Feasibility of diffusion weighting with a local inside-out nonlinear gradient coil for prostate MRI.

PURPOSE
Prostate cancer remains the 2nd leading cancer killer of men, yet it is also a disease with a high rate of overtreatment. Diffusion weighted imaging (DWI) has shown promise as a reliable, grade-sensitive imaging method, but it is limited by low image quality. Currently, DWI quality image is directly related to low gradient amplitudes, since weak gradients must be compensated with long echo times.


METHODS
We propose a new type of MRI accessory, an "inside-out" and nonlinear gradient, whose sole purpose is to deliver diffusion encoding to a region of interest. Performance was simulated in OPERA and the resulting fields were used to simulate DWI with two compartment and kurtosis models. Experiments with a nonlinear head gradient prove the accuracy of DWI and ADC maps diffusion encoded with nonlinear gradients.


RESULTS
Simulations validated thermal and mechanical safety while showing a 5 to 10-fold increase in gradient strength over prostate. With these strengths, lesion CNR in ADC maps approximately doubled for a range of anatomical positions. Proof-of-principle experiments show that spatially varying b-values can be corrected for accurate DWI and ADC.


CONCLUSIONS
Dedicated nonlinear diffusion encoding hardware could improve prostate DWI.


I.A. Prostate DWI
The lifetime risk of being diagnosed with prostate cancer is approximately 1/9 for American men 1-3 , but because of slow progression and the significant morbidities associated with treatment, many men should consider declining, or at least delaying, treatment 4,5 . On the other hand, prostate cancer remains the 2 nd leading cancer killer of men (following only lung cancer), so early detection of aggressive cancers remains an urgent problem. Therefore, one of the greatest needs in prostate cancer is a monitoring method that distinguishes benign or low grade prostate lesions from aggressive ones. Standard template biopsy, which blindly samples 10 + cores in a regular pattern, risks overdiagnosis, underdiagnosis, and urinary and erectile dysfunction, making it a poor option for diagnosis or surveillance 6,7 .
Diffusion-weighted MR imaging (DWI) does noninvasively detect prostate cancers [8][9][10] and the images can be quantified into maps of apparent diffusion coefficient (ADC), which consistently reflect Gleason score and aggressiveness. Therefore, DWI has potential to specifically improve the detection of deadly cancers. Brightness in a DWI image (or darkness in an ADC map, with inverted contrast) reflects restricted water diffusion, so it is sensitive to the decreased luminal space and increased epithelial volume that accompanies malignancy. For men with high PSA (prostate specific antigen), prior MRI, typically can reduce unnecessary biopsies by a third 11 . In addition, these images can be used for biopsy guidance, increasing detection of high grade cancer and reducing biopsies of low grade disease [12][13][14][15] . However, clinical adoption has been limited by the poor image quality observed in routine prostate DWI, which is a key part of the multiparametric protocol. Fig. 1 shows typical image quality from a prostate imaging protocol. While the structural T2w image (Fig. 1a) has good SNR and resolution, it shows little contrast between cancerous and benign tissue. This is not uncommon for peripheral zone lesions, which make up ≈ 75% of cancer cases 8 . The ADC map ( Fig. 1b), though very noisy and low resolution, does show markedly lower ADC in what was later confirmed to be an aggressive lesion. However, it was initially misread as artifact, due to the poor image quality. Images like these result in inconsistent interpretations, with high interreader variability, and limited clinical deployment 16 .
Last edited Date : N ovember25, 2020 Figure 1: Images from a typical prostate MRI underscore the need for better imaging. (a) While a T2w image has high signal and resolution, it often fails to show contrast for peripheral zone tumors, which make up ≈ 75% of cases. (b) In the same patient, a hypointense region in the ADC map (i.e. a region of low diffusion) does distinguish the cancerous region from healthy prostate. But the overall image quality is low due to the low SNR of conventionally encoded diffusion MRI. Because of these shortcomings in prostate imaging, most clinicians rely on blind biopsies of the prostate, which can lead to over-and under-diagnosis of lethal disease.
Prostate DWI images are noisy because diffusion weighting depends exponentially on the square of gradient strength, and limited gradient strength must be compensated for by applying the gradient for very long times. The most widely used and simplistic model of diffusion is for Gaussian unrestricted diffusion, where: where S DW I and S 0 are the signal intensities with and without diffusion weighting, respectively, D is the intrinsic diffusion rate of the spins, analogous to the apparent diffusion coefficient (ADC) observed in vivo, γ is the gyromagnetic ratio, G is the local gradient

