First-Order Spatial Encoding Simulations for Improved Accuracy in the Presence of Strong ${\mathbf{\text{B}}}_0$ and Gradient Field Variations

Abstract

Purpose:

Low-cost and application-specific scanners with unconventional designs can entail substantial inhomogeneities of the main magnetic field $B_0$ and non-linearities of the gradient fields, which challenge assumptions made in MRI simulators. This work aims to develop a simulation framework that captures the encoding effects of strong field variations, accurately and efficiently, to enable the assessment of effects such as geometrical distortions, signal dropout, and foldover artifacts.

Methods:

Like many other MRI simulators, ours discretizes magnetic fields in space. However, we extend the MR signal simulation at each grid point from the 0^th-order approximation, which assumes piecewise constant fields, to a 1^st-order approximation, which assumes piecewise linear fields. We solve the signal equation by analytically integrating over each grid cube, assuming linear field variations, and then summing over all cubes. We provide analytical integrals for several pulse sequences.

Results:

The 1^st-order approximation captures strongly varying fields and associated intravoxel dephasing more accurately, avoiding severe “ringing” artifacts present in the usual 0^th-order simulations. This enables simulations on a much coarser grid, facilitating computational feasibility.

Conclusion:

The first-order simulator enables the evaluation of unconventional scanner designs with strongly varying magnetic fields.

1 |. INTRODUCTION

The goal to expand MRI’s reach has driven recent efforts to build low-cost and application-specific scanners,¹ including scanners specialized for point-of-care imaging with unconventional scanner geometries, such as single-sided or open-access magnets.^2,3 These designs utilize flexible field configurations, where the patient is positioned adjacent to rather than inside the magnet, which typically requires weaker and less homogeneous magnetic fields. Magnetic field inhomogeneities complicate spatial encoding, and our work aims to enable the assessment of the spatial encoding properties of scanner designs with strong field variations in silico.

Most current simulation tools rely on mostly homogeneous magnetic fields.^4–7 They discretize the imaged object (e.g., a body model or a high-quality MR image) and the magnetic fields, assuming constant magnetic fields within each cube. This assumption is adequate for sufficiently homogeneous $B_0$ and linear gradient fields (see Fig. 1). In the following, we will demonstrate that this assumption is inadequate in the presence of strong magnetic field variations. We propose to improve the simulations by approximating the $B_0$ and gradient field variation as linear within each cube, which allows a more accurate estimation of intravoxel dephasing. For several pulse sequences, we derive analytical solutions to the MR signal equation integrated over locally linear fields, which then become new base elements for summation over the grid. We demonstrate the efficacy of this approach in two test cases: 1) a saddle shape ${\text{B}}_0$ field distribution as a simple test bed, and 2) more realistic magnetic fields resulting from simulations of a prostate scanner design. Our simulation platform captures the effects of strong variations of magnetic fields on spatial encoding considerably better than the common piecewise constant approximation.

FIGURE 1.

Sketch of the total magnetic field variations ($B_0\left(\mathbf{\text{r}}\right)+G_x\cdot x$) when superimposing a saddle-shaped main magnetic field $B_0\left(\mathbf{\text{r}}\right)$ with a nominal field strength $B_0\left(\mathbf{0}\right)=99\text{mT}$ and a strictly linear gradient field of strength $G_x=5\text{mT}/\text{m}$, which represents a typical field during k-space readout. The main magnetic field is aligned along $z$ and its strength is given by $B_0(x,y,z)=B_0^{\text{nominal}}+\alpha \left(x^4-y^4\right)$ where $B_0^{\text{nominal}}=99\text{mT}$ is the nominal field and $\alpha =3\text{mT}/{\text{m}}^4$ introduces a saddle-shaped inhomogeneity. As indicated by the central cube, the magnetic field in the isocenter is well-described by the 0^th-order approximation. In contrast, the 0^th-order approximation is insufficient in areas with strong field variations, as indicated by the cube on the top-right.

2 |. METHODS

2.1 |. Signal equations

Our simulation framework focuses on accurately modeling intravoxel dephasing arising from highly non-linear fields and ignores contrast-generating effects such as relaxation and diffusion. Therefore, we scale the longitudinal magnetization $m_0\left(\mathbf{\text{r}}\right)$, where $\mathbf{\text{r}}$ denotes space, to match the intensity of an input MR image. This effectively imprints the contrast of the input image (here, a 3T clinical $T_2$-weighted image) on the output image, neglecting differences in relaxation characteristics between different field strengths.

After exciting the longitudinal magnetization $m_0$ with a radio frequency (RF) pulse and assuming no initial transversal magnetization, the MR signal is given by the integral of the excited magnetization over space:

$$s\left(t\right)=\int m_0\left(\mathbf{\text{r}}\right)\text{sin}\left[\alpha \cdot B_1^{+}\left(\mathbf{\text{r}}\right)\cdot f\left({\omega }_0\left(\mathbf{\text{r}},0\right)\right)\right]B_1^{-}\left(\mathbf{\text{r}}\right)e^{-i\varphi \left(\mathbf{\text{r}},t\right)}d\mathbf{\text{r}},$$ (1)

where $\alpha$ denotes the nominal flip angle, and $B_1^{+}\left(\mathbf{\text{r}}\right)$ the transmit RF field, which is assumed to be real-valued and is normalized to 1 at the isocenter (variations in the phase of the transmit field correspond to changes in flip axis, and can be easily incorporated as a starting phase distribution $\varphi (\mathbf{\text{r}},0)$). For simplicity, we assume instantaneous RF pulses, which ignores subtle variations of spin relaxation during the pulse. Dephasing during excitation pulses is much more pronounced and we use the small-flip angle approximation to describe dephasing with the spectral response function $f\left({\omega }_0(\mathbf{\text{r}},0)\right)$, where ${\omega }_0(\mathbf{\text{r}},0)$ is the spatially varying Larmor frequency at the time of the excitation pulse, here defined as 0. Throughout this paper, we operate in the rotating frame, i.e., ${\omega }_0$ denotes the laboratory Larmor frequency minus the RF operating frequency ${\omega }_0^{\text{RF}}$. We note that we ignore dephasing during refocusing pulses. $B_1^{-}\left(\mathbf{\text{r}}\right)$ denotes the complex-valued RF receive coil sensitivity.

