As a mature imaging modality, magnetic resonance imaging (MRI) is an established, valuable tool in the health care delivery system. The strength of the modality clearly lies in the versatility of mechanisms that can be used to create contrasts in images. Those contrasts have proven to deliver a high diagnostic value. However, there are areas in which MRI remains behind expectation. Two major such areas are speed and quantification. Although there are many innovations that improve the measurement speed such as compressed sensing or deep learning reconstruction, the quantification is still waiting for a disruptive breakthrough.

These 2 challenges are of different nature. The broad availability of MRI is hindered by the comparably high cost that is still inherent to the method, partially due to the technical setup needed, but also due to the relatively low throughput, for example, when compared with computed tomography. Quantification is different, as it primarily promises to further increase the overall value of MRI, independently of the cost. One could argue that it also leads to lowering cost; however, those effects will probably be secondary when compared with the added value.

We focus here on quantification, which is actually the key word for the need to resolve a number of problems such as the data incoherence in multicenter studies, result dependence on the scanner manufacturer, the technical design, the data processing pipeline, and so on. “Quantification” is nearly synonymous to “data harmonization,” “measurement reproducibility,” and “calibration.”^{1} As the problem of quantification persists from the first days of MRI on, it might be timely to recall that a persistent practical difficulty is often a manifestation of a fundamental problem.^{2} In this article, we try to reveal the problem behind the quantification challenge and use this knowledge as the scene for discussing 2 rapidly developed branches of MRI: microstructure MRI and fingerprinting (Fig. 1 ).

FIGURE 1: The exponentially increasing number of publications on fingerprinting and microstructure. Plotted are the numbers of PubMed entries found with both corresponding key words in the title or abstract. Left, “magnetic resonance imaging (MRI)” and “microstructure.” Right, “MRI” and “fingerprinting.” The smooth black lines show the fitted exponential functions that reveal the number of publications having doubled every 2.9 and 1.5 years, respectively, with the larger literature pool on the microstructure. The insets show the same data in the semilogarithmic scale in which the exponential function looks straight. Right plot is adapted from Novikov et al^{2} with data for 2018 and 2019 added.

We start with the question “what is the outcome of an MRI scan?” For decades, this answer was self-evident and commonly assumed in the clinical practice; it is a grayscale picture “weighted” with some contrast achieved by manipulating nuclear magnetization in human body. The fact that this picture is made out of numbers and the related question of what these numbers mean has recently come into focus of the scientific and imaging community.

For a physicist, an MRI scanner is a measurement device. The meaning of the measured numbers and their precision, accuracy, and reproducibility all fall into the realm of a physical measurement. Such measurements cover a broad spectrum from direct to very indirect ones. To give examples, measuring the voltage on a battery is a direct measurement, whereas the discovery of the Higgs boson was an ultimately indirect measurement for which a complicated theory had to be involved to provide a reliable interpretation of the particle counts obtained on cutting-edge detectors.

Returning to MRI, measurements of the proton density are rather direct. One can even quantify them in absolute number using a small water reference. However, the relation between weighted MRI scans and the biophysical tissue properties is very indirect. The indirectness is a blessing and a curse. The blessing of MRI is its immense multiparametric nature, that is, its sensitivity to a plethora of physiology- and pathology-related properties of nuclear spin environments. To give a few examples, it is the sensitivity to the stochastic magnetic field at the Larmor frequency resulting in a specific T1, at nearly zero frequency resulting in T2, the sensitivity to motion in the presence of external field gradients underlying diffusion imaging, 4-dimensional flow measurements in cardiac MR, and measurements of the stiffness for MR elastography.^{3} The curse of MRI is its immense multiparametric nature, that is, its sensitivity to a plethora of tissue properties that can affect the previous contrasts. Poorly understood or uncontrolled, these parameters induce unwanted bias and variability in the measurement results.

To address this fundamental challenge head-on, we need to accept that truly quantitative MRI (qMRI) is necessarily model based, more similar to the Higgs boson discovery than it might seem. This is because the nuclear magnetization has no direct physiological meaning.^{*} The models (actually synonymous to “theory”) are necessary to relate the magnetization to the tissue parameters we want to measure. Even if we do “simple” T1/T2 mapping, it assumes a model, which is the exponential approach to the equilibrium magnetization. If the model is right, and if its parameter estimation is unbiased, there is no reproducibility crisis and no need for harmonization; different platforms, vendors, and protocols would still yield the same genuine tissue parameters, much like voltmeters made by different vendors show the same voltage on your battery. Obviously, the state of affairs in MRI is very different. Models are generally oversimplified, which leads to bias and apparent variability in their parameter estimation.

