1 Introduction

Magnetic resonance spectroscopy (MRS) has been widely used in medical diagnosis due to its advantage to provide diagnostic information about the biochemical content of the human tissues in a noninvasive manner. Among the several processing steps (in’t Zandt et al., 2001) applied to the MRS data, the quantification of the metabolites presented into the retrieved spectrum constitutes a significant and challenging scientific research field. The quantification of the metabolites’ concentrations is achieved by measuring the area under their peaks presented into the MRS spectrum.

Depending on the processing domain where the quantification algorithms are applied, they are classified into two categories: time-domain (Vanhamme et al., 2001) and frequency-domain (Mierisova and Ala-Korpela, 2001) methods. While in the time-domain algorithms, the acquired signal (MRS) is processed in its physical domain, in the frequency domain, the signal is transformed into its frequency spectrum by applying a Fourier Transform (FT). Although the proposed methodology is independent of the processing domain, the overall technique is described in the time domain.

In this work, a novel technique that encounters the quantification procedure of in vivo MRS metabolites as an optimization problem, which is solved by using a simple Genetic Algorithm (GA), is proposed (Weber et al., 1998). The parallel nature of the GA gives it more chances to converge in a global optimum at the expense of speed. Moreover, evolutionary optimization permits the usage of objective minimization functions without the requirement to be differentiable and allows prior knowledge of the metabolite properties (such as the frequency and phase of each metabolite peak) to be easily incorporated into the overall quantification procedure.

Mainly, there are three open issues to cope with regarding the quantification of the metabolites by using evolutionary optimization: (1) the high overlapping of the metabolites’ peaks that makes the separation of these peaks difficult, (2) the low convergence rate of the GA as the number of unknowns increases, due to high complex search space and (3) the low performance of the GA in noise conditions, where the presence of a noisy signal’s samples increases the search space virtually.

As far as the first issue is concerned, our previous studies (Papakostas et al., 2009, 2010a, 2010b) have shown that the GA is able to separate the peaks satisfactorily, and its performance is highly dependent on the noise level of the acquired MRS signal.

On the other hand, the capabilities of the GA to find a global optimum set of peak parameters that are necessary to quantify them decrease as the unknown variables are increased. One advantage of the GA-based quantification is that it permits the easy incorporation [in the form of constraints in the fitness function definition (Papakostas et al., 2010a)] of any prior knowledge into the quantification procedure. In this way, the number of unknowns can be reduced by boosting the overall performance of the GA.

This phenomenon of the high complex search spaces becomes more difficult in noisy conditions. The presence of noisy samples further increases the complexity of the search space since the correlation of the desired approximated signal with the noisy one is significantly lower.

However, while the incorporation of as many as possible prior knowledge enforces the GA based optimization procedure, this amount of knowledge is not always available. In this case, the performance of the GA-based quantification is quite poor.

The main contribution of this chapter focuses on the increasing of the GA quantification performance, by alleviating the influence of the noise with the use of less prior knowledge simultaneously. This is achieved by applying an additional processing stage of noise removal and signal smoothing. For this purpose, along with a wavelet denoising by applying discrete wavelet transform, the signal separation by Singular Value Decomposition (SVD) analysis is also examined (Zhu et al., 2003). These two processing steps are investigated separately and in a combinational fashion by formulating a novel, two-stage evolutionary quantification technique with improved performance.

The organization of this chapter is as follows. The main processing steps of the proposed methodology are analyzed in detail in section 2. An extensive study regarding the performance of the introduced method is described in section 3. Finally, the main conclusions are summarized in section 4.

2 Proposed methodology

In this section, a model-based quantification method that tackles the problem as a curve-fitting procedure is proposed. The curve-fitting procedure is implemented by considering an optimization problem of finding the model parameter set that best fits the measured curve.

Mainly, three different line shapes are commonly used to model the magnetic resonance (MR) signals—that is, Free Induction Decays (FIDs) (Vanhamme et al., 2001): the Lorentzian, Gaussian, and Voigt model line-shapes. The proposed quantification methodology considers the first line-shape model, while all the processing is taking place in the time domain, although it also can be applied with the other line shapes and in the frequency domain.

An MRS signal is composed of a set of metabolites that are exponentials, and its description according to the Lorentzian line shape is as follows:

$y_{MRS}\left(t\right)={\displaystyle \sum _{k=1}^K{a}_k{e}^{j{\phi }_k}{e}^{\left(-d_k+j2\pi f_k\right)t}},$

where K represents the number of different resonance frequencies. Moreover, α_k is the amplitude, d_k the damping factor, f_k the frequency, and φ_k the phase of the kth peak component.

