Introduction

There is growing evidence the pseudo-first order, homeostatic cellular water efflux rate constant [k_io] has a significant contribution from the membrane Na⁺,K⁺-ATPase metabolic rate [MR_NKA].¹ This vital enzyme activity has not been accessible in vivo, and is thus a very powerful new biomarker. The rate constant k_io can be precisely measured with tissue ¹H₂O MR when the longitudinal shutter-speed [к₁] is sufficiently large, which occurs only when the extracellular contrast agent (CA) concentration is very high¹ or the magnetic field strength (B₀) very low.²

Methods

Longitudinal tissue ¹H₂O relaxation data can be analyzed for steady-state trans-cytolemmal water exchange kinetics using the Bloch-McConnell-Woessner [BMW] two-site-exchange [2SX] equations.^3,4 In these, it is important to appreciate the intra- and extra-cellular ¹H₂O signal intrinsic relaxation rate constants [R_1i and R_1o, respectively] differ from the apparent values [R′_1i and R′_1o] determined in an experiment. The same is true for the intrinsic compartmental population fractions [p_i and p_o] vs. the apparent, experimental values [p′_i and p′_o]: p_i = 1 – p_o and p′_i = 1 – p′_o. The BMW equations relate the apparent experimental parameters to their intrinsic counterparts.^3,4 The equations include к₁, defined^1-4 as abs[R_1i – R_1o]_. We use this aspect to elucidate the entire experimental range – from ultra-low B₀ to high [CA_o] concentration. We stipulate a representative intrinsic parameter set {R_1i, R_1o, p_i, and k_io} and then use the BMW equations to “reverse engineer” expected experimental values.

Results

Figure 1 shows the stipulated R_1i (blue) and R_1o (red) values. As NMR properties, they depend on B₀ and [CA_o]. Figure 1(left) varies log B₀ (proportional to log ν_L, the Larmor frequency) with [CA_o] = 0. Figure 1(right) varies [CA_o] with B₀ = 1 T. The R₁ values on the left are from an in vivo tumor,² and those on the right are calculated with an experimental CA relaxivity, 3.8 mM^-1s^-1.⁴ If there was no exchange [k_io = 0], these would be the measured values. The к₁ variation is also shown, as well as the point where к₁ = 0, the vanished-shutter-speed [VSS] condition. The other intrinsic tissue properties, p_i = 0.8 and k_io = 1 s^-1, are also representative,^1,2 and B₀‑ and [CA_o]-invariant in this isothermal plot. This is shown in Figure 2, where k = k_io + k_oi = k_io[1 + (p_i/p_o)], p_i (blue), and p_o (red) exhibit horizontal lines: the abscissa is the same as Fig. 1. The exchange kinetics vary with only temperature and/or metabolism. The plot of k is reproduced in Figure 3, along with the к₁ trace. When k_io is finite, the BMW equations yield experimental R′₁ and p′ behaviors very different from their intrinsic counterparts. Figure 4 shows the B₀- and [CA_o]-dependences of R′_1,fast (above) and R′_1,slow (below), while Figure 5 shows p′_fast and p′_slow. The apparent population of the faster relaxing intrinsic component (p′_fast) must vanish in the VSS [the experimental relaxation goes from non-mono-exponential to mono-exponential].^5,6 Since this is R_1i on the left and R_1o on the right, the compartmental assignments of R′_1,fast and R′_1,slow must switch between R′_1i (blue) and R′_1o (red), and p′_fast and p′_slow between p′_i (blue) and p′_o (red), upon VSS crossing.⁷ The only BMW term changing sign passing through VSS is the к₁ argument (R_1i – R_io), and this has considerable consequence.

Discussion

Figure 5 has important implications. A thin horizontal dashed line at p′ = 0.1 represents the generous hope that a 10% minority component could be detected. Even if so, there is a considerable range – in fact spanning the entire current clinical enterprise - in which experimental relaxation is effectively mono‑exponential, and precise k_io measurement is difficult.¹ When clinical instruments with B₀ < ~ 0.1 T under construction are realized, one can expect to find non‑mono-exponential relaxation. However, down to ~ 0.01 T the experimental apparent minority component will correspond to the actual majority component, p′_i. At B₀ ~ 0.01 T, there is apparent population equality [APE; p′_fast = p′_slow],⁵ and thus between APE and VSS, there is an apparent population inversion [API] regime. An empirical bi‑exponential analysis of relaxation decay – as is often done – would yield a complete compartment miss-assignment. One must use the BMW equations to extract the true p_i and k_io parameters that will be of great biomedical import.

Acknowledgements

Grant Support: NIH: R44 CA180425, Brenden-Colsen Center for Pancreatic Care.

References

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