NMR shutter‐speed elucidates apparent population inversion of ¹H₂O signals due to active transmembrane water cycling

Abstract

Purpose

The desire to quantitatively discriminate the extra‐ and intracellular tissue ¹H₂O MR signals has gone hand‐in‐hand with the continual, historic increase in MRI instrument magnetic field strength [B ₀]. However, recent studies have indicated extremely valuable, novel metabolic information can be readily accessible at ultra–low B ₀. The two signals can be distinguished, and the homeostatic activity of the cell membrane sodium/potassium pump (Na⁺,K⁺,ATPase) detected. The mechanism allowing ¹H₂O MRI to do this is the newly discovered active transmembrane water cycling (AWC) phenomenon, which we found using paramagnetic extracellular contrast agents at clinical B ₀ values. AWC is important because Na⁺,K⁺,ATPase can be considered biology’s most vital enzyme, and its in vivo steady‐state activity has not before been measurable, let alone amenable to mapping with high spatial resolution. Recent reports indicate AWC correlates with neuronal firing rate, with malignant tumor metastatic potential, and inversely with cellular reducing equivalent fraction. We wish to systematize the ways AWC can be precisely measured.

Methods

We present a theoretical longitudinal relaxation analysis of considerable scope: it spans the low‐ and high–field situations.

Results

We show the NMR shutter‐speed organizing principle is pivotal in understanding how trans–membrane steady–state water exchange kinetics are manifest throughout the range. Our findings illuminate an aspect, apparent population inversion, which is crucial in understanding ultra‐low field results.

Conclusions

Without an appreciation of apparent population inversion, significant misinterpretations of future data are likely. These could have unfortunate diagnostic consequences.

1. INTRODUCTION

1.1. Discerning tissue water compartmentalization

The major water compartmentalization in tissue is intra‐ and extracellular (“inside”/“outside”). In almost all parenchymal tissue, the vascular space comprises a small volume fraction. Thus, in the simplest approximation, this is a two site situation. A reliable way to quantitatively discriminate in vivo ¹H₂O_i and ¹H₂O_o NMR signals has been a very long quest.

1.2. Active transmembrane water cycling

Recently, this pursuit has gained much greater importance. It has been discovered that the pseudo‐first‐order rate constant for homeostatic cellular water efflux (k_io) has an energetically active component, k_io(a), as expressed in Equation 1, and

$$k_{\mathit{io}}=k_{\mathit{io}}(p)+k_{\mathit{io}}\left(a\right)$$ (1)

elaborated in Equation 2. The passive component, k_io(p), is 〈A/V〉P_W(p), where: 〈A/V〉 represents

$$k_{\mathit{io}}=⟨\frac{A}{V}⟩P_W\left(p\right)+\left(\frac{x}{\left[H_2{O}_i\right]⟨V⟩}\right){}_{}^{c}M{R}_{\mathit{NKA}}$$ (2)

the voxel average or region of interest average 〈cell surface area/volume〉 ratio, and P_W(p) is the diffusive (“passive”) cell membrane water permeability coefficient. In this study, quantities in brackets, 〈〉, represent voxel or region of interest averages. All symbols and acronyms are defined in the Appendix. It was previously thought k_io = k_io(p): there was no active component. However, this is not the case: k_io(a) is always present, and often dominant.1, 2, 3, 4 It is elaborated as (x/([H₂O_i]〈V〉))^cMR_NKA, where ^cMR_NKA is the cellular metabolic rate of the cell membrane Na⁺,K⁺‐ATPase (NKA) (fmol(ATP)hydrolyzed/cell/s), [H₂O_i] is the intracellular water concentration, and x is the stoichiometric mole ratio of water actively cycled to ATP hydrolyzed by NKA [fmol(H₂O)/fmol(ATP)]. Thus, an enzymatic activity generates a membrane permeability. Active transmembrane water cycling (AWC) is a fundamental aspect of water biology not previously described.

This is important because NKA can be considered biology’s most vital enzyme, but its in vivo homeostatic activity has never been measurable or amenable to mapping.1, 2, 3, 4 Significantly, it has been found that k_io correlates with metastatic potential in breast5 and melanoma6 cell lines, and with neuronal firing in brain tissue.4 This is a new imaging biomarker with potentially great power.

Previously, the main candidate techniques for ¹H₂O_i/¹H₂O_o signal discrimination required the use of an exogenous, extracellular paramagnetic contrast agent (CA_o) to increase the ¹H₂O_o R_1o (≡ 1/T_1o) value selectively.3, 7, 8 This will be detailed below. Indeed, studies on model systems,1, 2, 3, 4, 9, 10 in which constant [CA_o] ≥ 5 mM can be sustained during complete relaxation recovery measurement, were required to confirm the (x/([H₂O_i]〈V〉))^cMR_NKA term in Equation 2. However, this approach is problematic for in vivo human study.3

Since the beginning of NMR, there has been a seemingly inexorable march to instruments with higher magnetic field (B ₀) values.11 This has been driven by the increased signal/noise ratio and spectral dispersion. However, this trend has not been particularly helpful for the discrimination of ¹H₂O_i and ¹H₂O_o. Even though CA detectability increases slightly with increasing B ₀,12 the diminished relaxivity (r₁) of approved CAs at current clinical B ₀ values13 requires large CA doses. Consequently, safety and environmental regulatory restrictions preclude achieving the high, sustained [CA_o] values sufficient for precise k_io determination in vivo.3 Recently, Aime and co‐workers have demonstrated CA‐free ¹H₂O_i/¹H₂O_o discrimination, again in an animal model and cell suspensions, at ultra‐low B ₀ values.5, 10 Here, we present a comprehensive analysis of the fundamental principles spanning the high and low field experiments.

2. METHODS

2.1. Intrinsic sample or voxel compartmental properties

There are two intrinsic NMR properties of interest: R_1i and R_1o, R_1i is the ¹H₂O_i R₁ value. For CA‐enhanced MRI, the ¹H₂O_o R₁ value is given by Equation 3,

$$R_{1o}=r_{1o}\left[C{A}_o\right]+R_{1o0}$$ (3)

where r_1o is the extracellular CA longitudinal relaxivity [CA_o], the extracellular CA concentration, and R_1o0 the R_1o value in the absence of CA. In addition to being temperature–dependent, the R_1i, r_1o, and R_1o0 properties are also B ₀–dependent.

