Mapping human brain capillary water lifetime: high‐resolution metabolic neuroimaging

Abstract

Shutter‐speed analysis of dynamic‐contrast‐agent (CA)‐enhanced normal, multiple sclerosis (MS), and glioblastoma (GBM) human brain data gives the mean capillary water molecule lifetime (τ _b) and blood volume fraction (v _b; capillary density–volume product (ρ ^† V)) in a high‐resolution ¹H₂O MRI voxel (40 μL) or ROI. The equilibrium water extravasation rate constant, k _po (τ _b ⁻¹), averages 3.2 and 2.9 s⁻¹ in resting‐state normal white matter (NWM) and gray matter (NGM), respectively (n = 6). The results (italicized) lead to three major conclusions. (A) k _po differences are dominated by capillary water permeability (P _W ^†), not size, differences. NWM and NGM voxel k _po and v_b values are independent. Quantitative analyses of concomitant population‐averaged k _po, v_b variations in normal and normal‐appearing MS brain ROIs confirm P_W ^† dominance. (B) P _W ^† is dominated (>95%) by a trans(endothelial)cellular pathway, not the P _CA ^† paracellular route. In MS lesions and GBM tumors, P_CA ^† increases but P_W ^† decreases. (C) k _po tracks steady‐state ATP production/consumption flux per capillary. In normal, MS, and GBM brain, regional k _po correlates with literature MRSI ATP (positively) and Na ⁺ (negatively) tissue concentrations. This suggests that the P_W ^† pathway is metabolically active. Excellent agreement of the relative NGM/NWM k _po v_b product ratio with the literature ³¹PMRSI‐MT CMR_oxphos ratio confirms the flux property. We have previously shown that the cellular water molecule efflux rate constant (k _io) is proportional to plasma membrane P‐type ATPase turnover, likely due to active trans‐membrane water cycling. With synaptic proximities and synergistic metabolic cooperativities, polar brain endothelial, neuroglial, and neuronal cells form “gliovascular units.” We hypothesize that a chain of water cycling processes transmits brain metabolic activity to k _po, letting it report neurogliovascular unit Na⁺,K⁺‐ATPase activity. Cerebral k _po maps represent metabolic (functional) neuroimages. The NGM 2.9 s⁻¹ k _po means an equilibrium unidirectional water efflux of ~10¹⁵ H₂O molecules s⁻¹ per capillary (in 1 μL tissue): consistent with the known ATP consumption rate and water co‐transporting membrane symporter stoichiometries. © 2015 The Authors NMR in Biomedicine Published by John Wiley & Sons Ltd.

Abbreviations used:

ADP

adenosine diphosphate

ASL

arterial spin labeling

ATP

adenosine triphosphate

CA

contrast agent

CBV

cerebral blood volume

CESL/CEST

chemical exchange spin lock/saturation transfer

CMR_oxphos

cerebral metabolic rate of ATP synthesis from oxidative phosphorylation

DCE‐MRI

dynamic‐contrast‐enhanced magnetic resonance imaging

F

cerebral blood flow (CBF)

FDG

fluorodeoxyglucose

Fe‐tol

ferumoxytol (Feraheme)

FOV

field of view

FXL

fast‐exchange limit

FXR

fast‐exchange regime

GBM

glioblastoma multiforme

GBM‐NA

GBM normal appearing

GdHPDO3A

gadoteridol (ProHance)

GM

gray matter

HP

hyperpolarized

IR

inversion recovery

MR_glc

metabolic rate of glucose (consumption)

MRSI

magnetic resonance spectroscopic imaging

MS

multiple sclerosis

MS‐NA

MS normal appearing

MT

magnetization transfer

NAGM

normal‐appearing gray matter

NAWM

normal‐appearing white matter

NGM

normal GM

NKA

Na⁺,K⁺‐ATPase

NWM

normal WM

PCr

phosphocreatine

PET

positron emission tomography

P_i

inorganic phosphate (PO₄ ³⁻)

ROI

region of interest

RRMS

relapsing–remitting MS

SPECT

single photon emission computed tomography

SSP

shutter‐speed paradigm

SXR

slow‐exchange regime

TI

inversion time

TP

tracer paradigm

TSC

tissue sodium concentration

WEI

water exchange index

WM

white matter

2SX

two‐site exchange.

Introduction

Every eukaryotic cell has a plasma membrane P‐type ATPase ion pump; for animals, this is Na⁺,K⁺‐ATPase (NKA) 1, 2. Since its activity maintains the transmural K⁺ and Na⁺ gradients and thus, respectively, the membrane potential and secondary active transport, NKA is vital for life. The normal “forward” reaction is the following, where the i and o subscripts indicate intra‐ and extra‐cellular, respectively.

$${\mathrm{A}\mathrm{T}\mathrm{P}}_{\mathrm{i}}+{{2\mathrm{K}}_{\mathrm{o}}}^{+}+{{3\mathrm{N}\mathrm{a}}_{\mathrm{i}}}^{+}\stackrel{\text{NKA}}{\to }{\mathrm{A}\mathrm{D}\mathrm{P}}_{\mathrm{i}}+{\mathrm{P}}_{\mathrm{i}}+{{2\mathrm{K}}_{\mathrm{i}}}^{+}+{{3\mathrm{N}\mathrm{a}}_{\mathrm{o}}}^{+}$$

Even in homeostasis, the NKA pump experiences continual turnover (there are return pathways). It has been estimated that it consumes over 50% of brain adenosine triphosphate (ATP) 3. Methods for measuring NKA activity have been adapted to the experimental sample. For solubilized, purified enzyme or tissue homogenate preparations, spectrophotometric 4 or radiolabeled (³²P) 5 ATP hydrolysis rate assays suffice. For intact cells in culture or in tissue preparations, voltage clamp current, ion‐selective (Na⁺/K⁺) microelectrode response, radioisotope (²²Na⁺/²⁴Na⁺/⁴²K⁺/⁸⁶Rb⁺) uptake/release 6, 7, 8, 9, or ²³Na⁺/⁸⁷Rb⁺ MRS 10, 11 methods measure NKA‐driven trans‐membrane ion transport kinetics. Phospholipid vesicles reconstituted with purified NKA allow measurement of both ATP hydrolysis and ion transport kinetics 5.

Each of these methods is best suited to macroscopically homogeneous samples. None are particularly appropriate for normally heterogeneous tissue, since there is no spatial encoding. Furthermore, many of these methods directly measure only net NKA activity, not homeostatic turnover. The radioisotope approach has been generally abandoned for about 20 years; deemed too problematic for even tissue preparations 12. As far as we are aware, NKA turnover has never been measured, let alone mapped, in a living animal or human subject. Doing so would provide a very fundamental view of ongoing metabolism, a new form of metabolic imaging. Since metabolic thermodynamics and kinetics have no necessary relationship (important example below), it is imperative to distinguish these aspects in imaging.

Mapping metabolic thermodynamics

Restricting ourselves essentially to human studies, so far metabolic imaging has been mostly accomplished by positron emission tomography (PET) 13 and magnetic resonance spectroscopic imaging (MRSI) 14, 15, with some single photon emission computed tomography (SPECT) 16. Very often, this is mapping of metabolic molecule tissue concentrations (“levels”) – the thermodynamic dimension of metabolism. For example, ³¹PMRSI gives the distribution of high‐energy phosphate brain levels 17. Such data could be used to estimate the cerebral distribution of the free energy for ATP hydrolysis 10, 18. ²³NaMRSI maps tissue sodium concentration (TSC) 19. If a TSC increase reflects an intracellular sodium, Na_i ⁺, increase, it signifies a decrease in the transcytolemmal sodium electrochemical potential gradient 19. ¹HMRSI can map the lactate, N‐acetylaspartate, and choline‐containing 20 metabolites. Other thermodynamic aspects assessed include H₃O⁺ (pH), O₂ (diminished in hypoxia), and redox level distributions 21. Newer metabo‐CESL/CEST (chemical exchange spin lock/saturation transfer) methods use RF pulses to detect metabolite (creatine 22, glucose 23, 24) resonances via the ¹H₂O MR signal. Though receptors are not metabolites, PET and SPECT can be used to map their tissue concentrations 25, 26. For example, the [¹¹C]cocaine tracer was used to determine the dopamine transporter concentration, 700 nM, in the abuser striatum 25. Receptors very often catalyze, or trigger by signaling, metabolic reactions.

Mapping metabolic kinetics

For all the power of thermodynamics, kinetic metabolic aspects – enzyme‐catalyzed fluxes (rates) – can be more informative. With PET and hyperpolarized ¹³CMRSI (HP‐¹³CMRSI), a non‐equilibrium isotope distribution (positron emitting isotope, e.g. in ¹⁸fluorodeoxyglucose (¹⁸FDG); HP (stable) ¹³C isotope magnetization) is introduced exogenously, with minimal invasion. Then, the (relatively slow) regional approach to isotope equilibrium is mapped using spatially encoded detection. The evaluation of generally much faster steady‐state (“equilibrium”) unidirectional fluxes must be accomplished with proper modeling 14, 15, 17, 27, 28. By far the most common example is the inference of the metabolic rate (consumption) of glucose, MR_glc (in µmol min⁻¹ g⁻¹), from the net metabolic rate (uptake) of ¹⁸FDG 27. In the human prostate tumor, the [1‐¹³C]pyruvate to [1‐¹³C]lactate conversion rate constant is 0.045 s⁻¹ 15. With rigorous modeling, this can yield the lactate dehydrogenase flux 14. In particular, the ³¹PMRSI modality offers the unique opportunity to measure steady‐state unidirectional fluxes completely non‐invasively. This takes advantage of the magnetization transfer (MT) phenomenon. An RF pulse is used to selectively perturb only certain molecular nuclear magnetization from equilibrium. Monitoring the (relatively slow) magnetization recovery with spatial encoding allows mapping of faster metabolic fluxes, again with proper modeling 17, 29, 30. For example in the brain, the integrated cellular creatine kinase flux and ATP production/ consumption rates have been determined 17.