The quantification procedure includes the determination of the parameter set {α_k, d_k, f_k, φ_k} for each peak comprising the MRS signal. Under these circumstances, the quantification procedure can be described as an optimization problem according to the following definition:

In this study, this optimization problem is addressed by the usage of a GA, the main operational principles of which are analyzed hereafter.

2.1 Methodology description

The proposed methodology consists of two distinctive processing stages, each having a specific contribution in achieving the target, which is the accurate quantification of the metabolites comprising the MRS signal under process. A block diagram of the introduced method is illustrated in Figure 30.1.

Based on this diagram, the acquired MRS signal is preprocessed initially, before being inserted to the main quantification mechanism of the method that gives the final solution (set of model parameters).

The two processing stages of the method, along with their specific subfunctions, are described in the next sections.

2.2 Stage 1: MRS signal preprocessing

The main role of this preprocessing stage is the smoothing of the analyzed MRS signal by applying wavelet denoising, SVD signal separation, or both. This stage is of major significance since it aims to reduce the influence of the noise to the quantification procedure by providing the second stage (GA quantification) with a signal that is very close to the desired line-shape model. In this direction, three different approaches are studied, as depicted in Figure 30.2.

Initially, a typical wavelet denoising procedure is applied on the real part of the time-domain MRS signal, while in the second approach, the SVD signal separation is used to discard the noisy singular values. Finally, these two methods are combined in a back-to-back operation to improve the final noise reduction.

It is worth mentioning that the order the wavelet denoising and SVD separation are applied is important. The former task gives a description of the signal in many frequency bands, and therefore, the denoising in the wavelet domain can remove more high-frequency components of the signal (and thus noise), while the coarse characteristics of the signal remain unchanged. On the other hand, SVD separation gives a more coarse description of the signal, so a possible first application of SVD could remove useful information of the signal and not as much noise. In the proposed scheme, the wavelet denoising removes many noisy components, and then, in a second phase, the SVD further smooths the already-denoised MRS signal.

2.2.1 Wavelet denoising

Wavelet analysis constitutes an advanced signal processing tool that enables the breaking up of a signal into shifted and scaled versions of the base wavelet, called mother wavelet. This description has the advantage of studying a signal on a time-scale domain by providing time and frequency (there is a relation between scale and frequency), which both are useful pieces of information about the signal, simultaneously.

The procedure of decomposition (analysis) into several resolutions and reconstruction (synthesis) of a signal f (MRS signal in this case) can be described by Eqs. (30.2) and (30.3), respectively:

$f\left(t\right)={\displaystyle \sum _k{u}_{j_0,k}\cdot {\phi }_{j_0,k}\left(t\right)}+{\displaystyle \sum _{j=j0}^{\infty }{\displaystyle \sum _k{w}_{j,k}\cdot {\psi }_{j,k}\left(t\right)}}$

  (30.2)

$u_{j,k}=W_{\varphi }\left(f\left(j,k\right)\right),w_{j,k}=W_{\psi }\left(f\left(j,k\right)\right),$

  (30.3)

where u_j,k, w_j,k are the scaling and wavelet coefficients, respectively; j,k are indices of the translation and dilation parameters; j₀ represents the coarsest scale; and W_ψ, W_φ are the mother wavelet (ψ) and scaling function (φ) wavelet transforms of the signal, defined as

$W_{\psi }\left(f\left(k{2}^{-s},2^{-s}\right)\right)=2^{s/2}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f\left(t\right)\cdot \psi \left(2^{s}t-k\right)\mathrm{d}t}$

  (30.4)

The scaling and wavelet coefficients are considered as the coefficients of a low-pass (signal approximation—low-frequency components) and a high-pass (signal details—high-frequency components) filter, respectively. These filters are the part of the quadrature mirror filter (Strang and Nguyen, 1997) that describes the one-level decomposition and reconstruction of the signal. The decomposition process can be iterated by decomposing the approximations of each level to lower resolution components, a procedure known as multiresolution analysis (MRA).

Wavelet denoising applied in this research includes the one-level decomposition of the real part of the MRS signal by using the one-dimensional (1D) discrete wavelet transform (DWT), subject to a specific mother wavelet, and the thresholding of the detail coefficients by applying soft thresholding (Donoho, 1995) according to the following formula:

${\hat{c}}_i=\left\{\begin{array}{cc}\hfill \mathit{sign}\left(c_i\right)\left(\left|c_i\right|-thr\right)\hfill & \hfill ,\left|c_i\right|>thr\hfill \\ \hfill 0\hfill & \hfill ,\left|c_i\right|\le thr\hfill \end{array}\right.$

  (30.5)

where c_i is the ith detail coefficient, ĉ_i is its compressed version, and thr the threshold. Soft thresholding is an extension of hard thresholding, which first sets to zero the elements whose absolute values are lower than the threshold, and then shrinks the nonzero coefficients toward 0.

