Insight into RF power requirements and B1 field homogeneity for human MRI via rigorous FDTD approach

Authors

  • Tamer S. Ibrahim PhD,

    Corresponding author
    1. Department of Radiology, University of Pittsburgh, Pittsburgh, Pennsylvania, USA
    2. Department of Bioengineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA
    3. School of Electrical and Computer Engineering, The University of Oklahoma, Norman, Oklahoma, USA
    4. Bioengineering Center, The University of Oklahoma, Norman, Oklahoma, USA
    • B800 Presbyterian University Hospital, 200 Lothrop St., Pittsburgh, PA 15213, USA
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  • Lin Tang MS

    1. School of Electrical and Computer Engineering, The University of Oklahoma, Norman, Oklahoma, USA
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Abstract

Purpose

To study the dependence of radiofrequency (RF) power deposition on B0 field strength for different loads and excitation mechanisms.

Material and Methods

Studies were performed utilizing a finite difference time domain (FDTD) model that treats the transmit array and the load as a single system. Since it was possible to achieve homogenous excitations across the human head model by varying the amplitudes/phases of the voltages driving the transmit array, studies of the RF power/B0 field strength (frequency) dependence were achievable under well-defined/fixed/homogenous RF excitation.

Results

Analysis illustrating the regime in which the RF power is dependent on the square of the operating frequency is presented. Detailed studies focusing on the RF power requirements as a function of number of excitation ports, driving mechanism, and orientations/positioning within the load are presented.

Conclusion

With variable phase/amplitude excitation, as a function of frequency, the peak-then-decrease relation observed in the upper axial slices of brain with quadrature excitation becomes more evident in the lower slices as well. Additionally, homogeneity optimization targeted at minimizing the ratio of maximum/minimum B1+ field intensity within the region of interest, typically results in increased RF power requirements (standard deviation was not considered in this study). Increasing the number of excitation ports, however, can result in significant RF power reduction. J. Magn. Reson. Imaging 2007;25:1235–1247. © 2007 Wiley-Liss, Inc.

SINCE HUMAN MRI was introduced as a clinical diagnostic tool, many safety concerns have been raised regarding the extent of the associated radiofrequency (RF) power deposition in tissue (1–10). Particularly, characterizing the dependence of the RF power deposition on the frequency of operation, i.e., Larmor frequency or B0 field strength, has been a topic of research interest over the last half century (3, 9, 10). Since many advancements in human MRI have been linked with higher B0 field strengths, predicting the necessary RF power absorption in tissue to attain a particular flip angle in all or in part of the human head/body is essential since higher field strengths are typically associated with increases (3, 9) in RF power requirements. The interest in identifying RF power requirements at very high fields, however, has been more academic than practical, since the technology to build ultra-high-field (≥7 Tesla) human systems did not exist. As human MRI is currently performed at field strengths reaching 7 (11, 12), 8 (13, 14), and 9.4 (15) Tesla, accurately predicting the RF power absorption associated with such operation has become essential to classify the potential clinical practicality of these systems as well as future ones. Furthermore, the critical health concern is not only associated with RF power deposition in the whole head, but the rise in local temperatures as well. In practice, one of the important concerns is the RF “hot spots” produced by the rise due to specific absorption rate (SAR) as well. It is therefore crucial to determine whether the limitations of total and local RF power absorption in tissue would impede the advancement (at least from a field strength perspective) of human MRI.

Many electromagnetic methods have been used in studying the relationship between RF power absorption in tissue and field strength or frequency of operation. The bases of these methods fall into two main categories, namely quasistatic (2, 3) and full wave (9, 10) models. Originally, quasistatic models have predicted square dependence, i.e., the RF absorbed power required to attain a fixed flip angle in the tissue is dependent on the square of the operating frequency (3). Full wave models, however, have predicted otherwise. For example, one model that utilized idealized current sources (9) has shown that the RF power depends initially on the square of frequency, then reduces to linear dependence at higher field strength (9). For other models that utilized rigorous modeling (10) of the coil source and treated the coil and the load as a single system, the results illustrated that the power increases with frequency, peaks at a specified value, and then drops as the frequency increases beyond that value (10).

Operation at ultra-high-field human MRI, however, has been associated with inhomogeneous RF field distributions (11, 12–18), resulting in significant inhomogeneity in flip angle distribution across the human head/body (11, 12, 16–18). As a result, predicting or even defining the RF power needed to attain a particular flip angle in all or part of the head/body is difficult and potentially unclear. Consequently, comparative studies between low- and high-field imaging with regard to RF power requirements become somewhat meaningless due to significant difference in the homogeneity of the field distribution at different field strengths. Recently, however, several techniques have been proposed to achieve homogenous excitation at ultra-high-field operation, including the use of transmit arrays with variable phase and variable amplitude driving mechanism (19–25) transmit sensitivity encoding (SENSE) (26–30) and tailored pulses (31, 32). Many works have demonstrated numerically (20, 23, 30, 31, 33, 34) and experimentally (22) the potential of such methods in achieving highly homogenous slice excitation. The power absorption associated with such methods, however, has not been fully investigated.