There are two intrinsic cell biology properties of interest: p_i, and k_io. The quantity p_i is the mole fraction (“population”) of water that is intracellular. The intracellular volume fraction, v_i, is given by Equation 4, where f_M is

$$v_i=\left(1-f_M\right)p_i+f_M$$ (4)

the tissue volume fraction inaccessible to mobile aqueous solutes.2, 8 When the vascular fraction is neglected, Equations (5a), (5b)a and 5b obtain, where p_o and v_o are the respective extracellular mole and volume fractions.

$$p_o=1-p_i$$ (5a)

$$v_o=1-v_i.$$ (5b)

The pseudo‐first‐order rate constant for homeostatic cellular water efflux is k_io. (The parameter k_io is the reciprocal of the mean intracellular water molecule lifetime, 1/τ_i. The τ_i value is often reported in the literature.) Because we assume a steady‐state, k_io is also given by Equation 6, where k_oi is the corresponding influx rate constant. While still

$$k_{\mathit{io}}=\left(p_o/p_i\right)k_{\mathit{oi}}$$ (6)

temperature‐dependent, p_i, f_M, k_io, and k_oi do not depend on B ₀ or [CA_o].

Figure 1 illustrates the isothermal behavior of realistic, representative intrinsic NMR properties over a very large range. The ordinate measures R₁: the R_1i curves are blue, while the R_1o curves are red. The abscissa is bifurcated: the left side measures an increase in log v _L, the Larmor resonance frequency (proportional to log B ₀: the ¹H magnetogyric ratio is 0.023 T/MHz), with [CA_o] = 0; while the right side measures an increase in [CA_o], with v _L constant at 43 MHz (B ₀ = 1.0 T), the largest v _L reached on the left. The smooth R_1i and R_1o curves on the left pass through fitted values for a murine xenograft TS/A breast cancer tumor.5 They exhibit the familiar dispersive shape [an R_1i inflection point near 0.1 MHz (0.002 T)]. The R₁ B ₀‐dependence has long been referred to as NMR dispersion. The R_1o values on the right are calculated using r_1o = 3.8 s⁻¹(mM)⁻¹, typical for approved Gd(III) chelates.7 All magnitudes are for T = 37°C.

Figure 1.

The dependences of the stipulated intrinsic compartmental longitudinal relaxation rate constants, R₁, on the: magnetic field, B ₀ (left), and extracellular CA concentration, [CA_o] (right). The left abscissa has a log B ₀ scale, with fixed [CA_o] = 0. The right abscissa is linear in [CA_o], with fixed B ₀ = 1.0 T. The intracellular R_1i (blue) and extracellular R_1o (red) rate constants on the left include those reported in Ruggiero et al.5 The R_1o values on the right were calculated from Equation 3 with extracellular CA relaxivity, r_1o = 3.8 mM⁻¹s⁻¹. The longitudinal MR shutter–speed, к₁ (Equation 9), and the VSS condition are indicated. It is important to note these would be the experimentally measured R₁ rate constants if there was no trans‐cytolemmal water exchange [k = 0]

Figure 2 illustrates the behavior of the intrinsic cell biology properties: we take representative values of p_i (0.8) and k_io (1 s⁻¹).3, 5 (Thus, p_o = 0.2, and k_oi = 4 s⁻¹ [k_io + k_oi = 5 s⁻¹]). The left ordinate measures the p values (p_i, blue; p_o, red), while the right ordinate measures the overall exchange rate constant, k, given by Equation 7. The abscissa is

$$k=k_{\mathit{io}}+k_{\mathit{oi}}$$ (7)

Figure 2.

The non‐dependences of the stipulated intrinsic compartmental mole fractions (“populations”), p, and intercompartmental exchange rate constant, k (Equation 7), on: (left) the B ₀ and (right) the [CA_o]. The abscissa is the same as in Figure 1. The intracellular p_i (blue) and extracellular p_o (red) populations are measured on the left ordinate, while k is measured on the right ordinate

the same as for Figure 1. Because these are isothermal plots, these properties exhibit horizontal lines.

2.2. Magnetic field‐dependence

It has long been known the tissue ¹H₂O ${{\text{R}}^{\prime }}_1$ (the observed, approximated monoexponential R₁ value) increases with decreasing B ₀ (reviewed in Rooney et al11). In the absence of CA_o, the ¹H₂O longitudinal relaxation mechanism is generally dominated by water intramolecular ¹H – ¹H magnetic dipole fluctuations at the v _L frequency. As B ₀ decreases, the v _L value decreases toward the increased probability of experiencing such fluctuations (“spectral density”) found in tissue.14 In contrast, the ¹H₂O R₁ value of pure water is not very B ₀–dependent: the inherent fluctuations have much greater frequency [reciprocal of the molecular rotational (“tumbling”) correlation time constant, τ_r ⁻¹ ≅ 10¹² s⁻¹] than v _L, and thus are too fast for efficient relaxation.11, 14 Therefore, it has long been suspected the inverse R₁, v _L relationship is due to the presence of macromolecules in tissue. Macromolecular tumbling is much slower than that of the molecules in pure water.15 The fluctuation is τ_r ⁻¹ ≅ 1.7/MW, where MW is the macromolecular mass in kDa. (Thus, even a smallish 100 kDa macromolecule has τ_r ⁻¹ ≅ 1.7 × 10⁷ s⁻¹.) To our knowledge, there is no simple physical model that predicts the B ₀‐dependence of tissue ¹H₂O ${{\text{R}}^{\prime }}_1$. Any attempt to match data with, say, a superposition of Lorentzian functions requires an empirical distribution of τ_r values.11

The extra‐ and intracellular tissue spaces both contain macromolecules,3 and one could not know for certain if extra‐ or intracellular macromolecules, or both, dominated the effect. One avoids this uncertainty by writing Equation 8 for iron‐free tissue.11 In this expression: r_1M is the tissue macromolecular relaxivity, and R_1H2O is

$$R_1^{\prime }=r_{1M}{f}_M+R_{1H2O}$$ (8)

the pure water R₁ at physiological temperature. (An extra term can be added if a tissue contains a sufficient amount of paramagnetic iron.11) The greater r_1M decrease with increasing B ₀ compared with that of r_1o is the cause of the aforementioned slight CA detectability increase.12

