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which is already five orders of magnitude below the radiation wavelength. Nevertheless, even though MRI records radio frequency emissions, this is done almost exclusively through near-field interactions, i.e. by Faraday induction, and not from a beam or ray that requires a lens for focusing.

One of the limitations in NMR is certainly that a single quantum emission event is not yet readily observable as it would be in photonics, even though Dan Rugar showed that a single spin can be observed [7]. Thus all Faraday induction-acquired MRI images have to resort to averaging of a vast number of emission events, and over extended time, to yield useful information. If detection sensitivity were to be increased, fewer emitters could be used, and could perhaps be averaged over shorter times.

      1.1.2 Limit of Detection

The statistical polarization level, i.e. that proportion of the total spin population that is available for quantum emission, is an additional cause of lack of signal. Proton spins for example are indistinguishable fermions, with a level occupation that follows

      

(1.1)

      and which collapses to Maxwell–Boltzmann statistics when e^{(εi−µ)/kT} ≫ 1, because the energy of a proton flip γħB₀ = 3.3 × 10⁻²⁵ J is tiny compared with the thermal energy kT = 4.11 × 10⁻²¹ J. Thus at typical equilibrium polarization levels at 11.7 T, a factor of only 10⁻⁴ in excess in population difference with respect to the Fermi level µ can contribute to the signal. An imaging voxel size is therefore principally limited by polarization, because we find – for microcoils at their limit of detection – a sample containing around 10¹³ spins is needed to form an observable signal. Clearly, this sets a lower concentration limit once the voxel size is specified. For example, at the average size of a single eukaryotic cell of (10 µm)³, containing the required nuclei, implies a concentration of at least 1.66 µM. By increasing polarization, the voxel size is thus principally reduced, or the lower concentration limit is reduced, which could be achieved by resorting to out-of-equilibrium polarization techniques such as parahydrogen-induced polarization (PHiP), signal amplification by reversible exchange (SABRE), or dynamic nuclear polarization (DNP), all of which are rather hard to perform noninvasively, and hard to selectively localize too. We will return to this point shortly. One of the key advantages of MR-based microscopy is the ability to noninvasively reveal molecular composition, correlated with morphology. From the perspective of biological systems, this can be leveraged to monitor, for example, spatially resolved metabolism. To estimate the best achievable spatial resolution (voxel size), signal-to-noise ratio (SNR) should be considered in the context of the metabolically active system. Key parameters are the molecule abundances (concentrations) and timescale that are targeted. Consider a spatially resolved fluxomic investigation: can one estimate a realistic MRI spatial resolution taking into consideration the expected biological dynamics? Alternatively, what is the smallest biological structure with which metabolic flux can be measured – thus, is it possible to monitor flux at the level of an organelle, single cell, cell cluster, or tissue?

Water is the most abundant molecule in biosystems and can be used as a useful reference from which scaling based on metabolite concentrations can be made. Using only the physical volume of a water molecule (0.03 nm³), and an optimistic limit of detection (LOD) of 10¹³ spins, then an order of magnitude estimate of the smallest voxel is 0.1 pl (approximately 4.5 µm isotropic resolution). This is approximately the volume of a single red blood cell. Intracellular metabolite concentrations vary over several orders of magnitude, with the most abundant molecules typically in the tens of millimolar regime. The best-case scenario scaling factor is then 10⁴ relative to water (assuming [water] = 55 M), and thus the smallest voxel volume increases to 1000 pl (100 µm isotropic resolution). For reference, this would correspond to 10 000 red blood cells or 2 fat cells (volume 600 pl per cell). Can the resolution be improved by signal averaging as a means to enhance SNR? Assuming metabolism is active during the measurement then one must consider the turnover rate of the target metabolite(s) relative to the time over which signal averaging is performed. Enzyme catalytic (second-order) rate constants span several orders of magnitude (k_cat/K_M ~10¹–10⁹ s⁻¹ M⁻¹), with a median of ~10⁵ s⁻¹ M⁻¹ [8]. If the metabolite concentration is 100 mM, then the metabolite will encounter the “median enzyme” with a rate of 10⁴ s⁻¹. At this concentration and a volume of 1000 pl, the metabolite concentration would drop below the LOD (~10 mM) in 6000 days giving more than sufficient time for signal averaging. At the diffusion limit 10⁹ s⁻¹ M⁻¹ then, 6 days are required before the signal is not observable. This rough estimate takes many liberties in the assumptions (catabolic and anabolic reactions, multiple pathways, enzyme performance, cell cycle, etc. are neglected) and simply suggests that signal averaging is reasonable, most likely limited by technical factors like long-term sample maintenance and spectral resolution accounting for magnetic susceptibility effects. Interesting, it is revealed that single cell metabolic monitoring is challenging yet possible as long as (i) large cells are selected; (ii) the metabolite is among the most abundant in the cell at millimolar concentration; and (iii) the cell can be maintained in an active state during the measurement. Spatial resolution and/or detected concentrations can be improved if hyperpolarization strategies are used where the effective LOD can be improved by orders of magnitude with percent (instead of ppm) levels of polarization. Using the same set of assumptions, now with percent levels of polarization, it is estimated that on the order of a few minutes is required (“median enzyme” kinetics) before the observation of hyperpolarized metabolic products, consistent with observations [9].

Currently, the only other means known to further increase resolution in cell biology, is based on the higher sensitivity and simultaneously higher localization that arises from proximity to the spin. When detector and spin are separated on the order of 1 nm, the dipolar coupling is strong. The techniques, such as magnetic resonance force microscopy [7], or nitrogen-vacancy (NV) centers in nanodiamonds [10], are either invasive (magnetic resonance force microscope [MRFM]) or require similar sample preparation as for STED (NV centers), namely to introduce a mobile quantum emitter that is tagged to a molecule to wander through the cell.

      1.1.3 Limit of Imaging Resolution

Paul Callaghan [1] and others [11–14] showed that the imaging resolution of MR microscopy is fundamentally limited by three factors: the diffusion coefficient of molecules within the sample, the line broadening due to magnetic susceptibility effects, and the specified SNR per voxel. While the limits placed by the first two factors can be pushed by stronger field gradients and dedicated pulse sequences [15], the rather poor SNR of the MR signal remains as the ultimate fundamental limit of resolution. This can be clearly seen from the following equation [16,17], which summarizes the factors that determine the achievable resolution for a specified SNR of the image:

      

(1.2)

where V_voxel is the voxel volume, d is the coil diameter, t_acq is the total acquisition time, and B₀ is the field strength. According to this equation, maintaining the image SNR while, for instance, halving the isotropic resolution (reducing the voxel volume by a factor of 8) necessitates either miniaturizing the coil by a factor of 8, increasing the acquisition time by a factor of 64, or increasing the B₀ field by a factor of 3.28. This explains why high-resolution MR images take excruciatingly long to acquire, and why most groups decrease coil diameter. However, at room temperature, coil diameter cannot be reduced indefinitely without disadvantageously increasing coil resistance, so that quality factor Q will ultimately limit this strategy.

      1.2

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