Spatial resolution

Extant metabolic imaging has revealed a tremendous amount about normal and pathological biochemistry, as it actually exists in vivo. However, new approaches can be attractive. The PET and HP‐¹³CMRSI modalities are costly. Typical nominal spatial resolutions and voxel volumes for human modalities are the following: ³¹PMRSI, (1.3 cm)³ = 2.2 mL 17; ¹HMRSI, (1 cm)³ = 1 mL 20; SPECT, (1 cm)³ = 1 mL 16; HP‐¹³CMRSI, (7 mm)³ = 340 μL 15; PET, (5 mm)³ = 125 μL 16; ²³NaMRSI, (4 mm)³ = 64 μL 19. These are often insufficient for discriminating significant human anatomy. For example, the cerebral gray matter (GM)/white matter (WM) boundary usually cannot be clearly distinguished. In comparison, MRI – generated from the relatively strong ¹H₂O signal – commonly provides higher spatial resolution: (1 mm)³ = 1 μL or better. Metabolic images are almost always accompanied by high‐resolution MRI (sometimes computed tomography) views of the same tissue. Therefore, though MRI is relatively inexpensive (compared with PET and HP‐¹³CMRSI) and employs no ionizing radiation, it is understandably often thought of as providing only anatomical and/or vascular information. Of course, it has long mapped some tissue functions, as in ciné cardiovascular MRI and functional MRI. The new metabo‐CESL/CEST techniques approach anatomical ¹H₂O resolution 22, since they employ this strong signal for indirect metabolite detection.

¹H₂O mapping of NKA flux

Here we introduce a method exploiting a newly discovered aspect of the biology of water itself – active trans‐membrane cycling. The (dynamic‐contrast‐enhanced) DCE‐MRI ¹H₂O method is in wide clinical use. It employs any of a number of approved paramagnetic, monomeric Gd(III) chelates as contrast agents (CAs). For tissues manifesting extensive CA extravasation, a proper pharmacokinetic analysis of the CA bolus DCE‐MRI time‐course yields the mean lifetime (τ _i) of water molecules inside the cells within a voxel 31. The reciprocal (τ _i ⁻¹) is the first‐order rate constant (k _io) for the unidirectional, equilibrium cellular water molecule efflux. We have recently documented the evidence, from enzymatic manipulations spanning a number of different cell types and models (from cells to animals to humans), that the magnitude of k _io is proportional to P‐type ATPase turnover 32. This is likely due to active transmembrane water cycling that accompanies the osmolyte cycling driven by the membrane ion pump 32, 33.

Because the normal blood–brain barrier is CA impermeable, cerebral DCE‐MRI data do not directly yield τ _i. However, the mean capillary water lifetime (τ _b) is readily determined. Fortunately, the metabolic activity of cerebral neurons is exquisitely symbiotically connected with those of neuroglia (oligodendrocytes, astrocytes) and thence capillary endothelial cells 34, 35, 36, 37 – all within synaptic proximities. The terms “neurovascular unit” and “gliovascular unit” have been coined 36 to connote this. Here, we present results from normal and multiple sclerosis normal‐appearing (MS‐NA) brain, along with MS lesion and glioma tumor, indicating that τ _b ⁻¹ (k _po) is proportional to metabolic turnover within the neurogliovascular unit. We show the first human brain k _po maps (40 μL voxels), and present evidence that these are metabolic flux maps at ¹H₂O resolution.

Technical background

Since many different principles are involved here, we present some technical background material.

Mean brain intra‐capillary blood water molecule lifetime (τ _b)

The average erythrocyte speed through cerebral cortical capillaries is 2 mm s⁻¹ or less 38, 39, 40: this measures blood velocity. In a common (1 mm)³ high‐resolution human ¹H₂O MRI voxel, a conservatively small estimate of the average, tortuous capillary path‐length is 2 mm 40, 41. Thus, the mean voxel transit time for a blood water molecule is at least 1 s. Many things happen to the molecule during this period. The mean lifetime inside an erythrocyte (τ _i) is 10 ms 42, 43, 44, 45. By equilibrium mass action, for a 40% hematocrit the mean plasma lifetime before entering a red cell (τ _p) is 15 ms. Thus, any given water molecule enters and leaves erythrocytes about 40 times during its voxel passage (also, the entire red cell water content is exchanged ~100 times). The 3D Einstein diffusion equation, ⟨r ²⟩ = 6Dt _D 46, allows estimation of H₂O molecule capillary wall encounter frequency: r is the capillary radius, D the water diffusion coefficient, and t _D the average time to diffuse a distance r. Inserting the mean feline r value, 2.6 µm 40, and a conservatively small D value (1.5 µm² ms⁻¹, half the pure water D), we obtain t _D = 0.8 ms. Even an H₂O molecule in the center of the capillary lumen would encounter the capillary wall more than 1000 times during its voxel passage. It is a very good approximation that capillary blood water is “well mixed.” Thus, we can estimate the probability of a water molecule escaping the capillary. The τ _b inverse, τ _b ⁻¹, is the unidirectional first‐order rate constant, k _po, for water extravasation 47. For a well‐mixed lumen, this is τ _b ⁻¹ = P _W ^†[A _ca/V _ca], where P _W ^† is the transendothelial water permeability coefficient, A _ca the individual capillary surface area, and V _ca the individual capillary lumen volume 48. For a cylindrical microvessel: τ _b ⁻¹ = 2 [P_W ^†/r]. Some time ago 49, we noted that an r = 2.6 µm and a primate P _W ^† value of 2 µm s⁻¹ 50, 51 predict 650 ms for τ _b. The k _po (τ _b ⁻¹) value (1.5 s⁻¹) corresponds to about 78% probability (=100[1 − exp(−k _po t)]: t is the capillary transit time (~1 s)) that any given water molecule will exchange out of the blood space (to be replaced by an extravascular H₂O molecule) during its capillary passage. Because blood velocity causes no net change in the number of (indistinguishable) capillary H₂O molecules, the τ _b quantity is not influenced by the blood flow (F; CBF) magnitude. This contrasts with the situation for the extraction of labeled water 50, which is surely “perfusion limited.”

It is important to note that P _W ^†/r is also independent of the intensive capillary density (ρ ^†) property. With tracer studies (e.g. intracarotid ¹⁵OH₂ 50) and sacrificial autoradiography (e.g. IV ³HOH 52), one obtains the intensive P _W ^† S product, where S is the total region‐of‐interest (ROI) vascular surface area per unit tissue volume – dependent on the vascularity. The latter is measured by the blood volume fraction (v _b, CBV): the ρ ^† V product (V is the mean V _ca). In P _W ^†/r, r is the mean ROI vascular radius, and is related only to the V factor of the ρ ^† V product. Thus, theory demands that τ _b is independent of ρ ^†, and therefore a potentially powerful new type of imaging biomarker: we characterize it as supra‐intensive. We show below that k _po is also experimentally independent of v _b, a very meaningful finding, and that it can distinguish cerebral pathology undetectable with ordinary intensive biomarkers.

Inter‐compartmental ¹H₂O exchange effects in in vivo MR

It has been known for 40 years that a sufficiently concentrated paramagnetic solute localized in a cell suspension extracellular space can cause non‐mono‐exponential longitudinal and/or transverse ¹H₂O relaxation. A two‐site exchange (2SX) analysis of the recovery yields τ _i, the mean intracellular water molecule lifetime (reviewed in 32, 43, 44, 45). This is equilibrium transcytolemmal water exchange. Longitudinal relaxation for yeast cell samples 33, 49, 53 exemplifies this. The extracellular CA increases the intrinsic outside water proton signal (¹H₂O_o) longitudinal relaxation rate constant, R _1o (≡(T _1o)⁻¹). Though we generalized and systematized the exchange picture (and introduced the term) only in 1999 54, this approach increases the longitudinal transcytolemmal “shutter‐speed,” т_1c ⁻¹ (≡|R _1o − R _1i|), sufficiently that this water exchange NMR system is moved out of its fast‐exchange‐limit (FXL) condition (т_1c ⁻¹ « (τ _i ⁻¹ + τ _o ⁻¹); R _1i is the intrinsic inside (¹H₂O_i) relaxation rate constant). A sufficient outside CA concentration, [CA_o], allows the NMR system to reach the slow‐exchange‐regime (SXR) condition. This is characterized by non‐mono‐exponential magnetization recovery, but is distinct from the slow‐exchange‐ and no‐exchange‐limit conditions 31, 33, 43, 47, 55. It is quite customary to achieve the SXR condition with cell suspensions 33, 43, 44, 45, 49, but there is no convincing evidence that the SXR can be reached in vivo with approved CAs. However, it has been shown that when the [CA_o] value is only modest, and the system can attain only the fast‐exchange‐regime (FXR) condition, it is still possible to measure τ _i by varying т_1c ⁻¹ (by varying [CA_o]) 49. The FXR condition features mono‐exponential longitudinal recovery: i.e., R ₁ is single valued, but with a non‐linear [CA_o] dependence. In the FXL condition, this dependence is linear 31, 33, 43, 49, 54. The in vivo implementation of these principles generally employs some variant of DCE‐MRI, the serial acquisition of T ₁‐weighted images before, during, and after a bolus CA injection. The common tracer paradigm (TP) pharmacokinetic analysis of an ROI or voxel signal intensity time‐course imposes the assumption that all exchange systems remain in their FXL conditions. This denies access to τ _b, or τ _i (each is held effectively zero in the FXL), and causes systematic changes in other pharmacokinetic parameters, such as v _b. However, shutter‐speed paradigm (SSP) pharmacokinetic analysis allows that τ _b and τ _i are finite, and relieves the systematic distortions of other biomarkers. Overviews of these concepts have been published 31, 33, 43, 55.