However, Ruggiero and co‐workers used a Matrigel phantom as a model for extracellular space.5 They found the R_1o value, so approximated, exhibits only a very slight NMR dispersion, as seen in Figure 1 (left). This supports a long‐held suspicion that the ${{\text{R}}^{\prime }}_1$ B ₀‐dependence is dominated by an increasing R_1i value with decreasing B ₀.16 This is attributed to sites for water molecules that are characterized as “buried” within macromolecules (H₂O_bu), and the surmise that such sites are more abundant in intracellular than in extracellular macromolecules.16 Water molecules in such sites can more fully experience the slower macromolecule rotation, and thus particularly effective slow intramolecular ¹H – ¹H fluctuations. Such macromolecules are endogenous, intracellular contrast agents, CA_is. Although their concentrations do not increase with decreasing B ₀, their relaxivities do. (Relaxivities and concentrations always appear together as products,2, 17, 18 as in Equation 3.) Nonetheless, the miniscule number of H₂O_bu molecules are still in rapid exchange (rate constant > 10⁴ s⁻¹)19 with the vastly greater number of all other H₂O_i molecules, certainly as compared with the k_io magnitude. They could even use the Grotthuss proton hopping mechanism.15 In any case, the cytoplasm is “well–mixed.” This is depicted in the Figure 3 cartoon. Some very large heterogeneous compartments (e.g., Xenopus oocyte [〈V〉 = 840 nL] cytoplasms)20 can exhibit inhomogeneous ¹H₂O resonances. However, most tissue cell 〈V〉 values range from hundreds of fL to a few pL.21 In such small cells, even a conservatively small diffusion coefficient leads to good water mixing in any NMR experimental time period.9

Figure 3.

A stylized cartoon depiction of the steady‐state water exchange processes that dominate the tissue ¹H₂O MR signal longitudinal relaxation at ultra‐low‐field. The trans‐cytolemmal process has a k_io (Equation 2) and a k_oi (Equation 6) rate constants. The exchange of water out of and into macromolecular buried sites, H₂O_bu, is much faster than k = k_io + k_oi. For the considerations here, cytoplasmic water is “well‐mixed.” This figure was prepared with the help of Gangxu Han

The steady‐state transmembrane water molecule exchange process, k = k_io + k_oi, is rate‐limiting; i.e., slower than essentially all other water molecule interaction kinetics in tissue. And, this has been the source of considerable confusion in the in vivo MRI literature. If the exchange kinetics were exceedingly slow, or exceedingly fast, the interpretation of experimental ¹H₂O ${{\text{R}}^{\prime }}_1$ data would be straightforward. But slow and fast are rather misleading adjectives. Figure 2 shows the k does not depend on B ₀ or [CA_o]. This is why we introduced18 the concept of the longitudinal NMR relaxation shutter‐speed, к₁,3, 8 the absolute value function defined in Equation 9. The comparison of k with к₁

$${\kappa }_1\equiv |R_{1i}-R_{1o}|$$ (9)

determines the “exchange condition” of the tissue ¹H₂O MR signal. If k is insufficiently greater than к₁, a “slow” condition obtains: but if it is sufficiently larger, the system is in a “fast” condition. In experimental terms, a slow condition means the longitudinal relaxation is non‐monoexponential, and a fast condition means the relaxation is monoexponential. The k_io measurement precision depends on the extent κ₁ exceeds k. Obviously, the greater the k value, the larger the к₁ required.

As Figures 1 and 4 indicate, in biological tissue it is к₁ that can be manipulated by the investigator [by means of B ₀ and/or [CA_o]], usually not k (almost all in vivo studies are isothermal). Because k does not change, the exchange reaction does not go “faster” or “slower.” Using these terms is meaningful only if one thinks of к₁ changes as “warping” time. Thus, a slow condition is more profitably understood as a “large‐shutter‐speed” (LSS) condition; and a fast condition as a “small‐shutter‐speed” (SSS) condition. These distinctions are important because the nature of the exchange condition strongly influences the correct interpretation of the experimental result. This is described by the well‐known Bloch–McConnell‐Woessner (BMW) Rate Law Equations18, 22, 23, which elaborate the phenomenological signal equation.

Figure 4.

The dependences of the stipulated MR shutter‐speed, к₁ (Equation 9), and intercompartmental exchange rate constant, k (Equation 7), on (left) the B ₀ and (right) the [CA_o]. The abscissa is the same as in Figure 1, and the ordinate the same as on the right in Figure 2. The vanished shutter‐speed VSS condition is indicated. The rate constant k_io can be determined with precision only when к₁ >> k, the large shutter‐speed regime LSSR period

2.2.1. Apparent sample or voxel compartmental properties

The experimental relaxation decay is often fitted with an empirical bi–exponential expression, with apparent fast and slowly relaxing components that are not coupled by molecular exchange, Equation 10, where: S can be a recovery time‐course signal or a multi‐pulse MR steady‐state

$$S/S_0=p_{\mathit{fast}}^{\prime }{\mathbf{S}}_{\mathbf{fast}}^{\prime }+{\mathbf{p}}_{\mathbf{slow}}^{\prime }{\mathbf{S}}_{\mathbf{slow}}^{\prime }$$ (10)

signal, S₀ is the Boltzmann signal, ${{\text{p}}^{\prime }}_{\text{fast}}$ and ${{\text{p}}^{\prime }}_{\text{slow}}$ the apparent populations, and ${{\mathbf{S}}^{\prime }}_{\text{fast}}$ and ${{\mathbf{S}}^{\prime }}_{\text{slow}}$ are fractional recovery or longitudinal and transverse relaxation saturation factor functions (running from 1 to 0). When transverse relaxation cannot be ignored, the saturation factor functions can be complicated. These are left implicit in Equations 11 and 12,