I.B. Nonlinear Gradients
Existing gradient hardware generates a Bz magnetic field whose magnitude changes linearly with position (Fig. 2a, red line), so the local field slope or gradient is uniform across the field of view. Nonlinear gradients similarly generate a Bz whose magnitude varies in space (Fig. 2a  Bz , by mathematical definition, provides less gradient in some regions but more gradient in others. For part of the field of view, this results in a much stronger gradient that can be used to impart diffusion weighting. In practice, the advantages of a nonlinear diffusion gradient are even greater, including fewer cancellations which would normally be needed to achieve linearity and potential for inside-out single-sided gradient designs.

APPENDIX I).
Designing a gradient specifically for diffusion weighting, rather than repurposing hardware primarily designed for spatial encoding, brings additional advantages for increasing gradient strength. The usual requirements on slope uniformity, both in magnitude and direction, require many field cancellations which ultimately waste amplifier energy. Reducing the volume over which the field must be generated may also reduce power needs 29 .
In fact, nonlinearity allows for hardware designs that do not encompass a volume at all, like the one presented here. Finally, since diffusion weighting requires just a few ramps per TR, short rise times are less critical. This allows higher inductance, more turns and therefore more field per amp. Alternatively, less voltage is required to achieve the slew rate, Gradients which reduces requirements on the amplifier and lowers the voltage breakdown strength requirements in the gradient coil 29,30 . Linear gradients, which can provide fast switching times and homogeneous spatial encoding over the entire FOV, can still be used for the rest of the pulse sequence, including slice selection, frequency and phase encoding, or any kind of preparation module. Therefore, standard DWI pulse sequences can be modified by simply adding a synchronization pulse that turns the nonlinear gradient on and off during diffusion encoding time. Notably, the nonlinear gradient can be played simultaneously with linear gradients, either for spatial encoding purposes like a prephaser, or to further add to the diffusion encoding gradient. When the spatial encoding is done by conventional means, the reconstruction is similarly conventional and can use regular FFT.

I.C. Nonlinear Gradients for Prostate DWI
Anatomically, DWI encoding with nonlinear gradients (NLG) is most naturally suited to an organ that is (i) small, so that each experiment yields an acceptable range of b-values across the region of interest (ROI), (ii) close to an accessible surface of the body, since gradients are typically strongest near the device, and (iii) structurally isotropic, so that variation in gradient direction is not a major confound. In practice, to justify additional hardware, the DWI contrast would also need to be a critical image for diagnosis. Thus, prostate cancer imaging is an ideal initial target for this type of hardware. The prostate is a small 4cm organ located only ≈ 6cm from the perineal surface with relatively isotropic diffusion, and prostate cancer is a widespread and deadly disease which is best imaged by DWI MRI.
Furthermore, prostate MRI is a rapidly growing application, as it is increasingly recognized as a cost-effective step following elevated PSA [31][32][33][34][35][36][37] . Therefore, this was chosen for a first and ideal organ to benefit from this class of device.
Putting these insights together, we designed an inside-out gradient that provides very high diffusion weighting at the prostate. Standard gradient hardware is generally built into a volume-encompassing former, typically a tube-like shape, and the relevant fields are generated inside the volume of the hardware. In contrast, nonlinear gradients windings can be placed in a wand-type former, generating the relevant fields outside the hardware. The proposed device has a diameter of 10cm, comparable to a typical coffee mug, and can be placed between the upper thighs to achieve close proximity to the prostate. (Fig. 3

) This
Last edited Date : N ovember25, 2020 I.C. Nonlinear Gradients for Prostate DWI geometry also minimizes interference with the standard linear gradients, which can be used for spatial encoding in the usual way, and standard receiver geometry, which is typically a set of anteriorly and/or posteriorly positioned surface coils.