It was gratifying when, in 1997, multiple infusions of an intravascular CA were used to vary the murine brain transendothelial shutter‐speed, т_1e ⁻¹ (≡|R _1b − R _1exv|) (¹H₂O_b and ¹H₂O_exv are the intra‐ and extravascular signals, respectively) and reach the FXR condition (т_1e ⁻¹ → (τ _b ⁻¹ + τ _exv ⁻¹); τ _exv is the mean extravascular water molecule lifetime) for this water exchange system. Variation of the plasma CA concentration, [CA_p], allowed the exchange kinetics to be measured 56. The cortical τ _b value we calculate from these results is 295 ms – of the magnitude we had earlier predicted from literature parameters (see above). In 2002, an approach using a single intravascular CA injection was demonstrated in the rat 57. The [CA_p] value was not varied, but the acquisition was combined with an arterial spin labeling (ASL) variant. This approach yields only P _W ^† S and, because ASL is used, the F value must be included in order to obtain P _W ^† S correctly 57. In 2003, we introduced an SSP DCE‐MRI method whereby a single CA bolus injection can be used to determine human brain τ _b 58. This approach is used here. Seven years ago, the water exchange index (WEI), an approximate, dimensionless, non‐linear τ _b ⁻¹ estimate, was demonstrated in the mouse, an approach also requiring a single intravascular CA injection 59. Recently however, the authors of Reference 59 themselves showed that, unfortunately, the WEI approximation formally depends on the v _b value 60. As noted above, an important feature of the actual τ _b ⁻¹ (k _po) biomarker is its v _b independence if r does not vary.

Indirect detection

There are crucial differences between the tracer and shutter‐speed paradigms. Classic tracers (radiolabeled molecules, electron‐dense compounds, etc.) are detected directly: the tracer molecule is also the signal molecule. Though the CA of DCE‐MRI plays the tracer pharmacokinetic role, it is detected indirectly – via its effect on the ¹H₂O signal. Thus, the CA is the tracer molecule but water is the signal molecule. These species are never distributed equally in tissue: water is in every compartment, each of which contributes to the ¹H₂O signal. For the classic solute TP, water is not molecular: it is a continuum filling tissue spaces. Furthermore, the compartmentalization of the classic tracer is not intrinsic to its signal: one cannot tell if the molecule is intra‐ or extravascular. However, CA compartmentalization is inherently encoded in the DCE‐MRI time‐course. Three simultaneous signals (¹H₂O_b, ¹H₂O_o, and ¹H₂O_i) and their sequentially varying T ₁ values report the time‐varying [CA] values in each compartment CA enters 31, 55. Thus, the imposition of a tracer analysis on DCE‐MRI data joins contradictory postulates: unknown versus known CA compartmentalization. The only reconciliation is by the assumption that all water exchange systems are in FXL conditions. If τ _b is assumed to be effectively zero, the vascular CA compartmentalization is “short‐circuited,” as if CA is both intra‐ and extravascular – a tracer‐like ambiguity. These principles also apply to metabo‐CESL/CEST 61.

Chemical equilibrium measurement

Another important difference is experimental. For classic water tracers (e.g. ¹⁵OH₂ 50), the study is initiated with a non‐equilibrium isotope compartmental distribution, and the kinetics of the tissue's approach to equilibrium are monitored by detecting only the labeled water. In DCE‐MRI, there is no compartmental selection in the initial water proton magnetization perturbation: all ¹H₂O signals are (usually) inverted, and the return to magnetic equilibrium of each of them is monitored simultaneously. Only magnetization equilibrium is perturbed. These methods can lead to common parameters (e.g. P _W ^† here), and thus can support each other. However, the experimental results must be analyzed with different paradigms as appropriate. For example, though the SSP assumes that brain capillary water is “well mixed” – justified above – it does not require the same assumption of extravascular water, which is in fact not well mixed. Although τ _i and τ _o values are typically hundreds of milliseconds, that of τ _exv is typically tens of seconds (see later) – because of the relatively sparse microvessel density. Yet the extravascular (“parenchymal”) MR system is in the fast‐exchange‐limit condition even though transcytolemmal water exchange is not particularly fast. With no extravascular CA, the т_1c ⁻¹ values are much smaller than (τ _i ⁻¹ + τ _o ⁻¹). Thus, the non‐well‐mixed nature of the parenchyma is of no consequence to the DCE‐MRI experiment, but “unstirred layer” effects can be significant for ¹⁵OH₂ re‐intravasation kinetics, or for any tracer study 12.

Active trans‐membrane water cycling

There is obviously considerable interest in the brain capillary τ _b (k _po ⁻¹) quantity. From the above, we see that τ _b variation can reflect a change in capillary r, in P _W ^†, or in both. Vasodilation or vasoconstriction (r alteration) would respectively decrease or increase k _po (τ _b ⁻¹). However, the rate constant for r changes is orders of magnitude smaller than k _po itself 33, 62. Changes in k _po not attributable to capillary size alteration are ascribed to P _W ^† variation and, until recently, this has been conceived as resulting from alterations in passive molecular processes (P _W ^†(passive)). These include (a) paracellular water passage through endothelial tight junctions, (b) simple, transcellular water diffusion across cell membrane lipid bilayers, and (c) transcellular transport through membrane aquaporin protein water channels 63, 64 and/or transcellular leakage through membrane transporters 65. However, NMR studies have recently revealed the cell membrane water permeability coefficient (P _W) to have an active component (P _W(active)) that dominates over the passive component 32, 33. This is due to active transmembrane water cycling accompanying active transmembrane osmolyte cycling, which is paced by the driving cell membrane P‐type ATPase ion pump 32, 33. For animal cells, this is NKA 32, 33, 66, 67, 68. The molecular mechanism likely involves water cotransporting membrane symporters 69, 70, and supports a cycling flux of 10¹² water molecules s⁻¹ per cell.

Experimental

Subjects

Healthy (2M/4F, 30 (±10) years), relapsing–remitting‐MS (RRMS) (2M/4F, 46 (±7) years, 18–55 years), and glioblastoma multiforme (GBM) (3M/2F, 19–57 years) subjects gave informed consent to OHSU Institutional Review Board approved protocols. The MS group was early in disease, but with positive MRI findings. An additional 52 year old female late‐stage RRMS subject was also studied. The GBM subjects had prior surgical biopsies or resections and chemo‐radiation therapy.

Healthy and MS subject DCE‐MRI

A 7 T whole‐body MRI instrument (Siemens, Erlangen, Germany), with quadrature transmission and 24‐channel phased‐array receive head RF coils, was used. Dynamic measurements employed a single‐slice inversion recovery (IR) turboFLASH technique 71, sampling magnetization at eight post‐inversion times (TI values): the inversion pulse was non‐selective. The 128 × 96 image matrix covered a (256 × 192) mm² field of view (FOV) (nominal pixel, (2 mm)²), and a 6° flip angle RF pulse selected a 10 mm transverse slice superior to the lateral ventricles (nominal voxels, (2 × 2 × 10) mm³ = 40 μL). Gadoteridol (ProHance; Bracco Diagnostics, Cranberry, NJ) was injected into an antecubital vein catheter at 1.0 mL s⁻¹ using a power injector (Medrad, Warrenville, PA) to deliver a dose of 28 µmol kg⁻¹ (typically ~5 mL), followed by a 20 mL saline flush at the same rate. For each CA injection, 50 IR image sets were collected with 2.3 s temporal resolution. Total acquisition time was 1.9 min. Parametric R ₁ maps were calculated on a voxel basis by fitting the signal magnitude at each TI with a full Bloch simulation incorporating all RF pulses and delays. The IR was modeled with a two‐parameter single exponential, using a gradient expansion algorithm.

GBM subject DCE‐MRI

Data were acquired using a 3 T Tim Trio (Siemens) instrument, body transmit and 12‐channel phased‐array head receive RF coils, and a full volume 2D gradient‐recalled‐echo echo‐planar imaging sequence. Non‐selective IR pulses were sampled at 24–36 TI values. The 128² image matrix covered a 256² FOV (nominal (2 mm)³ isotropic resolution; 8 μL nominal voxels) 72. ProHance DCE‐MRI was similar to the controls and MS subjects. The next day, the intravascular FeO nanoparticle CA ferumoxytol (Fe‐tol; Feraheme; AMAG Pharmaceuticals, Waltham, MA) was used. Points were obtained in the CA steady states before and after three IV Fe‐tol injections 73, fractionated into doses (1:2:4) totaling 4 mg(Fe) kg⁻¹ (72 µmol(Fe) kg⁻¹, ~12 nmol(Fe‐tol) kg⁻¹), each at 3 mL s⁻¹. The 5 min acquisitions were initiated 120 s after each Fe‐tol injection, during the steady‐state period when [CA_p] was uniform and constant. (The 14 hour Fe‐tol plasma t _1/2 ensures [CA_p] is larger after each successive injection.) All four acquisitions were completed in 40 min. Pre‐ and post‐CA session R ₁ maps were co‐registered to pre‐CA T ₁‐w MPRAGE maps using rigid body transformations. This protocol yielded four pharmacokinetic time points, sufficient to characterize high quality R _1t versus R _1b measurements such as those in NA brain (see Fig. 5 later). This sparse temporal sampling approach was originally designed for optimization of brain coverage and spatial resolution. Unlike gadoteridol (GdHPDO3A) DCE‐MRI, in which [CA_p] reaches large values only transiently – during the first pass (Fig. 1), Fe‐tol steady‐state [CA_p] can attain quite high and sustained levels.

Figure 5.

GBM ¹H₂O DCE‐MRI data. These arise from a 52 year old male glioblastoma subject. The axial R ₁ map in the center was obtained 30 min after injection of a monomeric Gd(III) chelate CA. The enhancing tumor is clearly visible in the image bottom left. The insets show R _1t versus R _1b plots (as in Fig. 1(C)) before and after injections of an intravascular CA, administered well after the Gd(III) CA had cleared. The ROIs giving rise to the plots are indicated by circles and an oval. The plots from frontal WM, putamen, and thalamus ROIs have hyperbolic shapes similar to that in Fig. 1(C). The points exhibit clear deviations from the linearity demanded by the TP. However, the tumor plot exhibits even sharper hyperbolic behavior. The points are well fitted with the SSP (Equation [1]; solid curves): parameter values are given in the text. Population‐averaged parameter values are presented in Table 1.

Figure 1.