The phase $\varphi (\mathbf{\text{r}},t)$ is accrued between the time of the excitation pulse and the time $t$ at which the signal is acquired, through spin evolution in the spatially varying ${\mathbf{\text{B}}}_0\left(\mathbf{\text{r}}\right)$ field as well as the spatially and temporally varying gradient fields ${\mathbf{\text{B}}}_{G_{x,y,z}}(\mathbf{\text{r}},\tau )$:

$$\varphi (\mathbf{\text{r}},t)=\underset{0}{\overset{t}{\int }}\gamma {‖{\mathbf{\text{B}}}_0(\mathbf{\text{r}})+{\mathbf{\text{B}}}_{G_x}(\mathbf{\text{r}},\tau )+{\mathbf{\text{B}}}_{G_y}(\mathbf{\text{r}},\tau )+{\mathbf{\text{B}}}_{G_z}(\mathbf{\text{r}},\tau )‖}_2-{\omega }_0^{\text{RF}}d\tau .$$ (2)

Here $\gamma$ is the gyromagnetic ratio. Note that the vectors ${\mathbf{\text{B}}}_0,{\mathbf{\text{B}}}_{G_x},{\mathbf{\text{B}}}_{G_y}$, and ${\mathbf{\text{B}}}_{G_z}$ account for the directions of the fields and, thus, also for concomitant fields, by taking the vector sum of fields in Eq. (2). Thereafter, however, we take the ${\ell }_2$-norm of the total field, which is equivalent to neglecting the—usually small—spatial variations of the magnetization’s precession axis.

Toward the periphery of k-space, the phase $\varphi (\mathbf{\text{r}},t)$ contains high-frequency oscillations over space, which challenge the numerical solutions of the integral in Eq. (1).⁸ Many MRI simulators overcome this issue by approximating the integral with a discrete sum:^4–7

$$s\left(t\right)\approx \sum _j{m}_0\left({\mathbf{\text{r}}}_j\right)\text{sin}\left[\alpha \cdot B_1^{+}\left({\mathbf{\text{r}}}_j\right)\cdot f\left({\omega }_0\left({\mathbf{\text{r}}}_j,0\right)\right)\right]B_1^{-}\left({\mathbf{\text{r}}}_j\right)e^{-i\varphi \left({\mathbf{\text{r}}}_j,t\right)},$$ (3)

where the index $j$ runs over all simulation grid elements and ${\mathbf{\text{r}}}_j$ points to the center of respective element. Replacing Eq. (1) with Eq. (3) entails the approximations that $m_0\left(\mathbf{\text{r}}\right)$, $B_1^{+}\left(\mathbf{\text{r}}\right)$, and $B_1^{-}\left(\mathbf{\text{r}}\right)$ are piecewise constant—approximations we will maintain throughout this paper. It also approximates $f\left({\omega }_0(\mathbf{\text{r}},0)\right)$ as piecewise constant, which we will discuss in more detail below. Lastly, it approximates the magnetic fields, including the gradient fields, as piecewise constant—a limitation we aim to overcome with the proposed approach.

For a homogeneous $B_0$ field and linear gradients, the impact of the last assumption can be estimated straightforwardly. In this case, the accumulated phase $\varphi (\mathbf{\text{r}},t)$ is just $\mathbf{k}\cdot \mathbf{\text{r}}$, where $\mathbf{k}$ indicates a particular location in k-space associated with the application of corresponding gradients. Assuming piecewise constant fields ignores the integration of actual field variations over each grid element:

$$\underset{C_j}{\int }{e}^{-i\mathbf{k}\cdot \mathbf{\text{r}}}d\mathbf{\text{r}}=l^3\text{sinc}\left(\frac{k_{x}l}{2}\right)\cdot \text{sinc}\left(\frac{k_{y}l}{2}\right)\cdot \text{sinc}\left(\frac{k_{z}l}{2}\right)$$ (4)

where $C_j$ indicates an integral over the volume of the $j^{\text{th}}$ grid element, assumed to be a cube of side length $l$. Thus, a 0^th-order approximation of a constant $B_0$ and linear gradient fields equates to neglecting a simple k-space filter. The relative approximation error varies smoothly from zero in the center of k-space to $1-(2/\pi {)}^3$ at the k-space periphery (assuming 3D k-space encoding). For many applications, this error is acceptable and Eq. (3) provides a reasonable approximation of Eq. (1).