The remaining coefficients are used to reconstruct the initial MRS signal by applying the 1D inverse discrete wavelet transform (IDWT). In this way, the noise components, which affect the detail part of the signal (high-frequency components) are discarded.

The wavelet denoising procedure can be described as follows:

Step 1: Apply one-level decomposition (1D DWT) on the real part of the MRS signal

Step 2: Apply soft thresholding to the detail coefficients

Step 3: Reconstruct the MRS signal by applying 1D IDWT

It must be noted that according to the MATLAB implementation (http://www.mathworks.com/) used in this study, the threshold used in step 2 is equal to

$thr=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|c_i\right|},$

  (30.6)

or if this value is 0, the threshold is set to

$thr=0.05\times max\left\{\left|c_i\right|\right\},i\in \left[1,N\right],$

  (30.7)

where c_i is the ith detail coefficient.

After the denoising of the MRS signal using the previous procedure, the signal is expected to be smoother, with reduced noisy components able to be quantified more accurately by the GA of the second stage.

It is noted that the prescribed wavelet denoising procedure is applied by using the MATLAB implementation, consisting of the functions dencmp() and wdencmp().

2.2.2 SVD signal separation

SVD has been applied successfully in MRS (Pijnappel et al., 1992; Stamatopoulos et al., 2009), to separate the signal into MRS and noise components. According to the SVD decomposition for any matrix $\mathbf{A}\in R^{m\times n}$, it takes the following form:

$\mathbf{A}\mathbf{=}\mathbf{US}{\mathbf{V}}^{\mathbf{T}},$

  (30.8)

where U is an $m\times n$ orthogonal matrix, S an $n\times n$ diagonal matrix, and V an $n\times n$ orthogonal matrix. The diagonal values of S are called singular values of A and correspond to the square roots of the eigenvalues of A^TA and AA^T.

In this work, the application of the SVD analysis aims to separate the MRS signal into the pure MRS signal and its noise component based on the following formula (Stamatopoulos et al., 2009):

$\begin{array}{c}{\mathbf{A}}_{MRS}={\mathbf{A}}_{MRS\mathit{signal}}+{\mathbf{A}}_{MRS\mathit{noise}}\\ ={\displaystyle \sum _{m=1}^M{u}_m{s}_m{v}_m^T}+{\displaystyle \sum _{m=M+1}^N{u}_m{s}_m{v}_m^T},\end{array}$

  (30.9)

where N is the number of the singular values of the A matrix. In this representation, it is assumed that the original part of the MRS signal is described by the first M (with the higher value) components, while the other value, N − M, corresponds to the noise components of the acquired MRS signal.

As it can be seen from this discussion of the SVD analysis, this procedure is applied to a matrix representation of a signal. Therefore, its application directly on the time-domain MRS is not possible, and an intermediate transformation of the signal to a matrix form has to be performed first.

For this purpose, a matrix consisting of the coefficients derived by applying a continuous wavelet transform (CWT) on the MRS signal for various scales and positions (dilations and translations) is constructed. This signal representation is in some sense the wavelet power spectrum (the squared absolute values |.|² of the coefficients) of the MRS signal over the time and for different scales and is computed as follows:

$C\left(s,p\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{y}_{MRS}\left(t\right)\psi \left(s,p\right)dt},$

  (30.10)

where s and p are the scale and position, respectively.

Advantages of this signal representation include that it captures the time and frequency variations of the signal and it permits the reconstruction of the initial image by applying the inverse continuous wavelet transform (ICWT). The reconstruction of the signal by its wavelet coefficients is necessary since after the rejection of the noisy singular values (m > M) and the SVD composition of the modified C^mod(s,p) coefficients, the denoised MRS signal has to be reconstructed in order to be quantified during the second stage of the methodology.

Moreover, the determination of the M value usually is performed empirically, by a trial-and-error procedure. However, in the proposed methodology, an automatic procedure for the determination of the number of the singular values that better describe the useful part of the MRS signal is also applied. The number of singular values that correspond to the useful part of the MRS signal is defined by applying cluster analysis on the overall singular values derived by the SVD process.

