: Design and Testing of a Radiofrequency Ellipsoidal Helmholtz Coil in Contrast with a … Design and Testing of a Radiofrequency Ellipsoidal Helmholtz Coil in Contrast with a Circular Pair for Nuclear Magnetic Resonance

Al-Snaie


INTRODUCTION
Magnetic Resonance Imaging (MRI) is a non-invasive imaging technique that does not use harmful radiation, unlike other imaging modalities.MRI is based on the phenomenon of Nuclear Magnetic Resonance (NMR), a phenomenon independently discovered by two research teams [1,2].NMR exploits the resonant behavior exhibited by certain materials when placed in a strong magnetic field.These materials selectively absorb electromagnetic energy at a frequency specific to their constituent cores and which varies with the strength of the externally applied static magnetic field.The absorption of electromagnetic energy positions moves the nuclei of the materials in a state of resonance, after which they return to their original state of thermodynamic equilibrium through a process known as relaxation with release of energy.In addition, there are a host of tissue-specific properties that interact with applied magnetic fields, the two most important of which are relaxation longitudinal (spin-lattice) and transverse relaxation mechanisms (spin-lattice).The use of signal coding techniques based on a spatial arrangement of magnetic field gradients allows obtaining information on the internal structure of biological samples.This information carried by the energy emitted by biological tissues, constitutes the basis of the MRI imaging modality [3][4][5].
In MRI, the need to make coils with the most uniform radiofrequency (RF) field B 1 is necessary, hence the use of volume RF coils.Among these volume coils, there are the "birdcage" type coils, with several types of operation and structures.They give good results, but they still have some drawbacks and constraints related to their construction.Indeed, the realization of this type of coils becomes complex because the most homogeneous RF field is obtained with a high number of bars, moreover, the excessive metal mass constituting the base rings of the coils loops out to be a seat of Foucault induced currents and causes disturbances during the switching of the gradients and visible trailing phenomena on the images.Moreover, the homogeneity of the B 1 field and the quality factor depend on the symmetry of the coil, hence the need for more precision at the manufacturing level [6][7][8].
In the case of simpler coils, such as Helmholtz coils which generate a field B 1 along the axis of the coil, the improvement in the homogeneity of the field is clearly greater.Thus, unlike "birdcage" type coils, the optimization of the structure in the absence of periodicity is very significant [7][8][9].
The main objective of this paper is the design and testing of circular and ellipsoidal Helmholtz coils that can produce a relatively, high electromagnetic field homogeneity to be used for NMR applications.

II. CALCULATION OF THE B 1 MAGNETIC FIELD
This section is devoted to the calculation of the field produced by the circular and ellipsoidal structures.The objective is to obtain an RF induction B 1 of better homogeneity so as to produce a homogeneous field of order 2 in the optimum case.The structure has a reduced dimension compared to the wavelength, so it is possible to apply the law of Biot-Savart and the principle of superposition to obtain the total field [9] on their axis [10].The magnetic field of a circular loop of N turns whose plane of the coil is centered at z=+R/2 (field B 1 (z)), is written [11][12] as: while the magnetic field of a loop of N turns whose plane of the coil is centered at y= -R/2 (field B 2 (z)), is: The field on the axis of a circular Helmholtz coil is the combination of the two previous magnetic fields, the superposition theorem which is validated by the linearity of Maxwell's equations gives the total field: We proceed by similarity with a structure consisting of two ellipsoidal loops, as shown in Figure 1.Each single loop has a semi-axial dimension x 1 and y 1 .To analyze this situation, a formula for the field produced around the origin (x = y = z = 0) by any loop traversed by a current I and of declination with respect to Oz is proposed [13][14]: where P nm (cosα) are the associated Legendre polynomials and P n (cosα) are the Legendre polynomials.
For the field on the O z axis, P n (cosθ) =1, and r = z, and the previous equation becomes: The total field is the sum of the elementary inductions.It is possible to use the first terms of development.Thus an appropriate spacing of the two loops allows a better homogeneity of the magnetic field with a cancellation of derivatives of order two of the field B 1 along the axis Oz and fortuitously the derivative of order four is relatively weak.For this, the two loops must be positioned at an azimuthal angle of 0.734 rad (about 42°).There is a unique solution for which the second order derivative of the field B 1 is zero.For this optimal structure (E opt ), the secondary geometric data are shown in Table I.The eccentricity of the ellipsoid is: The value of the magnetic field at any point in the space is the sum of the fields created at this point by each of the single ellipsoidal loops and the approximated expression is [15][16]: with:

