Synopsis

Dynamic-contrast-enhanced MRI (DCE-MRI) has been widely used to characterize microvasculature permeability. Recently, it was shown to reveal metabolic activity using the shutter-speed pharmacokinetic paradigm (SSP), in which steady-state intra/extracellular water exchange kinetics was incorporated into DCE-MRI data analysis. Interesting insights into DCE-MRI signals come from modeling the extravascular tissue MR signal. The questions addressed here are, “When can extravascular ¹H2O longitudinal magnetization recovery from inversion/saturation still be described by a single-exponential process, and when can the intra/extracellular water exchange kinetics be accurately determined?”

Purpose

Dynamic-contrast-enhanced MRI (DCE-MRI) is a widely used clinical imaging tool.¹ A quantitative DCE-MRI protocol is a pharmacokinetic study. A paramagnetic contrast agent (CA) is injected intravenously and transiently extravasates only to the extracellular tissue spaces, a process described by Kety-Schmitt (KS) pharmacokinetic law (Figure 1). Interesting aspects of the analysis of DCE-MRI signals come from modeling the extravascular tissue MR signal. Typically, a tracer pharmacokinetic paradigm (TP) has been used,² where longitudinal magnetization, M, recovery from inversion/saturation is assumed to be described by an empirical single exponential process with apparent relaxation rate, . However, this ignores an important feature of water compartmentalization, i.e., finite steady-state exchange of intra- and extracellular water molecules.³

In 1999, two-site-exchange (2SX) expressions for steady-state intra/extracellular water exchange kinetics (Figure 1) were incorporated into DCE-MRI data analysis, via the shutter-speed pharmacokinetic paradigm (SSP).³ SSP-based analysis not only characterize microvasculature, like TP, but also reveal cellular metabolic activity.^4,5 In SSP models, M is described with a bi-exponential function, which could admit two MR signals with different apparent relaxation rate constants. The questions addressed here are the conditions when M relaxation can still be described as a single-exponential process and when the intra/extracellular water kinetics can still be accurately determined under SSP.

Methods

To illustrate the effects of varying [CA_o] during DCE-MRI, simulations with the following 2SX parameters (Figure 1): f_i = 0.80, R_1o0 = 0.55 s^-1, and r_1o = 3.94 s^-1mM^-1. The values were varied from 0 to 3 s^-1, with 0.5 s^-1 steps, and the [CA_o] values were varied from 0 to 6 mM. The simulations were run at two different intrinsic intracellular ¹H2O relaxation rate constants: R_1i = 0.55 and 2.00 s^-1. In all simulations, the small microvascular plasma (and blood) signal was ignored.

The 2SX model describes intra- and extracellular M with an empirical bi-exponential function,

$$$\frac{M_{0}-M(t_{1})}{M_{0}}=(1-\cos\alpha)\left[f_{ sm}^{'}e^{-R_{1sm}^{'}t_{1}}+(1-f_{sm}^{'})e^{-R_{1lar}^{'}t_{1}} \right]$$$ (1)

where$$$M(t_{1}) $$$ is the magnetization at recovery time $$$t_{1}$$$, $$$M_{0}$$$, at equilibrium, α the effective flip angle of the inversion/saturation pulse, and $$$R_{1sm}^{'}$$$ and $$$R_{1lar}^{'}$$$ are the small and large apparent relaxation rate constants, respectively, and $$$f_{ sm}^{'}$$$ is the apparent fractional intensity of the signal with $$$R_{1sm}^{'}$$$. The analytical expressions for Eq. (1) quantities given in terms of physical quantities are described in Figure 2.⁶

Results

Discussions

Figure 3 illustrates important theoretical features of the 2SX model. The abscissa is a measure of the longitudinal shutter-speed ($$$\kappa_{1}\equiv \mid{R_{1i}-R_{1o}}\mid$$$) for this system.⁷ For simulations at $$$R_{1i}-R_{1o0}=0$$$ and 1.45 s^-1, $$$f_{lar}^{'}$$$ approaches 0 as $$$\kappa_{1}$$$ approaches zero. This has been traditionally called the fast‑exchange-limit [FXL]. However, the FXL term comes from NMR in chemistry, where reactions can be accelerated or slowed, i.e., $$$k_{io}$$$ can be increased or decreased, respectively. Figure 3 makes clear the $$$f_{lar}^{'}$$$ vanishing is independent of the $$$k_{io}$$$ value at finite $$$k_{io}$$$. Thus, the FXL label is misleading. It is more descriptive to refer to the left ordinate as the vanishing-shutter-speed-limit [VSSL]. This is important because the TP represents a special case of the SSP – in the limit of a short SS. It has been shown algebraically that as $$$\kappa_{1}$$$ vanishes, $$$R_{1sm}^{'}$$$ approaches the f-weighted R_1i, R_1o average $$$[\equiv R_{1}^{'}]$$$.⁸ Any DCE-MRI model within the TP is the special VSSL case of the analogous shutter-speed model.^7,9

In most practical situations, ($$$R_{1i}-R_{1o0}$$$) is small in tissue but > 0 and [CA_o]_max rarely exceeds 2 mM.^8,10 In these cases, $$$f_{lar}^{'}$$$ is very small, and its signal also likely suffers disproportionate transverse relaxation quenching $$$(R_{2lar}^{*}>R_{2sm}^{*})$$$.⁷ Thus, the component can reasonably be neglected. In this very common regime, the recovery is mono-exponential, but the relaxation rate constant is $$$R_{1sm}^{'}$$$ (Figure 2), not $$$R_{1}^{'}$$$ defined in TP model

$$$R_{1}^{'}$$$ = r_1o[CA_o] + $$$R_{10}^{'}$$$ (2)

This can be called the vanishing shutter-speed regime [VSSR]. Measurements in blood suggests the VSSR extends to [CA_o] past 20 mM; most likely due to transverse quenching.⁸ This is important because $$$k_{io}$$$ is only accessible in the VSSR but not the VSSL.

Acknowledgements

References

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