The misunderstood meander: Redesigning MRI meander-line surface coils to reduce noise, increase uniformity, and eliminate image artifacts

Abstract

Meander-line, or zig-zag, MRI surface coils theoretically promise spatially uniform fields with optimal field localization close to the coil. In reality, they suffer poorer than expected field localizations and acquired images are often highly inhomogeneous, plagued by repeating stripe-like signal-loss artifacts. We show that both these detrimental effects arise from coil design based on the same invalid approximation in the underlying theory. Here, the conventional approximation is corrected, yielding a modified coil design that validates the new theory by rectifying the above problems. Specifically, an easily implementable coil correction, which amounts to the addition of a single extra turn of wire, is introduced and shown to increase signal uniformity by an order of magnitude, eliminate image artifacts, and reduce unwanted signal interference from deeper within the sample by tightening the coil field localization to close to the coil, as intended for zig-zag designs. With independent optimization of coil size and imaging depth possible, such corrected meander-lines surface coils may be well suited for large area, near-surface imaging and spectroscopy applications.

Graphical Abstract

Introduction

Surface coils surpass volume coils for imaging and spectroscopy applications near surfaces, affording better sample accessibility and improved local signal-to-noise ratios (SNR) with inherently greater fill factors[1,2]. But planar surface coils lag volume coils in terms of B₁ field uniformity. Since surface coils are often intended for imaging near the surface, lack of B₁ uniformity normal to the plane of the coil is not necessarily problematic; it may even be advantageous for sample slice selection without imaging gradients or special pulse localization sequences[3–6], or for maximizing sensitivity by restricting B₁ to only near the plane of interest. B₁ non-uniformity within imaging planes parallel to the plane of the coil, however, can be a serious drawback, introducing undesired spatially varying excitations and receive sensitivities. Optimal imaging and spectroscopy with most simple loop coils has thus often been restricted to a region, of the order of the coil diameter, that is offset from the coil by a distance also of the order of the coil diameter where B₁ is locally more uniform. However, this coupling of image area and depth impairs sensitivity very close to the coil for near-surface MRI imaging and NMR spectroscopy. One solution is the use of dense arrays containing many small loop or microstrip coils, with multi-channel coil arrays achieving good sensitivity both close to and far from the overall array,

A simpler alternative to a massive array of small coils, however, is the single-channel, single-wire meander-line coil. Such planar meander-line, or “zig-zag” or “serpentine” coils[7–12], are designed to achieve highest sensitivity close to the coil, decoupling imaging area and imaging depth, with imaging area given by overall coil size, but imaging depth set independently by the spacing between the neighboring, alternating, current paths comprising the meander (as detailed below). In theory, such coils should also offer high fill factors and highly uniform B₁ fields over flat planes of effectively unlimited area parallel to the plane of the coil. This planar, large-area uniformity should make meander-line geometries ideal choices for coils intended to image structures close to the surface. In reality, however, meander-line coils have yet to gain much acceptance in NMR or MRI, arguably because existing designs achieve neither their promised field uniformities nor fill factors.

While favorable theoretical predictions regarding such coils’ uniform fields exist[12,13], corresponding experimentally measured B₁ fields that support this theory are generally absent. As explained below, in addition to poorer than expected fill factors, the actual B₁ uniformities of typical planar meander-line coils are no better --- and are often worse --- than alternative, more common surface coil geometries. This paper details (i) theoretical predictions of meander-line coils and why they have been misinterpreted or misapplied, (ii) how present designs of meander coils adversely affect imaging or spectroscopy, and (iii) how meander coils can be redesigned to better match theory and increase B₁ field uniformity and localization close to the coil. It is suggested that such corrected meander-line geometries may form ideal elements for large-area NMR / MRI detectors optimized for near-surface imaging or spectroscopy applications including, for example, MRI of the skin or NMR spectroscopy of layered cell cultures, or various single-sided magnet applications.

