Metabolic activity diffusion imaging (MADI): I. Metabolic, cytometric modeling and simulations

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 H₂ 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, D₀ ) 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.

Keywords: DWI; Monte Carlo permeability; Voronoi cells; membrane; random walks; stochastic; water.