Xin Li^{1}, Silvia Mangia^{2}, Jing-Huei Lee^{3}, Ruiliang Bai^{4}, and Charles S. Springer^{1}

^{1}Advanced Imaging Research Center, Oregon Health & Science University, Portland, OR, United States, ^{2}Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN, United States, ^{3}Biomedical Engineering, University of Cincinnati, Cincinnati, OH, United States, ^{4}Interdisciplinary Institute of Neuroscience and Technology, Zhejiang University, Hangzhou, China

### Synopsis

**The homeostatic cellular water efflux rate
constant, k**_{io}, has a significant contribution from cell membrane
sodium pump activity previously unmeasurable. With high extracellular contrast agent concentration or ultra-low
magnetic field, k_{io} can be precisely determined by two-site-exchange
analysis of *in vivo* ^{1}H_{2}O
longitudinal relaxation data. With the
low field case, there is an inversion of the *apparent* tissue compartmental contributions from the *true*
values. The NMR shutter-speed
organizing principle informs an analysis spanning the entire range of conditions.

### 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 ^{1}H_{2}O MR when the longitudinal
shutter-speed [к_{1}] is sufficiently large, which occurs only when the
extracellular contrast agent (CA) concentration is very high^{1} or the magnetic
field strength (B_{0}) very low.^{2} ### Methods

Longitudinal
tissue ^{1}H_{2}O 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 ^{1}H_{2}O 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 к_{1}, defined^{1-4}
as abs[R_{1i}
– R_{1o}]_{. }We use this aspect to elucidate the entire
experimental range – from ultra-low **B**_{0}
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 **B**_{0} and [CA_{o}]. Figure 1(left) varies log **B**_{0} (proportional to log
ν_{L}, the Larmor frequency) with [CA_{o}] = 0. Figure 1(right) varies [CA_{o}] with **B**_{0} = 1 T. The R_{1} 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^{-1}s^{-1}.^{4}
If there was no exchange [k_{io} = 0], these would be the
measured values. The к_{1}
variation is also shown, as well as the point where к_{1} = 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 **B**_{0}‑ 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 к_{1} 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 **B**_{0}- 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 к_{1}
*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 **B**_{0}
< ~ 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 **B**_{0} ~ 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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