ΔR₂^* Gd-DTPA relaxivity in venous blood

Abstract

The accuracy of perfusion measurements using dynamic susceptibility contrast (DSC) MRI depends on estimating contrast agent concentration in an artery, i.e., the arterial input function (AIF). One of the difficulties associated with obtaining an AIF are partial volume effects (PVEs) when both blood and brain parenchyma occupy the same pixel. Previous studies have attempted to correct AIFs which suffer from PVEs using contrast concentration in venous blood. However, the relationship between relaxation and concentration (C) in venous blood has not been determined in vivo. In this note a previously employed fitting approach is used to determine venous relaxivity in vivo. In vivo relaxivity is compared to venous relaxivity measured in vitro in bulk blood. The results show that the fitting approach produces relaxivity calibration curves which give excellent agreement with arterial measurements.

Introduction

Relaxivity, the relationship between relaxation rate (R₂^*) and tracer concentration (C), is an important function in every DSC MRI analysis. It is normally assumed that the relaxivity of paramagnetic tracers such as Gadolinium-diethylenetriaminepentacetic acid, (Gd-DTPA) is linear; however this is only true in simple solutions. In tissues – both brain parenchyma and blood – the relationship is distinctly non-linear due to secondary magnetic field perturbations created by the tissue microenvironment (1–5). Studies have shown neglecting the non-linear nature of relaxivity in vivo substantially affects the accuracy of perfusion measurements such as cerebral blood volume and flow (6).

The arterial input function (AIF, i.e., the concentration of tracer in arteries) is generally used in DSC measurements and the relaxivity of Gd in arterial blood has been well characterized empirically (4,7). Measurements of venous concentration can be used to automate arterial pixel selection and correct for partial volume effects in the AIF (8–10). These methods have assumed that the relaxivity of arterial and venous blood are the same. However, this is not generally the case as the presence of paramagnetic deoxyhemoglobin in will contribute to venous relaxivity. A recent study by Blockley et al. empirically determined the relaxivity of venous blood in vitro at different field strengths using a blood phantom doped with a Gd based tracer (5). This study found that relaxivity was quadratic with the minimum at some positive value of Gd concentration, C. i.e., that relaxivity initially decreases with increasing C, as the presence of paramagnetic Gd decreases field differences between diamagnetic plasma and paramagnetic deoxygenated red blood cells. This would result in an increase in signal intensity at the beginning of the bolus but this behavior does not appear to be observable in vivo.

The aim of this note is therefore to determine the relaxivity of venous blood by a fitting approach previously used to determine brain parenchyma relaxivity (11). The approach determines the quadratic venous relaxivity function that gives the best agreement between the areas under arterial and venous concentration-time curves (CTCs).

Methods

Imaging Parameters

The study was approved by the Institutional Review Board of this institution. DSC data was acquired from 6 meningioma subjects at 1.5T (TR 1s; TE 47ms; FA 40°; matrix 128×128; FOV 228×228 mm²; 7×5mm slices) and 3T (TR 1s; TE 32ms; FA 30°; matrix 128×128; FOV 230×230 mm²; 10×5mm slices) using a gradient echo EPI sequence during injection of 0.1mmol/kg Gd-DTPA.

Signal Measurements

Arterial and venous pixels are detected semi-automatically using adaptations of techniques first suggested by Rempp et al. (12). First, signal was averaged over all images at each time point to obtain an average brain signal. Although it might be preferable to extract the brain first, in practice this average is dominated by brain pixels. The following parameters are then measured: fractional signal drop (FSD), full width at half drop (FWHD) and arrival time (AT, defined as the time at which signal is decreased by 10% of the maximum FSD). Candidate arterial pixels were defined as those with a FWHD at least 1s less than the average brain value and an AT at least 1 second before the average brain value. Similarly candidate venous pixels were defined as those 1s wider and 1s later than the average. Candidate pixels were visually inspected and the ten with the greatest FSD that showed no evidence of “spiky” boluses, indicative of phase cancellation with surrounding parenchymal pixels (4), were averaged to produce arterial and venous measurements for each patient.

Signal in an EPI sequences (neglecting T₁ effects) is given by

$$S=S_{\mathit{pre}}exp(\mathrm{\Lambda }(0)-\mathrm{\Lambda }(C))$$ [1]

where S_pre is the pre-bolus signal and Λ is a function of concentration. The relaxivity of arterial bulk blood has been measured empirically at multiple field strengths and has been found to follow a quadratic

$$\mathrm{\Lambda }(C)=({qC}^2+pC)T_E.$$ [2]

At 1.5T, q = 0.74 s⁻¹mM⁻² and p = 7.62 s⁻¹mM⁻¹ (4); at 3T, q = 2.61 s⁻¹mM⁻² and p = 0.5 s⁻¹mM⁻¹ (7). (Hematocrit was 40% in both cases.)