The field that has arisen to address the modeling challenge is currently referred to as tissue microstructure imaging with MRI (microstructure MRI). This field accepts the physicist's point of view from the outset, taking the approach to model the tissue microstructure and physiology. This yields the MRI signal, given a specific acquisition scheme, and helps finding the microstructural tissue parameters given a measured signal. Such an approach to biological MR was used even before the inception of imaging.^{4–6} Although microstructure mapping is now usually associated with diffusion MRI, it actually originated from relaxometry, yielding the microstructural explanation to functional MRI based on the blood oxygen level dependent contrast in the 1990s. The number of publications on microstructure MRI grows exponentially ever since, doubling every 3 years (Fig. 1 ) with hundreds of abstracts at recent meetings of the International Society for magnetic Resonance in Medicine.

Any measurement of tissue properties cannot happen without a well-thought-out acquisition. In fact, the more physiologically adequate and robust we want our metrics to become, the more synergies should occur between the modeling and acquisition communities. This conversation is only at its inception. We can, roughly, outline a few different “types” of acquisitions. A classic type is to keep everything fixed and to vary a single acquisition parameter, such as echo times (TEs) or b-value. The multimodal one is to vary a few parameters in a controlled way, for example, by combining diffusion measurements with relaxometry.^{7–9} And finally, an “ultimate” acquisition is magnetic resonance fingerprinting (MRF), where all acquisition parameters are varied in a complex pseudorandom sequence to turn MRI acquisitions into greedy information gathering. The expected signal is modeled for a broad range of the targeted tissue properties, such as T1 and T2, with the results stored in a dictionary. Using this dictionary, exhaustive search algorithms point on the tissue parameter combination giving the best match between the modeled and the actually measured signals. Having started in 2013 in the arena with a long history of T1 and T2 quantification, MRF has successfully shown how it can provide robust and repeatable results.^{10,11} Being a promising method, MRF has also enjoyed an exponential growth in the number of publications with doubles each 1.5 years albeit with yet an order-of-magnitude smaller overall number of articles (Fig. 1 ).

In this article, we aim at giving a brief overview on the imaging paradigm so excitingly offered by MRF and microstructure MRI. We sincerely hope that we can offer the reader a key to these 2 dynamically developing research fields. We hope that we can seed some motivation to contribute in understanding their relations to the tissue structure and physiology as well as the growing proximity to each other.

MAGNETIC RESONANCE FINGERPRINTING
The result of an MRI measurement is in general affected by a multitude of parameters. For a technique that claims to quantitatively measure a parameter, this implies that the used sequence needs to be robust against variations of other parameters while being sensitive to the targeted one. Traditionally, methods for qMRI try to follow this concept by sampling exponential recoveries/decays or using steady-state signals. However, deviations of the real measurement from the model such as interdependencies with other tissue or system parameters such as the static magnetic field B0, the transmit field B1+, diffusion, or magnetization transfer remain to a certain extent and impair quantification accuracy and precision. The aforementioned concept also connotes that only a single parameter is quantified per qMRI measurement. For any multiparametric analysis, the resulting parameter maps—potentially even of different spatial resolution—need to be registered due to subject movement. In addition, longer scan times compared with qualitative imaging are required to collect the necessary data for the reconstruction of quantitative information. The entirety of these limiting factors has restrained widespread use of qMRI in clinical routine.