Theory

A meander-line coil consists of a continuous wire looped back and forth to create a number of connected parallel wire segments in which the current flows in alternate directions (Fig. 1). Such meander-line wire geometries are not unique to NMR or MRI; they have been used in related domains including nuclear quadrupole resonance (NQR)[13] and ferromagnetic resonance[14] as well as in other fields including electromagnetic surface acoustic wave transduction[15] and atomic physics where they are used as magnetic mirrors[16]. The fields of such coils are generally approximated by the quasi-static fields of coils with infinitely many wire segments. For such infinite meander-line coils, the fields decay exponentially away from the coil surface and, in planes parallel to the plane of coil, become increasingly homogeneous with increasing distance from the coil. Both properties are ideal for a surface coil. The exponential field decay implies high filling factors for imaging close to the surface while the homogeneous field implies uniform excitation and reception sensitivity for imaging over large areas. The natural assumption that these properties apply at least approximately to meander-line coils with finite, rather than infinite, numbers of wire segments, is often implicit in meander-line coil designs. However, in many situations this assumption is untrue. In particular, unlike for an infinite coil, the field homogeneity of a finite meander-line coil can worsen with increasing distance from the coil.

Figure 1. Schematic of a meander-line coil geometry.

A typical planar coil comprises an even number of lines, here N = 10, connected in series with current flowing antiparallel in neighboring wire segments. Wire-to-wire spacings and spatial periodicity of the current are a and 2a, respectively. Depending on operating frequency, possible required capacitance splits are omitted for simplicity.

Although such initially counterintuitive effects of finite coil size have been recognized in atomic physics fields[17–18], these problems have not been taken into account in the design of NMR or MRI meander-line coils. The theory presented here adapts the commonly applicable atomic physics insights into terms relevant to radio-frequency (RF) coil design and expands upon them in areas unique to NMR/MRI.

While exact current distributions in the coil wires, and hence the exact fields of the coil, will depend on the operating NMR/MRI RF frequency, the mismatch between infinite and finite coils is dependent only on overall coil geometry. Therefore, we present only an abbreviated, quasi-static derivation for a general infinite system of antiparallel conductors before addressing why a finite one differs so substantially.

In the free space surrounding the coil, the vector field of an infinite coil $\stackrel{\rightharpoonup }{B_{\infty }}\equiv \left\{B_{\infty ,x},B_{\infty ,y},B_{\infty ,z}\right\}$ can be represented by a magnetic scalar potential, $\stackrel{\rightharpoonup }{B_{\infty }}=-\nabla {\phi }_{\text{m}}$ that satisfies the Laplace equation, ${\nabla }^2{\phi }_{\text{m}}=0$. Assuming a coil oriented with its main current lines parallel to the scanner B₀ field along the z-direction (and ignoring small contributions from short turning points at the end of each straight wire section), $B_{\infty ,z}=0$ and the Laplace equation simplifies to

$$\frac{{\partial }^2}{\partial x^2}{\phi }_{\text{m}}=-\frac{{\partial }^2}{\partial y^2}{\phi }_{\text{m}}$$ (1)

Given the spatially periodic current flow in a meander-line coil, φ_m must be periodic in x. Fourier expanded, the scalar potential therefore takes the form:

$${\phi }_{\text{m}}=\sum _{n=1}^{\infty }\mathrm{Re}\left[{\varphi }_n{e}^{-iknx}\cdot \text{Φ}(y)\right]$$ (2)

where the wavenumber $k\equiv \frac{2\pi }{2a}$ is inversely proportional to the spatial period, 2a, of the coil wires and where for simplicity any phase angle dependence on the relative x-origin position is omitted. The Fourier coefficients, φ_n, depend on the current distributions within the coil wires and therefore on RF effects, but their exact values are unimportant to the overall field scaling, allowing a simpler quasi-static analysis throughout. The above complex notation helps expose the intrinsic connection between in-plane sinusoidal, and out-of-plane exponential fields because the Laplace equation is immediately satisfied if the y-dependence, Φ(y), mimics the x-dependence except for an $i\equiv \sqrt{-1}$ factor. The scalar potential φ_m and vector B_∞ field therefore become

$$\begin{gathered} {\phi }_{\text{m}}=\sum _{n=1}^{\infty }\mathrm{Re}\left[{\varphi }_n{e}^{-inkx}\cdot e^{-nky}\right] \\ \stackrel{\rightharpoonup }{B_{\infty }}=\sum _{n=1}^{\infty }{\varphi }_{n}nk\cdot \{\sin (nkx),\cos (nkx),0\}\cdot e^{-nky} \end{gathered}$$ (3)