Normal ¹H₂O DCE‐MRI data. These arise from a 22 year old female control subject. Panel D shows axial R ₁ maps pre‐ and at one point post‐CA administration. The CA injection is indicated by blue arrows in the R ₁ time‐courses in panels A and B (A, blood ROI R _1b response function; B, WM ROI R _1t response function). The WM ROI is indicated by the yellow circles in panel D. Panel C plots the R _1b dependence of R _1t from A and B. The points exhibit a clear deviation from the linearity demanded by the TP, which embodies the FXL constraint (dashed line). The points are well fitted with the SSP (solid curve), Equation [1], with parameters given in the text. Population‐averaged parameter values are presented in Table 1.

DCE‐MRI pharmacokinetic modeling

Non‐linear modeling (IDL; Exelis, Boulder, CO) was used to extract accurate estimates of R _1exv, v _b, and τ _b for selected ROIs (and also on a voxel‐wise basis) using the equation

(1)

p _b is the mole fraction of tissue water in blood (v _b = p _b f _W, where f _W is the tissue volume fraction accessible to mobile aqueous solutes 55). The independent modeling measure is R _1b(t), which is determined by [CA_b](t) (defined by R _1b(t) = R _1b0 + r _1b[CA_b](t), with R _1b0 the pre‐CA R _1b and r _1b the CA relaxivity). R _1b(t) was measured directly from an R ₁ map ROI placed fully within the sagittal sinus. Equation [1] describes a 2SX NMR system spanning the FXL and FXR conditions, depending on [CA_b]. For elaboration, see References 31 and 42.

Results

k _po and v _b values in normal and normal‐appearing MS brain

Figure 1 displays 7 T ¹H₂O IR turboFLASH DCE‐MRI data, and SSP analysis 58, for a 22 year old female control subject. Panels A and B show R ₁ time‐courses: GdHPDO3A was injected at about 35 s (arrows). The Figure 1(A) data are from a blood (sagittal sinus) ROI, those in Figure 1(B) from the normal white matter (NWM) ROI indicated by circles in panel D; axial R ₁ maps before and after CA administration. The CA passes through the brain without appreciable extravasation. Panel C shows the R _1b dependence of R _1t from Figure 1(A), (B). If the TP obtained, the R _1t vs. R _1b plot would be linear. This is indicated by the dashed line in Figure 1(C). The non‐linear data are well fitted with the 2SX SSP expression of Equation [1] (f _W = 0.8), spanning the FXL and FXR conditions 31, 42, 55, 58 (solid curve)] with τ _b = 560 ms, v _b = 0.018, and R _1exv = 0.85 s⁻¹ (the intrinsic extravascular ¹H₂O R ₁). If the TP is forced to the data, the dashed FXL line must pivot about their origin (R _1b(0), R _1t(0)) and its slope, v _b, is significantly decreased. In this case, the TP gives v _b = 0.015, a 17% underestimation. Of course, TP also denies access to τ _b, since it assumes it zero.

Figure 2 displays axial voxel‐by‐voxel parametric maps for the subject of Figure 1. The biomarkers are (a) R _1exv, (b) v _b, and (c) k _po (τ _b ⁻¹). As expected, R _1exv is greater in NWM than in normal gray matter (NGM), and the v _b map exhibits greater NGM (0.03) than NWM (0.01) values. The latter approximate rather well absolute CBV fractions, not relative values. Though such maps are quite important, they exhibit the natures of the R _1exv and v _b properties. The larger NWM R _1exv values reflect the greater macromolecular volume fractions of NWM 74 and the larger NGM v _b values reflect the well‐known greater NGM vascularity. As far as we are aware, Figure 2(c) is the first k _po map. As discussed above, τ _b is a supra‐intensive parameter. It is very interesting that the k _po map (Fig. 2(c)) exhibits greater intensity in NWM, averaging 3.0 s⁻¹, than in NGM, 2.5 s⁻¹.

Figure 2.

Normal parametric maps. SSP maps for the healthy control subject of Figure 1. (a) R _1exv (intrinsic, extravascular R ₁), (b) v _b (blood volume fraction), and (c) k _po (τ _b ⁻¹) (τ _b is the mean capillary water molecule lifetime). The k _po rate constant map displays ongoing resting‐state metabolic activity.

Figure 3 shows the voxel‐by‐voxel k _po versus v _b scatter plot of many of the data of Figure 2(b), (c). The voxels were chosen from 50² in a square slab ROI centered on and covering about 75% of the brain image slice of Figures 1 and 2. The R _1exv relaxation rate constant spectrum (histogram) was used to assign the voxels 74. The 649 voxels with R _1exv values between 0.80 and 0.92 s⁻¹ were identified as NWM, and yield the pink points in Figure 3. The 670 voxels between 0.62 and 0.72 s⁻¹ are labeled NGM, and give the olive points in Figure 3. As expected, the NWM points cluster below v _b = 0.02. The NGM points cluster about a v _b value (0.06) somewhat greater than expected, because a number represent voxels with some partial‐volume averaging of vessels larger than capillaries – especially near the cortical surface (Fig. 2(c)). Interestingly, the NGM k _po (τ _b ⁻¹) values are essentially independent of v _b, and the NWM v _b values are essentially independent of k _po. The basically horizontal and vertical orthogonal NGM and NWM clusters are consistent with parameters not numerically correlated by data fittings. The interesting trends seen in Figure 3 are physiological. The parameter v _b = ρ ^† V. By definition, k _po is ρ ^† independent, and is dependent on only V ^−1/2. Since k _po is experimentally independent of v _b, k _po variations must be due to P _W ^† variations (Background). In NWM voxels these are large but v _b is small and apparently regulated (likely ρ ^† regulation), while in NGM voxels k _po seems regulated. The very slight downward slope of the green point cluster at larger v _b is due to the partial‐volume averaging mentioned above. Larger v _b values reflect larger mean r values, and there is a slight k _po decrease due to this, but mostly k _po is constant in NGM.

Figure 3.

Normal scatter plot. The plot of k _po (τ _b ⁻¹) values versus v _b values for the pure NWM (pink) and NGM (olive) voxels from Figure 2(b), (c). v _b is the capillary density–volume product, ρ ^† V. The NWM and NGM assignments were made from the R_1exv (Fig. 2(a)) value histogram (not shown; see text). Parametric interdependence would be manifest as sloped clusters. The basically orthogonal NWM and NGM clusters signify the k _po and v _b biomarkers are essentially independent of each other, and this means k _po is dominated by capillary wall water permeability (see text).

Now, we turn to population averages. Table 1 presents (about 3.6 mL) ROI biomarker values averaged for six healthy controls and six RRMS subjects with “non‐enhancing” WM lesions. Precision is generally quite good: the SEMs listed are due mostly to inter‐subject variation. The fact that the k _po values in NWM and NGM are more similar than the v _b values is due to the aforementioned supra‐intensive nature of τ _b. The R _1exv values are reduced in NGM, normal‐appearing gray matter (NAGM), and MS lesions because of decreased macromolecular volume fractions 74. The v _b values are increased in normal‐appearing white matter (NAWM) and NAGM, and decreased in MS lesions. More interesting is the fact that k _po is decreased in MS‐NAWM and MS‐NAGM, and even more‐so in lesions.

Table 1.

Population‐averaged resting‐state human brain ROI parameter values

	SSP DCE‐MRI (1H2O)
	R 1exv (s−1)	v b	τ b (s)	k po (τ b −1) (s−1)c
Healthy controls (n = 6)a
NWM	0.831 (±0.021)	0.014 (±0.002)	0.35 (±0.04)	3.2 (±0.56)
NGM	0.679 (±0.015)	0.031 (±0.004)	0.41 (±0.06)	2.9 (±0.59)
Relapsing–remitting MS (n = 6)a
NAWM	0.810 (±0.022)	0.019 (±0.002)	0.48 (±0.05)	2.2 (±0.20)
NAGM	0.672 (±0.009)	0.045 (±0.004)	0.50 (±0.03)	2.0 (±0.13)
Lesion	0.624 (±0.009)	0.012 (±0.003)	0.59 (±0.14)	1.8 (±0.45)
Glioblastoma (n = 5)b
NA‐frontal WM	1.10 (±0.027)	0.008 (±0.001)	0.44 (±0.04)	2.6 (±0.31)
NA‐thalamus	0.90 (±0.009)	0.017 (±0.001)	0.38 (±0.05)	2.9 (±0.37)
NA‐putamen	0.78 (±0.013)	0.012 (±0.005)	0.43 (±0.03)	2.5 (±0.22)
Tumor	0.67 (±0.013)	0.046 (±0.013)	≥5.6	≤0.18

^aProHance, 7 T.

^bFerumoxytol, 3 T.

^c⟨k _po⟩ (≠⟨τ _b⟩⁻¹). Uncertainties are ±SEM.

For a cylindrical capillary, k _po = 2P _W ^† r ⁻¹: the quantity r is a 1D measure of capillary size. With a conservatively large r value (3 µm 39, 40), k _po = 0.7P _W ^† (k _po in s⁻¹, P _W ^† in µm s⁻¹). With a typical P _W ^† value (2 µm s⁻¹ 50, 51), k _po = 4r ⁻¹ (r in µm). Thus, k _po is linearly related to both P _W ^† and r ⁻¹, with different coefficients. In the Discussion section, we compare concomitant relative (%) changes in the population‐averaged v _b and k _po parameters for the NGM → NAGM (Table 1) transition. A deductive quantitative analysis shows that the k _po decrease is dominated by a P _W ^† decrease. The analogous exercise indicates an even greater P _W ^† decrease in MS‐NAWM. k _po (τ _b ⁻¹) is dominated by the P _W ^† factor, not the r ⁻¹ factor.

Decreased k _po in the MS lesion

In non‐enhancing MS lesions, the k _po value is decreased even further: the average for the six Table 1 RRMS subjects is 1.8 s⁻¹. However, these represent relatively early‐stage disease. Figure 4 shows results for a 52 year old female late‐stage RRMS subject. Quite large chronic demyelinated WM lesions appear hypointense in the R _1exv map (Fig. 4(a)) – indicating extensive macromolecular loss, consistent with demyelination and gliosis. These lesions are many months past their last high CA‐enhancement stage. The v _b map (Fig. 4(b)) is rather similar to that of the control (Fig. 2(b)), but reduced (<0.01) in lesion areas and NAWM. However, the k _po map (Fig. 4(c)) is dramatically altered. Unlike NWM (Fig. 2(c)), the WM region is extremely hypointense. The k _po values in the lesions themselves (~1.5 s⁻¹) are decreased below the MS‐NAWM mean (2.2 s⁻¹) and even the RRMS lesion mean (1.8 s⁻¹) (Table 1). Furthermore, compared with the RRMS NAGM mean (2.0 s⁻¹) (Table 1), the k _po values (~2.9 s⁻¹) are considerably increased in this advanced subject NAGM. (It is hard to discern the NAWM situation because the lesions are so large.)