The well-known k-means (Kuncheva, 2004) clustering algorithm is used in order to group the singular values of the acquired MRS signal into two separate clusters: one with the useful signal and the other with the noise. Those singular values (M values) that belong to the signal cluster is used in the first summation of Eq. (30.9), while the remaining values (N − M) are used in the second summation of the same equation. The SVD signal separation procedure is shown in detail in Figure 30.3.

The SVD signal separation procedure described in this section is applied on the two approaches of the first stage, illustrated in Figure 30.2. While the role of this procedure is clearly defined in the second approach (signal denoising), its usage in combination with the wavelet denoising in the third approach aims to improve the accuracy of the k-means clustering, and therefore the fidelity of the separated signal. The produced denoised MRS signal is then fed to the second stage of the GA-based quantification process.

2.3 Stage 2: GA quantification

As already mentioned in the previous section, the MRS metabolites quantification procedure can be defined as an optimization problem. In order to solve it, a simple GA is used to create a possible solution by applying repetitively a set of genetically inspired operators.

These operators try to mimic the process that characterizes the evolution of living organisms (Holland, 2001). This theory is based on the mechanism of survival of the fittest individuals in a population. In fact, some specific procedures are taking place until the predominance of the fittest individual.

In the sequel, the used terminology in the field of genetic methods for optimization and searching purposes is given (Coley, 2001):

1. Individual (chromosome) is a solution of a problem satisfying the constraints and demands of the system in which it belongs.

2. Population is a set of candidate solutions of the problem (chromosomes), which contains the final solution.

3. Fitness is a real number that characterizes any solution and indicates how appropriate the solution is for the problem under consideration.

4. Selection is an operator applied to the current population, in a manner similar to the one of natural selection found in biological systems. The fitter individuals are promoted to the next population, and poorer individuals are discarded.

5. Crossover is the second operator that follows Selection. This operator allows solutions to exchange information in such a way that the living organisms use in order to reproduce themselves. Specifically, two solutions are selected to exchange their substrings from a single point and afterward, according to a predefined probability (Pc). The resulting offspring carry some information from their parents. In this way, new individuals are produced, and new candidate solutions are tested in order to find the one that satisfies the appropriate objective.

6. Mutation is the third operator that can be applied to an individual. According to this operation, its single bit of an individual binary string can be flipped with respect to a predefined probability (Pm).

After the application of these operators to the current population, a new population is formed and the generational counter is increased by 1. This process will continue until a predefined number of generations are attained or some form of convergence criterion is met.

While the incorporated genetic operators are almost the same for each application, where a GA is applied to solve an optimization problem, the module of Fitness Calculation is application dependent and needs particular formulation. For the needs of metabolite quantification, the fitness of each candidate solution, which corresponds to a model parameter set, is measured by comparing the constructed MRS signal with the real acquired signal by means of a predefined objective function.

The general form of the chromosome coding the parameters sets of k metabolites peaks used in the case of MRS quantification is depicted in Figure 30.4.

The fitness function used to measure the strength of each candidate solution of a current population has the following form:

$\mathit{fitness}={\displaystyle \sum _{i=1}^N{\left(F\left(i\right)-\hat{F}\left(i\right)\right)}^2}$

  (30.11)

where N is the number of the MRS signal’s samples, and F and $\hat{F}$ the original and estimated MRS signals, respectively. Taking into account that the GA usually performs the minimization of an error function, the chromosome with the lowest fitness according to Eq. (30.11) is the optimal solution of the problem. Besides the line-shape model parameters that need to be optimized according to this definition, the number of peaks is also unknown and has to be found. To this end, the GA is applied iteratively in an early stage for different numbers of peaks (e.g., 1–20) and selecting that number that shows the lowest final fitness value (Papakostas et al., 2009, 2011).

3 Experiment

In order to investigate the performance of the proposed quantification methodology, a set of appropriate experiments was arranged. For the experimental purposes, specific software was developed in MATLAB, while all experiments were executed on an Intel i5 3.3GHz PC with 8 GB random access memory (RAM).

For experimental purposes, an artificial MRS signal was used; the MATLAB source code that generated it came from http://www.esat.kuleuven.be/sista/members/biomed/data005.htm. The signal consisted of 11 exponentials derived from a typical in vivo ³¹P spectrum measured in a human brain. The ³¹P peaks from brain tissue, phosphomonoesters, inorganic phosphate, phosphodiesters, phosphocreatine, gamma-ATP, alpha-ATP, and beta-ATP were presented in this simulation signal. The time-sampling interval is 0.333 ms, and the number of samples in the signal is 256. The real and imaginary parts of the artificial MRS signal are depicted in Figure 30.5, while its corresponding frequency representation (amplitude spectrum) is derived by applying the Fourier transform and takes the form shown in Figure 30.6.