III. COMPARISON OF THE B 1 FIELD HOMOGENEITY
The performance in terms of homogeneity of the B 1 field of the ellipsoidal Helmholtz Coil (H E ) are compared with a circular Helmholtz Coil (H C ) of the same radial size (diameter d equal to 5cm). Figure 2 shows the simulation of the theoretical curves of the normalized magnetic induction B 1z produced along the axis Oz by both coils.

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Thus, the theoretical width (absolute or relative) of the profiles can be calculated at 1% and 10% of the maximum.The values are presented in Table II.Compared to the reference H C , the lobe width at 10% of the ellipsoidal coil is increased by 12%.This increase is also greater (20%) for lobe width at 1%.The performance stated above is very general and does not in fact depend on the size of the coil.Indeed, it suffices to apply a simple scale factor compared to the circular and ellipsoidal prototypes to obtain coils of greater or lesser size.

IV. ELECTRICAL MODELING
In this section, the complete electrical modeling of the structures, considering all couplings, is investigated.The complete electrical modeling of the coils is necessary to explain their operation and thereby demonstrate their feasibility.The general electrical model for the developed structures is represented schematically in Figure 3.Because of the symmetry, the input impedance Z can be expressed as [17][18][19]: The two electrically isolated loops are characterized by a set of two equations with two unknowns (I 1 and I 2 ): In fact, two resonance modes are distinguished: a co-current mode with a symmetric behaviour I 1 = I 2 , which is the mode of interest, and a counter-current mode with non-symmetric behaviour I 1 = −I 2 (gradient mode).In co-current mode, the system of (10) comes down to a single equation [20]: ba + ^ _ `a = 0 (11) Equation ( 10) corresponds to the resonance equation: If the coupling coefficient k is introduced (M 12 = k L 1 ), the resonance equation ( 12) becomes: efc + M Jc c g` = 1 (13) Equation ( 13) has a positive real solution [21]: For the counter-current mode (I 1 = −I 2 ), the value of frequency is: The two resonance modes are located on either side of the resonance frequency f 0 (f 0 = 1/(LC) and they are all the more distant as the coupling is strong.A representation of the input impedance amplitude Z e is given in Figure 4. Theoretical curve of the input impedance of two identical tuned circuits.
The current ratio is given by the secondary circuit equation, which is [22]: When there are no losses at the resonance frequency f 1 of the first mode Z=jX=−jMω and consequently I 2 /I 1 =+1, the mode is co-current.
For the mode at frequency f 2 , a change of sign takes place, so Z = jX = + jMω and the mode is counter current (I 2 /I 1 = −1).
In the case of low losses or high quality factor, i.e.Q 0 = RLω 0 >> 1 [23], these two modes retain their characteristics in terms of frequencies and currents, while the impedances at resonance remain low [24].The simulation allowed observing the variations of the currents and their relationship (Figure 5).The two loops are tuned to the same frequency f 0 = 105MHz, with k=0.05,L = 117.6nH,C = 19.47pF,R = 0.41Ω, and quality factor Q 0 = 189.
It can be seen that at f o , the transfer of power from the primary to the secondary coil is optimal.This is a particularity of the inductive coupling which is widely used in RF [25][26].For the other frequencies f 1 and f 2 , the currents I 1 and I 2 have a maximum value and the difference between the two modes remains at the phase level.

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Fig. 5. I1 and I2 variation curves and their ratio.