$${\mathbf{S}}_{\mathbf{fast}}^{\prime }=\mathbf{f}\left(\mathbf{time},\alpha ,\mathbf{TR},{\mathbf{R}}_{\mathbf{1},\mathbf{fast}}^{\prime },\mathbf{TE},{\mathbf{R}}_{\mathbf{2},\mathbf{fast}}^{\prime }\right)$$ (11)

$${\mathbf{S}}_{\mathbf{slow}}^{\prime }=\mathbf{f}\left(\mathbf{time},\alpha ,\mathbf{TR},{\mathbf{R}}_{\mathbf{1},\mathbf{slow}}^{\prime },\mathbf{TE},{\mathbf{R}}_{\mathbf{2},\mathbf{slow}}^{\prime }\right)$$ (12)

where time is the recovery period, and α, TR, and TE are the steady‐state acquisition pulse flip angle, repetition time, and echo time, respectively. (We have presented examples of recovery2, 9 and longitudinal steady‐state7 functions.) Each apparent component is defined to have single ${{\text{R}}^{\prime }}_1$ (${{\text{R}}^{\prime }}_{1,\text{fast}}$, ${{\text{R}}^{\prime }}_{1,\text{slow}}$) and complicated ${{\text{R}}^{\prime }}_2$ (${{\text{R}}^{\prime }}_{2,\text{fast}}$, ${{\text{R}}^{\prime }}_{2,\text{slow}}$) values.

However, in the vast majority of in vivo MR experiments, there is molecular exchange (steady–state) between the populations, and this makes the situation completely different from the empirical bi–exponential description. This is true even if, to simplify the subsequent derivation, we set ${\mathbf{S}}_{\text{fast}}^{\prime }$ = ${\mathbf{S}}_{\text{slow}}^{\prime }$ = 1. (As we will see below, however, this is probably, and importantly, not true for in vivo experiments, where multipulse acquisition, with incomplete recovery, is required for imaging purposes.) For nonimaging studies of model systems, S can be measured with α = 90° and complete recovery (large TR). Thus, the longitudinal saturation components in Equations 11 and 12 can be dealt with.

With these caveats, the apparent longitudinal relaxation rate constants, R₁′, and longitudinally “fully relaxed” (“unsaturated”) apparent mole fractions, p′, are expressed, in the isochronous (v _Lo = v _Li) BMW two‐site‐exchange (2SX) exchange Equations (13), (14), (15), (16) through (13), (14), (15), (16), as functions of the intrinsic system parameters.7, 18, 22, 23

$$R_{1,fast}^{\prime }=\frac{1}{2}\left[\left(R_{1i}+R_{1o}\right)+k\right]+\frac{1}{2}{\left\{{\left[\left(R_{1i}-R_{1o}\right)+\left(k_{\mathit{io}}-k_{\mathit{oi}}\right)\right]}^2+4k_{\mathit{io}}{k}_{\mathit{oi}}\right\}}^{1/2}$$ (13)

$$R_{1,slow}^{\prime }=\frac{1}{2}\left[\left(R_{1i}+R_{1o}\right)+k\right]-\frac{1}{2}{\left\{{\left[\left(R_{1i}-R_{1o}\right)+\left(k_{\mathit{io}}-k_{\mathit{oi}}\right)\right]}^2+4k_{\mathit{io}}{k}_{\mathit{oi}}\right\}}^{1/2}$$ (14)

$$p_{\mathit{fast}}^{\prime }=\frac{1}{2}-\frac{1}{2}\left\{\frac{\left(R_{1i}-R_{1o}\right)\left(p_o-p_i\right)+k}{{\left\{{\left[\left(R_{1i}-R_{1o}\right)+\left(k_{\mathit{io}}-k_{\mathit{oi}}\right)\right]}^2+4k_{\mathit{io}}{k}_{\mathit{oi}}\right\}}^{1/2}}\right\}$$ (15)

$$p_{\mathit{slow}}^{\prime }=\frac{1}{2}+\frac{1}{2}\left\{\frac{\left(R_{1i}-R_{1o}\right)\left(p_o-p_i\right)+k}{{\left\{{\left[\left(R_{1i}-R_{1o}\right)+\left(k_{\mathit{io}}-k_{\mathit{oi}}\right)\right]}^2+4k_{\mathit{io}}{k}_{\mathit{oi}}\right\}}^{1/2}}\right\}.$$ (16)

(These equations correct typographical errors in Equations 6 and 7 of Li et al.7 The term that was printed as p_i(1 – p_i)/τ_i in Equation 6 should have been p_i/[(1 – p_i)τ_i], and the square root should have been of the Equation 7 denominator.) Each apparent parameter has contributions from both analogous intrinsic parameters.

3. RESULTS

It is very important to recognize the experimentally measured relaxation rate constants and mole fractions (${\text{R}}_1^{\prime }$ and ${\text{p}}^{\prime }$) are not the same as the system R₁ and p values that are desired. We input the Figures 1 and 2 intrinsic parameter values into Equations (13), (14), (15), (16) to “reverse engineer” typical parameters from experiments. This is illustrated in Figures 5 and 6, where expected ${{\text{R}}^{\prime }}_1$ and ${\text{p}}^{\prime }$ values, respectively, are plotted with abscissae identical to that of Figures 1, 2, and 4. The Figure 5 ordinate is identical to that of Figure 1, while the Figure 6 ordinate is identical to the Figure 2 left ordinate. The empirical labels are given outside the right ordinates: ${{\text{R}}^{\prime }}_{1,\text{fast}}$ (Equation 13) and ${{\text{R}}^{\prime }}_{1,\text{slow}}$ (Equation 14) for Figure 5, and ${{\text{p}}^{\prime }}_{\text{slow}}$ (Equation 16) and ${{\text{p}}^{\prime }}_{\text{fast}}$ (Equation 15) for Figure 6. In Figure 1, the R_1i and R_1o values cross (point): R_1,cross = R_1i = R_1o = 1.1 s⁻¹ at [CA_o] = 0.17 mM. This would be the experimental result if there was no exchange: k = 0, the no‐exchange‐limit (NXL). However, when exchange kinetics are finite this crossing is avoided (Figure 5). (See also Figure 3 of Bai et al24 and Figure 4 of Labadie et al.25) This result is a property of the “mixing” inherent in any coupled differential equations, such as those giving rise to Equations (13), (14), (15), (16).