It is worth mentioning that, without loss of generality, the quantification methodology proposed in this work is making use of the real part of the MRS signal. However, the methodology can be applied successfully to the complex or imaginary signals unchanged. Moreover, the signal is processed in the time domain, although frequency-domain processing also can be considered.

The parameters of the 11 peaks comprising the artificial MRS signal of Figure 30.5 are summarized in Table 30.1.

In order to investigate the quantification performance of the proposed methodology under the three examined approaches shown in Figure 30.2, two different experimental scenarios were considered. In the first scenario, the parameters of the peaks were almost completely known, while in the second, there was only a little information about them.

The settings of the GAs for the two scenarios are summarized in Table 30.2.

Note that these settings were not optimal. Rather, they were found by trial and error, and therefore the following results can be improved significantly by applying a more sophisticated calibration procedure. Furthermore, the quantification accuracy of each studied method is measured through the relative error (RE) in percent, defined as

$RE\left(\%\right)=\frac{\left|V_{Est}-V_{\mathit{Act}}\right|}{V_{\mathit{Act}}}\times 100$

where V_Act is the actual and V_Est the estimated value of the parameter V, respectively.

In order to minimize the influence of the GA’s randomness on the experimental results, each experiment has been executed 10 times and the mean values of them are presented. Moreover, 10 different levels [having a standard deviation (Stdv) from 5 to 50 with step 5] of Gaussian noise have been applied to the original MRS signal to construct noisy signals.

3.2 Scenario 2: Limited prior knowledge

In the second experimental scenario, only limited prior knowledge is considered [namely, the frequencies and phases (f, φ)], while the other two parameters [namely, the amplitude and damping factor (α_k, d) of each peak] are unknown. The additional unknown parameter (d) causes an increase of the chromosomes’ length by a factor of 2. Therefore, the population size, mutation probability, and generation number of the algorithm need to be increased, as shown in Table 30.2.

3.2.1 Performance of the first stage

However, before studying the performance of the each incorporating the denoising preprocessing stage, it is useful to investigate the denoising operation itself. The performance of each denoising approach (wavelet, SVD, wavelet+SVD), measured by the signal-to-noise ratio (SNR) in decibels for different noise levels, is summarized in Table 30.4.

Table 30.4

Performance of Each Denoising Method in Terms of SNR (dB)

Signal	Gaussian Noise (Stdv)
	5	15	25	35	45
Noised	0.008	− 0.015	− 0.087	− 0.206	− 0.369
Wavelet	0.042	0.089	0.107	0.093	0.053
SVD	0.088	0.079	− 0.089	− 0.179	− 0.244
Wavelet + SVD	0.121	0.055	0.069	0.105	0.060

From all this, it can be deduced that all the denoising approaches improve the SNR of the MRS signal, since they remove sufficient noisy components. Of the three methods, the wavelet + SVD approach shows the best results in almost all the noise levels, behavior that helps the fitting mechanism of the second stage.

3.2.1.1 Overall Performance

The optimization problem that the algorithm is required to solve in this case is more difficult, also due to the noisy conditions. In this scenario, the preprocessing stage of the proposed algorithm is incorporated, and the performance of the three possible approaches is investigated. In order to understand the difficulty of this optimization problem, let us look at Tables 30.5 and 30.6, which show the performance of the single GA-based quantification procedure without the preprocessing stage.

Table 30.5

Quantification Results (% RE) of the Single GA When the Unknowns Are (α_k, d_k)—Values of (α_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	21.7	51.6	38.3	26.0	10.3	11.3
2	31.6	69.2	38.2	19.0	1.3	7.2
3	24.5	52.9	36.4	27.7	7.6	16.4
4	5.1	5.7	0.8	1.4	6.3	12.8
5	1.1	0.3	1.9	2.1	3.1	1.9
6	3.0	0.1	19.7	20.3	36.3	35.8
7	0.1	0.9	12.3	10.9	17.6	21.5
8	3.8	4.8	10.6	15.1	21.8	25.3
9	25.1	13.4	9.9	16.8	9.9	30.0
10	1093.5	491.0	240.2	482.5	10.5	939.7
11	54.5	23.1	8.9	17.5	12.3	38.0