Each sinusoidally varying component of the coil field decays exponentially normal to the plane of the coil with decay constant inversely proportional to its spatial period. Since the current flow is anti-parallel in neighboring wires, all n = even terms vanish, yielding a B_∞ magnitude:

$$\left|B_{\infty }\right|={\varphi }_{1}k{e}^{-ky}\cdot \sqrt{1+6\frac{{\varphi }_3}{{\varphi }_1}\cos (2kx)e^{-2ky}+\text{terms involving}\cos (4kx)e^{-4ky},\cos (6kx)e^{-6ky}\dots }$$ (4)

Note that even though the field direction rotates across the coil, for a coil that generates a pure sinusoidal field above its surface, only the first term remains, leaving a uniform field magnitude across any plane parallel to the coil. For more realistic coils that do not generate pure sinusoidal fields, the higher field harmonics decay increasingly rapidly leaving a field again dominated by the leading harmonic a short distance from the coil surface. This field magnitude is again completely uniform parallel to the coil. These flat isosurfaces of B_∞ field magnitude distinguish meander-lines from other surface coils and are a key motivation for their use. Another principal advantage over other single-channel coils is the ability to control image offset distances independently of coil size. With a coil field exponential decay length given by $\frac{1}{k}\equiv \frac{a}{\pi }$ (which equals the spatial period divided by 2π), wire spacings and numbers can be adjusted to match desired image depths while keeping coil size and field of view (FOV) constant.

Conventional theory implicitly assumes that the coil field, $\stackrel{\rightharpoonup }{B_1}$, is proportional to $\stackrel{\rightharpoonup }{B_{\infty }}$. The difference between a finite and infinite coil, however, is that a finite coil lacks the field contributions from additional alternating current-carrying wires that would have extended out to infinity from each of its edges. The missing field from these missing wires, $\stackrel{\rightharpoonup }{B_{\text{err}}}$, can be regarded as the error in approximating a finite coil as an infinite one. Or, mathematically, the field of an actual coil can be expressed as ${\stackrel{\rightharpoonup }{B}}_1\equiv {\stackrel{\rightharpoonup }{B}}_{\infty }-{\stackrel{\rightharpoonup }{B}}_{\text{err}}$. Because the field of a straight current-carrying wire varies inversely with distance, the magnitude of the “missing” field above the center of a finite coil with N wires due to the “missing” wires off one edge is

$$\left|B_{\text{err}}\right|=\left|\sum _{n=1}^{\infty }\frac{{(-1)}^n}{a\left(\frac{N}{2}+n\right)}\right|\cdot \frac{{\mu }_{0}i}{2\pi }$$ (5)

for free-space permeability, μ₀, and coil current, i. For large N this asymptotically approaches

$$\left|B_{\text{err}}\right|\approx \frac{1/2}{\frac{aN}{2}}\cdot \frac{{\mu }_{0}i}{2\pi }$$ (6)

Because aN/2 is half the coil width, for a coil with a large number of wires the infinite arrays of missing wires on each side of the coil contribute approximately the same field at the coil center as would two single missing wires at the coil edges that each carry half the coil current. The infinite theory therefore fails completely at distances, $y_{\text{finite}}$, from the coil where the edge error fields B_err overwhelm the B_∞ field of an infinite coil:

$$y_{\text{finite}}\approx \frac{1}{k}\ln \left(\frac{{\varphi }_1{\pi }^2}{{\mu }_{0}i}N\right)\approx (1+\ln N)\cdot \frac{a}{\pi }$$ (7)

Here, the logarithmic dependence results from the fast exponentially decaying B_∞ field competing against the more gradually decaying B_err fields. Full calculation would require the value of φ₁, which depends on wire widths, spacings and internal current distributions (and hence also on RF frequency, skin depth, and proximity effects), but numerical calculations show that $\frac{{\varphi }_1\pi }{{\mu }_{0}i}$ is always of order unity, affording the approximate analytic scaling above. Alternatively, recasting in terms of overall coil width adds the inverse dependence on coil wire number:

$$y_{\text{finite}}\approx \frac{1+\ln N}{N}\cdot \frac{\text{coil width}}{\pi }$$ (8)