The in vitro relaxivity of venous bulk blood with a 30% oxygen extraction fraction (OEF) follows a parabola but with a shift 0(a) along the concentration axis

$$\mathrm{\Lambda }(C)=\left(q{(C-a)}^2+pC\right)T_E.$$ [3]

At 1.5T, q = 6 s⁻¹mM⁻², p = −11 s⁻¹mM⁻¹ and a = −0.046 mM; 3T, q = 18 s⁻¹mM⁻², p = −35 s⁻¹mM⁻¹ and a = 0.037 mM (hematocrit 42%) (5).

In this study we assumed the same form for relaxivity as found in bulk blood, Eq. [3], but found the parameters q, p and a that gave the best agreement between arterial and venous CTCs by the following fitting procedure similar to that used previously used to determine parenchymal relaxivity (11).

Fitting Method

All CTCs were fitted to the single compartment recirculation (SCR) model (13)

$$C(t)=g(t)+\kappa {\int }_0^{t}g(\tau )d\tau$$ [4]

where κ is a ratio of the area under the bolus to the recirculation height of the curve and is a constant less than one (usually about 0.05) and g is the gamma variate function

$$\text{g}(t:y_{max},t_{max},\alpha )=y_{max}{t}^{\prime \alpha }exp\left(\alpha (1-t^{\prime })\right)$$ [5]

where

$$t^{\prime }=\frac{(t-t_0)}{t_{max}-t_0},$$ [6]

t₀ is the start if the bolus, t_max is the time when the bolus is at maximum height, y_max is the maximum height and α is a decay parameter. The first term in Eq. [4] (the gamma variate) describes the tracer bolus per se so that the area under the curve is given by

$$A=\int \text{g}(t)dt=y_{max}{\left(\frac{t_{max}}{\alpha }\right)}^{\alpha +1}\mathrm{\Gamma }(\alpha +1)$$ [7]

where Γ is the gamma function. The integral term describes recirculating contrast. The SCR model is preferable to the alternative of fitting a gamma variate to the bolus portion of the CTC alone for a number of reasons (13).

The procedure is based on the basic postulates of DSC MRI that 1) the area under the bolus, A, must be equal for arterial and venous (and all other vascular) CTCs and 2) the height of the recirculation portion of the CTCs must also be equal. The values of q, p and a that best realize these conditions were found by simultaneously fitting CTCs from 5 patients at both 1.5 and 3T. Briefly, for each patient A and κ are estimated in the artery by modeling the CTC as Eq. [4], converting to signal using Eq. [1] and the bulk arterial blood relaxivity curves (Eq. [2]) and fitting to the measured signal with S₀, y_max, t₀, t_max, α and κ as free parameters. Similarly each vascular CTC is modeled by Eq. [4], converted to signal using Eqs. [1] and [3] and fitted to the measured signal. However, the venous fit is constrained to give equal values of A (Eq. [7]) (and hence y_max) and κ to those found in the artery but with the relaxivity parameters q, p and a _as fitting parameters in addition to S₀, t₀, t_max and α. Fits to all 5 patients were performed simultaneously with different gamma variate parameters for each patient but the same relaxivity parameters for all. Since we are comparing two pools of blood with the same hematocrit, it is not necessary to correct for this. Fitting to signal rather than estimates of concentration avoids errors associated with the non-linear relationship between noise and concentration (14–15). Including S₀ as a fitting parameter is somewhat unusual but should provide the best overall fit of the data.

Results

Figure 1 shows measured (points) and fitted (line) venous signal measurements for 5 patients at 1.5 (a) and 3T (b). Table 1 gives the coefficients for venous relaxivity (Eq. [3]) derived from in vivo fitting. It should be noted the a term is very small and can in practice be neglected. This finding is in contrast to that in in vitro bulk blood (5) measurements. Figure 2 shows relaxivity calibration curves for arterial (red) blood, in vitro bulk venous blood (5) (black; coefficients are given in the methods section) and in vivo venous blood (blue) at 1.5 (a) and 3T (b).

FIG. 1.

Measured (points) and fitted (solid lines) venous signal intensity time curves from five patients at 1.5T (a) and 3T (b). The fitted curves were calculated using the relaxivity parameters that gave the best agreement between the area under the bolus, and steady state amplitude of arterial and venous concentration time curves.

Table 1.

