Robust Quantification of Contrast Agent (CA) Concentration with Magnetic Field Correlation (MFC) Imaging

Abstract

Contrast-enhanced perfusion studies of the brain by means magnetic resonance imaging (MRI) are used to estimate a number of important brain tissue parameters, including cerebral blood flow and volume. In order to calculate these parameters, the contrast agent (CA) concentration must first be estimated. This is usually accomplished by measurement of a nuclear magnetic resonance (NMR) relaxation rate with the assumption of a linear relationship between the rate and the CA concentration. However, such a linear relationship does not necessarily hold in biological tissues due to compartmentalization of the CA in either the intravascular or extracellular spaces. Here we propose an alternative MRI method of CA quantification based on measurement of the magnetic field correlation (MFC), which is theoretically predicted to have a robust quadratic dependence on the CA concentration even when the CA is compartmentalized. In this study, CA concentration estimation by means of MFC is shown to be more accurate than established methods based on relaxation rates in yeast cell suspensions.

Estimation of contrast agent (CA) concentration, C, is an essential first step in the measurement, of a number of different brain tissue parameters, such as cerebral blood flow, cerebral blood volume, and vascular transfer constant by contrast-enhanced perfusion MRI. To accomplish this, a linear relationship between C and the change in relaxation rate due to the presence of CA is usually assumed, i.e.,

$$R=R^{\mathit{\text{tissue}}}+rC,$$ [1]

where R is the relaxation rate (R₁, R₂, or $R_2^{*}$), R^tissue is the relaxation rate in the absence of CA and r is a constant expressing the relaxivity of CA (which in general is tissue-dependent). However, it is known that this linear relationship is only strictly valid for simple solutions and may not hold in tissues due to compartmentalization of the CA in either the intravascular or extracellular spaces (1–8). As a consequence, the application of Eq. [1] to the analysis of contrast-enhanced perfusion MRI data can lead to systematic quantification errors.

Another CA-sensitive parameter is the magnetic field correlation (MFC), which may also be estimated with MRI (9,10). The MFC is defined by

$$\mathrm{M}\mathrm{F}\mathrm{C}(|t_2-t_1|)={\mathrm{\gamma }}^2\langle \mathrm{\delta }B(t_1)\mathrm{\delta }B(t_2)\rangle ,$$ [2]

where δB(t) is the local magnetic field shift (relative to the uniform background field) experienced by a water proton at a time t and γ is the proton gyromagnetic ratio. That the MFC is a function of the time difference, |t₂ − t₁|, rather than t₁ and t₂ individually is a consequence of time translation invariance. The dependence of the MFC on the CA concentration is, to an excellent approximation, given by

$$\mathrm{M}\mathrm{F}\mathrm{C}(|t_2-t_1|)=a_1{(C+a_2)}^2+a_3$$ [3]

where a₁, a₂, and a₃ are tissue-specific parameters (that may also depend on the time difference |t₂ − t₁|; see Appendix for additional details). The principal assumptions required for the validity of Eq. [3] are that the local magnetic field shift is much smaller than the applied field strength and that the CA concentration can be regarded as constant during the interval between t₁ and t₂. Since both of these conditions are well satisfied for a typical MRI experiment, the accuracy of Eq. [3] is, in practice, likely to be higher than that of Eq. [1]. In particular, CA compartmentalization is expected to have a much smaller effect on the accuracy of Eq. [3] than it has on that of Eq. [1].

The parameter a₂ is often relatively small and may be neglected. One can then invert Eq. [3] to give

$$C\approx k\sqrt{\mathrm{M}\mathrm{F}\mathrm{C}-\mathrm{M}\mathrm{F}{\mathrm{C}}_0},$$ [4]

where $k\equiv 1/\sqrt{a_1}$ and MFC₀ is the MFC for C = 0. Eq. [4] shows explicitly how MFC measurements can be used to quantify a CA concentration up to an overall tissue-specific scaling factor.

The goal of this study is to test the accuracy of Eqs. [3] and [4] for yeast cell suspensions by measuring the MFC for a range of concentrations of gadopentetate dimeglumine (Gd-DTPA). The accuracy of Eq. [1] for R₁, R₂, and $R_2^{*}$ relaxation rates is also assessed in the same yeast suspensions as a comparison.