Magnetic resonance fingerprinting^{12} uses a different approach to overcome these limitations. The fact that a sequence is sensitive to multiple parameters is deliberately accepted and made part of the solution. The idea behind it is based on 3 key ingredients:

1. Exploiting freedom in sequence design to yield signals that are explicitly highly sensitive to several parameters.
2. Correlation of measured and theoretical signal using a precalculated set of expected signals stored in a dictionary. The dictionary includes the simulated signals for the anticipated parameters with the highest expected influence on the signal.
3. Sampling that produces incoherent artifacts, that is, subsampling k-space in a way that resulting artifacts do not impair the correlation process.
1) In MRF, sequence parameters such as flip angles, pulse phases, repetition times (TRs) or TEs, as well as gradient pulses are varied to generate a complex signal response, in contrast to conventional sequences where steady-state signals are used. This freedom opens up the possibility to design more efficient sequences explicitly sensitive to several tissue or system parameters and thereby to intrinsically resolve potential interdependencies of parameters by simultaneous quantification.
2) Instead of fitting the measured signal response to a parameterized signal equation, the measured signals (“fingerprints”) are compared with a precalculated dictionary of potentially observable signals. The dictionary can be simulated numerically using the Bloch equations (Equation 1) or semianalytically using the extended phase graph formalism.^{13}
$\frac{d{M}_{x}\left(t\right)}{\mathit{dt}}=\gamma {\left(\mathit{M}\left(t\right)\times \mathit{B}\left(t\right)\right)}_{x}-\frac{{M}_{x}\left(t\right)}{{T}_{2}}$

$\frac{d{M}_{y}\left(t\right)}{\mathit{dt}}=\gamma {\left(\mathit{M}\left(t\right)\times \mathit{B}\left(t\right)\right)}_{y}-\frac{{M}_{y}\left(t\right)}{{T}_{2}}$

$\frac{d{M}_{z}\left(t\right)}{\mathit{dt}}=\gamma {\left(\mathit{M}\left(t\right)\times \mathit{B}\left(t\right)\right)}_{z}-\frac{{M}_{z}\left(t\right)-{M}_{0}}{{T}_{1}}$

EQUATION 1. Bloch equations that describe the evolution of macroscopic nuclear magnetization M exposed to a magnetic field B . Transversal magnetization (M _{x} _{, } _{y} ) decays exponentially with T _{2} , whereas longitudinal magnetization (M _{z} ) approaches its equilibrium value exponentially with T _{1} .

By comparing a measured fingerprint with the dictionary of simulated fingerprints, the most similar simulated fingerprint is identified. This identification process reveals the signal's properties, which are the parameters that were used for the simulation of the best matching fingerprint. With the dictionary-based approach, more complex signal models taking more effects into account can be used because computational demanding tasks can be carried out before the measurement and not after, such as in fitting methods. Note that, although using a dictionary is a necessary part of MRF, it is not sufficient to classify a method as such. See for example microstructure mapping, based on a traditional acquisition scheme.^{14}

3. Magnetic resonance fingerprinting acquisitions can be substantially shortened by means of spatial undersampling. Established techniques are single-shot spirals and radial sampling, but in principle, every sampling including conventional Cartesian sampling can be used. The undersampling factors can be considerably higher than the ones applied in conventional MRI. The basic assumption underlying the speed of MRF is the hypothesis that a temporal variation of the sampling pattern leads to incoherent aliasing artifacts in the time domain. Accordingly, the pattern matching process is assumed to be unaffected by these artifacts.
An example of the fingerprinting framework is shown in Figure 2 .

FIGURE 2: Exemplary schematic implementation of magnetic resonance fingerprinting. A FISP (fast imaging with steady precession) sequence with RF pulses of varying flip angles and varying repetition times between pulses is used to generate signals. The same sequence is simulated for different combinations of T1 and T2 relaxation times and stored in a dictionary. Sampling is performed with rotations of a spiral sampling pattern, which yields highly undersampling artifact afflicted images at every time point. The signal from a voxel is matched to the dictionary to find its most similar counterpart and thereby resolving T1 and T2. By repeating this pattern matching process for every voxel, parameter maps are generated.