In this work, a detailed study at the MRI RF power requirements is provided using a full wave model, namely the finite difference time domain (FDTD) method (35). The calculations are performed using two volume coils: a transverse electromagnetic (TEM) resonator (36) and a single element extremity coil (17), and two different coil loads: an 18-tissue anatomically-detailed human head model (20) and a cylindrical phantom filled with saline-like dielectric properties. As suggested in previous works (14, 16, 17, 19, 20), the approach utilized in these calculations employed rigorous modeling of the drive ports and treated the coil and the load as a single system. A roadmap is given for the RF power/MRI field strength dependence as follows. First, numerical analysis (up to 11.7 Tesla) is presented using the single-element coil and the cylindrical phantom that illustrates the regime when quasistatic approximation holds. This analysis is then followed with detailed studies of RF power requirements using a TEM resonator loaded with the human head model and under field strengths ranging between 4 and 9.4 Tesla. These studies focus on the RF power variations as a function of the number of drive ports (four, eight, or 16), the driving mechanism (variable or fixed phase/amplitude excitation), and the orientations and positioning of the slices of interest within the human head model. Since it was possible to achieve highly homogenous two-dimensional (2D) excitations in several slices across the human head model by varying the amplitudes and phases of the drive ports, relevant studies of the RF power/field strength dependence were achievable under well defined (specified ratio of maximum/minimum B1+ field intensity within the slice of interest), fixed (at different field strengths), and homogenous excitation fields.

MATERIALS AND METHODS

Coils and Loads

The simulations in this work were performed using numerical models of two coils and two loads: 1) a shielded single coaxial-element coil loaded with a cylindrical phantom; and 2) a TEM resonator loaded with an anatomically detailed human head mesh. The FDTD technique (implemented with an in-house package) was utilized in the calculations of the electric and magnetic fields and the power requirements. Following the method suggested by Chen et al (37) and developed and implemented by Ibrahim (10, 25), Ibrahim et al (20), and Ibrahim and Lee (38), both the RF coil and the load were modeled as a single system. The inclusion of the coil as well as the load in a simultaneous modeling (rigorous, as previously defined (17, 25)) approach relies on three main bases. The first basis is 1) an excitation of the coil (transmit array) with a voltage source with an appropriate bandwidth; followed by 2) an examination of the coil's frequency response (comparable to the network analyzer's S11 plot and Smith Chart) to determine if the coil's mode of interest is tuned to the appropriate frequency. If the coil in question is not tuned to the frequency of choice, numerical tuning of the coil is performed through adjusting the gap size between each pair of inner coaxial elements, and steps 1) and 2) are performed again. The second basis is carrying out all excitation and tuning steps while the load is numerically present in the coil. The third basis is an implementation of all aforementioned steps for all of the coil's applicable drive port(s).

The above-mentioned technique of modeling the coil and the load as a single system accounts for the electromagnetic effects on the load due to the coil and, reversely, on the coil due to the load, which are interdependent. Note that this technique is different from utilizing “idealized conditions” (9) in calculating RF power requirements. In such a modeling approach (9), the coil is assumed to function as an ideal transmission line (39, 40) and the electromagnetic effects of the load on the coil are not considered. This technique of rigorous volume RF coils full-wave modeling has demonstrated excellent agreement with experimental measurements in terms of predicting the transmit and receive magnetic fields (17, 41), and thus images (17, 41) and electric fields (38), and therefore power deposition.

The details of the FDTD models of coils and loads utilized in this work are given below.

Shielded Single-Coaxial Element and Cylindrical Phantom

The first coil considered was a shielded single-coaxial element coil typically used for extremity imaging at ultra high fields (42, 43). The coil length and diameter were set at 16.4 cm and 10 cm, respectively. The load in this coil was a 9.4-cm-long cylindrical phantom with a circular cross-section with a diameter of 4.6 cm. The electromagnetic dielectric properties of the phantom were assigned to have dielectric constant of 78 and conductivity of 1.154 second/m. A 3D FDTD model (2-mm3 resolution) was developed for the loaded coil. Figure 1 displays: 1) the FDTD grid of the loaded coil; and 2) a detailed 3D diagram of the coil's element. A stair-step approximation was used to model the shield of the coil. The coil element (as shown in Fig. 1) was modeled using a modified FDTD algorithm (discussed below). To resemble typical experimental settings and to break the symmetry associated with the placement as well as the geometry of the phantom, the coil strut was shifted 2 mm from the center axis.

Figure 1.

3D FDTD grid of: 1) the single element coil loaded with the small/symmetrical phantom, and 2) the coil element. The coil element, also used in the TEM resonator model (Fig. 2), is tuned by adjusting the gap between the two inner coaxial elements.

Similar to the TEM resonator (discussed below), the loaded coil was tuned by adjusting the gap between the inner coaxial elements (Fig. 1) until the coil's single mode was positioned at the frequency of interest. Using the FDTD simulations, the lower (gap size = 3 cm) and upper (gap = coil length – 3.6 cm) bounds of the tuning frequencies were found to be 254 MHz (6 Tesla for 1H imaging) and 485 MHz (11.7 Tesla for 1H imaging), respectively. Note that these values were obtained using a dielectric constant of 2.2 to resemble a Teflon filling between the inner and outer coaxial lines. Throughout this work, we will refer to this coil as “single-element coil” and this load as “small/symmetrical phantom.”

TEM Resonator and Anatomically-Detailed Human Head Model

The FDTD technique was also utilized to model a TEM resonator loaded with an anatomically-detailed 18-tissue human head mesh. The coil structure was composed of 16 elements; each element is identical to that associated with the single element coil (Fig. 1). The coil shield had a diameter of 34.6 cm and a length of 21.2 cm. The human head mesh was placed in the coil such that the chin was aligned with the coil's bottom ring.