Figure 5.

With the intrinsic parameters from Figures 1 and 2, we calculated the dependences of the expected apparent compartmental longitudinal relaxation rate constants, ${\text{R}}_1^{\prime }$ (${\text{R}}_{1,\text{fast}}^{\prime }$, Equation 13; ${\text{R}}_{1,\text{slow}}^{\prime }$, Equation 14) on (left) the B ₀ and (right) the [CA_o]. The ${\text{R}}_{1,\text{i}}^{\prime }$ and ${\text{R}}_{1,\text{0}}^{\prime }$ rate constant segments are colored blue and red, respectively. The colors switch when the curves pass through the VSS condition. The abscissa is the same as in Figure 1, and the ordinate the same as on the left in Figure 1

Figure 6.

With the intrinsic parameters from Figures 1 and 2, we calculated the dependences of the expected (longitudinally) fully relaxed apparent compartmental populations, ${\text{p}}^{\prime }$ (${\text{p}}_{\text{slow}}^{\prime }$, Equation 16; ${\text{p}}_{\text{fast}}^{\prime }$, Equation 15) on (left) the B ₀ and (right) the [CA_o]. The ${\text{p}}_{\text{i}}^{\prime }$ and ${\text{p}}_{\text{o}}^{\prime }$ population segments are colored blue and red, respectively. The colors switch when the curves pass through the VSS condition. The abscissa is the same as in Figure 1, and the ordinate the same as on the left in Figure 2. An apparent population equality APE point is indicated, as is the region of apparent population inversion API, along with a conservatively small monoexponential relaxation regime. Unfortunately, current clinical MRI protocols fall within this latter region: the system is constrained to the vast shutter–speed “wasteland” exhibited in Figure 4

At the smallest B ₀ (0.0002 T; v _L = 0.01 MHz) and largest [CA_o] (4 mM) values simulated, when к₁ is maximally different from k (Figure 4), the ${{\text{R}}^{\prime }}_{1\text{i}}$ and ${{\text{R}}^{\prime }}_{1\text{o}}$ values are just approaching the LSS limit (LSSL) condition (к₁ >> k) values: ${{\text{R}}^{\prime }}_{1,\text{fast}}$ = R_1i + k_io (28 + 1 = 29 s⁻¹) on the left, and ${{\text{R}}^{\prime }}_{1,\text{fast}}$ = R_1o + k_oi (15 + 4 = 19 s⁻¹) on the right; ${{\text{R}}^{\prime }}_{1,\text{slow}}$ = R_1o + k_oi (2 + 4 = 6 s⁻¹) on the left, and ${{\text{R}}^{\prime }}_{1,\text{slow}}$ = R_1i + k_io (1 + 1 = 2 s⁻¹) on the right (Figure 5). However, more importantly, the p′ values (Figure 6) have clearly not yet reached the LSSL p′ values (p_i = 0.8, p_o = 0.2) (Figure 2). For in vivo human studies, it is currently not very practical to work near the earth’s field (~0.0001 T),26 and it is essentially disallowed to achieve [CA_o] of even transiently 3 mM.27 Thus, realistic clinical MR examinations are constrained to never even approach the LSSL condition. The shutter‐speed cannot be increased sufficiently.

Therefore, one must account for steady‐state trans‐cytolemmal water exchange kinetics [k] if one wishes to extract accurate system p_i and p_o values. This has been experimentally demonstrated for myocardium,28 where p_o (extracellular volume fraction, ECF) is an extremely important biomarker.3 Changes in p_i and p_o report tissue edema (a net intercompartmental water transfer) because the mean cell volume 〈V〉 = v_i/ρ, where ρ is the cell (number) density (e.g., cells/μL): the relationship between v_i and p_i is given in Equation 4. Assuming either k = ∞ (as has frequently been done), or k = 0, is incorrect.

3.1. Relaxation exponentiality

Figure 6 is very informative. It is unproductive to consider only the empirical ${{\text{R}}^{\prime }}_{1,\text{fast}}$, ${{\text{R}}^{\prime }}_{1,\text{slow}}$, ${{\text{p}}^{\prime }}_{\text{slow}}$, and ${{\text{p}}^{\prime }}_{\text{fast}}$ parameters. What are important are the intrinsic parameters with which they correlate. Because we have simulated over such a wide range, we can assign segments of the apparent curves correctly (blue for ${{\text{R}}^{\prime }}_{1\text{i}}$ and ${{\text{p}}^{\prime }}_{\text{i}}$; red for ${{\text{R}}^{\prime }}_{1\text{o}}$ and ${{\text{p}}^{\prime }}_{\text{o}}$). When к₁ approaches zero, the faster relaxing apparent component vanishes (${{\text{p}}^{\prime }}_{\text{fast}}$ → 0; Figure 3 of Bai et al,24 Figure 2 of Vétek et al29), and the experimental relaxation time‐course becomes monoexponential (${{\text{R}}^{\prime }}_1$ is single‐valued). When к₁ is actually zero, the vanished shutter‐speed (VSS, Figures 1, 4, 5, 6) condition, R₁′ is given by Equation 17.30

$$R_1^{\prime }=p_o{R}_{1o}+p_i{R}_{1i}=R_{1,cross}.$$ (17)

The signal arises from all the water: the k_io parameter does not enter the equation, and thus is intrinsically indeterminate.

Practically speaking, however, it is hard to experimentally detect a small minority component even when it is present. We draw a horizontal line at p′ = 0.1 in Figure 6. It strikes the blue ${{\text{p}}^{\prime }}_{\text{i}}$ curve on the left at log v _L ≅ 0.7 (v _L ≅ 4.3 MHz; B ₀ ≅ 0.1 T), and the red ${{\text{p}}^{\prime }}_{\text{o}}$ curve on the right at [CA_o] ≅ 2.9 mM. Because all current human MR instruments have B ₀ > 0.1 T and (as suggested above) [CA_o] values > 3 mM cannot be sustained, clinical ¹H₂O data are constrained to exhibit apparent monoexponential longitudinal relaxation (single‐valued ${{\text{R}}^{\prime }}_1$). The accessible shutter speeds are too small (Figure 4): clinical MRI is trapped in a vast shutter‐speed “wasteland.” This regime indicated is conservatively small: it can be experimentally difficult to detect a minority component even with p′ somewhat greater than 0.1.

