The effects of equilibrium transcytolemmal water exchange on the determination of contrast reagent concentration in vivo

We appreciate the opportunity to reply to Dr. Buckley's letter (1), and to reemphasize our thoughts on accurate methods for converting 1H2O relaxation rate constants to in vivo contrast reagent concentration ([CR]) values (2-4). We completely share his feeling for the “importance of the material” (1). Unfortunately, part of Buckley's communication seems to stem from a misunderstanding of what we reported. This is reflected in the fact that he simulates a skeletal muscle experiment (fig. 1 of Ref. 1) that we did not do because it would be almost impossible to analyze accurately. Also, the omission of detail on Dr. Buckley's simulation obscured its incorrectness. Finally, a rate constant in our third paper (2) is misidentified in his letter. Simulated spoiled GRE signal intensities are plotted as functions of thigh muscle interstitial CR concentration. The solid curves were calculated as described in the text. Left panel: Simulations analogous to Buckley's (1) Fig. 1 (TR = 0.100 s, flip angle = 45°). The curve labeled “T1L, T1S (τi = 0)” is the same as his labeled “fast exchange,” “T1L, T1S (τi = 1 s)” is analogous to his “full two-site exchange,” and “T1L, T1S (τi = ∞)” is analogous to his “slow exchange.” The dotted curve labeled “T1L (τi = 1 s)” was obtained as described in the text, and is the same as Buckley's labeled “Landis model.” The order of our curves is different than the author's. Right panel: The spoiled GRE-equivalent of the actual slice-ROI PURR thigh muscle acquisition (2) (TR = 1.0 s, flip angle = 6°). The labels are the same. An MRI CR is detected indirectly, by its effect on the 1H2O NMR signal; most often, on the latter's longitudinal relaxation rate constant, R1 (≡ T). Since, in most tissues, most water is intracellular, and since CRs enter at most the extracellular space (interstitium), the determination of [CR] from R1 is not simple. Only if the equilibrium transcytolemmal water exchange appears infinitely fast on the NMR time-scale (the fast-exchange-limit, FXL), is it true that [CR] is linearly dependent on R1. At most field strengths, detectable levels of CRs change the time-scale such that the inherent frequency of this exchange (though itself unaltered and still fast) is revealed to be insufficient for the FXL to be sustained (3, 5). This causes the recovery or establishment of 1H2O longitudinal magnetization to be bi-exponential (3, 4). There are—in theory—three observable quantities: a large relaxation time, T1L, a small relaxation time, T1S, and the fractional contribution, aS, of the T1S component [≡ (1 − aL)] (3, 4). However, at low [CR] values the T1S component is vanishingly small (aS → 0, the FXL, as [CR] → 0). It can easily become undetectable in the practical sense. Furthermore, since R1S (≡ T) > R1L (≡ T), the R1S component contributes differentially early in the relaxation, where it can be readily contaminated with systematic errors due to macromolecular 1H2C signals and/or 1H2O signals affected by flow (2, 3), and random errors due to pulse imperfections (fig. 4k of Ref. 3). We refer to the situation in which T1S is effectively undetectable as the fast-exchange-regime [FXR], and this will obtain for almost all in vivo studies (2, 3, 5). Thus, 1H2O relaxation is often best described by an effectively single exponential function, with T1 = T1L. However, in the FXR the [CR]-dependence of R1L is decidedly non-linear, and this distinguishes it from the FXL (2-5). For the acquisition of CR bolus-tracking (B-T) data, we have advocated the most prudent approach of trying to accurately measure R1L, and minimizing contamination from R1S or the systematic and random errors. This is quite feasible since the analytical expression for R1L (eq. 3 of Ref. 2; eq. 6 of Ref. 3) contains all the two-site-exchange (2SX) model parameters. Standing alone, it does embody the “full two-site exchange model”: it is not “an invalid assumption” (1). This is the approach we took for the B-T study of the rat thigh muscle group reported in Ref. 2—the experiment that Dr. Buckley purports to examine in his figure. However, he simulates data that would have been obtained with “a spoiled gradient echo acquisition (TR 0.1 s, flip angle 45 degrees)” (1). In our experiment (raw data in fig. 3 of Ref. 2), we used a slice-ROI PURR (2, 3) acquisition. This