${\text{B}}_0$ inhomogeneities and non-linear gradients introduce a dependency of the approximation error in Eq. (3) on space, time, and the pulse sequence: the stronger the spatial variations of $\varphi (\mathbf{\text{r}},t)$ within a simulation cube, the less accurate is the 0^th-order approximation, as illustrated in Fig. 1. For a grid cube positioned in the isocenter, the visualized field is well-approximated as piecewise constant. In contrast, the example cube towards the top right contains a steep gradient of the magnetic field, which is poorly approximated by a constant field. To reduce the error, we propose to use a 1^st-order approximation that models the $B_0$ and gradient fields as piecewise linear, i.e., linear over the limited spatial extent of each grid cube:

$$s\left(t\right)\approx \sum _j{m}_0\left({\mathbf{\text{r}}}_j\right)\cdot B_1^{-}\left({\mathbf{\text{r}}}_j\right)\cdot \underset{C_j}{\int }\text{sin}\left[\alpha B_1^{+}\left({\mathbf{\text{r}}}_j\right)\cdot f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)+{\nabla }_{\mathbf{\text{r}}}{\omega }_0\left({\mathbf{\text{r}}}_j\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)\right]\cdot \text{exp}\left[-i\left(\varphi \left({\mathbf{\text{r}}}_j,t\right)+{\nabla }_{\mathbf{\text{r}}}\varphi \left({\mathbf{\text{r}}}_j,t\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)\right]d\mathbf{\text{r}}.$$ (5)

A numerical solution of the integral in Eq. (5) faces the same issues as Eq. (1): at the periphery of k-space, $\varphi (\mathbf{\text{r}},t)$ contains high-frequency oscillations, which prohibit an efficient solution. Under certain circumstances, however, we can derive an analytical solution. In this study, we examine two imaging scenarios for which we were able to find analytic solutions.

2.1.1 |. 3D imaging

First, we consider 3D imaging, assuming

1. a piecewise linear main magnetic field (${\mathbf{\text{B}}}_0$)

2. piecewise linear gradient fields (${\mathbf{\text{B}}}_{G_x},{\mathbf{\text{B}}}_{G_y},{\mathbf{\text{B}}}_{G_z}$)

3. non-selective RF pulses

4. a piecewise constant spectral profile of the transmit RF coil:

   $$f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)+{\nabla }_{\mathbf{\text{r}}}{\omega }_0\left({\mathbf{\text{r}}}_j\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)\approx f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)\right).$$

5. a piecewise constant magnetization ($m_0$)

6. piecewise constant transmit ($B_1^{+}$) and receive ($B_1^{-}$) RF fields

Under these assumptions, the signal contribution of the $j^{\text{th}}$ grid-cube in a 3D pulse sequence is given by

$$s_j\left(t\right)\approx m_0\left({\mathbf{\text{r}}}_j\right)\text{sin}\left[\alpha B_1^{+}\left({\mathbf{\text{r}}}_j\right)f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)\right)\right]B_1^{-}\left({\mathbf{\text{r}}}_j\right)\bullet \underset{C_j}{\int }\text{exp}\left[-i\left(\varphi \left({\mathbf{\text{r}}}_j,t\right)+{\nabla }_{\mathbf{\text{r}}}\varphi \left({\mathbf{\text{r}}}_j,t\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)\right]d\mathbf{\text{r}}.$$ (6)

This integral has the following analytical solution:

$$s_j\left(t\right)\approx m_0\left({\mathbf{\text{r}}}_j\right)\text{sin}\left[\alpha B_1^{+}\left({\mathbf{\text{r}}}_j\right)f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)\right)\right]B_1^{-}\left({\mathbf{\text{r}}}_j\right)\cdot \text{exp}\left[-i\varphi \left({\mathbf{\text{r}}}_j,t\right)\right]\cdot l^3\cdot \text{sinc}\left[l/2\cdot {\partial }_x\varphi \left({\mathbf{\text{r}}}_j,t\right)\right]\cdot \text{sinc}\left[l/2\cdot {\partial }_y\varphi \left({\mathbf{\text{r}}}_j,t\right)\right]\cdot \text{sinc}\left[l/2\cdot {\partial }_z\varphi \left({\mathbf{\text{r}}}_j,t\right)\right].$$ (7)

2.1.2 |. 2D multi-slice imaging

Second, we consider 2D (multi-slice) imaging, assuming

1. a piecewise linear main magnetic field (${\mathbf{\text{B}}}_0$)

2. piecewise linear gradient fields (${\mathbf{\text{B}}}_{G_x},{\mathbf{\text{B}}}_{G_y},{\mathbf{\text{B}}}_{G_z}$)

3. a Gaussian excitation pulse with the spectral response

   $$f\left({\omega }_0\left(\mathbf{\text{r}}\right)\right)=e^{-{\omega }_0(\mathbf{\text{r}}{)}^2/{\sigma }^2},$$

   where $\sigma =\gamma \Delta z{G}_z$ is the RF-pulse bandwidth associated with the nominal slice thickness $\Delta z$

4. a spectral profile of the transmit RF field much broader than the RF-pulse bandwidth.

5. the small tip-angle approximation:

   $$\text{sin}\left[\alpha B_1^{+}\left({\mathbf{\text{r}}}_j\right)\cdot f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)+{\nabla }_{\mathbf{\text{r}}}{\omega }_0\left({\mathbf{\text{r}}}_j\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)\right]\approx \alpha B_1^{+}\left({\mathbf{\text{r}}}_j\right)\cdot f\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)+{\nabla }_{\mathbf{\text{r}}}{\omega }_0\left({\mathbf{\text{r}}}_j\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)$$

6. a piecewise constant magnetization ($m_0$)