Table 30.6

Quantification Results (% RE) of the Single GA When the Unknowns Are (α_k, d_k)— Values of (d_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	15.4	38.5	31.0	26.1	21.5	24.6
2	25.7	57.8	32.2	14.3	3.6	4.9
3	19.7	40.0	28.2	26.1	14.5	24.8
4	4.8	4.7	0.5	1.5	2.5	8.0
5	2.0	1.7	4.4	6.7	3.5	7.2
6	2.9	1.2	21.6	24.9	40.7	40.9
7	0.2	0.2	10.2	10.6	16.7	22.0
8	4.0	6.2	15.4	21.2	32.0	37.9
9	6.8	4.9	5.1	9.1	10.3	16.2
10	1474.2	733.3	541.5	745.2	24.0	1548.8
11	24.0	8.9	1.4	3.4	8.6	7.9

From these tables, it can be recognized that the performance of the single GA has been degraded with the addition of an extra unknown parameter for each metabolite peak. The main difficulty of the algorithm is not only the noisy conditions, as in the case of the first scenario, but the large search space formed by the number of the unknown parameters. This is justified by the contents of the first column of Table 30.5, which corresponds to the noise-free case, showing that the derived amplitudes are far enough from the desired values.

As far as the peaks’ damping factors derived by the single GA method are concerned, the accuracy is better than those of the amplitudes, as can be concluded by looking at Table 30.6.

The corresponding quantification results for the case of the first approach of implementing the second stage of the proposed methodology are summarized in Tables 30.7 and 30.8.

Table 30.7

Quantification Results (% RE) of the First Approach (Wavelet Denoising) When the Unknowns Are (α_k, d_k)—Values of (α_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	50.8	41.2	60.1	21.9	41.9	17.6
2	63.1	57.5	72.9	23.7	37.3	8.6
3	52.5	42.7	61.3	16.9	42.7	19.7
4	4.1	2.3	5.4	1.8	7.8	12.2
5	7.2	6.0	0.1	5.2	0.3	5.5
6	1.7	1.0	14.9	20.1	36.5	38.1
7	0.8	1.6	8.3	13.4	23.7	19.3
8	2.9	5.9	9.3	14.7	21.4	25.0
9	11.3	7.3	5.6	22.5	25.2	14.5
10	437.8	194.7	0.3	770.5	736.2	141.2
11	21.1	11.2	0.0	44.6	42.0	14.7

Table 30.8

Quantification Results (% RE) of the First Approach (Wavelet Denoising) When the Unknowns Are (α_k, d_k)—Values of (d_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	34.3	32.2	42.9	1.3	38.5	27.6
2	52.9	47.5	64.1	24.1	32.7	6.0
3	35.7	32.0	42.7	1.2	36.4	26.1
4	3.5	2.2	3.4	0.1	4.2	6.3
5	6.0	5.5	3.1	9.1	6.3	3.5
6	1.5	1.8	18.3	24.7	40.7	43.4
7	1.0	0.5	6.8	13.0	22.9	18.9
8	3.9	8.1	12.4	21.1	31.0	36.5
9	3.3	2.8	5.5	8.3	12.3	11.3
10	697.2	433.4	3.3	1348.5	1355.0	368.6
11	6.5	0.3	2.0	13.6	10.1	2.3

By comparing the performance of the wavelet denoising (first approach) with that of the single GA quantification, it can highlight that it outperforms the previous one. The wavelet denoising helps the searching mechanism of the GA by reducing the influence of noise. In this way, more accurate peak parameters (amplitudes and damping factors), near the desired values, are found by the quantification stage of the proposed methodology. For example, while the mean RE values of computing α_k with the single GA method are 64.83% and 103.63% for the 5 and 45 noise levels, respectively, the corresponding errors of the first approach are 33.74% and 28.77%—significantly less.

The quantification results of the second approach, presented in Tables 30.9 and 30.10, are better than those of the wavelet denoising for the low noise levels. For example, the mean absolute error values (of α_k) for the 5 and 45 noise levels are 19.71% and 69.66%, respectively.