is quite different from a gradient recalled echo (GRE) pulse sequence. To edit the T1 values for T1L, we used the T1 distribution means measured (with 64 recovery points) during the CR bolus passage through the thigh. If we try to describe this acquisition with GRE-equivalent parameters, the TR value would be ∼ 1s (dashed curve in fig. 3d of Ref. 2), and the flip angle, α, would be 6° (3). We were explicit on this (p. 568 of Ref. 2). Thus, Buckley simulates an experiment that we did not do. Furthermore, the inverse of his laudable demonstration of signal intensity [CR]-dependence—[CR] determination from the signal intensity of his experiment (of course, the purpose of such studies)—is sufficiently fraught with errors to make it almost impossible. Unfortunately, from his letter one cannot know exactly what Buckley has done. He does not give the 2SX parameter values used, though he states: “The data are representative of rat thigh muscle and were calculated using the exchange parameters described by the authors (2)” (1). These parameters are: R1i and R1o0, the intra- and extracellular 1H2O rate constants, respectively, in the absence of exchange (and CR, for the latter); r1o, the interstitial relaxivity; τi, the mean intracellular water lifetime; and po, the extracellular water fraction. Thus, he presumably used our values (R1i = 0.69 s−1, R1o0 = 0.50 s−1, r1o = 3.77 s−1 (mM)−1, τi = 1.1 s, and po = 0.11 (p. 568 of Ref. 2)). Since the interstitial (“outside”) CR concentration, [CR0], never rises above 2.5 mM following almost any injection (figs. 5, 6 and table 1 of Ref. 2)), the value of aS is in most real situations “rendered minuscule by the exchange kinetics” (p. 568 of Ref. 2) (reduced from 0.11, here, to <0.03). Nonetheless, we further presume that Buckley used our eqs. 6–8 (p. 469 of Ref. 3) to calculate the [CR0]-dependences of R1L, R1S, and aS, respectively, out to 6 mM. The consequences of the author's ambiguity are significant. We can obtain Buckley's solid curves using the expression: S = S0 · sinα[1 − (aL · E1L + aS · E1S)]/[1 − (aL · E1L + aS · E1S) · cos α], where S and S0 are the signal intensities from spoiled GRE steady-state and fully-relaxed, equilibrium magnetization, respectively, and E1L,S ≡ exp(−TR · R1L,S). We reproduce his “fast exchange” (thick) curve with τi = 1 μs, his “full two-site exchange” (normal) curve with τi = 1.1 s, and his “slow exchange” (thin) curve with τi = 1 Ms. (Buckley's dotted curve arises from the parameter set given above and: S = S0 · sin α(1 − E1L)/(1 − E1L · cos α).) However, when relaxation is actually bi-exponential, the two contributions must be treated completely independently. They reach different steady states, which are disproportionately weighted depending on the TR and α magnitudes. The first equation above does not render the two signals sufficiently separate. The appropriate expression is: S = S0{[aL · sin α(1 − E1L)/(1 − E1L · cos α)] + [aS · sin α(1 − E1S)/(1 − E1S · cos α)]} (6). Exchange effects are accounted for in the equilibrium values; aL, R1L, aS, and R1S. Using the same parameter set, but this equation, we obtain the solid curves in the left panel of Fig. 1: “T1L, T1S (τi = 0)” (thick) with τi = 1 μs, “T1L, T1S (τi = 1 s)” (normal) with τi = 1.1 s, and “T1L, T1S (τi = ∞)” (thin) with τi = 1 Ms. The latter two are not the same as the analogous Buckley curves. In fact, the thin solid curve actually lies below his dotted curve, reproduced here but labeled “T1L (τi = 1 s).” Since the latter is Buckley's so-called “Landis model,” we find it to remain satisfyingly closer to that analogous to his “full two-site exchange” curve (normal) than either of those analogous to his “fast exchange” or “slow exchange” curves. Moreover, as stated above, these simulations do not correspond to the acquisition we actually conducted. The right panel of Fig. 1 shows a simulation of the GRE experiment (TR = 1 s, flip angle = 6°) approximating the slice-ROI PURR study we did do. The labels are the same, and it is quite clear that the so-called “Landis model” [T1L (τi = 1 s)] tracks the so-called “full two-site exchange” curve [T1L, T1S, (τi = 1 s)] very well, even out to 6 mM. Note that the expanded ordinate scale for this GRE-equivalent study illustrates the “dynamic range” penalty accruing if something approximating the PURR experiment is precluded by spatiotemporal resolution constraints (2). Since the dotted curve now lies