To extract k_io and p_i when the relaxation is effectively monoexponential, when one is in the wasteland, one must vary [CA_o] (as pharmacokinetically, after a bolus injection) or – now ‐ B ₀, and take advantage, to the extent possible, of the nonlinear ${{\text{R}}^{\prime }}_{1,\text{slow}}$ [CA_o]– or B ₀‐dependence in this regime (Figure 5). Neglecting the ${{\text{p}}^{\prime }}_{\text{fast}}$ (${{\text{R}}^{\prime }}_{1,\text{fast}}$) contribution is the most common shutter–speed (dynamic‐contrast‐enhanced) DCE–MRI version [fast‐exchange‐regime‐allowed (FXR‐a)].1, 7, 8, 17, 27, 31 One can see (Figure 6) the ${{\text{p}}^{\prime }}_{\text{fast}}$ term is mostly vanished by the exchange effect. However, precision can be poor, and k_io indeterminate in circumstances of insufficient CA extravasation, negligibly small [CA_o], as in the normal‐appearing brain.3

We neglect potential non‐monoexponential contributions from vascular H₂O or magnetization transfer (MT) from macromolecular ¹H resonances.32 The model study systems are avascular, or effectively so. In vivo, the vascular contribution is generally limited to the initial portion of the DCE time‐course.8 Also, the low α, on–resonance RF pulse in a 3D imaging acquisition is generally not very MT sensitive.33, 34, 35

The contention that k_io can never be accessed by DCE‐MRI is predicated on the supposed longitudinally fully relaxed ${{\text{p}}^{\prime }}_{\text{fast}}$ and ${{\text{p}}^{\prime }}_{\text{slow}}$ contributions.36 The longitudinal saturation expressions (the α, TR, ${{\text{R}}^{\prime }}_{1,\text{fast}}$, and ${{\text{R}}^{\prime }}_{1,\text{slow}}$ functions of Equations 11 and 12 have been assessed (Figure S1 of Li et al31; Figure 6 of Buckley36). If not unity, these weight ${{\text{p}}^{\prime }}_{\text{fast}}$ disproportionately relative to ${{\text{p}}^{\prime }}_{\text{slow}}$. (That is, they make ${{\mathbf{S}}^{\prime }}_{\text{fast}}$ and ${{\mathbf{S}}^{\prime }}_{\text{slow}}$ each less than unity, but ${{\mathbf{S}}^{\prime }}_{\text{fast}}$ > ${{\mathbf{S}}^{\prime }}_{\text{slow}}$.) However, taking this into account and forcing data fittings with the fully longitudinally relaxed biexponential expressions (Equations (10), (11), (12), (13), (14), (15), (16) has been found to introduce unacceptable systematic errors, cause k_io to become artificially indeterminate,7, 31 and sometimes to make fittings poorer.36 When the two components cannot be experimentally discriminated and separately fitted, the ${{\text{R}}^{\prime }}_{1,\text{fast}}$ and ${{\text{R}}^{\prime }}_{1,\text{slow}}$ B ₀‐ and [CA_o]‐dependences tend to counteract one another (the “avoided crossing,” Figure 5), and thus reduce k_io influence (Equations 13 and 14). It seems the ${{\text{p}}^{\prime }}_{\text{slow}}$ contribution is disproportionately (essentially exclusively) acquired. So, we consider the transverse saturation expressions (the TE, ${{\text{R}}^{\prime }}_{2,\text{fast}}$, and ${{\text{R}}^{\prime }}_{2,\text{slow}}$ functions of Equations 11 and 12). In DCE, TR is often < 5 ms, so TE must be very small. Although the saturation expressions can be very complicated,33, 34, 35 an infinitely small TE is equivalent to ${{\text{R}}^{\prime }}_{2,\text{fast}}$ = ${{\text{R}}^{\prime }}_{2,\text{slow}}$ = 0. We have proposed, however, it is quite plausible that ${{\text{R}}^{\prime }}_{2,\text{fast}}$ and ${{\text{R}}^{\prime }}_{2,\text{slow}}$ are sufficiently nonzero due to magnetic susceptibility gradients.7, 31

During the bolus CA passage, there are significant paramagnetic CA concentration gradients across capillary walls and cell membranes. Plasma [CA] can exceed 5 mM immediately upon CA arrival in the tissue.27, 36 Susceptibility gradients due to such concentration differences have been shown to significantly increase R′₂ values.37 The fully relaxed ${{\text{p}}^{\prime }}_{\text{fast}}$ contribution has already been rendered much smaller than ${{\text{p}}^{\prime }}_{\text{slow}}$ by exchange (it vanishes in the VSS condition), Figure 6. Thus, even if ${{\text{R}}^{\prime }}_{2,\text{fast}}$ and ${{\text{R}}^{\prime }}_{2,\text{slow}}$ are equally elevated, it is easy to imagine the fortuitous consequence that ${{\text{p}}^{\prime }}_{\text{fast}}$ is completely saturated (“quenched”), leaving partially saturated ${{\text{p}}^{\prime }}_{\text{slow}}$ as the meaningful component. Whatever the mechanism, considerable experimental evidence has accumulated that k_io can be usefully estimated in many DCE–MRI experiments (see the Discussion section)

A truly noninvasive diffusion‐weighted imaging (DWI) analysis that does not require a CA or a shutter–speed shows considerable promise in determining k_io.3 This novel DWI approach works at clinical B ₀ values, seems to measure large k_io values with more precision, and allows separation of the irreducible v_i cell biology factors, ρ and 〈V〉. These pathology‐related properties are very important in their own right, and in discriminating the 2 Equation two terms to access x^cMR_NKA itself: ρ and 〈V〉 are not accessible with DCE‐MRI.3