MATERIALS AND METHODS

Cell suspensions were prepared by mixing 63 g of yeast, Saccharomyces cerevisiae (Fleischmann’s Active Dry Baker’s Yeast) in 1 liter of distilled water and allowing the mixture to settle for 48 h at room temperature. After yeast activation, 500 ml of the supernatant was removed and the concentrated yeast suspension was mixed and aliquotted to six 60-ml plastic bottles. Gd-DTPA (Magnevist; Berlex Laboratories, Wayne, NJ, USA) was added to each nuclear magnetic resonance (NMR) bottle in varying amounts to yield concentrations of 0, 1, 2, 4, 7, and 10 mM. Six more 60-ml plastic bottles were filled with distilled water and the same concentrations of Gd-DTPA as the cell suspension bottles. A 10-ml sample of the cell suspension was centrifuged at 960 g for 3 min resulting in a cell fraction of 0.25. Gd-DTPA concentration in our phantom is equivalent to the total tissue concentration; however, MFC always shows a quadratic dependence regardless of investigating total tissue or plasma concentration. MFC values are expected to be close to zero, and show no dependence on Gd concentration, for simple solutions. Bottles were shaken vigorously before imaging to ensure uniform distribution of cells and Gd-DTPA through the suspension. Six bottles at a time were submerged in a corn syrup bath to minimize susceptibility effects and imaged as a group. Corn syrup was used because its extremely short T₂ eliminates its signal and any associated artifacts (9).

Imaging was performed on a Siemens 3T Trio MRI scanner (Siemens Medical Solutions, NJ, USA) with a multichannel head coil. Single horizontal slices were acquired through the center of the array of bottles with segmented (multishot) echo-planar imaging (EPI) sequences to decrease imaging time. Common image parameters were as follows: TR = 2000 ms; field of view (FOV) = 200 × 200 mm²; slice thickness = 1.7 mm, matrix size = 128 × 128; EPI factor = 13; number of excitations (NEX) = 1.

Raw signal intensities (S′) in each bottle and mean background noise (σ) were measured within regions of interest (ROIs) using ImageJ 1.40g (Wayne Rasband, National Institutes of Health, USA). Signal intensities were corrected for rectified noise using the formula

$$S\prime =\sqrt{S^2+{\mathrm{\sigma }}^2},$$ [5]

where S is the corrected signal intensity. MFC and relaxation time estimates were obtained by fitting the appropriate equations to the corrected signal intensities using nonlinear least squares fitting routines (LAB Fit Curve Fitting Software; Wilton Pereira da Silva).

MFC Measurement

MFC was measured with an asymmetric spin-echo sequence with Hahn echo time = 24 ms, and refocusing pulse time shifts t_s = 0, −1, −2, −3, −4, −5, −6, −7 and −8 ms (9). Negative values for t_s indicate that the interval between the initial 90° excitation pulse and 180° refocusing pulse is reduced from the usual spin-echo value of TE/2). ROI signal intensities were corrected for noise (Eq. [5]) and fitted to Eq. [6] (below).

It can be shown (9) that signal intensity is, to a good approximation, a Gaussian function of t_s

$$S(TE)=S_0(TE)\text{exp}(-2t_s^{2}MFC(TE/2)).$$ [6]

MFC may then be estimated by a nonlinear least squares fit of Eq. [6] to measured values of S at a particular TE value. Since all of our results are done for one echo time, all MFC measurements correspond to the same time. A detailed discussion of the formal conditions of validity and limits for Eq. [6] are given in Refs. 9 and 11. However, it should be noted that a small cell fraction is not a requirement for the validity of Eq. [7] (below), in contrast to some analytic forms that have been proposed for signal decay in the presence of magnetic field inhomogeneities (12,13). Since TE = 24 ms for all of our MFC measurements, our MFC values correspond to a time of 12 ms (i.e., TE/2). As discussed in Refs. 9 and 11, the time-dependence of the MFC is due to water diffusion. A similar Gaussian signal decay was also found in blood using an asymmetric spin-echo time sequence (14).

T₁ Measurement

T₁ measurements were performed using an inversion-recovery (IR) sequence with TE = 17 ms and 10 different inversion times, TI = 35, 55, 75, 95, 115, 200, 400, 600, 800, and 1000 ms. Since magnitude reconstruction was employed, negative ROI signal intensities appear positive. These were therefore corrected manually before correcting for noise. Corrected signal intensities were fitted to the following expression to estimate T₁.

$$S=S_0(1-2\text{exp}(-TI/T_1)+\text{exp}(-TI/T_1)).$$ [7]

T₂ and T₂* Measurement

T₂ measurements were obtained using a spin-echo sequence with six different values of TE (17, 27, 37, 47, 57, and 67 ms). Similarly, $T_2^{*}$ measurements were made using a gradient-echo (GRE) sequence with 12 different TE values (5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 60 ms). Noise-corrected signal intensities were then fitted to simple monoexponential decays.

$$S=S_0\text{exp}(-TE/T_D),$$ [8]

where T_D is T₂ or $T_2^{*}$ for spin and GRE sequences, respectively.