Table 30.9

Quantification Results (% RE) of the Second Approach (SVD Signal Separation) When the Unknowns Are (α_k, d_k)—Values of (α_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	61.9	51.1	26.7	17.5	18.7	1.2
2	58.7	46.5	9.6	2.3	8.2	9.8
3	57.3	49.1	18.1	12.3	16.3	2.1
4	4.0	2.1	0.8	3.5	9.1	17.7
5	2.1	1.5	2.0	3.9	5.5	11.5
6	4.1	3.8	6.9	11.4	17.5	33.5
7	16.1	14.5	3.4	0.2	9.7	23.7
8	8.9	10.0	14.7	21.5	27.5	29.9
9	11.3	8.3	14.0	10.6	18.9	21.8
10	202.5	27.7	296.3	0.0	341.7	504.0
11	13.5	2.4	12.4	11.1	3.6	12.1

Table 30.10

Quantification Results (% RE) of the Second Approach (SVD Signal Separation) When the Unknowns Are (α_k, d_k)—Values of (d_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	45.5	37.9	29.7	29.1	35.0	28.3
2	44.1	36.1	4.8	0.8	4.3	9.9
3	33.2	26.5	11.7	14.8	23.3	23.0
4	3.6	3.7	4.8	5.9	8.0	12.3
5	2.4	4.1	4.2	6.8	7.7	5.0
6	6.3	4.8	8.0	15.2	24.0	40.6
7	8.6	6.5	2.3	6.6	15.2	25.8
8	10.9	12.9	17.6	25.2	33.3	37.7
9	8.1	8.2	7.3	9.1	10.8	10.9
10	314.2	20.1	519.2	8.7	695.8	1350.0
11	14.0	6.8	7.6	1.8	3.2	9.2

It is worth pointing out that wavelet denoising shows better performance when the noise level is generally increased, as compared to the second approach. This means that the SVD signal separation discards fewer noisy signal’s components than the first approach. On the other hand, in conditions of low noise, the first approach seems to reject useful information, where the SVD is more accurate.

The third approach, consisting of the back-to-back operation of wavelet denoising and SVD signal separation, tries to make better use of their advantages by combining them. The wavelet denoising is applied first, in order to remove the noisy components of the MRS signal, so the SVD can work more accurately by keeping the useful parts of the signal. The less noisy the signal that is guided to the SVD module, the more accurate is the procedure (k-means clustering) of finding the parameter M of SVD decomposition. Tables 30.11 and 30.12 summarize the quantification results of the third approach.

Table 30.11

Quantification Results (% RE) of the Third Approach (Wavelet Denoising Followed by SVD Signal Separation) When the Unknowns Are (α_k, d_k)—Values of (α_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	29.1	38.8	24.0	19.7	2.5	6.3
2	16.1	27.7	10.1	12.4	12.7	19.3
3	18.8	35.2	16.0	16.5	2.1	6.4
4	6.5	6.3	2.8	12.3	9.2	16.5
5	3.7	5.5	2.3	9.8	5.9	11.7
6	0.6	3.6	10.9	13.5	18.4	27.1
7	10.9	12.1	0.3	1.0	4.9	8.7
8	7.0	8.9	14.5	19.9	27.1	34.3
9	7.2	15.1	9.4	11.5	17.9	16.4
10	42.5	429.3	21.8	13.5	239.8	7.0
11	1.0	24.1	1.2	1.2	15.6	0.8

Table 30.12

Quantification Results (% RE) of the Third Approach (Wavelet Denoising Followed by SVD Signal Separation) When the Unknowns Are (α_k, d_k)—Values of (d_k)

Peak (k)	Gaussian Noise (Stdv)
	0	5	15	25	35	45
1	23.6	31.3	27.2	29.6	24.3	22.9
2	8.6	18.9	4.7	7.6	15.4	21.8
3	5.3	20.8	11.0	18.3	13.4	17.2
4	0.5	1.0	6.3	13.1	8.1	13.9
5	3.8	7.8	3.1	0.2	5.5	2.9
6	3.0	3.9	12.0	16.5	23.2	33.7
7	5.2	6.2	4.8	7.0	11.3	13.9
8	8.9	10.6	18.8	25.7	34.8	44.9
9	7.2	6.8	8.7	10.3	12.3	15.5
10	37.2	605.3	14.3	4.0	458.9	9.9
11	8.7	15.4	6.4	5.1	7.4	4.5

In order to better understand the impact of the proposed methodology under the several approaches examined in this experiment, the data from Tables 30.5–30.12 have been combined and displayed in Figures 30.7–30.9.

Figures 30.7 and 30.8 illustrates the mean RE in percent of the proposed methodology compared with the single GA quantification method for the case of the amplitude (a_k) and damping factor (d_k) parameters, respectively. From these figures, it can be concluded that the proposed methodology in all three approaches significantly improves the searching capabilities of the GA, by converged to the best peaks’ parameter sets.

Moreover, the third approach, consisting of the combinative operation of the wavelet denoising and the SVD signal separation, shows the best performance in almost all the noise levels. This can be deduced by studying the mean RE in percent for both parameters (amplitudes and damping factors) of the methods, as depicted in Figure 30.9.