The original publication^{12} has stimulated plenty of research. Although the original MRF was based on a bSSFP (balanced steady-state free precession) sequence, able to quantify T1, T2, and B _{0} , several modifications and implementations using different basis sequences have been explored. A version termed pseudo steady-state free precession for MRF^{15} was proposed to reduce undesired effects of static magnetic field inhomogeneities and intravoxel dephasing on the bSSFP-based implementation. Here, relationships between the flip angles, TRs, and TEs were derived to better maintain the spin echo–like refocusing characteristic of bSSFP. Another popular approach^{16} to reduce the sensitivity to static magnetic fields and intravoxel dephasing was proposed, which is based on a FISP (fast imaging with steady precession) sequence. The FISP sequences have unbalanced gradient moments, which lead to fully dephased spin ensembles at the end of every TR. The FISP-MRF approach therefore exhibits no sensitivity to off-resonance; however, it has reduced signal-to-noise ratio compared with bSSFP. An implementation^{17} that considers constraints on specific absorption rates is based on the quick echo splitting NMR (QUEST) sequence. Only a few radiofrequency pulses are required in this implementation to produce a multitude of echoes. Furthermore, more than 1 sequence can be combined in 1 acquisition. In plug-and-play MRF,^{18} a fast low-angle shot sequence is combined with a FISP sequence in an interleaved manner, and 2 or more transmit coil modes are used for a more homogenous excitation over the imaged volume. Because MRF allows many degrees of freedom, a further research question is to optimize the sensitivity of an implementation to the parameters intended to be measured.^{19,20}

As well as the signal preparation, potential sampling patterns have been examined. For 2-dimensional implementations, most implementations use spiral sampling patterns due to their efficiency or radial sampling patterns. It is also possible to design the gradients such that the produced sound resembles music (music MRF^{21} ). For 3-dimensional implementations, for example, stack-of-spirals,^{22} spiral projection,^{23} and Cartesian^{24} trajectories have been proposed. Magnetic resonance fingerprinting reconstruction methods go beyond the simple pattern matching. A variety of iterative,^{25,26} low-rank,^{27,28} or machine learning^{29,30} reconstructions have been proposed to reduce the effect of undersampling artifacts and thereby improving parameter map quality.

Magnetic resonance fingerprinting has been applied in various body regions including the brain,^{31–33} abdomen,^{34} cardiac,^{35} prostate,^{36} and musculoskeletal.^{37} Besides the application in different body regions, the framework has been proven to successfully quantify various effects. This ranges from system parameters such as B0^{12} and B1+^{38} to a broad spectrum of tissue metrics. Sequences were designed to be sensitive to, for example, flow,^{39} diffusion,^{40} perfusion,^{41} CEST,^{42} or magnetization transfer^{43} and to quantify those effects. Moreover, effects such as intravoxel dephasing^{44,45} and vascular properties have been quantified with MRF methodology.^{46} Although the aforementioned effects can be described on a macroscopic level, a next step for MRF could be the extension to tissue microstructure.

MICROSTRUCTURE MRI
We now turn to microstructure MRI first describing its main idea, its promise, and challenges. Some differentiation of the conventional terminology will be necessary to develop a confusion-free picture of its relation to MRF and other MRI techniques.

Microstructure MRI is a rapidly developing offspring of MRI (Fig. 1 )^{2} with the ultimate goal, sometimes referred to as “in vivo MRI histology.” This is the promise and the challenge at the same time, today yet more the latter. In more modest words, the goal is to obtain information about the cellular tissue structures, that is from spatial scales of units on the order of tens of micrometers. The challenge is the fact that this scale is much finer than the available MRI resolution above a millimeter when imaging large parts of the human body. As a paradigmatic example, one can refer to the technique called vessel size imaging,^{47–51} a method for evaluating the mean capillary caliber in the human brain, which is approximately 10 μm in the norm, using MRI with 2 mm resolution.

Such a deeply subresolution nature of microstructure demands a mediator for the translation of cellular properties from the microscopic level to the macroscopic scale of MRI voxels. Such a mediator is physics-style theory, which is commonly called biophysical modeling in the MRI community.^{2} The “anatomy” of modeling is illustrated in Figure 3 . The central task of any theory is to develop a simplified picture of reality. The presence of such a picture is a sign of theory, regardless of the actually used technique, either analytical or numerical simulations.

FIGURE 3: The workflow of microstructure MRI illustrated with the contents in Dhital et al.^{52} Brain white matter (reality) is modeled as a bunch of cylinders with some orientation dispersion and a number of cylinders in the transverse plane representing incoherent axons and glia processes. This model is subjected to the planar diffusion weighting that suppresses (dark color) the signal of molecules, which are mobile in the transverse plane (model + measurement). This results in an analytical prediction for the signal, which is fitted to the measured data giving the intra-axonal diffusion coefficient. White matter image: Courtesy of Alejandra Sierra.^{53}

Forward Problem
The simplified picture of reality cannot be drawn arbitrarily. Finding such a picture is the essence of biophysical theory. In other words, it is singling out the cellular properties that can affect the MRI signal in an essential and specific way, the so-called relevant parameters. Numerous other properties have to be neglected as they do not essentially affect the signal and are thus inaccessible by such a measurement. It is not a rare situation when a target microstructural parameter turns to be irrelevant for a given measurement technique.