As was done in previous work (10, 20, 25, 38), an in-house rigorous 3D FDTD model was developed for the loaded coil as shown in Fig. 2. The FDTD domain was divided into approximately eight million cells with a resolution of 2 mm × 2 mm × 2 mm. Within each cell, the electric and magnetic fields were calculated using a leap-frog iterative scheme (35). For the boundary condition, perfectly matched layer (PML) (44) were used. A total of 16 PML layers were placed on six boundaries in the x, y, and z planes. A stair-step approximation was used to model the coil shield and the top and bottom rings of the coil. A modified FDTD algorithm was used to change the coaxial elements from squares into octagon shapes (25) to 1) minimize the errors caused by stair stepping in these critical tuning elements; and 2) achieve an eight-fold symmetry.

Figure 2.

3D anatomically-detailed human head model loaded within the 16-element TEM resonator. The six individual head cuts show the orientation of six slices (A1–A4, Sa, and Co) used in the B1+ field calculations. On each slice, five points are picked where the B1+ field intensity is recorded from the FDTD calculations. The symbols (X, □, ◊ *, and +) show the spatial positioning of these five points. The spatial area of each slice is shown under each head cut.

From analytical models based on the multiconductor transmission line theory (36, 39, 45), 16 modes exist for a 16-element TEM resonator, of which 14 modes are in pairs of degenerate frequencies, resulting in modes at nine distinctive frequencies. Mode 1, the second (measured from the zero frequency point) mode on the coil's frequency spectrum, was utilized for the power and field calculations. Using the FDTD simulations and Teflon material as the filler between each pair of inner and outer coaxial lines, the lower (gap size = 3 cm) and upper (gap = coil length – 3.6 cm) bounds of mode 1 tuning frequencies were found to be 171 MHz (approximately 4 Tesla for 1H imaging) and 406 MHz (approximately 9.4 Tesla for 1H imaging).

It is noted that all the electromagnetic analyses were performed in the specified frequency ranges (171–406 MHz for the head coil and 254–485 MHz for the single element coil). As idealized conditions (9) were not utilized for this study, frequencies outside of these ranges would constitute a nonphysical operation of both coils since these frequency ranges represent the physical as well as the numerical (because of the rigorous modeling approach) limits of the coils' operations.

Field Calculations and Optimizations

In addition to other components, the transverse magnetic (B1) field contains two circularly-polarized components, which can be defined here as B1+ and B1 fields and represented as follows (46, 47):

equation image(1)
equation image(2)

B1x and B1y are the x and y components of B1 field and are represented as complex values with amplitudes and phases. Since we are concerned with field excitation, only the B1+ field, which is the flip-inducing component, was considered in our homogeneity optimizations, while the total electric field was considered across the whole head model for the power calculations. In order to study or define a relation for the MRI RF power dependence on field strength, we investigated RF power requirements associated with numerous settings and homogeneities of the B1+ field.

For the single-element coil loaded with the small/symmetrical phantom, the excitation can only be performed at a single port as shown in Fig. 1. As such, one case was considered in which we investigated the B1+ field intensity within the volume of the small/symmetrical phantom. In terms of the TEM resonator loaded with the human head mesh, several cases were considered at five different frequencies/field strengths: 4 (lower bound), 5, 7, 8, and 9.4 Tesla (higher bound). Figure 3 describes the different excitations utilized in investigating the B1+ field intensity/distribution within the head coil, namely:

  • 1“16-Port Excitation,” in which all the coil elements were utilized,
  • 2“8-Port Excitation,” in which eight of the coil elements were utilized. Two different sets of elements were considered, as shown in Fig. 3, “Type AC” (eight elements denoted by A and C) and “Type BD” (eight elements denoted by B and D), and
  • 3“4-Port Excitation” (“Type A,” “Type B,” “Type C,” and “Type D”), in which the coil elements are denoted by A, B, C, and D as shown in Fig. 3.
Figure 3.

Axial slice of the FDTD grid of the 16-element TEM resonator loaded with the 18-tissue anatomically detailed human head model showing a cut of the coil where slice A1 (Fig. 2) is located. All coil ports are used in 16-port excitation; ports labeled A and C are used in eight-port excitation “Type AC”; ports labeled B and D are used in eight-port excitation “Type BD”; four ports labeled A are used in four-port excitation “Type A.” The same naming convention is used for four-port excitation “Type B,” “Type C,” and “Type D”.

At any of the above mentioned five field strengths, the multiport excitation was done as follows. The TEM resonator was tuned from each of the coaxial elements while using a uniform gap size between each pair (total of 16) of the inner coaxial elements. As has been observed at 340 MHz (20), the same/uniform gap size was sufficient to resonate the coil's mode 1 at the same frequency, regardless of which coil element is excited. For “N-Port Excitation,” the number of FDTD runs per frequency was N + additional runs in order to obtain the appropriate gap size between each pair of the inner coaxial elements such that mode 1 lies at the frequency of choice. Once the frequency of mode 1 was determined, the FDTD code was run again with Fourier transformation across the whole human head. The number of FDTD code runs per frequency in this step was N (through driving each element of the TEM coil). It is important to note that in these calculations, while every port is properly matched when individually excited; all the ports are not simultaneously matched to the same impedance. Nonetheless, the unique field distributions obtained using quadrature/optimized excitations are solutions of Maxwell's equations and are therefore physically realizable fields (considering the port-to-port coupling) with the same coil/load since the coil structure is not altered at any point in the simulations.

