Erratum to: MRI‐Based Transfer Function Determination through the Transfer Matrix by Jointly Fitting the Incident and Scattered B1+ Field (Magn Reson Med. 2020; 83:1081–1095)

This erratum aims to correct a couple of small errors in some of the equations in our original paper ‘MRI‐based transfer function determination through the transfer matrix by jointly fitting the incident and scattered B₁ ⁺ field’ by Tokaya et al. The paper makes use of a range of expressions to describe the background and scattered B₁ ⁺ field. These B₁ ⁺ fields can be described using e^iωt or e^−iωt time convention. Both approaches are correct but they result in different expressions for the field distributions.

The scripts that were used for the data processing contained correct expressions so all results and conclusions are correct and valid. However, for some equations in the paper the two time conventions have been mixed up resulting in some erroneous expressions. We apologize for this mistake and we hope that this document will avoid future confusion of potential readers. The following three equations are: incorrect.

Page 1083, left column, bottom line:

$$\zeta =\pm \sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2+i\sigma \omega },$$

should be

$$\zeta =\sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2-i{\mu }_0\sigma \omega }.$$

Page 1084, Equation (6):

$${\mathit{B}}_{\mathbf{1}\mathit{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)=\frac{{\mu }_0}{4\pi }\underset{C}{\int }I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)\mathit{d}{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)e^{i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}\right],$$

should be

$${\mathit{B}}_{\mathbf{1}\mathit{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)=\frac{{\mu }_0}{4\pi }\underset{C}{\int }I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)\mathit{d}{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)e^{-i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}\right].$$

This concatenates to two expressions in Equation (7):

$$\begin{array}{ll}{\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)& =\frac{{\mu }_0}{4\pi }\sum _{i=1}^{\frac{L}{\left|d{r}_i\right|}}\mathit{M}\left(c_k\right)E\left({\mathit{r}}_{\mathit{i}}^{\prime };a_{\mathit{mn}}\right)\mathit{d}{\mathit{r}}_{\mathit{i}}\times \left({\mathit{r}}_{\mathit{i}}-\mathit{r}\right)e^{i\frac{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^2}\right]\\ & =\frac{{\mu }_0}{4\pi }\sum _{i=1}^{\frac{L}{\left|d{r}_i\right|}}\mathit{M}\left(c_k\right)\frac{\mathbf{-}\cos {\theta }_i^{\prime }}{\sigma +\mathit{i}\omega \mathit{ϵ}}\left(2i{\partial }_x+2{\partial }_y\right)B_{1,\mathit{bg}}^{+}\left({\mathit{r}}_{\mathit{i}}^{\prime };a_{\mathit{mn}}\right)\mathit{d}{\mathit{r}}_{\mathit{i}}\times \left({\mathit{r}}_{\mathit{i}}-\mathit{r}\right)e^{i\frac{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^2}\right],\end{array}$$

should be

$$\begin{array}{ll}{\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)& =\frac{{\mu }_0}{4\pi }\sum _{i=1}^{\frac{L}{\left|d{r}_i\right|}}\mathit{M}\left(c_k\right)E\left({\mathit{r}}_{\mathit{i}}^{\prime };a_{\mathit{mn}}\right)\mathit{d}{\mathit{r}}_{\mathit{i}}\times \left({\mathit{r}}_{\mathit{i}}-\mathit{r}\right)e^{-i\frac{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^2}\right]\\ & =\frac{{\mu }_0}{4\pi }\sum _{i=1}^{\frac{L}{\left|d{r}_i\right|}}\mathit{M}\left(c_k\right)\frac{\mathbf{-}\cos {\theta }_i^{\prime }}{\sigma +\mathit{i}\omega \mathit{ϵ}}\left(2i{\partial }_x+2{\partial }_y\right)B_{1,\mathit{bg}}^{+}\left({\mathit{r}}_{\mathit{i}}^{\prime };a_{\mathit{mn}}\right)\mathit{d}{\mathit{r}}_{\mathit{i}}\times \left({\mathit{r}}_{\mathit{i}}-\mathit{r}\right)e^{-i\frac{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}_{\mathit{i}}-\mathit{r}\right|}^2}\right].\end{array}$$

For clarity, a more detailed derivation of these equations is given down below.

The solution of the Maxwell equations in a source‐free homogeneous medium is found through the Helmholtz equation ¹ :

$${\nabla }^2\mathit{B}(\mathit{r},t)={\mu }_0\sigma \frac{\partial \mathit{B}(\mathit{r},t)}{\partial t}+{\mu }_0{\epsilon }_0{\epsilon }_r\frac{{\partial }^2\mathit{B}(\mathit{r},t)}{\partial t^2}.$$ (1)

Separation of variables results in

$$\mathit{B}(\mathit{r},t)=\mathit{B}(\mathit{r})T(t).$$ (2a)

$$\frac{{\nabla }^2\mathit{B}(\overrightarrow{r})}{\mathit{B}(\overrightarrow{r})}=\frac{1}{T(t)}\left({\mu }_0\sigma \frac{\partial T(t)}{\partial t}+{\mu }_0{\epsilon }_0{\epsilon }_r\frac{{\partial }^{2}T(t)}{\partial t^2}\right)=-{\zeta }^2.$$ (2b)