While it is obvious that the infinite approximation must stop working at some distance from a finite coil, what can be initially surprising—and a key reason behind misapplication of conventional meander-line theory—is how near to the coil surface the infinite theory breaks down. Because of the non-linear decay of B_∞, the intuitive assumption that this critical distance from the coil should scale as some fraction of the coil size, fails. The dependence of $y_{\text{finite}}$ on only the number of coil wires and the wire-to-wire spacing, means that it can be different for coils of the same overall size. Additionally, the logarithmic scaling keeps $y_{\text{finite}}$ always very small. For example, for a typical NMR/MRI meander-line coil with, say, three or four back and forth turns of wire (N = 6 or 8), the infinite theory breaks down completely less than one wire spacing away from the surface. But it already begins to falter as soon as B_err becomes even appreciable to B_∞, which may be only a fraction of a single wire width away, as detailed experimentally below. Increasing the coil size by increasing the number of coil wires does little to postpone this breakdown. A coil with, for example, 100 wire segments (more than likely to be used given the increased wire length and resistance), fails to appear infinite less than one spatial period, 2, from its surface. Even a coil as a wide as a football field is long with fifty thousand 1 mm wide wires spaced 1 mm apart from one other, would fail to look infinite well before rising to the tops of the blades of grass. In other words, unless a coil possesses periodic boundary conditions (as for example a cylindrical coil [9]), the infinite theory very quickly fails.

Because of the different vectorial behaviors of the B_err and B_∞ fields, these rapid changes from effectively infinite to finite destroy the coil field uniformity, producing imaging artifacts and signal voids. Figure 2 compares the theoretical fields above the centers of an infinite coil, on which conventional meander-line theory is based, and a more realistic finite coil. As expected, for both coils the field uniformity begins to improve with increasing distance from the coil wires. But for the finite coil this improvement quickly switches to an increasingly less uniform field as distance increases and the infinite coil theory breaks down. The field reaches 100% non-uniformity around the distance $y_{\text{finite}}$ where the magnitudes of B_∞ and B_err become equal. Because the B_∞ direction rotates across the coil, while the B_err direction does not, the fields periodically reinforce and counteract each other. This interference produces the commonly observed sinusoidal intensity variations, or repeating stripes, across images acquired with meander-line coils, including even total signal voids due to regions of zero field that occur around the distance $y_{\text{finite}}$ from the coil.

Figure 2. Infinite versus finite meander-line coil fields.

Equispaced isocontours of magnetic field magnitude, B₁, above the centers of a theoretical meander-line coil with an infinite number of wire segments (A), and a coil with N = 8 wire segments (B). In contrast to the ever-improving field uniformity with increasing distance from the infinite coil, for a realistic coil, the field uniformity begins to improve but then quickly worsens with increasing distance. The closed contours encircle dead spots in the field of a finite coil (B) where B₁ goes to zero and relative field uniformity is at its worst.

These sinusoidal, stripe-like, signal variations can be easily mistaken for near-surface field roughness from the individual coil wires, particularly because they run parallel to the coil wires and already appear so close to the coil surface. They are, however, distinct from, and independent of, the field variations across each wire. The near-surface variations repeat with dominant spatial frequencies of at least 1/a, matching a that of the wires, or with multiples thereof, such as 2/a if wire widths and spacings are similar and one is close enough to the surface to discern the two edges of each wire, as seen in Fig. 2. However, because current (and hence field) alternates direction over adjacent wires, the field non-uniformity arising from the infinite to finite transition, which includes the zero field points, occurs instead at half the spatial frequency, ½a. This halving of frequency, also seen in Fig. 2, indicates where the infinite approximation a starts to fail and can be considerably less than a single wire spacing. Here, real coils directly oppose the infinite theory: the intuitive strategy of imaging further from the coil to improve field uniformity not only sacrifices sensitivity, but also forfeits precisely the field uniformity it was expected to achieve.