closest to the normal solid curve in both panels, it is encouraging that there will be GRE sequences with TR, α combinations that yield both good signal strength and good accuracy. Though we had 64 recovery points in our slice-ROI PURR muscle study (fig. 3 of Ref. 2), we still chose to use the “most robust portion” of the T1 distribution for “best approximating (T1L)−1” (p. 568 of Ref. 2). This way, we avoided the significant potential errors attendant to trying to accurately extract the T1S component; surely a problem for almost all in vivo cases. Realistic inclusions of the systematic errors described above would change the Fig. 1 solid line curvatures considerably. Noise from the random errors could obliterate the differences between the dotted and normal solid curves. If it is possible to accurately discriminate all three observables (our definition of the slow-exchange-regime, SXR (3)), then one can use the additional, different [CR0]-dependences of R1S and aS (eqs. 7 and 8, respectively, of Ref. 3) to extract the 2SX model parameters—as we did originally for yeast cell suspensions (4). In fact, this would be preferred because it “provides for more balanced sensitivities to the five 2SX model parameters” (p. 476 of Ref. 3). And, it may be barely the case at the higher [CR0] values achievable by CR injection directly into the rat brain interstitium (2, 7). However, one cannot expect that this will usually be practical. Furthermore, during B-T studies, the additional relaxation points required are usually obtained only at the expense of the data spatial encoding (2). Thus, if a single (or small number of) relaxation point(s) is all that is affordable, it is most important that this (these) be timed for best determination of T1L, the [CR0]-dependence of which bears all the information pertinent to the equilibrium transcytolemmal water exchange kinetics. In the case of thigh muscle, we deliberately used for the B-T studies (2) the same method (slice-ROI PURR) for T1 (T1L) determination that we used in the prior careful steady-state “CR titrations” of the rat thigh (3). Thus, even though we could well calculate the observed [CR0]-dependence analytically with 2SX parameters in good agreement with the literature (fig. 5 of Ref. 3), we described its additional empirical calibration merit explicitly (pp. 569, 570 of Ref. 2). We have carried this consistency further to demonstrate that the inappropriate use of the FXL to analyze data truly in the FXR (like those from the thigh muscle) propagates significant systematic errors to the pharmacokinetic parameters ultimately desired (8). These results show unequivocally that the exchange effects are quite sizable. Finally, Dr. Buckley believes that “meaningful examples of the error introduced by the Landis approach can be seen in the authors' own data” (1), referring to the rate constant values for the CR wash-out from the rat thigh muscle and brain. However, he misidentifies our rate constant, L, which is 0.32 (min)−1 for the thigh muscle (p. 572 of Ref. 2). We stated that L is a measure of the “first-order rate constant kpo indicated in Fig. 1.” (p. 572 of Ref. 2). In Buckley's (Toft's) notation, kpo is kpe, the unidirectional rate constant for uptake from plasma into extracellular space. Figure 1 of our paper (2) shows that the unidirectional rate constant for efflux from extracellular space into plasma, kop in our notation, is given by (Vp/Vo)kpo if the two unidirectional permeability · surface area products are equal (not unreasonable). Since (Vp/Vo) (the plasma to interstitium volume ratio) is likely no more than 0.25 in the case of skeletal muscle, kop is no more than 0.08 (min)−1. This is very consistent with the idea that “kep” (our kop) is “rate-limiting,” i.e. “less than the rate constant of extraction to the kidneys,” which is “of the order of 0.1 (min)−1 (9)” (1). Buckley has incorrectly taken L [0.32 (min)−1] to be equal to kep. In conclusion, Dr. Buckley has joined us in identifying as an important issue the proper characterization of tissue 1H2O longitudinal relaxation in the presence of CR. Our position is that, in most cases, one can most accurately succeed in evaluating [CR] by trying to discriminate and measure the R1L component, the major contributor. We continue to try to develop improved pulse sequences for doing this. We thank Drs. Thomas Barbara, Gregory Hall, and Christopher Sotak for insightful discussions.