Bland-Altman Plots

Equation [4] was fitted to a plot of MFC vs. Gd concentration. Estimated values of Gd concentration, C_est, were then obtained for each bottle using these values and the measured value of MFC. Similarly, Eq. [1] was fitted to plots of R₁, R₂, and $R_2^{*}$ vs. C by linear least squares fitting to yield estimates of relaxivities r₁, r₂, and $r_2^{*}$ and hence C_est for each bottle.

Bland-Altman plots (i.e., plots of C_est − C vs. [C_est + C]/2) were then generated to compare the agreement between true values of C, and estimates obtained by the MFC method and each relaxivity method.

RESULTS

Typical measured signal intensities and fitted curves are given in Fig. 1. Figure 2 gives plots of MFC and relaxation rates plotted against Gd-DTPA concentration ([Gd]) for both cell suspensions and water phantoms. MFC values were fitted to a quadratic model (Eq. [4]). This provided an excellent fit (R² = 0.9991) for the cell suspension (Fig. 2a). As expected, MFC values of the water phantoms are close to zero (−0.19 to 0.06) consistent with the absence of magnetic field inhomogeneities and show no variation with [Gd]. Figure 2b–d gives plots of relaxation rate against [Gd] for both yeast and water phantoms. As expected, relaxation rate varies linearly with [Gd] for the water phantoms (the dashed line gives the linear least squares fit). However, the response of the yeast phantom is decidedly nonlinear (the solid line is the fit through the first two data points for illustration).

FIG. 1.

Cell suspension signal intensities with 4 mM Gd-DTPA vs. (a) the MFC refocusing pulse shift (|t_s| ≤ 8 ms), (b) TI in T₁ measurements (TI ≤ 1000 ms), (c) TE in T₂ measurements (TE ≤ 67 ms), and (d) TE in $T_2^{*}$ measurements (TE ≤ 60 ms). The lines are fits to Eq. [7] (a), Eq. [8] (b), and Eq. [9] (c,d). Standard error estimates for all data points are shown with error bars.

FIG. 2.

MFC (a) and relaxation rates R₁ (b), R₂ (c), and $R_2^{*}$ (d) plotted against Gd-DTPA concentration for cell suspensions and water phantoms. MFC in the cell suspensions is fitted to a quadratic polynomial. In the relaxation rate plots, solid and dashed lines represent linear extrapolations, based on the [Gd] = 0 and [Gd] = 1 mM data points for the cell suspension and water phantom, respectively. Standard error estimates for all data points are shown with error bars.

Figure 3 illustrates the linear relationship between Gd-DTPA concentration and the right-hand side of Eq. [4] (R² = 0.9956). Figure 4 shows Bland-Altman plots of MFC and relaxation rates. The 95% limit agreements had ranges of 0.964 for MFC, 7.41 for R₁, 2.35 for R₂, and 1.25 for $R_2^{*}$.

FIG. 3.

Plot of MFC difference as a function of Gd-DTPA concentration (Eq. [4]).

FIG. 4.

Bland-Altman plots for MFC (a) and relaxation rates R₁ (b), R₂ (c), and $R_2^{*}$ (d). The y-axis represents the difference and the x-axis the mean of estimated and true Gd concentrations, respectively. The solid line indicates the mean difference and the dashed lines show ±95% limit agreements (i.e., approximately twice the standard deviation of the difference).

DISCUSSION

Estimating CA concentration is very important, and many factors contribute to the nonlinear dependence of relaxivity on CA concentration, such as compartmentalization, blood inflow effects, flip angle, system calibration, combination of R₁ signal enhancement with R₂ and $R_2^{*}$ decay, acquisition parameters, injection protocols, and macromolecular content (15–17). This study confirms that MFC measurements depend quadratically on Gd-DTPA concentration in yeast cell suspensions to a high degree of accuracy. This dependence is significantly better than the generally assumed linear dependence of relaxation rates on concentration, supporting the potential of MFC to more accurately quantify CA concentrations.