With a bird's-eye view of physics, the picture of reality always depends on the spatiotemporal scale of considered phenomena relative to the sensitivity scale of the measurement.^{54} For example, the extremely complex picture of quarks and gluons inside protons and neutrons becomes fully irrelevant for physics of atoms for which the only relevant parameters are the nucleus charge and—depending on the measurement—mass or spin. Increasing further the considered scale, we come to chemistry for which the most relevant are the occupied and vacant electron orbitals, although the deep electron shells of an atom are not so important. Jumping over very many orders of magnitude, we come for example to different moduli describing the mechanical properties of materials. They surely depend on the molecular composition, but engineers do not need to even know about the molecules. They just use tables of empirically defined parameters of construction materials.^{†}

A similar example within the field of MRI is the emergence of relaxation times and diffusion coefficients in pure water. Although the dynamics of water molecules on the nanometer scale is extremely complicated, the MRI-relevant parameters on the macroscopic scale are just 3 numbers: T1, T2, and D. The picture behind the relaxation theory is randomly rotating molecules with the rotational correlation time being the central parameter. The diffusion coefficient represents the variance of molecular displacement, although all other moments of the displacement distribution (the mean third, fourth, and so on powers) become irrelevant with time. The presence of only 3 relevant parameters with a large number of irrelevant ones is guaranteed by the central limit theorem. It predicts the Gaussian distribution for the sum of many independent random variables regardless of their original probability distribution. This is perhaps the most simple example how a large number of parameters that define the form of the original distribution become massively irrelevant. This theorem is applied to the spin phases and molecular displacements resulting in particular in the monoexponential signal relaxation in homogeneous media as described by Equation 1.

We now take an imaginary trip from the molecular scale to the millimeter-sized MRI voxels in a biological tissue (Fig. 4 ). This tissue possesses the cellular structure, which defines its function and makes it much more complex than merely the aqueous solution of proteins and other substances. This structure spans from the submicrometer scale up to tens of micrometers for large cells. At this scale, our cards are shuffled; simple signal forms become obsolete, giving place to complex, ill-studied functional forms. Diffusion becomes non-Gaussian, time-dependent, nonmonoexponential relaxation. In particular, the spin echo relaxation rate, the so-called R2, acquires a dependence on the TE, which is absent in structureless fluids.

FIGURE 4: Illustration of the spatial scales involved in the MRI signal formation. The basic relevant parameters, which are T1, T2, and the diffusion coefficient D, are formed on the molecular level and affect the contrast of MRI scans. Other, irrelevant parameters describing the molecules have no influence on the signal. The tissue microstructure is represented with symbols for a bunch of axons, neural cells, the vascular system, and the lung airways spanning several orders of magnitude. The large gap between the scales of molecules and imaging (MRI signal) guarantees that the central limit theorem takes effect with a high precision. Because the scale separation between the tissue microstructure and imaging is not so large, the central limit theorem does not fulfill so precisely; therefore, there are more relevant parameters that make diffusion and relaxation more complicated, in particular, time dependent.

This happens because time available for MRI measurements after a single excitation ranges from 1 millisecond to 1 second. During this time, water molecules diffuse over the distances from a few tens of micrometers. In the optimist's view, this is a gift from nature because this length coincides with the typical cell size, which is the reason for the sensitivity of diffusion MRI to the cellular structure. In the pessimist's view, there is not such a huge difference in the scales as between the molecules and the MRI voxels; therefore, the central limit theorem does not work out so well and we are exposed to the full overwhelming complexity of diffusion and relaxation in such a complex medium as a living tissue. Yet, with the understanding of this challenge and by adapting the measurement techniques to the measurement targets, the MRI community has already solved a number of problems, which are presented below.

Coming back to the workflow of modeling, the simplified picture of reality results in a prediction for the MRI signal when supplemented with a specific measurement technique. This prediction can be made in either form, analytical or numerical. The entire way from the real tissue to the signal prediction is commonly referred to as the solution to the forward problem.