Two different feed strategies that involve the phase and amplitude of the driving voltages were considered at the above mentioned five magnetic field strengths: 1) fixed (integer multiples of phase-shifts) phase and fixed (uniform) amplitude (FPA); and 2) optimized phase and optimized amplitude (OPA). The B1+ field optimizations/calculations were all performed upon four axial slices (A1–A4), one sagittal slice (Sa), and one coronal slice (Co), as shown in Fig. 2. The thickness of each slice is 2 mm, i.e., one cell of the FDTD grid. In the FPA condition, integer multiples of 2π/(number of excitation ports) were utilized; while the OPA condition was carried-out to achieve a more homogenous1B1+ field distribution within the slice of interest.

Homogeneity of the B1+ field distribution within any of the above mentioned six slices was chosen to be evaluated by a dimensionless factor, “max/min,” which is defined as maximum B1+ field intensity over minimum B1+ field intensity within the slice of interest. Therefore, the closer “max/min” is to unity, the more homogenous (from a B1+ field distribution point of view) the slice is considered to be. Since we are concerned with evaluation of power requirements/behavior, max/min is considered a more appropriate choice than using the standard deviation (SD) as done in previous work (20). Unlike the SD method (20), this guarantees that the B1+ field intensities within any slice of interest lie within a specific range (maximum and minimum) and therefore the reported power values would: 1) facilitate physical interpretation; and 2) represent more meaningful/definitive findings.

2D whole-slice uniformity optimizations were performed by varying the amplitudes and phases of the fields produced by exciting each element of the coil. The optimization routines were configured to determine the amplitude and phase that should be applied to the voltage driving each coil element in order to achieve better B1+ field distribution homogeneity (lower max/min). The optimization routines were comprised of a combination of both gradient-based and genetic algorithm functions, in which a single iteration may go through either of these two methods.

Power Calculations

Assuming the coil conductors and dielectrics are lossless (as was done numerically), the real RF power entering (after matching circuits) the coil can be approximated as the sum of the absorbed (in tissue) power and the radiated (exiting from the coil and not absorbed in the tissue) power. In determining the RF power requirements, we have deliberately only considered the absorbed (in the small/symmetrical phantom or in the human head mesh) power. This was done for two main reasons. First, unlike the radiated power, the absorbed power is associated with tissue dissipation and heating concerns. Second, the percentages of the coil's radiated powers vary at different frequencies, making the comparison of power requirements at different field strengths unclear even for the same (geometry and dimensions) RF coil. As a result, we have utilized the absorbed power rather than the total power entering the coil in determining the requirements for achieving specified flip angle(s) in all or part of the load. The power absorbed in the load is calculated as follows:

equation image(3)

where σ(i,j,k) (S/m) is the conductivity of the FDTD cell at the (i,j,k) location; Ex, Ey, and Ez (V/m) are the magnitudes of the electric field components in the x, y, and z directions, respectively; Δx(i,j,k) Δy(i,j,k), and Δz(i,j,k) are the dimensions of each FDTD cell at location (i,j,k) and the summation is performed over the whole volume of the load. Except for the single element coil loaded with the small/symmetrical phantom, the excitation amplitudes/phases are adjusted to provide an average B1+ field intensity of 1.174 μT, the field strength needed to produce a flip angle of π/2 with a 5-msec rectangular RF pulse, in each selected slice, and then the absorbing power is calculated from Eq. [3]. This can be clearly achieved with the linearity of Maxwell's equations since Driving_Voltage ∝ E ∝ H and therefore Power ∝ E2Bmath image, where E and H are the electric and magnetic field intensities, and ∝ indicates linear dependence.

RESULTS AND DISCUSSION

Single-Element Coil Loaded With Small/Symmetrical Phantom: A Glance Into the Quasistatic Predictions

In the quasistatic regime, the power dissipated in a cylinder (assumed for all practical purposes to be the RF power required to obtain a specified flip angle) with radius r and length l is given by (10, 48):

equation image(4)

where B is the magnetic flux density, σ is the conductivity, and ω is the frequency. Therefore, until recently (9, 10), it was established that the RF power needed to obtain a specified flip angle in an imaged object varies with ω2, or with the square of the B0 field strength (given that the frequency of the applied RF field equals the Larmor frequency). If we examine the quasistatic power relation given by Eq. [4], it is clear that B was assumed to be the field that excites the spins. This can only be applicable if the transmitted magnetic field that exists in the load is a circularly polarized transverse magnetic field in a specified sense of rotation, i.e., the B1+ field. With commonly/clinically used RF excitation methods, this is only and approximately valid when the transmit volume coil is excited in quadrature at low frequency, where the dimensions of coil and object to be imaged are small compared to operating wavelength; this is clearly not the case for high-and ultra-high-field human MRI. In addition for the ω2 dependence to hold, it is also assumed that: 1) homogeneity, and 2) the strength of the B field do not vary with frequency. If this assumption to the MRI excite (B1+) field, the fraction of the total transmitted field that contributes/projects to the B1+ field direction is would have to be constant under all B0 field strengths (frequencies). Again, this is not a valid assumption for a wide frequency range such as 64 MHz (1.5 Tesla) and 400 MHz (9.4 Tesla).