A. Prototype Dimensions
The results of the experimental tests obtained with the developed coils are presented in this section.An ellipsoidal Helmholtz coil and a circular Helmholtz coil were built for MRI (H1) at 100.241MHz.To facilitate comparisons, they have the same radial size d=5cm.The precise dimensions of the coils are given in Table III.Figure 6 shows a photography of both developed coils.The diameter of the wire constituting the loops is 1mm.

B. Tuning and Matching of the Coils
Each loop of the coil was tuned at the same self-resonance frequency f 0 , higher than the resonance frequency of the coil.By predetermination of f 0 , an empty preset of the coil can be accomplished.Fine tuning of the coil at the working frequency f 1 is then performed if necessary [27].Using the HP 4195A network analyzer, we were able to obtain the variation of the magnitude and the phase in air without load of the input impedance seen by a test loop in terms of frequency as shown for the ellipsoidal coil in Figure 7. Curve of the input impedance of the ellipsoidal coil seen by a test loop.
At resonance frequency f 1 , there is almost the same value for the first mode and a small error for the second mode.At f 2 , the difference is around 3% for the circular coil H C and around 1% for the ellipsoidal coil H E .The matching of the coils was carried out without load using a coupling loop (inductive coupling).The position of the coupling loop relatively to the coil determines the optimal distance to get an impedance of 50Ω for the matching [28].This is approximately 1.9cm for the H C (1.99cm in theory, this corresponds to an error of 5%) and 2.7cm for the H E (2.56cm in theory, corresponding to an error of 6%).This matching technique allows optimal energy transfer between the coils and the transmission and reception system.It should be noted that the matching and tuning of the coils depend on the nature of the sample and therefore adjustments should be considered for any new load.

C. Testing the Coils without Load
Off-field (in-air) tests were performed for both H C and H E coils.Thus, in the absence of a sample, relative field measurements along the axis of the coils were carried out using a small measurement loop connected to the HP 4195A network analyzer, and comparative readings of the intensity profile of the magnetic field B 1 were established as shown in Figure 8.The experimental curves of the normalized magnetic induction produced along the Oz axis by both coils are given in Figure 9. Axial profiles in terms of distance (cm) for coils HE and Hc.The obtained B 1 field profiles along the axis for the two coils (Figure 8) are in good agreement with the theoretical predictions of Figure 2. Compared to the H C (reference coil), the width at 10% of the lobe is 1.32 times greater for the ellipsoidal coil, which is comparable to the theoretical predictions.Another positive point for the ellipsoidal coil, its efficiency is better.Indeed, at identical absorbed power, the field emitted in the center is on average about 20% higher than that produced by the H E coil.Moreover, the quality factor of the ellipsoidal coil has also an interesting increment of 32%.

VI. CONCLUSION
The design and testing of two prototype probes for MRI comprising two circular and ellipsoidal free elements was presented.The two loops are correctly spaced and connected directly in series to a tuning capacitor.A complete electrical modeling of the coils taking into account all couplings (inductive and capacitive) was investigated.For a given resonant frequency, a predetermination of the tuning frequency of the tuned circuits is carried out, as well as the calculation of the associated tuning capacitors.
At the practical level, the prototypes built are in fact of the same radial dimensions to facilitate comparisons.Thus, compared to the reference circular Helmholtz coil, field measurements in free space with the ellipsoidal coil are in good agreement with the theoretical predictions.The width of the lobe at 10% is increased by 20% for the ellipsoidal coil.Moreover, a significant improvement in the quality factor is observed in free space for the ellipsoidal coil.Also, the efficiency is increased by 37% for the ellipsoidal coil.Finally, the signal to noise ratio is higher for the ellipsoidal coil compared to the circular coil.

Fig. 3 .
Fig. 3.General electrical model for the developed structures.

Fig. 4 .
Fig. 4.Theoretical curve of the input impedance of two identical tuned circuits.

Fig. 7 .
Fig. 7.Curve of the input impedance of the ellipsoidal coil seen by a test loop.

Fig. 8 .
Fig. 8.Axial profiles in terms of distance (cm) for coils HE and Hc.

Fig. 9 .
Fig. 9.Axial profiles in terms of distance (cm) for coils HE and Hc.

TABLE III .
COIL DIMENSIONS