Figure 4.

Late‐stage MS parametric maps. SSP maps for a 52 year old female advanced RRMS subject. (a) ¹H₂O R _1exv: extensive, chronic WM lesions are hypointense. (b) v _b shows substantial reductions (<0.01) in the lesion areas. (c) k _po (τ_b ⁻¹) shows striking contrast between lesions and NAGM. The k _po values are strongly depressed in lesion areas, and considerably elevated in cortical regions, compared with the control (Fig. 2(c)). The k _po map suggests low resting‐state metabolic activity in these significantly demyelinated lesions.

Decreased k _po in the GBM tumor

For GBM capillaries, clinical monomeric Gd(III) chelate CAs extravasate too rapidly to allow k _po determination. Thus, we used the intravascular, coated superparamagnetic iron oxide nanoparticle Fe‐tol as CA 73. This agent has a molecular mass of 750 000 Da (10³ times that of GdHPDO3A; 588 Da). Its K ^trans in normal and NA brain tissue is ~10^‐8 min⁻¹ 75. (The biomarker K ^trans ≈ P _CA ^† S, where P _CA ^† is the endothelial CA permeability coefficient 76.) During the first pass, it remains intravascular even in very advanced GBM tumors with extremely permeable capillaries. Figure 5 shows results from a 52 year old male GBM subject. In the center is an R ₁ map obtained 30 min after GdHPDO3A injection. The large CA‐enhancing tumor is clearly visible at the bottom left. Twenty‐four hours after GdHPDO3A, the subject received IV Fe‐tol. Inset are data (points) obtained from four representative ROIs (frontal WM, thalamus, putamen (white ellipses), and tumor (red circle)) during the Fe‐tol injections. Each plot shows the R _1b dependence of R _1t (as in Fig. 1(C)). If the TP (τ_b → 0) held, the R _1t vs. R _1b plots would be linear. None are, and all are well fitted by Equation [1] 2SX SSP expression curves with R _1exv, v _b, and τ _b varied. The R _1exv, v _b, and τ _b parameter values returned are 1.24 s⁻¹, 0.008, 0.29 s (frontal WM); 0.99 s⁻¹, 0.013, 0.32 s (thalamus); 0.94 s⁻¹, 0.014, 0.43 s (putamen), and 0.77 s⁻¹, 0.028, 1.52 s (tumor). For five subjects, the population‐ and ROI‐averaged parameter values are given in Table 1. For this 3 T study the R _1exv values are greater than the normal and MS brain 7 T entries. Tissue macromolecular relaxivity is greater at smaller field 74. In the GBM normal appearing (GBM‐NA) brain, the v _b values are generally smaller than normal. They are large in the tumor.

The k _po values in the GBM‐NA brain are fairly normal. This supports the general accuracy of the three Fe‐tol injection steady‐state protocol. (The latter favored spatial resolution over pharmacokinetic temporal resolution, just the opposite of the GdHPDO3A protocol used to obtain the Table 1 normal and MS‐NA values.) Importantly, the Figure 5 tumor tissue k _po is decreased by more than a factor of five. The example tumor ROI shows why: the R _1t vs. R _1b plot has a much sharper hyperbolic shape. Though the Fe‐tol protocol yielded only four pharmacokinetic points, this behavior is confirmed by the population (n = 5) averaging (Table 1), where the tumor k _po decrease is over an order of magnitude. Since there is likely a family of fittings of the four data points for which the k _po values are even smaller, it is best to consider the Table 1 GBM tumor k _po entry as an upper limit. The very large GBM tumor tissue τ _b values were unexpected. Future Fe‐tol protocols will acquire more than four points while still achieving full brain coverage and good spatial resolution.

The fact that k _po decreases in MS lesions and in GBM tumors is very significant. We show below that this is due to P _W ^† decreases. However, these are both pathological tissues well known to have leaky capillaries: K ^trans is clearly increased in each. Furthermore, the increase of the P _CA ^† factor dominates K ^trans. This is what is meant by the colloquial phrase “increased capillary permeability.” The facts that P _W ^† decreases while P _CA ^† increases mean that water and CA molecules do not exchange across the capillary wall by the same dominant mechanism. This important finding is elaborated in the Discussion section.

τ _exv values

Finally, equilibrium mass action demands τ _exv = τ _b[(1 – p _b)/p _b], where p _b is the mole fraction of water that is vascular (=v _b/f _W). Combining the Table 1 v _b and τ _b values, we obtain τ _exv values of 19 s and 10 s for NWM and NGM, respectively. These are comparable to the first‐order lifetime, 45 s, calculated 49 from the observed 31 s t _1/2 for brain parenchyma ¹⁵OH₂ intravasation 50. As detailed above, this 2SX expression does not require the assumption that parenchymal water is “well mixed,” only that its MR systems are in their FXL conditions.

Discussion

k _po variations are due to P _W ^† differences

As noted above, the essential independence of the experimental k _po and v _b parameters in the Figure 3 voxel scatter plot signifies that k _po is dominated by capillary wall water permeability. Further, quantitative deductive analyses of concomitant population‐averaged τ _b ⁻¹ ratios and v _b ratios in normal brain, MS‐NA brain, GBM‐NA brain, MS lesions, and GBM tumors show that variations in the capillary equilibrium water efflux rate constant (k _po) are dominated by differences in microvessel wall water permeability (P _W ^†), not capillary radius. Example analyses are detailed in Appendix A.

Briefly, we use the relationships k _po(A)/k _po(B) = [P _W ^†(A)/P _W ^†(B)] (r _B/r _A), and [v _b(B)/v _b(A)]^1/2 = (ρ _B ^†/ρ _A ^†)^1/2(r _B/r _A), for ROIs A and B. For example, for A = NAGM and B = NGM we plot in 3D capillary property space the trace of all points that simultaneously satisfy the experimental population‐averaged k _po(A)/k _po(B) and [v _b(B)/v _b(A)]^1/2 ratios (Fig. A1). The experimental data are incompatible with the mean capillary radius and density simultaneously remaining invariant from NGM to NAGM. The brain literature generally indicates it more likely that chronic v _b differences are due to capillary density (ρ ^†) differences than to capillary dilation or constriction (r changes) 38, 39, 40, 41. (Even in an acute hypercapnic perturbation, the microvascular radii for the dominant capillary volume fraction remain unchanged 41. The very smallest capillaries, normally effectively occluded, are opened during the hypercapnia – there is some “recruitment” – but in most capillaries there is a blood velocity increase 41.) Therefore, the Figure 1A results clearly indicate that the mean capillary water permeability in MS NAGM is reduced from its value in NGM (the NGM → NAGM transition). For equal mean capillary radii, P _W ^†(NAGM) = 0.8P _W ^†(NGM), P _W ^† is reduced by 20%. Recall that r is the average for a large number of capillaries. Only 100 capillaries μL⁻¹ means 4000 per 40 μL voxel. The Table 1 ROIs represent 80–100 voxels. 100 voxel ROIs in six subjects yield averages over 2 400 000 capillaries.

Capillary water exchange is dominated by transcellular pathways

There are many possible pathways water molecules can use for capillary egress and ingress. Figure 6 summarizes these. It depicts (a) paracellular water passage through endothelial tight junctions (endothelial cells are colored gray), (b) simple, transcellular water diffusion across cell membrane lipid bilayers, and (c) transcellular transport through membrane aquaporin protein water channels 63, 64, 77 and/or leakage through membrane transporters 65. (Inspirations for this diagram are found in References 35, 36, and 63. It emphasizes water equilibria, and is otherwise greatly simplified.) The Figure 6(d) transcellular process will be elaborated below.

Figure 6.

A neurogliovascular unit chain mechanism. Water exchange processes determine mean water molecule lifetimes in blood (τ _b, beige), interstitium (τ _o, aqua), endothelial (τ _i′, gray), neuroglial (τ _i, pink), and neuronal (τ _i″, blue) cell spaces. The equilibrium paracellular (a), simple diffusion (b), facilitated transcellular (c), and active water cycling (d, stars) pathways are indicated, as are “Magistretti steps” (e–g). We suggest the d steps couple unit metabolic activity to τ _b.

The facts that, in MS lesions and GBM tumors, P _W ^† decreases while P _CA ^† increases mean that water and CA molecules exchange via different pathways. CA molecules are universally thought to employ the para(endothelial)cellular pathway (Fig. 6(a)), and we previously thought that this would be a major mechanism for water as well. In Appendix B, however, we compare k _po with k _pe (for CA extravasation) to show that, for the normal brain, by far the vast majority (>95%) of capillary water efflux (and influx) occurs via one or more transcellular processes (Fig. 6(b)–(d)). This is a fundamental finding, and we are not aware that it has been previously known.

Equilibrium transendothelial water exchange is a metabolically active process

Which, if any, of the transcellular pathways (Fig. 6(b)–(d)) dominates k _po? Mechanisms 6(b) and 6(c) (bilayer diffusion, and passage through trans‐membrane aquaporin and/or protein channels, respectively) are passive, i.e. require no energy expenditure. However, comparison of our results with literature metabolic imaging studies of the same tissues indicates that the process measured by k _po is metabolically active.