Fortunately, there are ways to mitigate this loss, as shown in Fig. 3. The first exploits an even-odd parity effect related to the number of wires in the coil. A coil with an even number of wire segments has currents flowing in opposite directions at its two edges which, given that they are on opposite sides of coil center, leads to two B_err fields that add constructively in the center of the coil. Conversely, for a coil with an odd number of wire segments, the B_err fields cancel at coil center. The odd coil therefore more closely resembles an infinite system with a more uniform field and higher fill factor, than does one with an even number of wire segments. Because of the logarithmic dependences, this even-odd parity effect remains significant no matter the coil size. For example, a coil with 99 or 101 wires presents significantly better field uniformity and fill factor than a coil with 100. The odd coil does still fail to fully mimic an infinite system away from the center but can be further corrected. A second mitigation strategy involves adding two extra wires to the coil, each of which mimics the missing field, B_err, from the infinite array of missing wires off each edge. Because the required current to match the B_err fields is approximately one-half of the coil current, this can be done by splitting the return current path into two equal pieces that wrap around the edges of the coil. In addition to canceling the B_err fields from the sides, the half currents that these extra wires add along the top and bottom of the coil help cancel unwanted contributions from the connecting paths at the ends of the straight wire segments, which carry the full coil current but each over total lengths of only one half the coil width.

Figure 3. Meander-line coil field profiles.

Equispaced isocontours of B₁ magnitude above meander-line coils with an even number of wire segments (left), odd number of wire segments (middle), and an odd number of wire segments together with a wrap-around split current return (right). B₁ uniformity, and agreement with the infinite coil model, is poorest for an even coil, better for an odd coil, and best for an odd coil with split-current return. (Contour y-axes are expanded 4-fold relative to x-axes to better show field detail)

Results and Discussion:

To test the predictions of the theory presented above, two similar meander-line coils were fabricated (see Fig. 4). One comprised an even number of wires, as is common for meander-line coils in the literature. The other used an odd number of wires together with a split-current return that combine to give a coil with just one extra current line that looks similar to, but that the theory predicts should perform markedly differently from, the conventional even coil.

Figure 4. Experimental meander-line coils.

Planar copper coils with (A) an even number of lines (N = 8), and (B) an odd number (N = 7) with a wrap-around split-current return. Wire widths and gaps are 2 mm. Schematics show capacitance splits and tuning-matching circuit.

Figure 5 shows MRI slices acquired perpendicular to the plane of the coils within uniform agarose phantoms placed directly on top of the coils. The smoothly decaying field above the coil with a correcting split-current return is contrasted to that of the conventional coil. The uncorrected coil shows significant field inhomogeneity with repeating signal voids across the image as one moves from left to right across the coil (Fig 5A). The split-current return coil shows a much more smoothly decaying field above the coil (Fig 5C). To better visualize the coil fields, Figs. 5B and 5D shows MRI mappings of isocontours of B₁ field magnitude, which are similar to those of the theory plots presented in Fig. 3. Multiple contours lines were generated by deliberately over-powering the coil, producing repeated signal maxima at 2πn ± the Ernst angle (see Methods). Field isocontours are visibly more uniform for the corrected coil.

Figure 5. Signal intensities and magnetic fields of conventional and corrected coils.

(A) and (C), MRI signal acquired from within agarose phantoms above conventional and corrected transmit-receive meander-line coils, respectively. Image planes extend perpendicularly up from plane of coils. (B) and (D), MRI detected isocontours of B₁ field magnitude, acquired by over-powering the coils (see text), in same imaging planes as (A) and (C) for the conventional and corrected coils, respectively. For the corrected coil (C) and (D), field uniformity improves with increasing distance, y, from the coil. In contrast, the initial uniformity improvements in the uncorrected conventional coil (A) and (B), quickly reverse, yielding highly inhomogeneous fields and repeating signal void artefacts across the image.