Previous published results (Fig. 3 in Ref.17) demonstrated that good homogeneity of, and similarity between the B1+ and B1 fields' intensities/distributions are obtained for the small/symmetrical phantom loaded in the single element coil at 6 Tesla. As was deduced (17), these two facts indicate that the arrangement of this coil and phantom is effectively producing a close-to-ideal (quasistatically predicted) linearly polarized field since it can be evenly split into B1+ and B1 fields (17). The polarization of this linearly polarized field can potentially be altered to become circular if quadrature excitation were possible with this coil. The same figure (17) also showed that the similarities between, as well as the good homogeneity of the B1+ and B1 fields' intensities/distributions, were less apparent at 11.7 Tesla. Figure 4 (in this work) displays the RF absorbed power (as a function of frequency) required to obtain a fixed average intensity of either of the circularly polarized fields (B1+ and B1), within the volume of the small/symmetrical phantom loaded in the single element coil (shown in Fig. 1). Figure 4 substantiates the frequency/load regime when the square dependence between RF power and B0 field strength is applicable and when the quasistatic approximations of RF power requirements begin to fail. It is clearly shown that except near high frequency values, the required RF power for both field components is almost identical despite the slight asymmetry in the coil model. In addition, the expected power/frequency square dependence is clearly apparent at lower frequencies with deviations as the frequency increases.

Figure 4.

Plots of absorbed (in the small/symmetrical phantom loaded in the single element coil) RF power required in order to obtain a fixed average (over the volume of the phantom) B1+ or B1 field intensity as a function of frequency; the “o2” plot denotes the square dependence predicted from quasistatic approximations.

Therefore, based on: 1) the geometries and sizes of the single element coil and of the small/symmetrical phantom load, and 2) the frequency/power relation presented in Fig. 4; the power requirements for loads, such as the human head/body, are expected to deviate greatly from those predicted by quasistatic approximations, most especially with increasing the B0 field strength. The following section examines this issue.

TEM Resonator Loaded With the Anatomically-Detailed Human Head Mesh: Beyond the Quasistatic Predictions

B1+ Field Distributions Under Fixed Phase/Amplitude (FPA) and Optimized Phase/Amplitude (OPA) Conditions

Figure 5 provides the max/min values obtained for four-, eight-, and 16-port FPA conditions at 4, 5, 7, 8, and 9.4 Tesla, and Fig. 6 shows a sample of the B1+ field distributions within the above-mentioned six slices for 16-port FPA and OPA conditions at 7 and 9.4 Tesla. With OPA conditions, the B1+ field distribution in each slice (Fig. 6) was obtained by minimizing max/min through applying the most aggressive optimizations at 9.4 Tesla: by 1) sweeping through all possibilities of initial conditions/generations, and 2) continuously adding vibrations to stalled iterations; and then using the resulting max/min as the homogeneity target in the same slice at 7 Tesla. Therefore under both B0 field strengths, the B1+ field distribution within a slice is characterized by the same max/min, or according to this work's classification, the same homogeneity. As expected, Figs. 5 and 6 and other results (not shown) demonstrate that 16-port excitation (FPA or most-optimized OPA) at 9.4 Tesla provides less homogenous B1+ field distributions compared to that obtained with 7 Tesla. Subsequently, under lower than 9.4 Tesla B0 field strengths, different B1+ field distributions within a slice are likely to be obtained, yet still have the same homogeneity, i.e., the same max/min (obtained with the most optimized OPA conditions at 9.4 Tesla). This is demonstrated in Fig. 6 with the two distinct 7 Tesla B1+ field distributions, denoted by “Solution 1” and “Solution 2,” which have the same sets of max/min. As was demonstrated in earlier numerical work (19, 20, 23), Fig. 6 confirms that significant improvement in the homogeneity of the B1+ field distributions can be achieved in many slices and in all directions with OPA conditions. It is clearly shown that the homogeneity of the B1+ field distributions is better achieved within relatively smaller area slices, as shown with slices A4 and Co (see Fig. 2).

Figure 5.

Plots of max/min (maximum B1+ field intensity over minimum B1+ field intensity within the same slice) for each slice shown in Fig. 2 at different B0 field strengths and under four-, eight-, and 16-port fixed phase/amplitude driving conditions. The four-port excitations types, “Type A,” “Type B,” “Type C,” and “Type D,” and the eight-port excitation types, “Type AC” and “Type BD,” are defined in Fig. 2. Each subfigure corresponds to a different slice. For a particular driving condition, the max/min values are presented with the same symbol at all B0 field strengths.

Figure 6.

(Color Scale): B1+ field distributions for each slice shown in Fig. 2. The data are presented at 9.4 and 7 Tesla using 16-port excitation with fixed and optimized phase/amplitude (FPA and OPA) driving conditions. In the horizontal direction, every set of six subfigures are orderly positioned to correspond to slices A1–A4, Sa, and Co (Fig. 2), respectively. In the vertical direction, under “9.4 Tesla, 16-Port Excitation,” the top and bottom rows correspond to 16-port FPA and OPA driving conditions, respectively. Under “7 Tesla, 16-Port Excitation,” the top row corresponds to 16-port FPA driving conditions and the middle and bottom rows correspond to two possible solutions for 16-port OPA driving conditions, specifically two different B1+ field distributions yet having the same homogeneity, i.e., the same max/min (described in Fig. 5). The number above each subfigure corresponds to max/min within that slice. All the results are presented for the coil configuration shown in Fig. 2.

Power Comparisons Under 16-, Eight-, and Four-Port Excitations and Fixed Phase/Amplitude Driving Conditions

Figure 7 displays the required RF power (W) to achieve a value of 1.174 μT for the average B1+ field intensity in each slice shown in Fig. 2 as a function of B0 field strength under 16-, eight-, and four-port FPA excitations. For each slice and number of excited ports, the corresponding max/min value of B1+ field intensity is shown in Fig. 5. Figures 5 and 7 provide two main observations with regard to the RF power requirements under FPA driving conditions, namely

Figure 7.

