Introduction

Consideration of the longitudinal MR shutter-speed [SS = κ_1io] for steady-state trans-cytolemmal water exchange has shown the first-order cellular water efflux rate constant [k_io] to have significant contribution from homeostatic Sodium Pump, NKA, activity.^1,2 [Definitions, Figure 1.] Though the cell membrane NKA may be biology’s most vital enzyme, its cellular metabolic rate [^cMR_NKA] - has never been measurable in vivo. So far, accurate k_io from ¹H₂O MR has been possible only when κ_1io is sufficiently large - when the extracellular contrast agent (CA) concentration is very high, or B₀ is very low.² Though these can be attained in model systems, they cannot be achieved in vivo.² A new diffusion-weighted imaging [DWI] analysis requires neither CA nor κ_1io, and yields the fundamental, un-factorable cell biology properties: cell density [ρ] and mean cell volume <V>, as well as <k_io>.¹ ρ and <V> have also not been available in vivo, but they can be estimated from histopathology. These help considerably in constraining the DWI analysis for increased k_io precision.

Methods

We use Monte Carlo random walks in randomly sized/shaped [“Voronoi”] cell ensembles.³ Figure 2 presents simulations varying each parameter in turn. Each simulation results from 100,000 walks in each of ten different [~ 19,000 cell] ensembles with the same k_io, ρ, and V. All particles had the 37^oC pure water diffusion coefficient [D₀ = 3.0 μm²/ms] inside or outside cells. The ordinates measure log[S(b)/S₀], where S(b) and S₀ are the transverse signal intensities at b and b = 0. The abscissae report the single-diffusion-encoding sequence b [= (γGδ)²t_D: the b unit here, ms/(μm)², is 1000 times the common s/(mm)².]

Results

With ρ fixed at 180,000 cells/μL, <V> at 5.0 pL [the intracellular volume fraction, v_i = ρ<V> = 0.90], Fig. 2a shows increasing <k_io> accelerates the non-linear [non‑Gaussian] decay. With <k_io> fixed at 12.0 s^-1, <V> at 5.0 pL, Fig. 2b reveals increasing ρ decelerates the decay, as expected, but over a large, 50,000 cells/μL, range [140,000 to 190,000]. With ρ fixed at 180,000 cells/μL, <k_io> at 12.0 s^-1, Fig. 2c shows little V‑dependence, perhaps surprisingly, even with a 30% <V> increase averaged over 10⁵ cells. The large dynamic range for large <k_io> in Fig. 2a, is very encouraging. Cell suspension κ_1io studies^4,5 show that, for aggressive cancer, <k_io> can exceed 75 s^-1; totally unreachable with CA in vivo.

Discussion

In Fig. 2, we also display quality awake human cerebral cortex (1.5T, 3.3 mL ROI) experimental data⁶ (points). One simulation (green in each panel) matches these remarkably well. We enlarge this in Figure 3, and add murine colorectal cancer xenograft (4.7T, 31 nL ROI)⁷ and human acellular [ρ = 0] bladder (Li [OHSU]: 3T, 0.72 mL ROI) data. Blue and black [Gaussian decay; pure H₂O] simulations also respectively match these results. However, matching goodness is not the only criterion: there are many literature empirical DWI fittings.¹ Our simulation parameters are physical quantities that can be validated with independent measurements. Figure 4 shows a table comparing the most pertinent of these [citations given] with the color-coded simulation properties. The ρ and <V> measures are from ex vivo histopathology: k_io must be determined from CA studies of viable models that, as above, are likely underestimations. The near absolute agreement is very encouraging. The probabilities of encountering and of permeating cell membranes are the major diffusion determinants, not “viscosity” considerations. It seems <k_io>, ρ, and <V> are sufficient to characterize tissue water diffusion.
Our ultimate goal is fitting entire experimental voxel b decays to produce absolute <k_io>, ρ, and <V> parametric maps [during iterations, the variables will interact]. However, we can appraise clinical implications with the commonly employed empirical ADC values from only the initial decay. We use the conventional b = 1 [Fig. 2 vertical dashed lines] to determine the asymptotic {ln[S(b)/S₀]/b} slope [-ADC], for each Fig. 2 curve. Figure 5a shows ADC = f<k_io>_ρ,V; Fig.5b, ADC = f(ρ)_kio,V; and Fig. 5c, ADC = f<V>_kio,ρ [i.e., f<V>_kio,ρ is the ADC <V>-dependence at constant <k_io> and ρ]. In Fig. 5a, the ADC value increases rather linearly with <k_io>. Since k_io reflects a fast metabolic rate [^cMR_NKA],¹ this suggests acute ADC changes are dominated by fast ^cMR_NKA changes. The Fig. 5 table lists well-known acute ADC changes. Though these are often attributed to <V> changes (e.g., “swelling”), Fig. 5c indicates ADC has very weak <V>‑sensitivity, for a large <V> change. Figure 5b shows ADC decreases rather linearly with ρ. However, large cellularity changes, > 1000 cells/μL, are required. Thus, ρ effects could dominate chronic ADC changes: cancer cell bed ρ can exceed 10⁶ cells/μL, with very small <V> [Fig. 4]. Effective cytotoxic therapy slowly increases the small malignant tumor ADC.¹ But, we expect <k_io> should give a faster response [ADC decrease; table highlight] for new targeted, cell-sparing therapies. Fig. 5 is also in general agreement with literature values reported for in vivo human brain cortex, trace-averaged D tensor; 0.8 (μm)²/ms.⁸ It is important to note, our DWI approach is complementary to those focusing on the tissue water anisotropic D tensor nature.

Acknowledgements

Brenden-Colson Center for Pancreatic Care.

References

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