To demonstrate the differing signal acquisitions of these different coils, Fig. 6 shows MRI slices taken from a volumetric image of the same agar phantom under normal lower flip-angle conditions likely to be used for imaging. Image planes are taken parallel to the plane of the coil, as would be the case for normal use, at different distances from the coil scaled by the field decay length, $\frac{a}{\pi }$. Image slices include very close to the coil surface at a distance (1/4) ∙ (a/π) where the infinite theory still holds and local field roughness is dominated by that due to the individual wires, at intermediate distances of $(1/2)\cdot \left(\frac{a}{\pi }\right)$ and (a/π) where the infinite theory starts failing, and at a distance of $2\cdot \left(\frac{a}{\pi }\right)$, where the infinite theory has completely broken down and stripe-like intensity artifacts dominate the conventional coil image. Note that even this furthest distance shown is still less than a single wire-to-wire spacing away from the surface. Also shown are representative line-cuts to better compare signal uniformity. Finite size interference effects, which are manifested for the even coil through its worsening signal uniformity with distance, are significantly muted for the split-return odd coil where field uniformity remains high. For a conventional uncorrected coil, the transition from infinite to finite can also be seen to occur very close to the surface suggesting that the commonly applied infinite coil theory is likely inapplicable in many imaging situations where such traditional meander coils have been used.

Figure 6. Signal uniformity for conventional and corrected meander-line coils.

Images show 50-μm thick slices taken parallel to the plane of the coil through agarose phantoms placed atop conventional (left) and corrected (right) coils. Slice distances from coils, in units of a/π, are labeled in white. Very close to the coil there is little difference; further away, where the infinite coil approximation fails, the corrected coil images become more homogeneous than the uncorrected ones, which show worsening homogeneity with increasing offset from the coil.

Figure 7 summarizes the observed variation in signal intensity as a function of distance from the coil for the conventional and the corrected coils, taken from the same image data used for Fig. 6 but including many more intermediate image planes. The curves quantify signal variation across the coil (along the x-axis of Fig. 1) as a function of distance from the coil surface. Each data point in the figure is derived from a separate imaging plane, in each of which the fractional variation was measured over the same region of interest (ROI) centered above the coil center and of width 2a. This ensures representative capture of a full spatial period across the coil. Fractional variation was taken as the difference between the observed signal maximum and minimum in each ROI normalized to the signal maximum in that ROI (see Methods). As implied in Figs. 5 and 6, the conventional coil field uniformity worsens sooner and more rapidly than for the corrected coil. Specifically, finite-size effects of the conventional coil start appearing around half a decay length, or around one-third the width of a single wire in the coil; the finite size of the corrected coil, however, becomes only gradually evident and only at distances several fold further from the coil. The field variation of the conventional coil approaches 100% around 3 ∙ (aπ) in good agreement with the approximate analytic scaling derived above (Eqns. 7,8). For the coils shown in fig. 4, this occurs just one-tenth of the coil width away from the surface. Here, dead zones of zero field cause repeating signal void stripes across the image regardless of how much power is used to drive the coil. In contrast, at similar distances from the corrected coil, signal variation remains low. At present the correction is not perfect with residual field distortions still evident. These are likely due to (i) imperfect splitting of the return current, (ii) stray fields from current paths to coil tuning/matching circuits, and (iii) the relatively small number of coil wires, N, which renders the half-current correction less accurate. Still, Fig. 7 shows that adding even an imperfect split-return line already improves signal uniformity by as much as an order of magnitude.

Figure 7. Spatial signal uniformity of meander-line coils.

Fractional variation of acquired imaging signal intensity as a function of distance above a conventional (uncorrected) meander-line coil and a coil with a correcting split-current return, scaled to the coil field decay length, a/π. Inset shows theoretical prediction. The 100% signal variation of the conventional coil is due to regions of total signal loss arising from finite size effects. At equivalent distances, the corrected coil offers improved signal uniformity.

The above has focused on field and signal variations in slices parallel to the plane of the coil, parameters that are relevant to imaging applications. However, for slice-free NMR spectroscopic applications that may rely on coil fields to limit signal acquisition to close to surface, field fall-off perpendicular to the coil plane also matters. According to conventional infinite coil theory, this field dependence, which may be arbitrarily complicated directly at the surface, quickly transitions into a single exponential with decay length determined by the coil wire spacing. This exponential decay is expected to efficiently cancel fields away from the coil, preventing unwanted field penetration deeper into the sample and, by reciprocity, eliminating pickup of contaminating signals from such deeper locations. In reality, however, uncorrected conventional meander-line coil fields do not decay exponentially, resulting in poorer than expected fill factors. Although confounding effects of such poor field decays have been noted before[19,20], no explanation has yet been given for why the field does not decay as rapidly as anticipated.