This is shown in Table 2. The second and third columns repeat the Table 1 v _b and k _po entries and place them adjacent to results of pertinent quantitative ³¹PMRSI 78 and ²³NaMRSI 19, 79 studies. Because of ³¹PMRSI spatial resolution limitations, it is important that WM/GM image segmentation be accomplished with co‐registered ¹H₂O maps, and then applied to apportion the MRSI measurements 78. Such results are listed in Table 2. It is clear that k _po exhibits a positive correlation with tissue ATP concentration, [ATP_t], comparing NWM with NGM, NWM with NAWM, NGM with NAGM, or NAWM with NAGM. For example, [ATP_t] decreases from normal brain in both NAWM (13%) and NAGM (20%). The phosphocreatine concentration, [PCr_t], also decreases (not shown) in rough proportion to [ATP_t] 78, consistent with the maintenance of overall ATP/PCr equilibrium. In contrast, k _po correlates negatively with TSC, [Na_t], which increases in NAWM (39%), NAGM (17%) 19, and GBM tumor (51%) 79. Since there is insignificant extracellular ATP, [ATP_t] reflects [ATP_i]. An [Na_t], TSC, increase could reflect an [Na_i] increase. An [ATP_i] decrease and a possible [Na_i] increase signify compromised metabolism: decreased ATP hydrolysis chemical potential and trans‐mural Na⁺ gradient electrochemical potential, respectively. The brain k _po values are correlated with metabolic thermodynamic properties.

Table 2.

The biomarker k _po measures metabolic activity

	SSP DCE‐MRI (1H2O)		31PMRSI	23NaMRSI	SSP DCE‐MRI (1H2O)	31PMRSI‐MT
	v b	k po (τ b −1) (s−1)	[ATPt] (mM)	[Nat] (mM)	k po v b (s−1)	CMRoxphos (pmol(ATP) s−1 μL−1)
Healthy controls
NWM	0.014	3.2	2.43	19a	0.045	50
NGM	0.031	2.9	1.62	31a	0.090	160
NGM/NWM					2.0	3.2
Relapsing–remitting MS
NAWM	0.019	2.2	2.11	27a	0.042
NAGM	0.045	2.0	1.29	36a	0.090
Lesion	0.012	1.8		35a	0.022
Glioblastoma
NA‐frontal WM	0.008	2.6		↑3%b	0.021
NA‐thalamus	0.017	2.9		↓12%b	0.049
NA‐putamen	0.012	2.5			0.030
Tumor	0.046	≤0.18		↑51%b	≤0.008
References	this work		78	19, 79	this work	17

^aReference 19.

^bRelative to NWM, Reference 79.

However, k _po is a kinetic parameter (of dimension reciprocal time). In order to validate a flux measurement, one must compare it with the gold standard flux measurement. Normal homeostatic neuronal cell metabolic rates have been measured with ³¹PMRSI‐MT, using ¹H₂O segmentation 17. For NGM, the ATP synthesis (mostly by oxidative phosphorylation, CMR_oxphos; Fig. 6) flux is 0.16 nmol s⁻¹ μL⁻¹ 17, and of course the consumption rate is the same. The fluxes between ATP and phosphocreatine (PCr) are seven times larger, 1.15 nmol s⁻¹ μL⁻¹ in each direction 17: integrated over the neuron, ATP and PCr are in effective equilibrium. The NGM and NWM CMR_oxphos values, 160 and 50 pmol(ATP) s⁻¹ μL⁻¹, respectively 17, are entered in Table 2, as is the CMR_oxphos(NGM)/CMR_oxphos(NWM) ratio, 3.2. By definition, k _po is proportional to the H₂O flux per capillary and v _b to the number of capillaries per unit tissue volume. Since CMR_oxphos is an ordinary intensive property (all that is accessible by directly detected ³¹PMRSI), we must multiply the supra‐intensive k _po by the intensive v _b in order to compare. Thus, the k _po v _b products and the k _po v _b(NGM)/k _po v _b(NWM) ratio, 2.0, are displayed in Table 2. The agreement of the flux ratios for these two very different and independent techniques, ³¹PMRSI‐MT and DCE‐MRI (¹H₂O), is rather remarkable and strongly suggests k _po is proportional to CMR_oxphos, per capillary.

The Table 2 NGM and NWM [ATP_t] and CMR_oxphos values exemplify the thermodynamics/kinetics distinction. While the steady‐state ATP concentration (and thus free energy) per unit tissue volume is 1.5 times greater in NWM than NGM, the oxidative phosphorylation ATP flux in the same unit tissue volume is 3.2 times greater in NGM than NWM.

A neurogliovascular unit chain mechanism

Table 2 shows that k _po is proportional to the ATP consumption flux per capillary (the core of the neurogliovascular unit). What could be a mechanism for this? The clue is in a recent report on the heterogeneity, and response to therapy, of cellular τ _i values within human breast tumors in vivo 32. That paper also assembles the evidence from model studies that τ _i ⁻¹ is increased by the gene dosage of, and substrates for, the driving cell membrane P‐type ATP‐ase ion pump, and decreased by specific inhibitors 33, 66. Cellular k _io (τ _i ⁻¹) reflects P‐type ATP‐ase turnover, per cell 32. Figure 6(d) visualizes a cascade (or chain) of τ _i changes for cells within the neurogliovascular unit. These are τ _i (neuroglia, pink), τ _i′ (endothelial cells), and τ _i′′ (neurons, blue). In the Figure 6 diagram, these active processes are indicated by trans‐membrane water cycles (stars, 6(d)). The pink cells can be astrocytes, oligodendrocytes, pericytes, etc. 34, 35, 36, 37. Combinations of neurons, glia, and microvessels have been termed “gliovascular units” 36, because of their crucial, exquisite symbiotic metabolic and energetic interactions 34, 35, 37. The Figure 6(d) pathways represent processes driven by NKA turnover, perhaps the most crucial ongoing cellular metabolic activity in the brain. These would affect each other by changes in transporter substrate concentrations (“paracrine communication”). This is plausible because these cells are within synaptic proximities (<50 nm), and have asymmetric transporter distributions. Microjets of water and substrates are continually injected into these confined spaces as transporters turn over. Obligate active trans‐membrane water cycling 32 means that water is effectively a substrate for the reactions driven by NKA turnover – the steady‐state system for which the forward reaction is given in the Introduction. Thus, it is possible that τ _b in turn reflects ongoing neurogliovascular unit metabolic turnover, particularly Na⁺,K⁺ATP‐ase activity: the smaller τ _b the greater NKA turnover, and vice versa – a cascade of altered active trans‐membrane water cycling – a k _po decrease reflecting k _io decreases within the unit. For example, the “Magistretti mechanism” (Fig. 6(e)–(g)) has neuroglia essentially conducting most glycolysis and transferring lactate to neurons for mostly oxidative phosphorylation: the capillary is intimately involved in this intercellular metabolic cooperativity 34, 35. An increase in k _po would reflect a speed‐up of Magistretti‐type processes (Fig. 6(e)–(g)), an exciting hypothesis. The increased k _po in NWM (Figs. 2(c) and 3) may reflect increased metabolic activity in common tracts shared by fluctuating resting‐state neural circuits. For example, the Figure 2c asymmetry (left centrum semiovale WM hyperintensity (image right)) could reflect a “rich club” node WM connection active at the time of this acquisition 80. Consistent with this, in other subjects (not shown) WM k _po hot spot loci vary. The greater spatial resolution planned (see below) could prove informative in this regard. (We note that the BOLD effect reflects the coupling of metabolic activity with vascular properties.) The extravascular water lifetime τ _exv is a complicated function of water populations and lifetimes τ _i″, τ _o, τ _i, τ _i′, etc. (Fig. 6) 55.

Absolute quantitation

Our results allow calculation of the equilibrium brain capillary water efflux. In 1 μL tissue, we estimated the average capillary length and radius as 2 mm and 2.6 µm, respectively, above. For a cylinder, this gives a mean capillary volume (V) of 42.5 pL. A 50 M [H₂O] yields 1.3 × 10¹⁵ H₂O molecules per capillary. For NGM, k _po is 2.9 s⁻¹ (Table 1). This gives the equilibrium water efflux = 1.3 × 10¹⁵ × 2.9 = 3.8 × 10¹⁵ H₂O molecules s⁻¹ per capillary (and, of course, an equal influx). Now, consider the homeostatic NGM CMR_oxphos, 160 pmol(ATP) s⁻¹ μL⁻¹ 17. If 75% is used for NKA turnover 3, we have 120 pmol(ATP) s⁻¹ μL⁻¹ consumption. For 100 capillaries μL⁻¹ 81, this is 1.2 pmol(ATP) s⁻¹ per capillary. The NGM water flux estimated above corresponds to 6 nmol(H₂O) s⁻¹ per capillary, and yields 5 × 10³ H₂O molecules cycled per NKA turnover (one ATP molecule consumed). Some individual water co‐transporting membrane symporters have H₂O stoichiometries approaching this order of magnitude 69, 70, 82, and there are likely a number of different symporters involved in the neurogliovascular unit chain (Fig. 6(d)). Thus, in addition to the remarkable agreement with relative CMR_oxphos values in Table 2, it is possible that brain k _po values can be interpreted quantitatively. For a 44 μL rat brain ROI in vivo, k _io was measured as 1.8 s⁻¹ using a very invasive intracerebroventricular CA infusion 83. The chain mechanism (Fig. 6(d)) suggests that k _po should be similar to k _io.

Clinical implications

This paper does not focus on clinical aspects. However, the approach introduced here has much to offer in this regard.

MS

The longstanding MS imaging hallmark is the “enhancing” WM lesion. In Appendix B, we mention that enhancement (with CA) is transient, and not always “caught” in an MRI study. The Figure 4 late‐stage subject is an example. Though her WM lesions are chronic, and large, they were not especially CA‐enhancing at the time of acquisition, and their conspicuity in the R _1exv map (essentially a T ₁‐w image inverse) is poor. However, if we inspect the τ _b map (the Fig. 4(c) inverse) in Figure 7, the conspicuity is very high. Thus, the prospect for detecting lesions is much greater with a τ _b map.

Figure 7.

Late‐stage MS τ _b map. The τ _b map for the 52 year old female advanced RRMS subject in Fig. 4. This is the Fig. 4(c) k _po map inverse. The τ _b values are strongly elevated in lesion areas, and strongly reduced in cortical regions, giving these chronic lesions very high conspicuity. Lesion CA‐enhancement (not shown) is almost non‐existent. Elevated subarachnoid τ _b values are artifactual.