II.A. Simulations
From analyses of body MRI scans, the prostate is approximately a 4cm sphere whose lower edge is 6cm superior from the perineal surface.The presented prototype was designed by Tesla Engineering Ltd and simulated using OPERA Electromagnetic Simulation Software Due to the spatially varying b-value in nonlinear experiments, the raw data from these scans also exhibits spatially varying contrast and spatially varying SNR. While the nonlinear scans can be used to make DWI with uniform b-value (as shown in Fig. 7), the fidelity of those images is ultimately determined by the fidelity of the ADC map used for scaling. Therefore, is the mean and σ is the standard deviation over each ROI. In simulations, since the exact delineation of healthy prostate and lesions were known, these were used as the ROIs for each tissue type.

II.B. Experiments
To prove the feasibility of encoding diffusion with a nonlinear gradient, initial experiments were performed with an existing nonlinear gradient, a volume encompassing head coil that produces a C3 field shape Bz(x, y) = x 3 − 3xy 2 . These studies imaged a kiwi, which has been lauded as a highly concordant phantom for prostate DWI 38   Field mapping was performed by incrementing the duration of a single gradient lobe (1.5-2ms) at low amplitude to avoid intravoxel dephasing, holding TE fixed at 20ms. The observed phased progression was used to calculate local frequency as ω(x, y) = ∆Φ(x, y)/∆t, and this was used to generate a map of the static field, which was then masked and fit to ta polynomial to generate a field map across the entire FOV. The local gradient was calculated  Fig. 3a shows the general positioning and the field generated by the device, and Fig. 3b shows a mock-up of the device placed on the scanner bed. A patient would be positioned with the device externally nestled between the legs, and a receiver coil would be placed above these. Water and power cables would extend in the foot direction.

III.A. Simulation of Hardware Performance
Simulations were also performed on this design to assess thermal and mechanical properties. Because the device is designed for placement at isocenter in the homogeneous region of the magnet, net forces due to device operation are expected to be negligible. Even for Last edited Date : N ovember25, 2020 misplacement up to ±1cm, forces are less than 1N due to symmetry. Even in a maximally catastrophic case of complete symmetry loss at 100% current, forces on the device reach < 500N, which can be restrained with adequately designed scaffolding. Improper positioning could also affect image quality through coupling to the linear gradients, but preliminary analysis indicates that a positioning tolerance of ±2mm is both sufficient and achievable.
The device is cooled and insulated such that surface temperature at maximum duty cycle will not exceed 42 • C. Other key metrics are reported in Table 1.

Gradient Specifications
Diameter     With nonlinear gradients, position of the prostate relative to the hardware could potentially affect performance, as a longer diffusion time is required to achieve sufficient weighting. However, simulations show that this dependence is relatively minor, and CNR remains well above that observed with linear gradients.

III.C. Experimental Demonstration of DWI with NLG
One key to achieving an order of magnitude increase in gradient strength is accepting nonlinearity in the field, which introduces nonuniform diffusion weighting across the image and could theoretically introduce bias in an ADC map. Thus we sought to show initial feasibility of DWI with nonlinear gradients using nonlinear gradient hardware available at our center [44][45][46][47][48] . Though the geometry of this device is not suitable for prostate imaging, the degree of nonlinearity is comparable and these results provide proof of principle demonstrations for encoding diffusion with a nonlinear field.  shows DWI images using a nonlinear C3 gradient and scaled to a uniform b-value. Standard images acquired by using linear gradients at these b-values (b) are shown for comparison and show excellent agreement. In the raw NLG images (c), the nonuniform diffusion weighting can be observed in the corners highlighted by blue arrows. Fitting this data to a monoexponential model yields ADC maps (d) that are in good agreement with those from linear gradients (e). Mapping S 0 from these monoexponential fits (f,g) shows that the shorter echo time afforded by nonlinear gradients results in about a doubling of signal. Furthermore, a comparison of image uniformity over the ROIs shown in (h) suggests the higher SNR of nonlinear DWI translates to better precision (i,j).
(TE=98), the maximum b-values were less than 200s/mm 2 , so ADC could not be accurately calculated. However, for the remaining vials, ADC agreement was very high. In these studies, because of the phantom geometry and the shape of the field, most of the vials required longer gradient durations for nonlinear gradient experiments. Thus, SNR of the underlying images and accuracy of the measured ADC was not necessarily better. However, these results provide reassurance that the diffusion contrast from nonlinear gradients closely matches that of linear gradients.