Inverse Problem
The final and therefore the most visible step in the microstructure workflow (Fig. 3 ) is the solution to the inverse problem, with the estimation of the model parameters given measured data. Data science is always engaged at this step. The involved methods range from the simple least-square fitting to artificial intelligence. A misconception often seen in discussions and publications is the belief that trained algorithms implemented on powerful computers can find any microstructural feature. In fact, no microstructural parameters are accessible. Figure 5 illustrates 2 basics limitations. Irrelevant parameters (of course in the sense of physics, not biology) are those that have little influence on the signal.

FIGURE 5: Schematics of the forwards and inverse problems. Each value of parameter results in a value of signal. However, finding the parameter given a signal is not always possible. Left, Long arrows show the ideal case of the well-posed inverse problem in the domain with a one-to-one correspondence between the parameter and the signal. In the domain where the parameter is small (but strictly positive), the signal dependence on the parameter is weak; therefore, many parameter values can explain the same signal measured with some noise. The signal-to-noise ratio in this example is approximately 50, and signal is proportional to parameter^4 similar to the dependence on the axonal diameter. Right, Similar situation appears near the minimum of the signal dependence on the parameter. Another case of the ill-posedness of the inverse problem is when 2 essentially different parameter values explain the same signal, similar to Jelescu et al.^{55}

An example is the axonal diameter in the human brain for measurements with clinical MRI systems.^{2,56–58} The typical axonal diameter in the brain is just too small for such measurements, although larger axons, for example, in the spinal cord can be measured with powerful gradient systems of animal scanners (see review^{59} and references therein). Another example is the problem of finding the parameters of the so-called standard model of white matter,^{2,59} which includes the volume fractions and the diffusion coefficients of intra-axonal and extra-axonal spaces and the axonal orientation dispersion. The signal acquired with multiple diffusion-sensitized directions and b-values (multishell measurements) is still too featureless to enable finding all involved parameters. Fitting the model to such a signal fixes a few combinations of the model parameters, but not all. At least 1 combination is left nearly arbitrary, thus resulting in a continuous degeneracy of the result.^{55,59} It happens on top of this problem that 2 or more groups of microscopic parameters result in very close signals, practically indistinguishable in real measurements.^{55,59} All these problems are usually unified under the common notion of the ill-posedness of the inverse problem.

The Challenge of Microstructure MRI
The previous discussion can be summarized with the following formula^{2} :

$\text{biophysical modeling}=\text{theory}+\text{parameter estimation}$

Note that the word “model” is often used in a much narrower sense as a synonym for a mathematical expression, which is fitted to the measured data. It is a tradition in data science that focuses on the parameter estimation step, regardless of the origin of the fitted expression. In the multidisciplinary MRI community, this creates however a long-lasting terminological confusion.

The complex indirect relation between microstructure and the MRI signal has far-reaching consequences for the translation to preclinical and clinical research.^{2}

First, the absence of a simple and direct relation between the microstructure and the signal demands distinguishing biophysical models with the specific microstructural parameters (cell size, volume fraction, membrane permeability, etc) and the signal-characterizing parameters such as T1, T2, diffusion coefficient,^{‡} diffusion tensor, and kurtosis. Yet these parameters are physical, describing in many cases a reasonable approximation to the signal dependence on time or b-value. However, they result from the massive averaging within MRI voxels, so massive that they lose the specificity to the cellular properties. In that sense, they decouple from the actual microstructure. It was proposed to call the signal-describing mathematical expression involving such quantities “representations” to contrast them to the actual biophysical modeling.^{2} As a rule of thumb, representations are mathematical expressions for the signal, although models are pictures that predict the signal when supplemented with a measurement technique (Fig. 3 ). Accordingly, a model prediction in the mathematical form becomes a representation when detached from its modeling roots.

From this point of view, MRF is an efficient way to find the model parameters of the relaxation in homogeneous fluid, T1, and T2. When applied to biological tissues, these relaxation rates parameterize the exponential signal representation, which might be rather precise in some cases, but not always. To give an example when such a representation is insufficient, it is a typical beginner's mistake when doing vessel size imaging to replace the spin echo measurement with the turbo spin echo (also known as fast spin echo or RARE). This sequence is faster with its shorter TE and thus more convenient. The replacement is also justified by the experience gained from dealing with homogeneous fluid, which are described by Equation 1. However, the targeted microvessel contribution to the relaxation rate crucially depends on the pulse sequence including its TE,^{49} which is ignored by such a replacement, resulting in a strong bias in the results.