Plots of required absorbed (in the human head) power in order to obtain a fixed (1.174 μT) average (over the area of each of the six slices shown in Fig. 2) B1+ field intensity as a function of B0 field strength. The results are presented under 16-, eight-, and four-port fixed phase/amplitude driving conditions. From left to right, the three columns correspond to 16-, eight-, and four-port excitations, respectively. From top to bottom, the six rows correspond to the results for the six slices (orderly positioned as slices A1–A4, Sa, and Co, respectively). In each subplot, the X-axis is the B0 field (Tesla) and the Y-axis is the power absorbed in the head (Watts). The excitation types that denote the choice of the excited coil elements in eight- and four-port excitations are defined in Fig. 3. All the results are presented for the coil configuration shown in Fig. 2.

  • aThe fewer number of excitation ports utilized, the more power absorbed in the human head (in order to obtain the same average B1+ field intensity in each slice). The difference in power requirements, however, is minimal between the eight-port (“Type AC” or “Type BD”) and 16-port excitations and is almost indistinguishable at 4 and 5 Tesla. The noticeable, yet still marginal, difference is obtained between eight-port and four-port excitations at 7-9.4 Tesla.
  • bThe difference between using eight-port “Type AC” and “Type BD” is indistinguishable. Compared to the other three types of four-port excitations, “Type C” (Fig. 2) requires more absorbed power in the head (in order to obtain the same average B1+ field intensity in each slice), most especially at 7 to 9.4 Tesla.

It can also be observed that the dependence of RF power on the B0 field strength varies in a very similar manner regardless of the number of drive ports utilized. In agreement with previously published data obtained while assuming idealized coil conditions (9), Fig. 7 shows that the RF power continuously increases as a function of B0 field strength in the larger axial slices, i.e., the lower ones (A1–A3). In the upper axial slice A4, however, the results differ from those obtained while assuming idealized coil conditions. In accordance with the RF power/frequency relation observed with rigorous modeling of the coil and of the excitation port (10), the required RF power peaks at 8 Tesla for slice A4 and then decreases at higher B0 field strengths. Additionally, the peak-then-decrease behavior was also observed for several slices (not shown) above the axial slice A4. The peaks for these slices however occurred at 7 Tesla rather than 8 Tesla (A4.) In the Sa and Co slices, the RF power peaks at 7 Tesla, drops noticeably for slice Co and negligibly for slice Sa at 8 Tesla, then rises again noticeably for slice Co and negligibly for slice Sa at 9.4 Tesla. While prior RF power/frequency relation (10) obtained using rigorous modeling conditions showed peak-then-decrease behavior at all the displayed axial slices, the decrease in the lower axial slice was minimal (10). This possibly shows that the RF power required in order to obtain a fixed value of the average B1+ field intensity at even lower axial slices may continue to increase with frequency. Additionally, the RF power/frequency relation shown in Ref.10 was obtained with a model of an eight-element linearly (one port) excited TEM resonator loaded with the visible human project head model (http://www.brooks.af.mil/AFRL), which is 1) tilted and 2) a volumetrically much larger model than the one utilized in this work. These underlined conditions are different than the ones utilized to present the power calculations shown in Fig. 7.

While the above analysis provides insight into RF power/field strength dependence, it is somewhat vague, since the homogeneity of the B1+ field distributions is 1) significantly different at different B0 field strengths, and 2) poor at higher B0 field strengths. In the following sections, we will attempt to address these two specific issues by homogenizing the B1+ field distributions using OPA driving conditions to achieve the same homogeneity (max/min) in any targeted slice at all the B0 field strengths of interest.

Absorbed Power Comparisons for the Same Homogeneity of the B1+ Field Distribution Under Different Number of Drive Ports and Driving Conditions

Starting from FPA four-port excitation, the B1+ slice A3 (as described in Fig. 2) was most aggressively optimized to achieve the lowest possible max/min using OPA driving conditions at 9.4 Tesla. Note that the OPA driving conditions were tested with the four possible excitation types (Fig. 3) associated with four-port excitation. The resulting lowest max/min of 1.70 (achieved for four-port “Type A”) was then utilized as the homogeneity target at all B0 field strengths under four-, eight-, and 16-port excitations. Except for 4 Tesla, at which FPA driving conditions were sufficient for eight- and 16-port excitations, OPA driving conditions were required to achieve max/min of 1.70 with four-port excitaiton and under all other B0 field strengths with four-, eight-, and 16-port excitations. Figure 8 displays samples of the resulting B1+ field and total electric field distributions in slice A3 at 4, 7, and 9.4 Tesla and using four-, eight-, 16-port excitations, where max/min = 1.70. A very interesting point to note here is the fact that under any of the three B0 field strengths shown in Fig. 8, a remarkably similar (not only in terms of max/min but also in terms of the overall characteristics) B1+ field distribution was obtained with four-, eight-, or 16-port excitations.

Figure 8.

Top: Comparison of the power absorbed in the human head under 16-, eight-, and four-port excitations. The results are presented for slice A3 (Fig. 2). The values on each subplot correspond to the power required to achieve average (across slice A3) B1+ field intensity = 1.174 μT. The power values at all field strengths and under all the driving conditions are achieved for the same B1+ field homogeneity, i.e., max/min (defined in Fig. 5) = 1.70. This value represents the best possible homogeneity achieved using four-port optimized phase/amplitude driving conditions at 9.4 Tesla. Bottom (Color Scale): The corresponding B1+ field and total electric field distributions in slice A3 at 4, 7, and 9.4 with max/min = 1.70. All the results are presented for the coil configuration shown in Fig. 2.