Apart from the previous quantification results, a Monte Carlo study with many different noise realizations can construct a more statistically accurate image of the methods’ performances. However, due to computation limitations coming from the high overhead of the GA operation, a Monte Carlo analysis with hundreds or millions of noise realizations is computationally difficult. Therefore, the 10 noise levels of the previous experiments (having Stdv from 5 to 50, with step 5) can be used to perform an analysis in this framework. Some statistical results (i.e., mean, Stdev) of this analysis are summarized in Table 30.13.

Table 30.13

Statistical Results (Mean/Stdv) of Monte Carlo Analysis Quantifying the 11 Peaks for the Studied Approaches

Peak (k)	Method
	Single GA	Wavelet	SVD	Wavelet + SVD
1	29.2/14.5	35.3/19.4	79.3/75.7	62.8/68.8
2	26.9/20.8	44.5/28.3	32.6/27.0	26.7/20.5
3	29.2/14.0	35.5/20.3	34.7/22.3	29.3/24.1
4	5.9/4.8	5.6/4.1	12.2/9.8	16.2/11.8
5	4.3/3.1	5.0/3.0	13.6/14.6	11.1/10.9
6	23.2/17.1	23.5/16.3	21.1/12.8	15.7/9.6
7	12.4/8.5	12.4/9.0	11.8/9.2	9.5/8.0
8	21.1/11.7	18.5/10.9	23.1/10.2	23.2/11.2
9	14.5/6.1	11.0/5.5	14.6/5.4	14.4/5.4
10	686.7/476.3	444.1/417.7	212.4/297.1	106.0/168.0
11	15.9/11.4	11.9/10.5	8.4/4.1	11.4/8.6
Mean	79.0/53.5	58.8/49.6	42.2/44.4	29.7/31.5

By examining these results, the superiority of the wavelet + SVD methodology can be confirmed via the statistics. Moreover, it is worth pointing out that of the 11 peaks of the simulated MRS signal, the 10th peak was not quantified with any of the approaches. A brief look at the peaks’ locations (Figure 30.6) shows that the 10th peak was the highest, with two other peaks being very close. This means that all the approaches have difficulty identifying this peak due to the presence of other peaks with almost the same amplitude and frequency together.

From the precedent analysis, it can be concluded that the low quantification performance of the single GA-based quantification technique, for limited prior knowledge and noisy conditions, as proposed in previous studies (Papakostas et al., 2009, 2010a, 2010b), can be improved significantly by incorporating an additional processing stage (Figure 30.1 and 30.2). Among the three approaches investigated in the context of the proposed two-stage quantification methodology, the third one, consisting of the combination of wavelet denoising with SVD analysis, shows the best overall performance. The proposed methodology of the third approach can reduce the mean RE in finding the (α_k,d_k) parameters of the artificial signal by 50% over the performance of the single GA method.

Although the introduced methodology exhibits improved quantification capabilities, further research has to be performed in order to establish it as a reliable MRS metabolite quantification procedure. The optimization of wavelet denoising by finding the most appropriate wavelet family and level of decomposition (the coiflet of an order 3 wavelet with one-level decomposition was used in our experiments), along with the studying of other clustering algorithms (such as SOM clustering and Fuzzy c-means), are some of the future research issues that have to be studied.

Furthermore, future research issues include the redesign of the proposed methodology to involve concurrent and distributed GAs (Adeli and Cheng, 1994; Adeli and S. Kumar, 1995; Hung and Adeli, 1994) to be applied in magnetic resonance spectroscopy imaging (MRSI), as well as investigations of how to extract rules using specific GA techniques (Ballesteros, 2010) in the context of MRS and MRSI. Finally, instead of uniform sampling in the definition of fitness in Eq. (30.11). it might be advantageous to investigate hybrid sampling techniques (Seiffert et al., 2009) in metabolite quantification.

4 Conclusions

A novel MRS metabolites quantification methodology was proposed in this chapter. The introduced method consists of two processing stages: one for the preprocessing of the MRS signal and the other for the peaks’ parameter optimization. It has shown an improved performance compared with the single GA method. The experimental results on artificial data testified that the combination of wavelet denoising and SVD signal separation in the preprocessing stage of the technique can lead to significant improvement in the quantification procedure under noisy conditions.

While the preliminary results are very promising, further research on the directions of enhancing wavelet denoising and SVD analysis, along with the application of the method in real data, is needed in order to establish it as a generic quantification procedure.