Despite recent advances in parallel transmit technology, there is a critical need to improve the uniformity of $$$B_1^+$$$ in high field MRI. A traveling wave¹ in an empty magnet bore can produce a uniform amplitude magnetic field, but once a human body is positioned in the bore, refraction and reflection of the incident wave produce strongly varying field amplitudes in the body. This work describes a new approach that aims to induce a traveling plane wave inside the sample, subjecting all spins to the same field amplitude.

The electromagnetic field both inside and outside the body can be described as a superposition of modes expressed, for example, in terms of the Debye potentials, which are solutions to the scalar Helmholtz equation.² The field generated by the RF coil in an empty bore is termed the incident field. In spherical coordinates, the Debye potentials, $$$\Pi$$$, for both the incident field and the field within the body have the general form $$r\Pi=\sum_{p=1}^\infty\sum_{q=0}^p\psi_p(kr)P_{pq}(cos\theta)[a_{pq}cos(q\phi)+b_{pq}sin(q\phi)]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ where $$$k$$$ is the wavenumber, the radial functions $$$\psi_p(kr)$$$ are related to the spherical Bessel functions $$$j_p(kr)$$$ by $$$\psi_p(kr)=kr{\cdot}j_p(kr)$$$, and $$$P_{pq}$$$ are the associated Legendre polynomials. Each term in the series designates one mode of the field; transverse magnetic (TM) and transverse electric (TE) fields are described by independent series.

Each mode of the incident field generates multiple modes within the body. The (complex) amplitude of the $$$n^{th}$$$ within-body mode generated by the $$$m^{th}$$$ incident mode equals the element $$$c_{nm}$$$ of the coupling matrix, $$$\tilde{C}$$$. There are two steps in the technique. First, the coupling matrix is estimated by mapping the transverse magnetic fields within the body generated by each incident field mode. Since the transverse field for each within-body mode can be calculated from Eq (1), this allows the coupling matrix to be estimated. Second, the superposition of incident modes corresponding to an internal plane wave is determined. The amplitude of the $$$n^{th}$$$ internal mode in an ideal plane wave defines the element $$$b_n$$$ of the known column vector $$$\vec{b}$$$. The incident mode amplitudes that generate the internal plane wave are the elements of the unknown column vector $$$\vec{\beta}$$$. The coupling matrix relates the two vectors: $$$\tilde{C}\cdot\vec{\beta}=\vec{b}$$$. Solving for the incident mode amplitudes, we have $$$\hat{\vec{\beta}}=\tilde{C}^+\cdot\vec{b}$$$, where $$$\tilde{C}^+$$$ is the pseudo-inverse of $$$\tilde{C}$$$. $$$\hat{\vec{\beta}}$$$ defines the amplitude and phase of the RF coil modes that produce an internal plane wave. Hence the method is known as Traveling Internal Plane-wave Synthesis (TIPS).

The procedure was simulated for a uniform ellipsoidal sample (relative permittivity $$$\epsilon_r = 80$$$, relative permeability $$$\mu_r=1$$$, conductivity $$$\sigma=0$$$ and principal axes of length 12, 14, and 16 cm, rotated by $$$20^{\circ}$$$ and $$$10^{\circ}$$$ to simulate chin-to-chest and left-to-right head rotation, respectively) imaged at 7T. The transverse field within the sample generated by each incident field mode was calculated by matching the transverse electric and magnetic fields at the surface of the ellipsoid. The resulting coupling matrix was used to evaluate two cases. First, the internal field was calculated assuming incident plane wave excitation. Second, the TIPS superposition of incident field modes was used to calculate the internal fields. The appropriate incident fields can be generated by arrays of radial electric and magnetic dipoles distributed over the surface of a coil former. The radial electric dipole array creates 48 TM field modes, while the radial magnetic dipole array generates 48 TE modes (96 channels in total).The $$$B_y$$$ fields for both types of incident wave are shown in the figure (traveling waves propagate from left to right). The coefficient of variation of the internal $$$|B_y|$$$ for the incident traveling wave is 62.5%, compared to 0.7% in the TIPS case.While the simulation was performed for a simple ellipsoidal phantom, the method relies only on the existence of a linear relationship between incident and internal mode amplitudes. Even in the heterogeneous, multilayered structure of the head, the linearity of Maxwell’s equations guarantees such a relationship, and hence the technique is expected to yield nearly uniform traveling waves in the human brain as well. Work is underway to evaluate performance using more realistic models of the human head.The TIPS method induces a traveling plane wave within a volume of interest (VOI) in the body by driving an appropriate superposition of external, incident field modes. Spins within the VOI are subjected to a $$$B_1^+$$$ magnetic field with nearly uniform amplitude. This method promises to improve image uniformity in high field MRI.