IV. Discussion
Because the device is meant only to deliver diffusion weighting, image encoding and reconstruction are unaffected. By focusing on ∇Bz over the anatomy of interest, the concept presented here could be comparable in price to other anatomically specific MRI accessories, such as RF coils, and be portable between scanners. Yet this accessory would deliver an order of magnitude increase in gradient strength and double contrast to noise ratios in prostate cancer detection.
The presented design shows that by abandoning design specifications required for imaging, particularly unidirectional linearity of the gradient and cylindrical geometry, the local magnitude and diffusion encoding can be greatly improved. Recently, this potential has also been explored in a volume-encompassing design for breast imaging 49 . Our experimental results show that nonlinear gradients can produce diffusion weighting that is qualitatively and quantitatively comparable to standard methods, even for phantoms whose geometry is
Several features make this approach highly feasible with minimal changes to an existing MRI system. Because diffusion waveforms for prostate DWI do not require a high degree of flexibility, a dedicated DWI gradient would only require the addition of pulse sequence synchronization pulses which then trigger a very simple waveform generator. Furthermore, while initial implementations will feed these waveforms to a dedicated amplifier, a DWI gradient of this kind could share the amplifier of one of the linear gradients via a switch.
In either case, the rest of the pulse sequence, including excitation with the body coil, image encoding with linear gradients, reception with standard RF arrays, and Fourier transform image reconstruction, proceeds unmodified. During the diffusion encoding module, the linear gradients could either be turned off or run in tandem, to achieve still higher local gradient or effectively move the region targeted for optimal diffusion weighting.
One concern with nonlinear gradients is the sensitivity of CNR to position. If the prostate is too close to the device, there is higher gradient variability and additional DWI images might be needed to get adequate diffusion encoding across the entire prostate. Alternately, if the prostate is too far from the device, there may be concern that the gradient is too weak to yield a significant benefit. While more sophisticated algorithms are possible for optimizing gradient durations and amplitudes, the presented simulations show that even a rudimentary algorithm for modifying ∆ as a function of position can achieve very high CNR for a range of anatomical placements.
Another consideration is whether the nonlinearity in the gradient affects the calculated ADC value. In the present design, nonlinearity over a 1mm voxel with a median gradient of 500mT/m contains deviations of approximately ±10mT/m or 2%, but the effects of this deviation partially cancel across the voxel 46 . This amount of gradient variation across a voxel is comparable to that present in our experimental work, which achieved good agreement using just the midpoint gradient value for the b-value of each voxel. However, more realistic assignments of local b-value may show better performance.
In addition, the dependence of ADC values on TE, diffusion time and b-value is an active area of research, and several models have been proposed. We have tried to address some Finally, while mechanical and thermal safety concerns were discussed in the Results, one