7. piecewise constant transmit ($B_1^{+}$) and receive ($B_1^{-}$) RF fields

The approximate signal contribution of the $j^{\text{th}}$ grid-cube in a 2D pulse sequence under these conditions is

$$s_j\left(t\right)\approx m_0\left({\mathbf{\text{r}}}_j\right)\alpha B_1^{+}\left({\mathbf{\text{r}}}_j\right)B_1^{-}\left({\mathbf{\text{r}}}_j\right)\cdot \underset{C_j}{\int }\text{exp}\left[-{\left({\omega }_0\left({\mathbf{\text{r}}}_j\right)+{\nabla }_{\mathbf{\text{r}}}{\omega }_0\left({\mathbf{\text{r}}}_j\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)}^2/{\sigma }^2-i\left(\varphi \left({\mathbf{\text{r}}}_j,t\right)+{\nabla }_{\mathbf{\text{r}}}\varphi \left({\mathbf{\text{r}}}_j,t\right)\cdot \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_j\right)\right)\right]d\mathbf{\text{r}},$$ (8)

We note that all simulations of 2D and 3D imaging in this paper entail 3D integrals over the entire volume. The solution to Eq. (8) is rather unwieldy and can be found in the Supplementary Material. To provide some intuition into the results, we provide the result for the corresponding 1D integral here:

$$s_j^{1\text{D}}\left(t\right)\approx m_0\left(r_j\right)\alpha B_1^{+}\left(r_j\right)B_1^{-}\left(r_j\right)\cdot \frac{\sqrt{\pi }\sigma }{2{\partial }_r{\omega }_0\left(r_j\right)}\cdot \text{exp}\left[-i\varphi \left(r_j\right)\right]\bullet \text{exp}\left[-\frac{{\partial }_r\varphi \left(r_j\right)}{{\left(2{\partial }_r{\omega }_0\left(r_j\right)\right)}^2}\cdot \left(l^2{\partial }_r\varphi \left(r_j\right)-4i{\omega }_0\left(r_j\right)\cdot {\partial }_r{\omega }_0\left(r_j\right)\right)\right]\bullet \left(\text{erf}\left[\frac{l}{2\sigma }{\partial }_r{\omega }_0\left(r_j\right)+\frac{i\sigma }{2}\frac{{\partial }_r\varphi \left(r_j\right)}{{\partial }_r{\omega }_0\left(r_j\right)}+\frac{2{\omega }_0\left(r_j\right)}{\sigma }\right]+\text{erf}\left[\frac{l}{2\sigma }{\partial }_r{\omega }_0\left(r_j\right)+\frac{i\sigma }{2}\frac{{\partial }_r\varphi \left(r_j\right)}{{\partial }_r{\omega }_0\left(r_j\right)}-\frac{2{\omega }_0\left(r_j\right)}{\sigma }\right]\right)$$ (9)

where erf denotes the error function, and ${\partial }_r\varphi \left(r_j\right)$ denotes the partial derivative of $\varphi$ with respect to $r$, evaluated at $r_j$.

2.2 |. Simulations

Fig. 2 outlines the simulation process. 3D maps of the main magnetic field ${\mathbf{\text{B}}}_0\left(\mathbf{\text{r}}\right)$, gradient fields ${\mathbf{\text{B}}}_{G_{x,y,z}}\left(\mathbf{\text{r}}\right)$, transmit RF field $B_1^{+}\left(\mathbf{\text{r}}\right)$, and receive RF field $B_1^{-}\left(\mathbf{\text{r}}\right)$ are loaded, along with a high-quality input image volume which is used as $m_0\left(\mathbf{\text{r}}\right)$. The fields are interpolated linearly and the image is interpolated with cubic b-splines to provide access to the values at any ${\mathbf{\text{r}}}_j$. The interpolation package further provides access to the gradients of each field, which will be used to calculate the 1^st-order approximation.

FIGURE 2.

Overview of the proposed simulation process. 1. The inputs of the simulation consisting of source image volumes $m_0\left(\mathbf{\text{r}}\right)$ and 3D maps for the main magnetic ${\mathbf{\text{B}}}_0\left(\mathbf{\text{r}}\right)$, the three gradient fields ${\mathbf{\text{B}}}_{{\mathbf{G}}_{\mathbf{x}}}\left(\mathbf{\text{r}}\right)$, ${\mathbf{\text{B}}}_{{\mathbf{G}}_{\mathbf{y}}}\left(\mathbf{\text{r}}\right)$, ${\mathbf{\text{B}}}_{{\mathbf{G}}_{\mathbf{z}}}\left(\mathbf{\text{r}}\right)$ and transmit/receive $\text{RF}\left(B_1^{+}\left(\mathbf{\text{r}}\right),B_1^{-}\left(\mathbf{\text{r}}\right)\right)$ fields. 2. Slice (or non-selective) excitation and the phase evolution are simulated for a particular sequence. 3. For each k-space data point and each receive RF coil, the signal contributions of each cube are calculated with Eq. (8) for 2D sequences or Eq. (6) for 3D sequences. 4. The total signal for each k-space data point is calculated by summing over all grid cubes. This process is repeated for all k-space data points of the sequence and the final image is obtained using a fast Fourier transform (FFT) for each coil signal, followed by a root sum of squares coil combination.

We implemented simulations of 6 pulse sequences:

• 2D multi-slice gradient echo (FLASH)

• 2D multi-slice spin echo (Supplemental Material)

• 2D multi-slice turbo spin echo

• 2D multi-slice spin echo EPI

• 2D multi-slice gradient echo EPI (Supplemental Material)

• 3D gradient echo (FLASH)

The simulations are written such that the sequence timing can be flexibly calculated for a given set of protocol parameters, such as image resolution, field of view (FOV), readout bandwidth, and slice position. We note that the current implementation of our simulator assumes instantaneous switching of gradient fields. In other words, gradient pulses are simulated as boxcars instead of trapezoids. Furthermore, all RF pulses are associated with a spectral response function that is calculated separately by taking the Fourier transform of the pulse shape, and are otherwise assumed to be instantaneous.