Meanwhile, MS understanding is evolving. While long considered a WM disease, emerging data suggest that GM may be an early, or even the initial, disease target 84, 85. A recent concept is that MS disease activity originates in brain regions other than WM, perhaps GM; the “outside‐in” hypothesis 85. In early disease, pro‐inflammatory cytokines are chronically upregulated and can reduce oxygen utilization despite sufficient delivery (“metabolic hypoxia” 86), mediate mitochondrial function, and decrease neurogenesis, and may increase overall neurodegeneration risk. Metabolic deficits of MS‐NAGM are more extensive than those in MS‐NAWM and include decreased oxygen utilization 85, altered perfusion, and high‐energy phosphate depletion 78.

If MS‐NAGM and MS‐NAWM NKA turnover is diminished by metabolic hypoxia then, according to our mechanism, the supra‐intensive k _po will decrease. This is what we see (Table 2). The fact that, in RRMS, k _po decreases by the same amount (31%) from both NWM and NGM is strong evidence for whole brain involvement. Furthermore, our results predict this should not be detectable by ³¹PMRSI‐MT, which can access only the intensive CMR_oxphos. Table 2 indicates that k _po v _b – the CMR_oxphos analog – does not decrease in RRMS NAGM or NAWM: the recruited ρ ^† increase exactly compensates the P _W ^† decrease.

The TSC is the volume fraction‐weighted average of the (extra‐ and intra‐cellular) compartmental Na⁺ concentrations within the voxel or ROI: [Na_t ⁺] = v _e[Na_o ⁺] + v _i[Na_i ⁺]; where v _e + v _i = 1. Globally in the brain, v _e is usually thought to be near 0.2 (thus v _i = 0.8); [Na_o ⁺] is considered (highly regulated) near 140 mM; and [Na_i ⁺] near 12 mM in normal homeostasis 79. Thus, there are (at least) two general mechanisms for increasing [Na_t ⁺]. The first (anatomic) is a v _e increase (with concomitant v _i decrease; cell shrinking; cell density decrease) without changing [Na_o ⁺] or [Na_i ⁺]. This is often called the Hilal mechanism. The second (metabolic) is an [Na_i ⁺] increase due to a slow‐down of membrane NKA turnover without changing v _e, v _i, or [Na_o ⁺]. NKA activity can be regulated in several different ways 3. Of course, there are various possible combinations of the first and second mechanisms simultaneously. The opposite changes could decrease [Na_t ⁺]. The good (negative) correlation of k _po with [Na_t ⁺] in both normal and NA‐MS brain (Table 2) is suggestive of the second mechanism. However, since we do not have an independent measure of v _e (and thus v _i), its interpretation remains ambiguous.

Also exciting is the significantly increased MS‐NAGM k _po in the late‐stage disease (Fig. 4(c)). If this is borne out in more subjects, it means that neurogliovascular unit NKA activity is increased in advanced MS‐NAGM – a strong indication of global metabolic GM involvement, possibly indicating an RRMS to secondary‐progressing MS conversion. Access to a metabolic imaging biomarker for this stage change would be of tremendous benefit. Inspection of the same map (Fig. 4(c)) suggests that the demyelinated lesions in WM have greatly diminished resting‐state metabolic activity. Perhaps the NAGM activity is increased because of the necessity to employ “detour” circuitry because of blocked rich club node connections. Alternatively, perhaps neurogliovascular unit cells enter apoptosis in advanced MS disease. There is an interesting report that cells intentionally put into a defined apoptotic state exhibit a substantially increased k _io 87.

GBM tumor

In GBM tumors, k _po values less than 1 s⁻¹ (Table 2) suggest that NKA turnover is exceptionally slow. This is consistent with tumor [Na_t] being increased by 51% over its value in NWM 79 (Table 2). Our finding that k _po values in GBM‐NA tissue are similar to those in control brain is also consistent with the fact that tumor‐NA tissue [Na_t] values differ little from controls 79 (Table 2). A recent qualitative ³¹PMRSI investigation of human GBM in vivo suggests that [ATP_t] and [PCr_t] values are essentially the same in tumor tissue as in control brain 88. Also it has been found that, while MR_glc is only slightly elevated over adjacent WM in GBM tumor 27, there is extensive hypoxia 89. In this condition, ATP production shifts from oxidative phosphorylation toward glycolysis. Within the neurogliovascular unit, this means that the locus of ATP synthesis shifts from neurons toward neuroglial cells 34, 35. The consequence of this could be that a decrease in neuronal glucose consumption is slightly overcompensated by an increase in neuroglia, which also have proliferated in the tumor. However, net NKA turnover in the neurogliovascular unit, dominated by neurons, would decrease and consequently neuronal [Na_i] would increase. The substantial GBM tumor k _po decrease (Table 2) may be a hypoxia signature. Unlike ¹⁸FDG PET or gluco‐CEST/CESL, the activity we ostensibly measure is catabolically downstream of an oxidative phosphorylation → glycolysis shift. The turnover of NKA is a major end‐point of central (intermediary) metabolism.

The population‐averaged value of 0.18 s⁻¹ for k _po in GBM tissue (Table 1) is quite surprisingly small. We find the tumor k _io value is substantial in the epithelial cancers (>2.0 s⁻¹, breast carcinoma 32; >2.5 s⁻¹, prostate adenocarcinoma 90) we have examined (hot spots even larger). Perhaps k _io within the GBM tumor is also significant and the small k _po really reflects a breakdown of neurogliovascular unit k _io/k _po coupling. Preliminary analyses of Gd(III) CA DCE‐MRI time‐courses (not shown) suggest that the GBM tumor k _io values are not as small as k _po. Of course, the GBM tumor could also have just a decreased metabolic activity.

Stroke

The acute and significant cerebral water apparent diffusion coefficient (ADC) drop after an ischemic event is of considerable clinical importance, though the mechanism has remained elusive. We have suggested 33 that an active trans‐membrane water cycling decrease may contribute. Consistent with this, the ADC drops by 40% within 15 min of direct application of ouabain (a specific NKA inhibitor) to the striatum – before there is significant decrease of the NKA substrate ATP_i 91.

Concluding remarks

We have designed a new acquisition pulse sequence that incorporates the “multiband” approach 92, 93 and yields full brain coverage with nominal voxel volume less than half that in Figures 2, 4, and 7.

The existence of, or dominance of, an active cell membrane k _io has implications for many different types of in vivo MR experiment. In the metabo‐CEST experiment, The RF‐induced ¹H₂O intensity change depends on two factors: (1) the metabolite concentration, and (2) the probability per unit time of a water molecule encountering the metabolite molecule. Some water must cross a cell membrane to gain this access, and we have suggested 61 that this transport can contribute to the metabo‐CEST signal. In gluco‐CEST, for example, the contribution from intracellular glucose is “very small to negligible” 94. However, about 80% of water is intracellular. An aim of gluco‐CEST is to determine relative glucose concentrations 23. Since k _io likely changes during a glucose challenge, the probability of intra‐ and extracellular H₂O molecules crossing the membrane changes. The fact that this would also alter the temporal probability of water encountering glucose could affect the interpretation of gluco‐CEST changes. This phenomenon may have already been manifest in the gluco‐CESL experiment 24.

It is possible that water movement in living tissue is dominated by active trans‐membrane water cycling. Significant shutter‐speed effects are very common in cancer MRI 32. (Since the angiogenic microvessels of malignant tumors have larger intrinsic K ^trans values than benign tumors, the use of SSP DCE‐MRI makes it possible to contemplate eliminating most, if not all, unnecessary biopsies (i.e. those that find no malignancy) in breast 95, 96, 97 and prostate 31, 90 cancer. These comprise about 70% of all breast and prostate biopsies. The K ^trans values of malignant tumors are systematically suppressed by the TP.) These effects are now also being found in animal and human myocardium in vivo 66, 68. The NMR shutter‐speed concept has broad application.

Appendix A.

k _po Variations and Accompanying v _b Variations

Consider (conservatively) the MS‐induced GM k _po change – the smallest in Table 1. We have ⟨τ _b(NAGM)⟩⁻¹/⟨τ _b(NGM)⟩⁻¹ = 2.0 s⁻¹/2.5 s⁻¹ = 0.80: the NAGM ⟨τ _b⟩⁻¹ is reduced by 20% from that in NGM (⟨k _po⟩ is reduced by 31%). From the fundamental theoretical relationship, ⟨τ _b(NAGM)⟩⁻¹/⟨τ _b(NGM)⟩⁻¹ = [P _W ^†(NAGM)/P _W ^†(NGM)](r _NGM/r _NAGM), where r _NGM and r _NAGM are the mean capillary radii in NGM and NAGM tissues, respectively. (Capillaries dominate vascular volume in most voxels 39, 41.) Thus, the experimental relationship [P _W ^†(NAGM)/P _W ^†(NGM)](r _NGM/r _NAGM) = 0.80 must be satisfied. There is an infinite number of possibilities: if r _NAGM = 1.25r _NGM, P _W ^†(NAGM) = P _W ^†(NGM); if r _NAGM = r _NGM, P _W ^†(NAGM) = 0.8P _W ^†(NGM); etc.

It is extremely difficult, and invasive, to determine individual capillary radii in vivo 38, 40, 41. However, we can estimate capillary radius changes, if any, from our data. The blood volume fraction v _b = (n/V _T)V = ρ ^† V, where n/V _T is the number of capillaries in the voxel or ROI total volume, the capillary number density, ρ ^† (hundreds μL⁻¹ 81), and V is the mean individual capillary volume. Assuming cylindrical capillaries, r ~ V ^1/2 and the ratio [v _b(NGM)/v _b(NAGM)]^1/2 = (ρ ^† _NGM/ρ ^† _NAGM)^1/2(r _NGM/r _NAGM). Taking v _b(NGM) as 0.031 and v _b(NAGM) as 0.045 (Table 1) gives [v _b(NGM)/v_b(NAGM)]^1/2 = 0.83.