3.2. Apparent Population Inversion

Somewhere below 0.1 T, one can begin to detect non‐monoexponential T₁ relaxation caused by k_io.5, 10 Figure 6 exhibits further interesting features. When the field value is small enough to observe apparent biexponential relaxation, but not yet as small as ~0.01 T, the apparent minority component does not extrapolate to the true minority component, p_o (red), but to the true majority component, p_i (blue). Between B ₀ ≅ 0.01 T and the VSS, there is an apparent population inversion (API). This has been noted previously (Figure 4 of Lee and Springer38). If one conducts only an empirical biexponential analysis (Equations (10), (11), (12) of such experimental data, one would find the minority component (blue p′ in Figure 6) has the faster relaxation (blue ${{\text{R}}^{\prime }}_1$ in Figure 5). If the apparent minority p′ value is near 0.2, as is quite likely, one would be tempted to incorrectly assign it to ¹H₂O_o, or any population other than H₂O_i, because p_i is commonly understood to be near 0.8. This is a common problem with the inappropriate application of an empirical biexponential analysis to data that do not have an intrinsic biexponential nature.39 The condition of apparent population equality (APE) (p_i′ = p_o′ = 0.5; log v _L ≅ −0.35 in Figure 6; v _L = 0.43 MHz, B ₀ = 0.01 T), and thus API, occurs only when the true majority component (p_i here) has the faster relaxation (R_1i on the left), and at the point given by Equation 18 (derived from Equations 15 and 16), where the к₁ argument (argк₁) is (R_1i – R_1o).

$$arg{\kappa }_{1,APE}=k/(p_o-p_i).$$ (18)

4. DISCUSSION

4.1. Why does API happen?

The curves in Figures 5 and 6 are generated from the BMW 2SX Equations (13), (14), (15), (16). On their RHSs, the shutter‐speed argument (R_1i – R_1o) appears in many places, and it plays a pivotal role. It is the only factor that changes sign from the left to the right of Figures 5 and 6: being positive on the left, and negative on the right. As a consequence, it is always the population with the apparent faster relaxation that vanishes as the system approaches the VSS condition (two vertical dashed lines in Figures 5 and 6), which is where argк₁ changes sign. This behavior can be seen graphically in simulated (Figure 3 of Lee and Springer38) and experimental (Figure S1 of Zhang et al9) decay curves, and can be derived from Equations 15 and 16. Thus, if the VSS is approached from the left in Figure 6, by increasing B ₀, it is ${{\text{p}}^{\prime }}_{\text{i}}$ that goes to zero. On the other hand, if the VSS is approached from the right, by decreasing [CA_o], it is p_o′ that goes to zero. Our simulations here are all for k = 5 s⁻¹. However, the rate of vanishing does depend on k: all other parameters held fixed, the greater the k the more shallow the vanishing.24 As a corollary, when a system passes through the VSS condition, the assignments of the apparent relaxation rate constants (${{\text{R}}^{\prime }}_1$s) and populations (p′s) must be switched; from blue to red and vice versa in Figure 6. This can be observed in experimental data (Figure 3 of Zhang et al40).

The deviation of an observed p′ from its corresponding inherent compartmental p does have a physical basis. There are subcompartmental spin populations with different diffusion (phase diagram)41 and/or exchange31 histories. Thus, these can have different R₁ values, but surely they exist in continua.42 Given these complications, the emphasis must remain on the inherent p values, which enjoy the well‐mixed attribute, and these can be extracted only with 2SX analyses of experimental data. Furthermore, this is also the only way to determine k_io with a shutter–speed experiment.

4.2. Implications

Many results indicate the new metabolic k_io biomarker can be very powerful. As befitting the crucial NKA role in intermediary metabolism, k_io has been reported responsive to several different metabolic alterations (Table 1). For instance, entries a, d, l, and t suggest that k_io reports from ground zero of the oncogenic transformation: it may increase because of the very ion transporter up–regulation that triggers K‐Ras/rapidly accelerated fibrosarcoma kinase/mitogen‐activated protein kinase signaled uncontrolled cell proliferation.3 This suggests its potential for early cancer detection. These consequences are surely due to the ^cMR_NKA contribution to k_io(a). Because of its vital nature, ^cMR_NKA is likely to be altered in most, if not all, pathologies. Another example is systemic multiple sclerosis.52 The Table 1 entries arising from DCE‐MRI estimation are clearly marked with a superscript #. It is important to note that entries d and t show DCE‐MRI results that have been validated with more precise ultra‐low field model experiments not subject to the DCE uncertainties.

Table 1.

k_io Responsiveness to Metabolic Changes

kio increases with:	a	Increased NKA pump expression (5,9*)
	b	Increasing cytoplasmic ATP (9*)
	c	Increasing [Ko +] (at low [Ko +]), with an NKA Michaelis‐Menten signature (4,43)
	d	Hypoxia (10,44#)
	e	Cisplatin‐induced apoptosis (45)
	f	Xenograft tumor apoptotic regions (46#)
	g	Human brain metastasis radiosurgery (47#)
kio decreases with:	h	Ouabain NKA pump inhibition (2,5,9*,43)
	i	Increasing [Ko +] (at sufficient [Ko +] to cause membrane depolarization) (4)
	j	WZB117 glucose uptake inhibition (5)
	k	O2 → N2 switch (9*)
	l	Increasing mitochondrial reducing equivalents (6#)
	m	Intracellular lonidamine (48#) mitochondrial complex II inhibition (49)
	n	Extracellular tetrodotoxin voltage‐gated sodium channel inhibition (4)
	o	Extracellular AP5 plus DNQX† post‐excitatory neuronal activity inhibition (4)
	p	Glutamine deprivation (10)
	q	Hypertension in myocardium (28#)
	r	Chemotherapy of human breast tumors (50#)
	s	Phosphatase activation breast tumor therapy (51#)
kio correlates with:	t	Tumor metastatic potential (5,6#)
	u	Neuronal firing (4)
	v	Oxidative phosphorylation rate (52&)
	w	O2 consumption rate (4)
	x	Head and neck cancer mortality (53#)
	y	18Fluoro‐2‐deoxy‐D‐glucose breast tumor uptake (54#)

ATP, adenosine triphosphate; k_io, water efflux k (1/τ_i); NKA, Na⁺,K⁺‐ATPase (sodium pump).

^*For yeast, pump is PMA1, inhibitor is ebselen.