Figure 8 also provides required RF power (W) to achieve a value of 1.174 μT for the average B1+ field intensity in slice A3 as a function of B0 field strength. It is clearly shown that the use of a fewer number of drive ports results in more power absorbed in the human head while maintaining

  • 1the same average B1+ field intensity,
  • 2the same homogeneity of the B1+ field distribution, and
  • 3interestingly for the shown slice, almost indistinguishable B1+ field distribution as well.

Similar to Fig. 7, which displays the RF power requirements for FPA driving conditions, Fig. 8 shows that the significant difference in power absorption comes under high B0 field strengths and between four-port and eight-port excitations; for the max/min of 1.70 and an average B1+ field intensity of 1.174 μT, a significant (2.3 W CW) absorbed power difference is observed between four-port and eight-port excitations at 9.4 Tesla. The electric field distributions shown in Fig. 8 clearly distinguish brightness in the center of the coil/load at 9.4 Tesla and under four-port excitation. This possibly explains the significant power increase under this condition compared to the other eight cases, which experience relatively lower electric field intensities in the center of the coil/load.2

By comparing the power requirements to achieve a max/min of 1.70 (Fig. 8) to the corresponding results (Fig. 7) under FPA driving conditions, it is clear the improvement in the homogeneity (as denoted by the particular max/min criteria) of the B1+ field distribution with OPA driving conditions comes at the cost of more power absorption. For example, the four-port OPA 9.4 Tesla results show an approximately 50% increase in the power absorption compared to that obtained with four-port FPA excitation. However, this power increase diminishes at lower B0 field strengths. This can be attributed to the lower improvement in homogeneity of B1+ field distribution as demonstrated in the change of the max/min (Fig. 5) obtained with FPA driving conditions to the max/min of 1.70 (most optimal only at 9.4 Tesla).

While the results of this section were presented exclusively for slice A3, we observed similar patterns for the other axial slices shown in Fig. 2. For example, Fig. 9 describes the absorbed power values resulting from optimization on the B1+ field in slice A1. With 16-port OPA excitation, Fig. 9 shows that in order to attain an average B1+ field intensity of 1.174 μT, the absorbed power increases from 0.87 to 1.43 (W) with max/min decreasing from 2.13 to 1.74 at 4 Tesla, while the power increases from 3.13 to 7.81 W with max/min decreasing from 21.94 to 1.74 at 9.4 Tesla. Comparison between four-port excitation under OPA and FPA driving conditions reveals the same conclusions (Fig. 9.) As such: 1) reducing the number of drive ports, and/or 2) improving the homogeneity (as denoted by the particular max/min criteria) of the B1+ field distribution with OPA driving conditions results in increased power absorption. On the other hand, Fig. 9 indicates that at each particular B0 field strength, the absorbed power under 16-port OPA excitation is higher than that under four-port OPA excitation. Clearly, the homogeneity of B1+ field distribution is much superior under 16-port (max/min = 1.74) than under four-port (max/min = 3.03) OPA excitations. In this case, the resulting increase in power absorption due to improved homogeneity overcomes the decrease expected from increasing the number of drive ports from four to 16.

Figure 9.

Plot of the required absorbed power in order to obtain an average (across slice A1 shown in Fig. 2) B1+ field intensity = 1.174 μT under 16- and four-port fixed and optimized phase/amplitude driving conditions at different field strengths. The power values for the optimized phase/amplitude driving conditions are presented for the same homogeneity of the B1+ field distribution at all field strengths; i.e., max/min (defined in Fig. 5) = 3.03 and 1.74 corresponding to the most optimized homogeneity at 9.4 Tesla under four-port Type C, and 16-port excitations, respectively.

Dependence of Absorbed Power on B0 Field Strength Under Variable Phase/Amplitude Driving Conditions

Figure 10 provides plots describing the absorbed power required to achieve a fixed: 1) average B1+ field intensity of 1.174 μT, and 2) max/min (most optimized solution at 9.4 Tesla) within each slice shown in Fig. 2 as a function of B0 field strength. The results are presented utilizing 16-port OPA excitation. Other possible solutions (representing different B1+ field distributions yet possessing the same max/min) are also provided in some of the plots. When comparing these power results to those obtained with FPA driving conditions (left column in Fig. 7), a noticeable difference can be observed in the characteristics of the relationship between absorbed RF power and B0 field strength. In slice A1, similarly to FPA excitation, the absorbed power continuously increases as a function of B0 field strength with 16-port OPA excitation. In the upper axial slices A2–A4, however, the absorbed power peaks at 8 Tesla for slices A2 and A4 and at 7 Tesla for slice A3 and then decreases at greater than 8 (or 7) Tesla field strength; it slightly rises again, however, at 9.4 Tesla for slice A3. In this way, the nature of absorbed power dependence on B0 field strength is similar for slice A4 under both 16-port FPA and OPA excitations. The power dependence observed for slices A2 and A3, however, is different from that obtained with 16-port FPA excitation. Therefore, by optimizing the homogeneity of the B1+ field distributions while attaining a fixed average B1+ field intensity within axial slices, the relationship describing the dependence of the absorbed power on B0 field strength is somewhat constant: the power typically peaks at some B0 field strength then decreases at higher values. It is fair to assume that under 16-port OPA excitation, the absorbed power required to obtain a fixed average B1+ field intensity in slice A1 will also peak at a B0 field strengths higher than 9.4 Tesla and then decrease thereafter. Such assumption can not be verified with this coil model as the 9.4-Tesla Larmor frequency represents its upper operational frequency limit.