The simulations assume perfect spoiling, i.e., each excitation pulse exclusively excites longitudinal magnetization. Further, we assume thermal equilibrium or, equivalently, steady-state longitudinal magnetization before each RF pulse, i.e., we neglect any transition into the steady state. Last, simulations of spin echo pulse sequences assume perfect, instantaneous, and non-selective refocusing pulses, i.e., the pulses have a 180-degree refocusing flip angle regardless of $B_1^{+}$, spectral profile, or slice profile.

For each grid-cube, k-space point, and receive RF coil, we calculate $\varphi \left({\mathbf{\text{r}}}_j,t\right)$ with Eq. (2). The gradient ${\nabla }_{\mathbf{\text{r}}}\varphi \left({\mathbf{\text{r}}}_j,t\right)$ is calculated by taking the derivative of Eq. (2), assuming that the derivative and the integral commute. The signal contribution of each grid cube is calculated with Eq. (7) and Eq. (8) for 3D and 2D pulse sequences, respectively. The overall measured signal is calculated by taking the sum over all grid cubes. The process is repeated for all measured k-space points and all receive RF coil elements.

All simulated pulse sequences utilize Cartesian k-space sampling and Nyquist sampling, where the Nyquist sampling rate was calculated for nominal fields. Image reconstruction is performed with a fast Fourier transform of each coil signal, followed by a root sum of squares coil combination.

2.3 |. Validation

We validated the 1^st-order approximation with error analyses and simulations of different pulse sequences. All simulations were based on a standard clinical $T_2$-weighted 3T prostate MR image, which was interpolated using a cubic b-spline algorithm and used as $m_0\left(\mathbf{\text{r}}\right)$. In the superior-inferior direction, the image was extrapolated with a constant value to compensate for the limited coverage of about 90 mm. For simplicity, we analyzed the imaging properties of a saddle-shaped $B_0$ field, followed by a more realistic field distribution resulting from a prostate scanner design, as described in the following sections.

2.3.1 |. Saddle-shaped ${\mathbf{\text{B}}}_0$ field

In this pictorial (and un-physical) example, the main magnetic field is aligned along $z$ and its strength is given by $B_0(x,y,z)=B_0^{\text{nominal}}+\alpha \left(x^4-y^4\right)$ where $B_0^{\text{nominal}}=99\text{mT}$ is the nominal field and $\alpha =3\text{mT}/{\text{m}}^4$ introduces a saddle-shaped inhomogeneity (Fig. 1). The maximum field inhomogeneity is 1876.86 ppm over a 10 cm diameter spherical volume (DSV).

In this example, we assume linear imaging gradients with a maximum strength of 28 mT/m, as well as perfectly homogeneous RF transmit and receive fields, i.e., $B_1^{+}=1$ and $B_1^{-}=1$.

2.3.2 |. Prostate scanner

This scanner design is tailored to imaging the prostate. The magnet has a simulated field strength of 99 mT in the isocenter, and the design emphasizes homogeneity within a small spherical volume (~5 cm radius) at the expense of severe inhomogeneities outside of this volume. The maximum field inhomogeneity is 471.27 ppm over a 10 cm DSV. The imaging gradients, with cardinal directions and field distribution illustrated in Fig. 3, have maximum strengths ranging between 21 mT/m (x- and y-gradients) and 36 mT/m (z-gradient) in the isocenter. Combining the inhomogeneous ${\text{B}}_0$ field and the gradient fields results in non-bijective behavior outside the small homogeneous sphere, which commonly results in aliasing (Fig. 3).

FIGURE 3.

Sketch of the magnetic field variations $\left(\left|{\mathbf{\text{B}}}_0\left(\mathbf{\text{r}}\right)+{\mathbf{\text{B}}}_{G_x}\left(\mathbf{\text{r}}\right)\right|\right)$ when superimposing the main magnetic field ${\mathbf{\text{B}}}_0\left(\mathbf{\text{r}}\right)$ and the readout gradient field ${\mathbf{\text{B}}}_{G_x}\left(\mathbf{\text{r}}\right)$ of the prostate scanner design. Here, the coordinate system was rotated by 45 degrees from the inferior-superior direction. The two line plots on the right show the magnetic fields along the lines indicated in the heatmap and demonstrate the non-bijective imaging properties of these fields. The circles indicate a 10 cm diameter spherical volume that was used as the target homogeneity region during the magnet design.

For RF transmit and receive, we simulated two loop-butterfly quadrature surface coils. One was positioned on the anterior of the subject as if on their lap, and the other on their posterior. Each coil is configured with a 90-degree bend to contour to the shape of the subject. The size of each element was optimized for maximal SNR at the typical position of the prostate.⁹ $B_1^{+}$ and $B_1^{-}$ fields of each coil element were simulated with the HFSS (Ansys, Canonsburg, PA) software. For transmission, we assume a single RF amplifier with quadrature splitters that drives both coils. The $B_1^{+}$ fields are scaled to achieve the nominal flip angle in the isocenter.

Since $B_0$ of this scanner design is rotated by 45 degrees from the inferior-superior direction, we rotate the coordinate system accordingly (Fig. 3), and assume slice selection along the angled $z$-direction.

For 2D pulse sequences, we assume that the transmit coil’s bandwidth is broader than the bandwidth of the RF pulse, and the former is neglected. For the 3D pulse sequences, where we assume non-selective RF pulses, we model the transmit coil’s spectral response as a Lorentzian curve with a 42 kHz bandwidth at FWHM, emulating the response curve obtained in RF coil simulations.