The two time conventions for $T(t)$ result in:

$$\mathit{B}(\mathit{r},t)=\mathit{B}(\mathit{r})e^{+\mathit{i}\omega \mathit{t}},$$ (3a)

or

$$\mathit{B}(\mathit{r},t)=\mathit{B}(\mathit{r})e^{-\mathit{i}\omega \mathit{t}},$$ (3b)

and therefore the corresponding equations for the wave number $\zeta$ are:

$$-{\zeta }^2=-{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2+i{\mu }_0\sigma \omega ,$$ (4a)

or

$$-{\zeta }^2=-{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2-i{\mu }_0\sigma \omega ,$$ (4b)

$$\zeta =\pm \sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2-i{\mu }_0\sigma \omega },$$ (5a)

or

$$\zeta =\pm \sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2+i{\mu }_0\sigma \omega }.$$ (5b)

The expression in the paper is $\zeta =\sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2+i\sigma \omega }$ (bottom line of left column on page 1083). Here first of all the vacuum permeability ${\mu }_0$ was forgotten for the second term. In addition, the wave number $\zeta$ as presented with a plus sign in the square root corresponds to a time convention of $e^{-\mathit{i}\omega \mathit{t}}$ while on page 1084, right column, 6th line, the opposite time convention is indicated: $\mathit{B}(\mathit{r},t)=\mathit{B}(\mathit{r})e^{+\mathit{i}\omega \mathit{t}}$. Following this positive time convention, the expression for the wave number at the bottom of the left column on page 1083 should therefore be (note that the minus sign option has been discarded because of the convention to use only the positive real part of wave numbers):

$$\zeta =\sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r{\omega }^2-i{\mu }_0\sigma \omega .}$$

The choice for the time convention also has effect on the expressions for the Jefimenko equation ¹ :

$${\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r},t;a_{\mathit{mn}},c_k\right)=\frac{{\mu }_0}{4\pi }\underset{C}{\int }\frac{I\left({\mathit{r}}^{\prime },t_r;a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}+\frac{\frac{d}{\mathit{dt}}I\left({\mathit{r}}^{\prime },t_r;a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}.$$ (6)

Again, using positive time convention this results in:

$${\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r},t;a_{\mathit{mn}},c_k\right)={\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)e^{\mathit{i}\omega \mathit{t}}=\frac{{\mu }_0}{4\pi }\underset{C}{\int }\frac{I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}{e}^{\mathit{i}\omega t_r}+\frac{\frac{d}{\mathit{dt}}I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}{e}^{\mathit{i}\omega t_r},$$ (7)

where $I\left({\mathit{r}}^{\prime },t_r;a_{\mathit{mn}},c_k\right)=I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)e^{\mathit{i}\omega t_r}$.

Using retarded time $t_r=t-\frac{\left|{\overrightarrow{r}}^{\prime }-\overrightarrow{r}\right|}{c_m}$ this evolves into ¹ :

$${\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)e^{\mathit{i}\omega \mathit{t}}=\frac{{\mu }_0}{4\pi }\underset{C}{\int }\frac{I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}{e}^{\mathit{i}\omega \mathit{t}}{e}^{-i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}+\frac{\frac{d}{\mathit{dt}}I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}{e}^{\mathit{i}\omega \mathit{t}}{e}^{-i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}.$$ (8)

Now calculate the time differential

$${\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)e^{\mathit{i}\omega \mathit{t}}=\frac{{\mu }_0}{4\pi }\underset{C}{\int }\frac{I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}{e}^{\mathit{i}\omega \mathit{t}}{e}^{-i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}+\mathit{i}\omega \frac{I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)}{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}{e}^{\mathit{i}\omega \mathit{t}}{e}^{-i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}.$$ (9)

Now remove time exponential on both sides and collect terms in brackets:

$${\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)=\frac{{\mu }_0}{4\pi }\underset{C}{\int }I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)e^{-i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}\right].$$ (10)

Similarly, using negative time phasor convention, the same equations result in:

$${\mathit{B}}_{\mathbf{1}\mathbf{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)=\frac{{\mu }_0}{4\pi }\underset{C}{\int }I\left({\mathit{r}}^{\prime };a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)e^{i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}-\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}\right].$$ (11)

However, Equation (6) on page 1084 of the paper is a mixture of these two equations.

${\mathit{B}}_{\mathbf{1}\mathit{,}\mathit{sc}}\left(\mathit{r};a_{\mathit{mn}},c_k\right)=\frac{{\mu }_0}{4\pi }\underset{C}{\int }I\left({\mathit{r}}^{\prime },t_R;a_{\mathit{mn}},c_k\right)d{\mathit{r}}^{\prime }\times \left({\mathit{r}}^{\prime }-\mathit{r}\right)e^{i\frac{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|\omega }{c_m}}\left[\frac{1}{{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^3}+\frac{\mathit{i}\omega }{c{\left|{\mathit{r}}^{\prime }-\mathit{r}\right|}^2}\right]$ (6 paper)

Abiding positive time convention, Equation (6) in the paper should be adapted into Equation (10) of this document. Consequentially, the same adaptations need to be applied to Equation (7) in the paper.

STATEMENT OF AUTHORSHIP CHANGE

Because of the considerable contribution of M.A. Eijbersen to the Erratum to this paper, the authors have agreed to add him as an author.