## Abstract

Evidence mounts that the steady-state cellular water efflux (unidirectional) first-order rate constant (k_{io} [s^{−1}]) magnitude reflects the ongoing, cellular metabolic rate of the cytolemmal Na^{+}, K^{+}-ATPase (NKA), ^{c}MR_{NKA} (pmol [ATP consumed by NKA]/s/cell), perhaps biology's most vital enzyme. Optimal ^{1}H_{2}O MR k_{io} determinations require paramagnetic contrast agents (CAs) in model systems. However, results suggest that the homeostatic metabolic k_{io} biomarker magnitude in vivo is often too large to be reached with allowable or possible CA living tissue distributions. Thus, we seek a noninvasive (CA-free) method to determine k_{io} in vivo. Because membrane water permeability has long been considered important in tissue water diffusion, we turn to the well-known diffusion-weighted MRI (DWI) modality. To analyze the diffusion tensor magnitude, we use a parsimoniously primitive model featuring Monte Carlo simulations of water diffusion in virtual ensembles comprising water-filled and -immersed randomly sized/shaped contracted Voronoi cells. We find this requires two additional, cytometric properties: the mean cell volume (V [pL]) and the cell number density (ρ [cells/μL]), important biomarkers in their own right. We call this approach metabolic activity diffusion imaging (MADI). We simulate water molecule displacements and transverse MR signal decays covering the entirety of b-space from pure water (ρ = V = 0; k_{io} undefined; diffusion coefficient, D_{0}) to zero diffusion. The MADI model confirms that, in compartmented spaces with semipermeable boundaries, diffusion cannot be described as Gaussian: the nanoscopic D (D_{n}) is diffusion time-dependent, a manifestation of the “diffusion dispersion”. When the “well-mixed” (steady-state) condition is reached, diffusion becomes limited, mainly by the probabilities of (1) encountering (ρ, V), and (2) permeating (k_{io}) cytoplasmic membranes, and less so by D_{n} magnitudes. Importantly, for spaces with large area/volume (A/V; claustrophobia) ratios, this can happen in less than a millisecond. The model matches literature experimental data well, with implications for DWI interpretations.

Original language | English (US) |
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Article number | e4781 |

Journal | NMR in biomedicine |

Volume | 36 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2023 |

## Keywords

- DWI
- Monte Carlo permeability
- Voronoi cells
- membrane
- random walks
- stochastic
- water

## ASJC Scopus subject areas

- Molecular Medicine
- Radiology Nuclear Medicine and imaging
- Spectroscopy