2.3.3 |. Analysis

We validated the 1^st-order approximation with an error analysis for both field configurations. To this end, we calculated the signal of two individual k-space points with the 0^th- and 1^st-order approximation and various grid-cube sizes ranging between 0.2 mm and 10 mm. In the absence of a ground truth (a numerical solution of Eq. (1) is unfeasible), we considered the 1^st-order approximation at 0.2 mm as a reference and calculated the relative error of all simulations with respect to this reference. The error analysis was performed at the example of a 2D gradient echo sequence with a readout duration $t_{\text{ADC}}=3\text{ms}$, an in-plane resolution of 3 mm, a FOV = 420 mm, and a 5 mm slice cutting through the isocenter.

Next, we compared images resulting from simulations with 0^th- and 1^st-order approximation, using 1 mm and 3 mm grid-cube sizes for the simulation, while keeping the imaging resolution at 3 mm in-plane and the slice thickness at 5 mm. For this analysis, we also used a 2D gradient echo sequence. The corresponding k-space plots for these simulated images can be found in the Supplemental Material.

For the prostate scanner design, we additionally simulated other commonly used sequences, namely 3D gradient echo, 2D turbo spin echo (turbo factor = 2), and 2D single-shot spin-echo EPI sequences. For the 2D sequences we use FOV = 420 mm, $t_{\text{ADC}}=3\text{ms}$, slice thickness = 5 mm, slice z location = 0 mm (isocenter), in-plane resolution = 3 mm. For the 3D sequence, we used an isotropic resolution = 3 mm and the same parameters otherwise. The corresponding k-space plots for these simulated images can be found in the Supplemental Material.

Last, we used reduced-FOV imaging to highlight the 1^st-order approximation’s ability to capture intravoxel dephasing, which attenuates the signal in regions of strong magnetic field gradients. We used the example of a 2D gradient echo sequence with the FOVs of 420 mm and 120 mm, respectively, and the above-described parameters otherwise.

3 |. RESULTS

3.1 |. Saddle-shaped ${\text{B}}_0$ field

The error analysis in Fig. 4 indicates that the 0^th-order approximation requires a grid size of 0.3 mm or smaller for a simulation of the k-space center with a relative error smaller than 0.1 (10%). For coarser grids, the error is around 1 and above. By contrast, the relative error for the 1^st-order approximation is below 0.1 for grid-cube sizes up to 3 mm, and it stays below 1 for grid-cube sizes up to 10 mm. The limitations of the 0^th-order approximation are even more pronounced for signals in the periphery of k-space, where the relative error is well above 1 at all grid sizes. In contrast, the relative error for the 1^st-order approximation remains around 1 or less for all grid-cube sizes.

FIGURE 4.

Relative error of the simulated k-space signal for the saddle-shaped $B_0$ field. The relative error is given by $\left|s_{\text{simulated}}-s_0\right|/\left|s_0\right|$ where $s_0$ is the signal computed using the 1^st-order approximation with 0.2 mm grid-cube size, which we use as a reference. Here, we simulated a 2D gradient echo sequence with a 3 ms readout, 5 mm slice thickness, 3 mm in-plane resolution, and a 420 mm isotropic FOV. Note the logarithmic scaling.

Images reconstructed from the full set of simulated k-space data (Fig. 5) further confirm these trends. Using the 0^th-order approximations at a cube size of 1 mm results in noise-like artifacts that obscure the prostate area. These artifacts are amplified when simulating the signal with a 3 mm grid-cube size. The 1^st-order approximation resolves these artifacts, and the resulting images are characterized by a marked reduction of signal outside the central homogeneous field area due to intravoxel dephasing. In this particular example, the images with both cube sizes are very similar, which suggests that it is sufficient to use a grid-cube size equal to the imaging resolution when using the 1^st-order approximation.

FIGURE 5.

Effect of grid-cube size on simulated images at the example of the saddle-shaped $B_0$ field. Here, we simulated a 2D gradient echo sequence that encodes an axial slice that cuts through the isocenter. The imaging resolution is 3 mm in plane with a 5 mm slice thickness.

3.2 |. Prostate scanner

Fig. 6 repeats the error analysis for the prostate scanner. In the center of k-space, the relative error of the 0^th-order approximation exceeds 1 (100%) for grid sizes of 0.6 mm and coarser. By contrast, the 1^st-order approximation has a relative error below 0.1 for all grid-cube sizes tested. Once again, the error of the 0^th-order approximation is even larger at the periphery of k-space, exceeding 1 even at a grid size of 0.2 mm, and 10 for grid sizes above 0.6 mm. The 1^st-order approximation has a relative error of less than 0.1 for grid sizes below 3 mm, and less than or equal to 1 up to a grid size of 10 mm.

FIGURE 6.

Relative error of the simulated k-space signal for the prostate scanner design. The relative error is given by $\left|s_{\text{simulated}}-s_0\right|/\left|s_0\right|$ where $s_0$ is the signal computed using the 1^st-order approximation with 0.2 mm grid-cube size, which we use as a reference. Here, we simulated a 2D gradient echo sequence with a 3 ms readout, 5 mm slice thickness, 3 mm in-plane resolution, and a 420 mm isotropic FOV. Note the logarithmic scaling.

Images reconstructed from the full set of simulated k-space data (Fig. 7) confirm these trends. Using the 0^th-order approximations at a cube size of 1 mm results in noise-like artifacts that are substantially amplified when simulating the signal with a 3 mm grid-cube size. The 1^st-order approximation resolves these artifacts and the resulting images are characterized by marked intravoxel dephasing outside of the central sphere. Similar to the previous result in Fig. 5, the images with both cube sizes are virtually identical, which suggests that it is sufficient to use a grid-cube size equal to the imaging resolution when using the 1^st-order approximation.