Figure 10.

Plots of the absorbed power required in order to obtain a fixed (1.174 μT) average (over the area of each of the six slices shown in Fig. 2) B1+ field intensity under different B0 field strengths. The calculations are performed using 16-port optimized phase/amplitude driving conditions. From top to bottom, the six rows correspond to the results for the six slices (orderly positioned as slices A1–A4, Sa, and Co, respectively.) The number above each subfigure corresponds to max/min (defined in Fig. 5) for that slice. max/min was fixed (most optimized value solution obtained at 9.4 Tesla) at each field strength for each slice. At 4, 7, and 9.4 Tesla, two solutions of the power calculations (represented by “▹” and “○”) are presented for two different B1+ field distributions, yet have the same max/min.

As slices Sa and Co orient differently from axial slices, so does the relationship between the absorbed power and the B0 field strength. Under 16-port OPA excitation, slice Sa shows that the dependence of the absorbed power on B0 field strength functions somewhat in an oscillatory fashion. In slice Co for one of the solutions, the power peaks at 8 Tesla then decreases at >8 Tesla field strength.

Figure 11 displays the ratios of the B1+ field intensities at the five points positioned in each of the six slices shown in Fig. 2 at 9.4 Tesla before (FPA) and after (OPA) optimization using 16 ports. For the same absorbed power, as expected, the B1+ field intensities for the majority of displayed points (28 out of 30) decrease with OPA driving conditions. Furthermore, it is observed that the most significant decrease in the B1+ field intensity typically occurs at the center point of each slice (◊), which typically possesses the highest B1+ field intensity with FPA driving conditions (Fig. 6). With OPA driving conditions, however, within each slice, the center point (◊) is typically characterized by a lower B1+ field intensity compared to the other four surrounding points. Therefore, OPA driving conditions usually result in significant defocusing of the B1+ field intensities within the central portion of the human head. This is clearly opposite to the typically observed (11, 16, 20, 41) field focusing effect that occurs in the central portion of the head with quadrature excitation or FPA driving conditions.

Figure 11.

Plots of the ratios (before and after optimization at 9.4 Tesla) of the B1+ field intensities at the five points positioned in each slice shown in Fig. 2. All the B1+ field intensities were scaled to the same absorbed power. The symbols selection is the same as that used in Fig. 2.

In conclusion, the main difficulty in accurately describing a correlation between power requirements and operating frequency is due to the inhomogeneity of RF fields at high field strengths. As a result, there is an extreme ambiguity in obtaining well defined relations in regards to this issue. In this work, using a rigorously applied FDTD scheme, we closely studied the RF power/frequency dependence up to 11.7 Tesla on models of two coils and two loads: 1) a shielded single coaxial-element coil loaded with a cylindrical phantom, and 2) a TEM resonator loaded with an anatomically-detailed human head mesh. Power relations for potential head imaging were presented by means of varying the amplitudes and phases of the exciting voltages in order to achieve highly homogenous B1+ field distributions with fixed/defined uniformity criteria at different B0 field strengths up to 9.4 Tesla.

As expected for small electrical loads, the results show that the RF power required in order to achieve a fixed average B1+ field intensity within the load is proportional to the square of the operating frequency, with deviations as frequency increases. This verifies predictions obtained using quasistatic approximations for electrically small loads (dimensions << the operating wavelength). When examining the results in the human head, however, the power dependence is considerably different. Using quadrature excitation (i.e., fixed amplitude and progressive integer multiples of phase shifts), it was clearly shown that the square dependence of the power on frequency vanishes to become: 1) linear in axial slices towards the bottom of the brain, or 2) peak-then-decrease in axial slices towards the top of the brain. The results also show that the use of more drive ports with this excitation mechanism results in reduction of the power requirements, most especially between four and eight ports.

When utilizing optimized excitation, the nature of the power dependence is different than that with quadrature excitation. First, optimization of the homogeneity of the B1+ field distributions results in increased power requirements. On the other hand, the peak-then-decrease relation observed with the upper axial brain slices with quadrature excitation becomes more evident in the lower brain slices as well. The results clearly show that in order to achieve a highly homogenous B1+ field distribution with a specified criteria of homogeneity, the use of more drive ports, and therefore more phase-locked amplitude-attenuated transmit channels, will significantly reduce the RF power required to achieve a fixed average B1+ field intensity. Several numerical studies were conducted and verified these findings.

While RF penetration plays a major role in the physics of the MRI RF power behavior at different field strengths, the coil-specific induced electromagnetic waves play a major role as well. For example, different results are to be expected between the TEM coils (36, 49) with their axially propagating (39, 50) electromagnetic waves and birdcage coils with their azimuthally propagating (51, 52) electromagnetic waves. In addition, it is essential to note that the presented conclusions are valid only for the max/min criterion for assessing the homogeneity of the B1+ field distributions. Different results (where using OPA driving conditions, the toal absorbed RF power can be even lower than that obtained with quadrature excitation (53)) are probable if using coefficient of variation (SD) approaches (20, 25), which do not necessarily aim at minimizing max/min over the region of interest as was done in this work.

  • 1

    Several studies will be considered for improving the homogeneity of the B1+ field distribution. These efforts are described in details in the Results and Discussion section.

  • 2

    As the B1+ field represents only a component of the total magnetic flux density, similar B1+ field distributions will not necessarily produce similar electric field distributions.

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