MREIM Simulation Algorithm

Figure 59 shows a block diagram representing the MREIM simulation algorithm. The algorithm of the simulation consisted of the following steps:

Step 1: Input of Parameters. The following parameters were constant in MREIM simulation:

• N - number of PE and FE steps:

- phase encode gradient strength incrementation p and time incrementation t, N = Np=

Nt= 128;

- spatial x and y incrementation k, L = 4 · N = 512; • F OV - Field of View and dx - pixel resolution:

- dx = 1 mm, which for 128 pixels has fixed the field of view, F OV = N · dx = 12.8 cm; • T R = 2000 ms - repetition time;

• T E = 50 ms - echo time;

• G_xmax = 22 mT/m – maximum PE gradient strength.

- Phase Encode step can be found as ∆Gy = 2G_Nxmax_p−1 . Time of PE gradient application is

linked to the PE gradient strength as τ = 1

γ·∆Gy·f ov.

Variable parameters:

• f - signal generator driving frequency. Generator driving frequency was varied over the range of 20–1000 Hz considering the following constraints:

- suggested frequency from the TransScan TS-2000 performance (200 Hz),

- low frequency range due to the use of real component of conductivity (α region), - use of conductive coupling for the phantom-FSs system,

- from Eqs. (6.58) and (6.65), it follows that the aberrational term is larger at lower generator frequencies.

To observe the shift due to PE effect at T R = 2 s, the generator frequency is required to be fractional, ∆ypix = {f } · T R · Np (see Section 6.2.2). Generator frequency was

linked to frequency resolution df to obtain the shift due to FE effect, ∆xpix= _dff ;

- objective: to investigate generator frequency influence on producing the most conspic- uous MREIM effect for SDSE sequence for the parameters specified above;

• df - frequency resolution (bandwidth per pixel):

- df was varied to optimize the FE and PE effects. Frequency resolution is proportional to frequency gradient strength, df = γGxdx, and inversely proportional to signal-to-noise

ratio (SNR). Since we are interested in both stronger perturbation of native MR image (at weaker encode gradients) and high signal-to-noise ratio, df is preferred to be in a lower range.

- objective: to determine the frequency resolution at which the MREIM is the most conspicuous for the specified parameters;

• i - applied current density:

- experimental values: 100–170 mA for the contact area of 10×10 cm2_{, which is equivalent}

to 10–17 A/m2_;

- objective: to determine the minimum value of applied current for the minimum MRM detectable tumor (R = 1.5 mm) with the borderline conductivity ratio (σratio= 3);

• R - radius of spherical tumor:

- radius of a cancer surrogate in the breast phantom was R = 5 mm, which was used as default for MREIM effect study;

- R = 2.5 mm and R = 1.5 mm were used to find the minimum applied currents for MREIM effect detectability;

• contrast - initial contrast between the higher conducting and lower conducting medium: contrast = Sin−Sout

Sout ·100%, where Sinand Soutare the pixel intensities inside and outside

the higher conducting region, respectively:

- based on experimental images, contrast between the tumor surrogate and surrounding agar medium was approximately 10%.

- objective: to investigate MREIM signal detectability dependence on initial tumor contrast.

Step 2: Template Image Generation.

The template image for a simple tumor model of a higher conducting sphere embedded in a lower conducting homogenous medium was generated for the 2-D slice at z = 0, with the higher conductive disk (assuming slice thickness) in the center. The values for pixel intensities inside and outside the sphere, S_in and S_out, were assigned based on the experimental results to produce contrast = 10%. Random Gaussian noise with zero mean and standard deviation σ_SD = 2 (by default) was added to the template image intensity image to make the simulation realistic.

Step 3: Three Loops.

The realization of Eq. (6.54) is performed through the iteration over three variables: p - phase incrementation, t - time incrementation, and k - space incrementation. For each discrete value of PE gradient and time, the signal in x, y plane is presented as the sum over the spatial incremented x and y.

Step 4: Calculation of Aberrational Magnetic Field.

For a simple tumor model of a higher conducting spherical tumor embedded into lower conducting medium, the aberrational magnetic field is calculated according to analytical expressions in Eq. (6.34) for each coordinate x and y, with z = 0. The image mask is used to distinguish the regions inside and outside the disk.

For a realistic model of a breast with various tumor shapes and anisotropic conductivity, the numerical solution to aberrational magnetic field (Eq. (6.49)) were incorporated. An algorithm for the numerical field solutions is provided in the following Section.

MREIM SIMULATION

INPUT OF PARAMETERS

N, R, Ratio, dx, df, f, i, TR, TE

TEMPLATE IMAGE GENERATION im(N,N) PHASE INCREMENTATION p = 0, N-1 TIME INCREMENTATION t = 0, N-1 SPACE INCREMENTATION k = 0, N-1 CALCULATION OF ABERRATIONAL MAGNETIC FIELD h

CALCULATION OF PHASE CHANGE DUE TO THE PHASE ENCODE AND ABERRATIONAL MAGNETIC FIELD

phase

CALCULATION OF PHASE CHANGE DUE TO THE FREQUENCY ENCODE AND

ABERRATIONAL MAGNETIC FIELD freq

INTEGRATION OVER SPACIAL VARIABLES x(k) AND y(k) real, imag

SIGNAL GENERATION sig(t,p) = complex(real, -imag)

2-D FAST FOURIER TRANSFORM im_FFT = FFT(sig(t,p)) IMAGE GENERATION im_fin = abs(im_FFT) _ + _ _ + + END

Step 5: Calculation of the Phase Change due to the PE Gradient and Aber- rational Magnetic Field Perturbation in PE.

The phase change consists of three terms: the unperturbed image and two aberrational terms from Eqs. (6.70) and (6.71) scaled by the factor containing the aberrational magnetic field.

Step 6: Calculation of the Phase Change due to the FE Gradient and Aber- rational Magnetic Field Perturbation in FE.

The phase change consists of three terms (Eq. (6.58): the native image and the perturba- tional terms proportional to the aberrational magnetic field.

Step 7: Calculation of the Signal’s Real and Imaginary Parts for Every p and t Value.

Formation of the matrix sig(t, p) based on the summation over the spatial x and y compo- nents. The phase is separated into real and imaginary components. The real and imaginary parts are multiplied by the template image.

Step 8: Application of 2-D Fast Fourier Transform to the Signal.

Step 9: Taking the Magnitude of the Raw Image to Obtain the Final Result.