = Q K, and s m {0, 1,..., Q 1} N 1 be an m-sequence of length N 1. Suppose d is obtained by (i) relabeling the sequence with s m + j N 1 ; and (ii) adding an 1 to a run of (K 1) 1 s into the sequence in (i). Then d is optimal for comparing Q HRFs of length K. Remark. Circulant orthogonal arrays can also be used to generate optimal designs with N = Q 2 t.

37 Analytical results (Q 3) To construct an optimal designs, we consider to use the popularly used m-sequences: exist when N = s r 1 where s,the number of different symbols, is a prime power can be generated by a primitive polynomial of a finite field toy example (N = 3 3 1): Buračas & Boynton (2002) proposed to use m-sequences as fmri designs for estimating Q = s 1 individual HRFs Result (Design construction) Let N = Q K, and s m {0, 1,..., Q 1} N 1 be an m-sequence of length N 1. Suppose d is obtained by (i) relabeling the sequence with s m + j N 1 ; and (ii) adding an 1 to a run of (K 1) 1 s into the sequence in (i). Then d is optimal for comparing Q HRFs of length K. Remark. Circulant orthogonal arrays can also be used to generate optimal designs with N = Q 2 t.

38 Analytical results (Q 3) To construct an optimal designs, we consider to use the popularly used m-sequences: exist when N = s r 1 where s,the number of different symbols, is a prime power can be generated by a primitive polynomial of a finite field toy example (N = 3 3 1): Buračas & Boynton (2002) proposed to use m-sequences as fmri designs for estimating Q = s 1 individual HRFs Result (Design construction) Let N = Q K, and s m {0, 1,..., Q 1} N 1 be an m-sequence of length N 1. Suppose d is obtained by (i) relabeling the sequence with s m + j N 1 ; and (ii) adding an 1 to a run of (K 1) 1 s into the sequence in (i). Then d is optimal for comparing Q HRFs of length K. Remark. Circulant orthogonal arrays can also be used to generate optimal designs with N = Q 2 t.

39 Analytical results (Q 3) The above mentioned designs might unfortunately have a long design length (N = Q K ). Here, we proposed to consider the relabeled m-sequences d m of length N = Q r 1 with an integer r(< K). The two-tuple balance property of m-sequences guarantees N that, for k = 1,..., Q 1 1, n k,1,1 (d m ) = N + 1 Q 2 1; n k,p,q (d m ) = N + 1 Q 2, (p, q) (1, 1). d m maintains a very similar combinatoric structure as the previously mentioned optimal design whenerver K N/(Q 1). Thus, N is not as large as Q K. These designs are (at least) very efficient in most cases.

40 Analytical results (Q 3) Table: Lower bounds of A- and D-efficiencies (%) of d m for some cases Q = 3 Q = 4 Q = 5 K N = 80 N = 242 N = 63 N = 255 N = 124 N = 624 A D Note. the relative efficiencies are calculated w.r.t. possibly hypothetical optimal designs of the same length N whose information matrix attains (I Q Q 1 J Q ) [(N 2 1)/(NQ)]I K

41 Discussion

42 Discussion We derive optimal fmri designs for estimating the HRF with Q = 1 and for comparing the HRFs with Q 2. We not only analytically establish the optimality of some popularly used designs, but also propose new, high-quality designs for experimenters to select from For some cases where optimal design results are unavailable, there are some very efficient computational approaches available in the literature for finding very efficient or optimal designs. Analytically deriving optimal fmri designs for these cases is a future study of interest.

43 Some References Analytical results: Cheng, Kao, and Phoa (2016) Optimal and Efficient Designs for Functional Brain Imaging Experiments. under review Cheng, and Kao (2015) Optimal Experimental Designs for fmri via Circulant Biased Weighing Designs. Annals of Statistics Kao (2015) Universally Optimal for Comparing Hemodynamic Response Functions. Statistica Sinica Kao (2014) A new type of experimental designs for event-related fmri via Hadamard matrices. Statistics & Probability Letters Kao (2013) On the optimality of extended maximal length linear feedback shift register sequences. Statistics & Probability Letters Computational approaches: Saleh, Kao, and Pan (2016) Fast algorithms for designing D-optimal event-related fmri experiments, JRSSC Kao, and Mittelmann (2014) A fast algorithm for constructing efficient event-related functional magnetic resonance imaging designs. JSCS Kao, Majumdar, Mandal, and Stufken (2013) Maximin and Maximin-Efficient Event-Related fmri Designs Under A Nonlinear Model. AOAS Kao, Mandal, and Stufken (2012) Constrained Multi-objective Designs for Functional MRI Experiments via A Modified Nondominated Sorting Genetic Algorithm. JRSSC Kao, Mandal, Lazar, and Stufken (2009) Multi-objective optimal experimental designs for event-related fmri studies. NeuroImage Kao, Mandal, and Stufken (2009) Efficient Designs for Event-Related Functional Magnetic Resonance Imaging with Multiple Scanning Sessions. Communications in Statistics-Theory and Methods

44 Thank You