Figure 2 exhibits substantial deviations from linearity in the cell suspension phantom for relaxation rates in bottles with [Gd] over 7 mM. Also shown in Fig. 2 is the strong quadratic relationship between MFC and Gd-DTPA at all concentrations up to 10 mM in a cell suspension. It should be noted that Fig. 3 in Ref. 10 shows a linear relationship between MFC and iron (mg Fe/100 g fresh weight) when measuring MFC in various brain regions. This relationship may be attributed to the variable spatial distribution of iron within the brain region (i.e., the density of iron-rich cells, not necessarily a change in iron concentration inside the cells). This corresponds qualitatively with the theoretically predicted behavior of the MFC (i.e., linear dependence on density and quadratic dependence on susceptibility difference) for a random sphere model (9). In our experiment, the spatial distribution of CA was fixed, hence producing a quadratic relationship between MFC and [Gd].

Bland-Altman plots were used to compare each method of measuring [Gd]. These plots are preferable to comparing correlation coefficients, as a high correlation does not necessarily imply strong agreement (18). Figure 4 shows that MFC and $R_2^{*}$ give the best agreement between estimated and true [Gd], with R₁ and R₂ at least an order of magnitude worse. $R_2^{*}$ measurements are somewhat easier to make than MFC measurements but the images are highly susceptible to imaging artifacts due to macroscopic field inhomogeneities and low signal-to-noise ratios with increasing [Gd], both of which may be avoided by using MFC imaging. In particular, MFC imaging the effects of macroscopic gradients can be quantified and subtracted without using additional sequences (10), unlike $R_2^{*}$, for which the process becomes more complicated.

Yeast cells were used in the cell suspension phantom both for convenience and because their size and intrinsic properties mimic brain tissue and blood to some degree. Yeast cells are oblong-shaped, ranging from 7 to 10 µm in diameter, similar to microglial and red blood cells, and have R₁ (0.67 s⁻¹) and R₂ (12.3 s⁻¹) values that lie within the range of certain brain tissues and deoxyhemoglobin concentration (19–23)

The range of Gd concentrations used in this study is somewhat larger than would typically be found in tissues at equilibrium (an average of 0.1 mM with a standard dose) but might be seen during the first pass of a standard- or multiple-dose bolus. Moreover, errors in estimates are of similar size throughout the entire range of concentrations used here (Fig. 4).

This study used Gd-DTPA, a low-molecular weight (<1000 Da) paramagnetic CA that is the most widely used CA for human studies. However, similar results are expected with other paramagnetic CAs and also those based on iron oxide particles. In this study, the distribution of Gd-DTPA was modeled after the extravascular extracellular space, not the intravascular space. Other factors that contribute to the complexity of estimating [Gd] are compartmentalization of CA, geometry of the compartments, and the water diffusion rates associated with these compartments (24).

Finally, the effect that the errors in concentration estimates found here have on the accuracy of measured perfusion parameters is beyond the scope of this work and is the subject of ongoing investigation.

APPENDIX

Here we sketch the derivation of Eq. [3]. If the CA is distributed according to a fixed spatial pattern, then the local magnetic field shift for an individual water proton depends linearly on the CA concentration so that

$$\mathrm{\delta }B(t)=\mathrm{\delta }B\prime (t)+C·F(t),$$ [A1]

where δB′(t) is the field shift in the absence of the CA and F(t) is a function that depends on the water proton’s diffusion path, and where we have assumed a time-independent concentration. The validity of Eq. [A1] also requires that δB′(t) < B₀ and δB(t) < B₀, with B₀ being the applied field, but these two conditions are generally well satisfied for clinical scanners. From Eqs. [2] and [A1], one finds

$$\mathrm{M}\mathrm{F}\mathrm{C}(|t_2-t_1|)={\mathrm{\gamma }}^2\langle \mathrm{\delta }B\prime (t_1)\mathrm{\delta }B\prime (t_2)\rangle +{\mathrm{\gamma }}^{2}C\langle \mathrm{\delta }B\prime (t_1)F(t_2)+F(t_1)\mathrm{\delta }B\prime (t_2)\rangle +{\mathrm{\gamma }}^2{C}^2\langle F(t_1)F(t_2)\rangle .$$ [A2]

The identifications

$$a_1\equiv {\mathrm{\gamma }}^2\langle F(t_1)F(t_2)\rangle ,a_2\equiv \frac{1}{2a_1}{\mathrm{\gamma }}^2\langle \mathrm{\delta }B\prime (t_1)F(t_2)+F(t_1)\mathrm{\delta }B\prime (t_2)\rangle ,a_3\equiv {\mathrm{\gamma }}^2\langle \mathrm{\delta }B\prime (t_1)\mathrm{\delta }B\prime (t_2)\rangle -a_1{a}_2^2,$$ [A3]

then lead directly to Eq. [3].