The second consequence follows from the complexity of biophysical modeling. As a mediator between the order-of-magnitude different scales, it involves methods of statistical physics, gradually crystallized in a separate discipline within MRI. In practical terms, it means that it is recommended to cooperate intensively with the modeling community that writes the publications referred to in Figure 1 , with the aim of achieving a clearly characterized validity of a model before it is “exported” to preclinical or clinical research. This is especially important, as nonadequate modeling does not lead to apparent image artifacts, but results in problems such as missing accuracy and precision.^{2}

Current Progress In Microstructure MRI
We now turn to the positive side of microstructure MRI by discussing the current trends and already achieved progress. There are 2 fortunate cases, rather exceptions from the general rule, in which the microstructure is quite directly reflected in the signal. The first one is the increase in the diffusion coefficient in edema. The microstructural picture behind this effect is the increased amount of water in the tissue, roughly speaking the “dilution” of the microstructure with water. Although plausible, it is not easy without histology to answer the question about the localization of water by finding the fractional increase in its amount in the extracellular space and specific cell species. Another example is the strong anisotropy of diffusion in the brain white matter, which helps revealing the principal fiber direction in a given voxel. However, fiber crossing and even the orientation dispersion within a fiber bundle break down the simplicity. Detecting fiber directions requires advanced methods^{60,61} not quite identical to finding microstructural parameters such as compartments' volume fractions.^{14}

From now on, we focus on diffusion MRI that is the most promising method for accessing the tissue microstructure. In the retrospective view of the last 2 decades, diffusion MRI has evolved from measuring the diffusion tensor^{62,63} to more detailed sampling of the signal, typically with the so-called multishell acquisition that enabled evaluating the kurtosis tensor.^{64–66} It was however noticed that the acquired diffusion-weighted signal is “remarkably unremarkable” or “uninformative”^{67,68} ; beyond the diffusion tensor, the signal shows little dependence on the microstructure of actually measured tissue, thus rendering biologically important microstructural parameters physically irrelevant.

The current way to resolve this problem is developing new diffusion weighting techniques with more specificity to the targeted microstructural features and less sensitivity to others. In particular, the departure from the decades-accepted multishell measurements helps to resolve the degeneracy of the standard model parameter estimation problem.^{69,70} For the standard model, it was empirically found^{69} that the fit landscape becomes notably less degenerate by including the planar tensor encoding, whereas Reisert et al^{70} have proven this statement analytically, using an expansion at low b-values.

The current development trend in diffusion MRI is a move from a commonly applied universal tool to a toolbox of specialized instruments. Examples of this approach include the observation of effectively 1-dimensional diffusion in the brain gray matter,^{56} detection of anisotropic randomly oriented compartments in apparently isotropic tissues,^{71–77} measurement of the exchange rates between tissue compartments,^{78–81} evaluation of surprisingly low signal contribution from small compact cells in the brain,^{82} with the only noticeable fraction in the cerebellum,^{83} and singling out the contribution of axons and measuring the intra-axonal diffusion coefficient in vivo in the normal human brain.^{52}

In general, the diffusion MRI toolbox engages a few underlying principles. It is an active manipulation of the nuclear magnetization to obtain more signal features carrying microstructural information. Such features often rely on the geometry of the targeted cell species, for instance, addressing small compact or cylindrical cells versus unbounded extracellular space. An example of such a manipulation is illustrated in Figure 3 in which the signal is suppressed everywhere, but from inside axons in a narrow bundle. Another example is the previously cited water exchange measurement for which the signal is suppressed in large compartments with unbounded diffusion (eg, extracellular space), but not in small ones. The exchange is quantified by observing the recovery of the overall diffusion coefficient due to the penetration of water from small to the large compartments.

COMBINING MRF AND MICROSTRUCTURE MRI
Because the exponential signal form is the least useful for microstructure MRI, the current technical development aims at creating dedicated measurement techniques that yield more feature-rich signals. This is a key problem where MRF could aid in designing optimal experiments for microstructure MRI. Besides the optimal experiment design and potential speedup using MRF's undersampling strategies, another advantage is the accurate multiparametric modeling; the holistic signal description and storage in a dictionary or alternatively a neural net might help in precisely resolving multiple microstructural tissue parameters from a single measurement. This however requires extending Equation 1 to account for the microstructural parameters. This is hardly thinkable as a simple modification of these equations. Combining MRF and microstructure MRI remains a challenge to the method developers.