It follows immediately from the above analysis, however, that poorer than expected fill factors are another consequence of the uncompensated error field due to the “missing” wires off each edge of the coil. Since the coil field B₁ is not equal to B_∞, but instead to B_∞ ― B_err, the finite coil field transitions from the expected exponential decay to a more gradual one dominated by the edge error fields at any distances of order $y_{\text{finite}}$ or beyond. These slowly decaying error fields reduce the filling factor because they extend further from the coil than the exponentially decaying fields of an idealized infinite coil. Therefore, just as odd wire numbers and split-return currents correct field non-uniformity by counteracting B_err, they also help recover the expected exponential field decay.

This sharper cutoff in the corrected coil field is evident in Fig. 5. A more quantitative analysis is also shown in Fig. 8, comparing signal fall offs from the conventional and the corrected coils. As in Fig. 7, each data point is derived from a separate imaging plane, except that Fig. 8 records the average in-plane signal intensity rather than signal intensity variation. The same ROI’s that extend over a full spatial period of the coil are also used here. While average signal intensities for conventional and corrected coils fall off similarly close to the coil, they soon diverge with the corrected coil falling off faster. On the figure’s logarithmic scale, an exponential decay should yield a straight line; the roll-over in the curve is an artifact of the fast-low-angle-shot (FLASH) pulse sequences used that led to larger steady-state signal suppression closer to coil where fields and flip angles are larger.

Figure 8. Signal decay away from coil.

Normalized signal intensity as a function of distance perpendicular to plane of coil, scaled to the coil field decay length, a/π. Inset shows theoretical prediction. Conventional coil signal does not decay exponentially, leading to extended stray fields and signal interferences from deeper within the sample. Corrected coil exhibits truer exponential field decay, improving field localization and suppressing signal interference.

By reciprocity, reducing the depth that the unwanted stray field extends into the sample should also reduce the noise pickup from deeper within the sample. Close to the coil surface the corrected coil design should therefore offer higher SNR than the uncorrected one even when there is no difference in signal strengths close to the coil. This is seen experimentally in the SNR maps of Figs. 9a and 9b. Because of the periodic field inhomogeneities of the uncorrected meander coil, SNR can be more quantitatively compared by averaging over the same ROI’s as before, which extend across one spatial period of the coil. This shows improved SNR close to the coil for the corrected coil design (Fig. 9c); only far from the coil (where the coils are never intended to be used) does the uncorrected coil achieve better relative SNR, simply because its stray field still persists at these distances whereas the corrected coil field has already been made to vanish.

Figure 9. Signal-to-Noise ratios of conventional and corrected meander-line coils.

SNR maps, to same scale, for conventional (A) and corrected coil (B). (C) Spatially averaged SNR as function of distance from the coil showing higher SNR for corrected coil.

Not only does a corrected meander-line coil offer higher field homogeneity and SNR, it also compares favorably to more common single-channel surface coils in terms of achievable sensitivity. As for any coil, a meander-line’s maximum field and sensitivity occur at the wire surface. Imaging directly next to the coil is generally precluded, however, due to sample geometries and/or wire susceptibility mismatches. Instead, since the meander-line coil field scales as ke^―ky, for any desired image depth from the coil, d, differentiating shows that field and sensitivity are maximized for a wire-to-wire spacing, a = πd. That is, SNR can be optimized if the coil is designed such that the given imaging depth corresponds to one coil field decay length. To be sure, in terms of overall SNR this optimum may shift somewhat on a case-by-case basis, sacrificing some sensitivity for either reduced resistance noise with a larger wire spacing, or reduced sample noise with a smaller wire spacing. Nevertheless, this remains a more favorable optimization problem than that faced by other surface coils intended for near-coil applications. Specifically, optimal meander-line coil sensitivity and SNR depend on wire spacing and number, but not on the overall size of the coil; for other single-channel surface coils, such as common loop or spiral coils, however, optimization for a given imaging depth requires changing the coil size, incurring unavoidable trade-offs between SNR, coil FOV, and image depth. This makes meander-line coils especially favorable for applications requiring planar FOV’s that are large compared to the imaging depth. The ability to select optimal depth independent of coil size may also benefit single-sided NMR applications where the non-uniformity of single-sided magnets introduce their own image offset requirements based on the magnet’s own homogeneity sweet spot. Note also that in addition to tightly confined fields, which lessen interference from deeper within a sample, the meander-line’s unique alternating current flow geometry also reduces other non-local noise signals, which would induce cancelling voltages of opposite phase between neighboring coil segments.