^#Employed shutter‐speed (к₁) dynamic‐contrast‐enhanced‐MRI.

^†(2R)‐Amino‐5‐phosphonovaleric acid plus 6,7‐dinitroquinoxaline‐2,3‐dione.

^&Indirect.

Damadian’s early ex vivo 0.6 T ¹H₂O MR study to discriminate malignant and normal tissue55 was cited by Lauterbur56 as part of the motivation for developing MRI.57 The fact that subsequent research showed sensitivity and specificity are insufficient for robust cancer detection has been one of the major drivers for the seemingly inexorable increase in B ₀ strength (moving to the right in the figures) in clinical MRI.11, 58 So, particularly the fact that Ruggiero and co‐workers’ results (although nonimaging) were obtained at very small B ₀ values, with a fast field cycling study (modest detection B ₀) of an in vivo murine xenographic tumor model,5 will stimulate renewed interest in ultra–low field MRI (moving to the left in the figures). It is exciting to find AWC is the molecular process that dominates the ultra‐low field ¹H₂O signal.

Ruggiero and co‐workers’ finding5 that it is mainly the k_io increase with malignancy that leads to decreasing ${{\text{R}}^{\prime }}_1$ provides an explanation for Damadian’s classic observation of the latter phenomenon.55 It suggests his ex vivo NMR acquisitions within five min of rat euthanasia were soon enough to retain most of the in vivo cellular ATP. Also, the increase of k_io with concomitant p_o increase in malignancy5 is consistent with metabolic competition between cancer cells.3 The greater the cell density, ρ, the slower the NKA activity per cell.

There has been only a small amount of fast field cycling work in human studies: a recent report at 0.06 T has been published.59 However, significant efforts to produce such human‐sized instruments are under way. The same is true for “portable” low‐field scanners, with possibly B ₀ < 0.1 T (Garwood MG, personal communication). ⁶⁰ These could become very valuable metabolic instruments.

Figure 6 illustrates, however, that B ₀ values sufficiently small to access the LSSL condition are quite unlikely to be reached. This emphasizes the importance of the BMW 2SX analysis detailed here. Otherwise, the bi–exponential relaxation results that will be obtained can be very easily miss–interpreted. This would cause quite unfortunate confusion, and actually represent a medical set–back.

APPENDIX 1. ACRONYMS AND SYMBOLS

APE	apparent population equality
API	apparent population inversion
ATP	adenosine triphosphate
〈A/V〉	mean cell area/volume ratio
AWC	active water cycling
α	read pulse flip angle
B 0	main magnetic field
BMW	Bloch‐McConnell‐Woessner
CA	contrast agent
CAi	intracellular CA
CAo	extracellular CA
[CAo]	CAo concentration
DCE	dynamic‐contrast‐enhanced
DWI	diffusion‐weighted imaging
ECF	tissue extracellular volume fraction
FXL	fast‐exchange limit
FXR	fast‐exchange‐regime
fM	tissue macromolecular volume fraction
H2Obu	buried water
H2Oi	intracellular water
H2Oo	extracellular water
[H2Oi]	H2Oi concentration
1H2O	water proton MR signal
1H2Oi	H2Oi MR signal
1H2Oo	H2Oo MR signal
k	steady‐state water exchange rate constant
kio	water efflux k (1/τi)
kio(a)	active kio contribution
kio(p)	passive kio contribution
koi	water influx k
K‐Ras	Kirsten rat sarcoma virus oncogene
к1	longitudinal MR shutter‐speed (SS)
LSS	large SS condition (formerly, SXR)
LSSL	large SS limit (formerly, SXL)
cMRNKA	cellular NKA metabolic rate
MW	molecular mass
NKA	Na+,K+‐ATPase (sodium pump)
NXL	no‐exchange‐limit
v L	Larmor frequency (often v 0)
v Li	1H2Oi v L
v Lo	1H2Oo v L
p	tissue water mole fraction (“population”)
pi	intrinsic H2Oi p
po	intrinsic H2Oo p
p′	apparent p
p′fast	fast‐relaxing component p′ (formerly, aS)
p′i	H2Oi p′
p′slow	slow‐relaxing component p′ (formerly, aL)
p′o	H2Oo p′
PMA1	plasma membrane H+‐ATPase
PW	membrane water permeability coefficient
PW(p)	passive PW
R1	longitudinal relaxation rate constant
R1,cross	= R1i = R1o
R1i	intrinsic 1H2Oi R1
R1H2O	pure water R1
R1o	intrinsic 1H2Oo R1
R1o0	R1o in the absence of CAo
R′1	apparent, approximate single R1 value
R′1,fast	fast‐relaxing component R′1 (formerly, R1S)
R′1i	apparent R1i
R′1o	apparent R1o
R′1,slow	slow‐relaxing component R′1 (formerly, R1L)
R2	transverse relaxation rate constant
R′2	apparent R2
R′2,fast	fast‐relaxing component R′2
R′2,slow	slow‐relaxing component R′2
RHS	right‐hand side
r1	longitudinal relaxivity
r1M	macromolecular r1
r1o	CAo r1
ρ	cell (number) density
S	tissue 1H2O signal strength
S0	Boltzmann S
S′fast	apparent fast‐relaxing saturation factor
S′slow	apparent slow‐relaxing saturation factor
SS	shutter‐speed (к1)
SSS	small SS condition (formerly, FXR)
SXL	slow‐exchange‐limit
SXR	slow‐exchange‐regime
TE	pulse sequence magnetization echo time
TR	pulse sequence repetition time
T1	longitudinal relaxation time constant
T1o	intrinsic 1H2Oo T1
τi	mean H2Oi molecule lifetime (1/kio)
τr	molecular rotational correlation time
〈V〉	mean cell volume
VSS	vanished SS condition (formerly, FXL)
v	tissue volume fraction
vi	intracellular v
vo	extracellular v (formerly, ECF)
2SX	two‐site‐exchange

Li X, Mangia S, Lee J‐H, Bai R, Springer CS Jr. NMR shutter‐speed elucidates apparent population inversion of ¹H₂O signals due to active transmembrane water cycling. Magn Reson Med. 2019;82:411–424. 10.1002/mrm.27725