A useful first step toward such a combination may be using MRF for multicompartment mapping. For example, 2 species with different T2 in a voxel can classically be resolved using a spin echo acquisition with varying TE. The observed biexponential decays are however very similar to deviations from the monoexponential decay due to the microstructure.^{84} By using an optimized MRF acquisition, the more complex and thereby unique signal evolution from a voxel containing several tissues can more likely be distinguished from other observed signals.^{85,86}

DISCUSSION: TOWARD CLINICAL IMPLEMENTATIONS
The 2 quite different approaches discussed previously are complementary in today's MRI universe. Magnetic resonance fingerprinting strives to find the most efficient ways to quantify voxel-averaged characteristics of the nuclear magnetization (eg, T1, T2, off-resonance) as well as the density and motion of the MR-reporting nuclei (flow and diffusion). These parameters are well familiar to radiologists due to their correlation with several pathologies. From the modern point of view, they specify the signal representations as follows: mathematical expressions, typically exponential ones that approximate the signal dependencies on time, and the strength of applied gradients. Such dependencies are indirectly related to the tissue microstructure, but can correlate with specific pathologies.^{2} For example, T1 was found to be an in vivo biomarker for higher-grade renal cell carcinomas^{87} or T1, T2, and ADC can be used to identify peripheral zone prostate cancer, which was validated by targeted biopsies.^{88} The efficiency of MRF for delivering such representation parameters is high,^{89} although yet to be compared with the theoretical limit on the information yield per unit of time^{90} for various implementations. Further, the penalty of undersampling should be better understood.

In contrast to the representations, microstructure MRI aims at the genuine tissue microarchitecture, often approached via a long chain of insights. The microstructural parameters available today are not defined by the clinical needs yet, rather by the technical feasibility of the measurements. Translation toward meeting clinical needs requires a slight change in the common researchers’ mindset. Although correlating parameters with pathologies remains inevitable in the clinical research, microstructure MRI can make the search for such parameters more targeted, in particular guided by the known histopathology. Such search can lead to unexpected parameters, irreducible to conventional measures such as T1, T2, and diffusion tensor. Although not easy to find, they promise to be more sensitive and specific, in statistical terms, better correlated with the disease.

The way to higher clinical efficiency of MRI is likely to lead through a very high sensitivity and specificity in a very narrow use case. For example, instead of looking at an examination with multiple contrasts and predefined parameter configurations for each pulse sequence, such as the multiparametric examination for the diagnosis of prostate cancer, one could apply a single yet unknown pulse sequence that is highly optimized for this case. Microstructure MRI and MRF can address this problem from 2 opposite ends: from the histopathology-based model of prostate cancer and from the development of a fast and efficient measurement for defining the model parameters, respectively. The MR signal prediction, which is necessary for MRF, is to be made based on the tissue properties, thus replacing Equation 1 with a microstructure-based signal description. It is then likely that the sequence with best contrast to noise with respect to the clinically relevant parameter can be computed by solving an optimization problem. Such a sequence would then serve as a virtual histology tool that describes the disease as accurately as it can be described by means of MRI. In more general terms, although MRF has the ability to drastically boost the relevance of MR quantification, microstructure MRI can lead it to specific clinical targets. From our point of view, this combination promises to lift clinical MRI into a new era.

On one hand, this hope motivates the development in both research fields. On the other hand, the novelty of the 2 methods requires a pioneering spirit on the clinical side, quite similar to the establishment of MRI as such. Where today’s quantification in MRI mostly addresses the properties of the magnetization and concludes diagnostic procedures from correlations with biology, the 2 methods discussed here promise to go beyond that. In other words, the promise lies in addressing selected biological properties of the examined tissue directly. In this scenario, the magnetization only acts as a mediator and is disappearing from the clinical stage. Building this bridge will not be a simple task. In particular, Microstructure MR would profit heavily from accurate and suitable pathological descriptions. If we start thinking of cells, blood vessels, and their geometry convolved with motion, the clinical applicability may sound far off. However, the promise of success may provide the needed energy.

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