FIGURE 7.

Effect of grid-cube size on simulated images using the prostate scanner design. The imaged slice cuts through the isocenter at a 45-degree angle (axial to coronal) in anatomical coordinates. We simulated the same 2D gradient echo pulse sequence (cf. Fig. 5).

Fig. 8 translates the above findings to a few exemplary pulse sequences (additional results from the 2D Spin Echo and 2D Gradient Echo EPI are in the Supplemental Material, Fig. S1). Using the 0^th-order approximation consistently results in noise-like artifacts. The 1^st-order approximation overcomes these artifacts and reveals key properties of each pulse sequence and their manifestation in the presence of severe field inhomogeneities. For example, the spin echo sequence refocuses the outer-volume signal, which results in strong artifacts, including both geometrical distortion and aliasing associated with the non-bijective encoding fields. The 1^st-order simulation of a spin echo EPI sequence exhibits severe blurring along the phase encoding direction, which reflects dephasing and signal attenuation while acquiring the outer lines of k-space.

FIGURE 8.

Comparison of different pulse sequences, simulated for the prostate scanner design. Here, we depict a slice that cuts through the isocenter at a 45-degree angle (axial to coronal) in anatomical coordinates. The 2D sequences were simulated with a 3 mm in-plane resolution, a 5 mm slice thickness, and a 3 mm simulation grid-cube size. The 3D gradient echo was simulated with a 3 mm isotropic resolution and the same parameters otherwise. Simulations of a 2D spin echo and gradient echo EPI sequence can be found in the Supporting Fig. 1.

Fig. 9 demonstrates that the outer-volume suppression observed in gradient echo pulse sequences can be utilized for reducing the field of view in the pulse sequence without fold-over artifacts. While the 1^st-order approximation allows for an assessment of the minimal FOV, the noise-like artifacts in the 0^th-order approximation hamper such an assessment substantially.

FIGURE 9.

Simulations of zoomed-FOV imaging experiments with the prostate scanner. We simulated a 2D gradient echo sequence with a 3 mm in-plane resolution, a 5 mm slice thickness, and a 3 mm simulation grid-cube size.

4 |. DISCUSSION

In this work, we expand established MRI simulation approaches to efficiently capture the effects of strong magnetic field variations. We showed that the presence of highly inhomogeneous magnetic fields and non-linear gradients results in severe artifacts when approximating the magnetic fields to the 0^th-order (piecewise constant). The 0^th-order approximation does not capture the severe intravoxel dephasing in regions with steep field gradients. Instead, it entails spurious constructive interference of signals from neighboring voxels, manifesting as “ringing” or noise-like artifacts in the simulated images. Importantly, these artifacts are not expected to appear in actual MR images; they are merely a consequence of an oversimplified simulation. In contrast, the 1^st-order approximation models the signal more adequately, capturing intravoxel dephasing and yielding simulations with physically plausible outer-volume signal suppression (cf. Figs. 5 and 7).

When using the 0^th-order approximation, the errors in the simulated k-space signal can be substantial: in the prostate scanner example, over 100-fold relative error was observed for the signal at the edge of k-space when simulated with a 2 mm cube size. Finer grids entail a severe computational burden, which grows with the cubic inverse of the grid size. For example, our simulations with 3 mm resolution, a 3 mm grid size, and using the 0^th-order approximation, took approximately 6 CPU-hours. At a 1 mm grid size, they took approximately 158 CPU-hours, which corresponds to the expected 3³-fold increase. Extrapolating to a grid size of 0.2 mm results in 820 CPU-days. Parallelizing, for example, on 400 CPUs, the simulation of a single image would require approximately 20 days, rendering accurate simulations with the 0^th-order approximation infeasible for many applications.

We overcome this challenge by approximating the $B_0$ and gradient fields to 1^st order (piecewise linear). We showed that the signal equation for each grid cube has an analytic solution under certain conditions (cf. Section 2.1). The 1^st-order approximation captures intravoxel dephasing and allows for a more accurate simulation of spatial encoding in the presence of significant magnetic field variations. In the analyzed examples, the 1^st-order approximation with a grid size equal to the imaging resolution (here 3 mm) had a lower error than the 0^th-order approximation with a 0.2 mm grid size (Fig. 6). Further, we observed no visible image degradation in the 1^st-order approximation between 1 mm and 3 mm grid size (Fig. 7). The latter took approximately 7.6 CPU-hours (11 minutes on a 40-core server), i.e., approximately 25% longer compared to the 0^th-order approximation at the same grid size.

Previous Bloch-based simulators that included a first-order approximation of intravoxel ${\text{B}}_0$ variation demonstrated signal mismapping and dephasing near air-tissue interfaces, but did not explore signal behavior in the presence of highly inhomogeneous applied magnetic fields and non-linear gradient fields, or non-standard field configurations^10,11. Although they allow for the additional simulation of contrast and/or the calculation of specific absorption rate (SAR), these simulations focus on inhomogeneities due to the imaging object itself, demonstrated for inhomogeneities up to 116 ppm.^10,11 Our simulations were built specifically for simulating spatial encoding effects due to highly inhomogeneous fields and nonlinear gradients that could be present in nonstandard, accessible scanners, and we demonstrate their feasibility for inhomogeneities up to 1870 ppm. The recently proposed phase distribution graph¹² framework also allows for the simulation of intravoxel dephasing, capturing the key features of our 3D imaging solution (Eq. 6) where non-selective RF pulses are assumed. The phase distribution graph is, however, currently not capable of capturing selective excitation pulses, which lie at the core of our signal equation for multi-slice 2D imaging (Eq. 8).