IV. DISCUSSION
important safety concern may be the potential for peripheral nerve stimulation. Achievable gradient moments in clinical imaging are often restricted by peripheral nerve stimulation (PNS) limits on the slew rate, but those limits are based on studies with full body gradients.
More recent studies of PNS show that the overlap of anatomy with induced currents is the real key to PNS stimulation, and due to their highly localized nature, nonlinear gradients have long been hypothesized to produce less PNS for a given dB/dt 26,58 . Furthermore, for this particular design, the most serious potential outcomes of PNS (i.e. stimulation of currents across the heart) are avoided, since the field and gradient are very small at that distance.
While true PNS limits are complex phenomena and will require additional testing for confirmation, the geometry of this device also provides important new degrees of freedom for mitigating PNS. In addition to lengthening ramp times, PNS can be further mitigated by shifting device placement a few centimeters. Last edited Date : N ovember25, 2020 Figure 9: dB/dt for nonlinear gradient. Peripheral nerve stimulation is a complicated phenomenon which is made more difficult to predict in the presence of nonlinear gradients. However, one advantage of the hardware proposed here is that PNS can be mitigated by device placement. This plot shows dB/dt for a 1ms ramp time with an offset of 120mm from the prostate. The standard 20T/s slew is exceeded over a very small volume of the anatomy, reaching a maximum of 43T/s along the x=0 line, with reductions in leg stimulation achievable by spacers or positioning. Notably, simulations for this positioning (Fig. 5) show that CNR is still approximately twice what is achievable with linear gradients.

V. Conclusion
We present a new approach to improving DWI of prostate, introducing a new type of MRI accessory that delivers high diffusion weighting at short echo time, resulting in much stronger gradients and higher SNR. Prostate is anatomically ideal for diffusion weighting via a nonlinear inside-out gradient, and the clinical significance and increasing ubiquity of prostate MRI is likely to justify the additional hardware, which could be shared across many scanners. Our results predict a doubling of CNR, which could improve the reliability of MRI for prostate cancer imaging, which in turn could reduce both the high mortality and overtreatment rates for this widespread disease.

VI.A. Theory
The simplest calculation of diffusion weighting is to consider two infinitely short gradient pulses separated by a delay τ . To simplify notation and without loss of generality, we consider spin with initial location x(t = 0) = φ(t = 0) = 0, so that ∆x = x and ∆φ = φ 0 . Over the delay, the spins have localized to a Gaussian distribution of positions, where σ is related to the rms distance travel in time τ . For linear gradients, G l , after the refocusing gradient lobe, the phase of each spin depends on its positions at t = τ , φ(t = τ, x) ≡ φ(x) = k l τ x = k l x and dφ = k l dx. We can therefore calculate the ensemble average of magnetization M . Assuming uniform spin density, the ensemble average of M is proportional to the ensemble average of e iφ , which can be reformulated as an integral over x, as follows.
The product σ 2 k 2 l reflects gradient winding, diffusion time, and diffusion rate. Thus the exponent is analogous to the product b ADC in the well-known Stejskal-Tanner equation 18 .
Using the same simplified approach for a spin in the presence of both linear and a second order nonlinear field, the difference is in the relationship between φ and x: φ(t = τ, x) ≡ φ(x) = G l τ x+G nl τ x 2 = k l x+k nl x 2 and dφ = (k l +2k nl x)dx Now the integral to be evaluated is, σ 2 e i(k l x+k nl x 2 ) (k l + 2k nl x)dx. Rearranging, one can obtain: Simplifying and removing complex values from the denominators, this becomes: As expected, when k nl is zero this matches the linear gradient expression. More, generally, neglecting nonlinear winding will cause a slight underestimation of the effective diffusion weight, which will lead to a slight overestimation of ADCs by a scaling factor of (1 + σ 4 k 2 nl ). This could bias ADCs by a few percent, if not corrected. The imaginary term in the exponent indicates that nonlinear winding can also add a periodic phase shift in the signal, similar to what is observed with dephasing under a nonlinear gradient 41 .
The complex factor in braces can alter both the magnitude and phase of the signal, in principle. However, in practice σ ≈ 10 −2 cm and k nl ≈ 10 2 -10 3 Hz/cm 2 , over which the magnitude of this term ranges from 1 to 0.993 and its phase changes by less than 9 degrees. These corrections are highly unlikely to have clinical significance even in high precision work.