The solid red curve in Figure A1 is the trace of all points that simultaneously satisfy the experimental relationships [P _W ^†(NAGM)/P _W ^†(NGM)](r _NGM/r _NAGM) = 0.80 and (ρ ^† _NGM/ρ ^† _NAGM)^1/2 (r _NGM/r _NAGM) = 0.83 (from the population‐ and ROI‐averaged Table 1 biomarker values), in a 3D space of tissue microvascular properties. The axes are P _W ^†(NGM)/P _W ^†(NAGM) (vertical), r _NGM ⁻¹/r _NAGM ⁻¹ (inverse capillary radius ratio, =r _NAGM/r _NGM), and ρ ^† _NAGM/ρ ^† _NGM (capillary density ratio). This is very informative. The dot–dashed red projection does not pass through the r _NGM = r _NAGM, ρ ^† _NGM = ρ ^† _NAGM point. The experimental data are incompatible with the mean capillary radius and density simultaneously remaining invariant from NGM to NAGM. The brain literature generally indicates it is more likely that chronic v _b differences are due to capillary density (ρ ^†) differences than to capillary dilation or constriction (r changes) 38, 39, 40, 41. (Even in an acute hypercapnic perturbation, the microvascular radii for the dominant capillary volume fraction remain unchanged 41. The very smallest capillaries, normally effectively occluded, are opened during the hypercapnia – there is some “recruitment” – but in most capillaries there is a blood velocity increase 41.) When the (solid) red curve passes through r _NAGM = r _NGM (black point), the other coordinates are ρ ^† _NAGM = 1.44 ρ ^† _NGM and P _W ^†(NGM) = 1.25 P _W ^†(NAGM) (dashed black lines). A conservatively large area for r _NAGM from 0.75 r _NGM to 1.25 r _NGM is shaded gray in the Figure A1 bottom plane. Over this area, the red curve ρ ^† _NAGM coordinates range from 2.56 ρ ^† _NGM to 0.92 ρ ^† _NGM, and the P _W ^†(NGM) coordinates range from 1.67 P _W ^†(NAGM) to 1.0 P _W ^†(NAGM). These results clearly indicate that the mean capillary water permeability in MS NAGM is reduced from its value in NGM. For equal mean capillary radii, P _W ^†(NAGM) = 0.8P _W ^†(NGM), P _W ^† is reduced by 20%. Recall that r is the average for a large number of capillaries. Only 100 capillaries μL⁻¹ means 4000 per 40 μL voxel. The Table 1 ROIs represent 80–100 voxels. 100 voxel ROIs in six subjects yield averages over 2 400 000 capillaries. The analogous exercise indicates an even greater P _W ^† decrease in MS‐NAWM. k _po (τ _b ⁻¹) is dominated by the P _W ^† factor, not the r ⁻¹ factor.

Figure A1.

Capillary property space. A plot of biomarker inter‐relationships for population‐averaged NGM and NAGM ROI values in Table 1. Thus, ⟨τ _b(NAGM)⟩⁻¹/⟨τ _b(NGM)⟩⁻¹ = 0.80, and v _b(NAGM)/v _b(NGM) = 1.5, where τ _b ⁻¹ ≡ k _po (τ _b is the mean capillary water lifetime) and v _b the capillary volume fraction. The red curve is the trace of points that satisfy these experimental relationships simultaneously, in a 3D space of brain tissue microvascular properties. The vertical axis is P _W ^†(NGM)/P _W ^†(NAGM), the capillary water permeability coefficient ratio. The oblique axes are the inverse capillary radius ratio, r _NGM ⁻¹/r _NAGM ⁻¹ (=r _NAGM/r _NGM), and the capillary density ratio, ρ ^† _NAGM/ρ ^† _NGM. The coordinates of the curve in regions of reasonable r _NAGM/r _NGM (gray shading) are consistent only with τ _b being dominated by the P _W ^† factor. When r _NAGM = r _NGM (black point), ρ ^† _NAGM = 1.44ρ ^† _NGM, and P _W ^†(NGM) = 1.25P _W ^†(NAGM). P _W ^† is reduced in resting‐state NAGM from its value in resting‐state NGM.

For the GBM tumor, we make the same analysis, using tissue ROI‐ and population‐averaged parameter values from Table 1. There is a 93% decrease in tumor (T) k _po relative to putamen (P). If the permeability coefficients were equal (P _W ^†(T) = P _W ^†(P)), then r _T = 14r _P (with r _T and r _P the mean capillary radii in tumor and putamen tissue, respectively). It is even more unlikely than in the MS‐NAGM tissue that capillaries would dilate by more than an order of magnitude in the tumor tissue. As above, the ratio [v _b(P)/v _b(T)]^1/2 = (ρ _P ^†/ρ _T ^†)^1/2(r _P/r _T). Unlike the MS lesions (Table 1), however, the tumor v _b(T), 0.046, is increased over normal‐appearing tissue, v _b(P) = 0.012. (This reinforces that k _po is indeed independent of v _b. It is decreased in MS lesions and in GBM tumors, though v _b is decreased in the former and increased in the latter.) This gives [v _b(P)/v _b(T)]^1/2 = 0.51. Though extremely unlikely, if r _T were actually 14r _P, then ρ _T ^† = 0.020ρ _P ^†: the tumor capillary density would be less than 3% of that of normal brain – even more unreasonable. An unchanged P _W ^† value is incompatible with both the k _po and v _b changes. If r _T = r _P, then ρ _T ^† = 3.8ρ _P ^†, and capillary density is increased almost fourfold in the tumor. This is plausible, and explains the v _b ratio. (However, capillary density does not affect the supra‐intensive P _W ^† value.) If r _T = r _P, P _W ^†(T) = 0.072P _W ^†(P). Whatever the actual capillary radius change, if any, it seems certain that overall P _W ^† is much decreased in the tumor.

Appendix B.

Calculation of Capillary Trans‐Cellular Water Flux

We pursue the significance of decreased MS lesion k _po – decreased trans‐endothelial water exchange. Consider lesion K ^trans values. The biomarker K ^trans ≈ P _CA ^† S, where P _CA ^† is the endothelial CA permeability coefficient 76. The DCE‐MRI enhancement of MS lesions is transient during disease progression, increasing and decreasing with time‐constants of months 98, 99, making them hard to “catch.” When measurable, however, the K ^trans values exhibit intra‐lesion heterogeneity, and can reach magnitudes over 10⁻² min⁻¹ 96, 98, 99. Thus, active MS lesions exhibit significantly increased K ^trans values, and these remain elevated above NWM values (10^‐5 min⁻¹) for at least six months after maximum enhancement 98. Though the chronic Figure 4 lesions are no longer “active” in the clinical sense, their K ^trans values are still ten times normal 98. Since v _b is decreased in the Figure 4 MS lesions, the S quantity must be as well. Thus, P _CA ^† must be significantly increased. There is little doubt that CA employs the para(endothelial)cellular pathway (Fig. 6(a) for capillary extravasation: the endothelial cell junctions must open somewhat 36 in an MS lesion. The paracellular pathway must also constitute a component of the passive water permeability, P _W ^†(passive), contribution. Thus, it is highly likely that, in an MS lesion, P _W ^†(passive) is increased. However, we see that in the lesion overall P _W ^† is decreased (k _po is decreased). The conclusion is that P _W ^†(active) is significantly decreased in an MS lesion.

Also, for monomeric Gd(III) chelate CAs, P _CA ^† S (K ^trans) is greatly increased (four orders of magnitude) in GBM tumors (see Fig. 5, center) 75, mostly due to the P _CA ^† factor (since the v _b increase is less than one order of magnitude). There is little doubt that this is due to widened para(endothelial)cellular pores. Thus, paracellular water extravasation (a P _W ^†(passive) pathway) must also increase. However, this would make k _po increase. Once again, we are left with the essentially inescapable conclusion that P _W ^†(active) is very much decreased in the brain pathology.

Previously, we calculated the equilibrium brain capillary water efflux in 1 μL tissue for an average capillary length and radius of 2 mm and 2.6 µm, respectively. For a cylinder, this gives a mean capillary volume (V) of 42.5 pL. A 50 M [H₂O] yields 1.3 × 10¹⁵ H₂O molecules per capillary. For NGM, k _po is 2.9 s⁻¹ (Table 1). This gives the equilibrium water efflux = 1.3 × 10¹⁵ × 2.9 = 3.8 × 10¹⁵ H₂O molecules s⁻¹ per capillary (and, of course, an equal influx). Extravasating CA molecules surely use the paracellular pathway (Fig. 6(a); for H₂O). A typical maximum plasma CA concentration is 3 mM 100. The blood [CA_b]_max = (1 – h)[CA_p]_max. A 0.4 hematocrit (h) yields [CA_b]_max = 0.6 × 3 = 1.8 mM. Thus, at maximum, there are 1.8 × 10⁻³ × 42.5 × 10⁻¹² × 6.0 × 10²³ = 4.6 × 10¹⁰ CA molecules per capillary. The CA extravasation first‐order rate constant (k _pe) is K ^trans/v _p 31, 32, 43, 47, also supra‐intensive. A large K ^trans value, 0.1 min⁻¹, say for a GBM tumor capillary, and v _p = (1 − h)v _b = 0.6 × 0.03 = 0.02, yields k _pe = 8.3 × 10⁻² s⁻¹. This gives a maximum efflux = 4.6 × 10¹⁰ × 8.3 × 10⁻² = 3.8 × 10⁹ CA molecules s⁻¹ per capillary. Thus the minimal H₂O efflux/CA efflux ratio is 1 000 000. Even if 28 000 H₂O molecules accompanied each CA molecule ([H₂O]/[CA_b]_max) through the paracellular tight junction pore (Fig. 6(a)), there would be 972 000 H₂O molecules simultaneously exiting by transcellular pathways (Fig. 6(b)–(d)). Only 3% of water employs the paracellular pathway (Fig. 6(a)); 97% of equilibrium water flux is transcellular (Fig. 6(b)–(d)). This is for quite leaky capillaries: less permeable vessels would give an even greater transcellular percentage. (Though the K ^trans for head muscle tissue is greater, 0.15 min⁻¹ 96, k _po may be more than 10 s⁻¹ 56. Consequently, even in that case less than 1% of the steady‐state water flux is paracellular (Fig. 6(a)).) Thus, for the normal brain (K ^trans ~10⁻⁵ min⁻¹) by far the vast majority of capillary water efflux occurs via one or more transcellular processes.

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