4.1 |. Applications

The proposed simulation framework and the provided code (cf. Section 6) have a number of potential applications. First, the 1^st-order approximation allows for an in silico assessment of novel magnet and gradient designs. Capturing intravoxel dephasing effects is essential for avoiding spurious noise-like artifacts that can overshadow true image features and encoding artifacts.

Second, the 1^st-order approximation can be used to test pulse sequences for hardware configurations with severe field inhomogeneities. For example, our results for the prostate scanner design show that a gradient echo sequence results in sufficient signal dropout to enable imaging with a field of view smaller than the body dimensions. In contrast, the spin echo sequence refocuses some of the signal, requiring a larger field of view to avoid aliasing. Using the standard 0^th-order approximation, the difference between the gradient- and spin echo sequences is obscured since intravoxel dynamics are not captured.

Third, our simulator can be used to degrade existing clinical MR images and explore machine-learning approaches for the reconstruction, processing, and interpretation of low-cost MR images.¹³ Building low-cost scanners includes challenges in spatial encoding in addition to reduced SNR. Our simulator uses existing images as inputs instead of body models to predict the effects on spatial encoding. This choice enables us to leverage existing databases of MR images rather than building large sets of labor-intensive and generally resolution-limited body models. As a result, the simulator can be used to generate training datasets of appreciable size and diversity, for machine learning and other applications. Such datasets have the advantage of being derived from a clinical population with natural anatomic variation as well as a realistic prevalence of pathology, which can be challenging to simulate accurately in purely synthetic body models. While our framework, as presented here, does not explicitly include SNR effects associated with low-field operation, such effects have been studied extensively and are comparatively straightforward to include in an additional simulation step.^2,13

4.2 |. Limitations

The proposed 1^st-order approximation would be comparatively easy to incorporate into a range of existing sequence simulators that currently employ the 0^th-order approximation. While the 1^st-order approximation substantially improves accuracy by better capturing intravoxel dynamics, its efficiency currently relies upon the existence of analytic solutions to the signal equation. This requirement limits the scope of the simulations by requiring some approximations and certain assumptions about the pulse sequence. Here we provided practical solutions for two cases. For 2D sequences, we found an analytic solution for Gaussian excitation pulses, and an analytic solution for more widespread pulses, such as filtered sinc pulses, might be unattainable. Further, the solution requires either a small-tip angle approximation (used here for gradient echo pulse sequences) or a uniform excitation field (used here for the refocusing pulses in spin echo pulse sequences). For 3D sequences with non-selective excitation pulses, the analytic solution has fewer requirements, the most important being the assumption of a piecewise constant spectral response of the RF coil. This approximation is sufficient in many cases, given the inherent smoothness of RF fields and the comparatively slow variations of RF sensitivities in space and frequency.

Our simulation framework ignores contrast-generating effects and, instead, we use existing, high-quality MR images with a predefined contrast as inputs. We note that some contrast-generating effects might manifest differently in inhomogeneous magnetic fields. Foremost, diffusion in strong gradient fields could result in further outer-volume suppression, in particular in the spin echo pulse sequences where we currently assume perfect refocusing. (cf. Fig. 8). In order to provide an order of magnitude for the diffusion effects, we can extract a representative gradient amplitude of 0.3 T/m from the right panel in Fig. 3 (dotteted line between −20 cm and −15 cm) and assume this gradient to be constant over time. In a spin echo sequence with $T_E=10\text{ms}$, this would equate to a b-value of approximately 650 s/mm².

While we capture the effect of concomitant fields^14–19 on the Larmor frequency, we currently neglect spatial and temporal variations in the precession axis by taking the ${\ell }_2$-norm in Eq. (2). Such variations could be incorporated with a vector notation of ${\mathbf{\text{B}}}_1^{-}\left(\mathbf{\text{r}}\right)$ and by projecting this vector onto the precession axis (cf. Eq. (2)). Note that temporal variations of the axis entail the necessity of a time-dependent projected $B_1^{-}(\mathbf{\text{r}},t)$.

Further, we assume gradient switching and RF pulses are instantaneous, and we assume perfect spoiling after each TR, i.e., we do not simulate coherence pathways. Such sequence details foremost affect the contrast and have only a secondary effect on spatial encoding, which motivated us to neglect them. Last, we do not simulate changes in the SNR. However, noise at the appropriate level can simply be added to the simulated k-space signal.

Lastly, while the 1^st-order approximation substantially improves accuracy compared to the conventional 0^th-order approximation, we note that actual magnetic–field distributions——especially in nonstandard configurations or fringe fields—may exhibit significant higher-order terms. In principle, extending the model to 2^nd-order terms would further improve the accuracy. However, such an extension renders the signal equation analytically intractable for most practical pulse sequences.

5 |. CONCLUSION

We presented a novel method simulating the spatial encoding with highly variable ${\text{B}}_0$ and gradient fields. We derived analytical solutions to the MR signal equation under the assumption of piecewise linear fields, as opposed to the conventional piecewise constant approximation. We demonstrated improved accuracy in capturing intravoxel dynamics and exemplified the benefits with the example of two magnetic field configurations. Our results underscore that accounting for intravoxel field variation facilitates simulations on a coarser grid, which can improve computational feasibility.

6 |. DATA AVAILABILITY STATEMENT

The simulation software presented in this work is available on https://github.com/radhikatibrewala/1stOrderMRISims. The package is implemented in the Julia language. The repository includes the saddle-shape $B_0$ field configuration and the prostate image file used in this work.