Conclusion

In summary, if not corrected for finite-size effects, meander-line coil performance diverges detrimentally from that predicted by conventional theory. Not only are fill factors poorer due to fields that extend further out from the coil than expected, but field uniformity worsens instead of improving with increasing distance from the coil, leading to banding or stripe-like image artifacts. Such coils can be corrected, however, with a wrap-around split-current return to better mimic the desired fields of infinite coil geometries. This correction strategy, which yields higher fill factors and B₁ field uniformities, is preferable to adding more wire segments, which would provide only a slow logarithmic approach to the infinite approximation at the expense of a linearly increasing coil resistance. It was also shown that a coil with an even number of wire segments, which is the most commonly used geometry because its input and output are conveniently on the same side, is ironically the worst possible design. Instead, combining the split-return with an odd number of coil wires can delay the adverse transition from infinite to finite long enough to yield infinite-like coil fields even for coils with only a few turns of wire. With optimal imaging depths that can be chosen independently of the planar image or coil size, such corrected meander-line coils may prove ideal for large-area, near-surface imaging and spectroscopy.

Methods

Coil fabrication and technical details:

Coils were milled from copper clad Teflon boards to leave copper wire traces 2 mm wide and 30 μm deep (the thickness of the copper cladding). For simplicity, gaps between all wires were also all set to 2 mm. Capacitive splits were added throughout as shown in Fig. 4 and coils tuned to resonate at 400 MHz (9.4 T). Unloaded to loaded Q’s were 199/172 for the conventional meander and 206/183 for the split return path. A cable trap was used to decrease shield currents. We employ a virtual ground method to mitigate the effects of common mode currents. No difference was detected in preliminary experiments to assess whether a balanced input using a balanced matching scheme had an effect.

Agarose phantom:

The agarose phantoms used for imaging comprised 2 % w/w agarose dissolved in water and doped with 50 micromolar manganese to reduce the phantom T₁ to around 1 s for faster image acquisition. To limit evaporation, the agarose was sealed in a home-made container approximately 4 cm x 2cm x 1cm. Phantoms were placed directly on top of the coils. To enable imaging close to the coil the base of the agarose container was formed from a thin microscope coverslip (approximately 150 micrometer thickness)

Imaging:

All imaging was performed on a Bruker 9.4 T scanner, using FLASH imaging sequences with TE = 7ms, TR = 30 ms, total imaging volume 5.12 cm x 2.56 cm x 1.92 cm and at 50 μm isotropic resolution to enable scanning close to coil and to allow high resolution characterization of both field homogeneity and fall off, parallel and perpendicular to coil plane respectively.

Data analysis:

Theoretical coil fields (Figs. 2 and 3) are calculated assuming uniform current distributions across the wire strips. Actual current distributions within wire strips will depend on RF skin and proximity effects and are therefore unlikely to be uniform, but the errors incurred only affect the calculated fields very close to the coil surface, far closer than any distances at which the difference between an infinite and finite coil become apparent. Quantification of observed signal variation (Fig. 7) was first corrected for the background noise level before calculating the signal variation ratio (signal maximum – signal minimum)/(signal maximum). The background correction was done separately for each imaging plane based on average noise level observed in image regions surrounding the phantom. Background noise was also first corrected for before taking the logarithms of signal intensity in Fig. 8. SNR maps were calculated based on average signal intensity divided by standard deviation in that intensity at each point in the image, with averaging performed in the z-direction (into the plane of Figs 9a and 9b, from data taken from image planes parallel to the plane shown but offset in front of, and behind, the image plane). The quantitative SNR comparison between coils as a function of distance from the coils, Fig 9c, was then calculated by averaging, for each distance y from the coil, the SNR maps in the x-direction across one spatial period of the coils.

Highlights:

• Meander-line, or zig-zag, surface coils have poorer field uniformity than predicted

• Invalid approximations in existing coil theory revealed and new theory presented

• Simple new coil design gives order-of-magnitude field uniformity improvement