Relaxivity coefficients for Eq. [3] (curves shown in fig. 2). Arterial coefficients have been determined by references 4 and 7 for 1.5 and 3T measurements, respectively, and in vitro bulk blood coefficients from reference 5.

		q	p	a
Arterial	1.5T	0.74	7.62	0
	3T	2.61	0.5	0
In vivo fitting	1.5T	0.0018	19.0	0.0086
	3T	0.74	12.1	0.0095
In vitro bulk blood	1.5T	6	−11	−0.046
	3T	18	−35	0.037

FIG. 2.

Arterial (red) and venous (in vitro – black; in vivo – blue) relaxivity calibration curves at 1.5T (a) and 3T (b). Curve coefficients are given in table 1.

Figure 3 shows CTCs derived for arterial signals (red), and venous signals using in vitro bulk blood relaxivity (black) and in vivo venous relaxivity expressions (blue) at 1.5 (a) and 3T (b). (The discontinuities in the in vitro curves are due to the ambiguity in relaxivity estimation caused by the non-mononotic nature of the in vitro relaxivity curve. Values between zero concentration and the first non-ambiguous concentration have therefore been derived by interpolation.) With the in vivo relaxivity, the relative widths, heights and arrival times of the arterial and venous curves are consistent with what would be expected from delay and dispersion: the arterial bolus starts before and is narrower and higher than the venous, the recirculation portion of the curve is of similar height. The venous CTC derived by the in vitro calibration curves are in very poor agreement with what would be expected from the arterial curves. The recirculation portion of the curve is grossly overestimated at both fields; at 1.5T the bolus is both higher peak and broader than the arterial curve which is not physiologically feasible.

FIG. 3.

Arterial and venous CTCs at 1.5T (a) and 3T (b). The discontinuities in the in vitro venous curves are because the same value of ΔR₂^* corresponds to two different concentrations below about 2 mM (see Fig. 2). The curve was therefore obtained by interpolation between zero and the first unambiguous value of ΔR₂^*.

Discussion

Theoretical modeling of Gd relaxivity in blood is a difficult problem and, as far as we are aware, has not yet proved amenable to analysis. Relaxivity expressions have therefore been found empirically using phantoms and been shown to resemble a quadratic for both arterial and venous blood. In this study we therefore empirically determined the quadratic function that best describes relaxivity in in vivo venous blood at 1.5 and 3T. The excellent fits in Fig. 1 confirm that quadratic functions adequately describe venous relaxivity although they do not prove that this is the true functional form. The agreement between arterial and venous CTCs derived from the in vivo relaxivity expressions (Fig. 3) further confirm this.

The quadratic term in the 1.5T in vivo relaxivity expression is very small, unlike that in other measurements. Again, theoretical modeling is difficult and we have no good explanation for this finding. However, we do note that similar findings are seen in arterial measurements.

The in vitro relaxivity curves (Fig. 2) give negative ΔR₂^* over a critical range of concentrations that include those seen during recirculation (between 0.5 and 1 mM – Fig. 3). If the arterially derived concentrations are correct the in vitro curves would predict a distinct signal increase at the beginning of the bolus and a recirculation portion greater than S_pre. Neither of these are seen in any venous measurements (Fig. 1).

Relaxivity is largely determined by magnetic field inhomogeneities caused by susceptibility differences between diamagnetic plasma, paramagnetic deoxyhemoglobin confined to the red blood cells (RBCs), paramagnetic contrast agent within the plasma and the diamagnetic surroundings (tissue in vivo, air or water in vitro). In large phantoms, inhomogeneities will be dominated by differences between plasma and RBCs. As the concentration of contrast agent increases these inhomogeneities will initially decrease until the susceptibility of the plasma equals that of the RBCs, creating a minimum in the relaxivity curve shifted to the right along the concentration axis. In small vessels, however, susceptibility differences between the lumen and surrounding tissue will also contribute to inhomogeneity and hence relaxivity. The initial dose of contrast agent will increase these inhomogeneities thus potentially negating the effect of RBC/plasma differences. This could explain the discrepancy between in vitro and in vivo relaxivities. If this explanation is correct it suggests that the relaxivity parameters derived here will be somewhat dependent of vascular diameter and should therefore be used with caution in veins different in size from cerebral veins. (Note that this effect would have less effect on in vitro measurements of arterial blood since oxygenated RBCs are diamagnetic.)

DSC data must be acquired with sufficient temporal resolution to capture the passage of the bolus so relatively low spatial resolution acquisitions are employed. Blood vessels are therefore likely to be smaller than the dimensions of pixels so that vascular measurements are contaminated by partial volume effects (PVEs). PVEs can result in phase cancellation of signal between vessels and surrounding tissue (17–18). Such phase errors can introduce distortions in bolus shape and errors into the calibration. These errors also usually cause characteristically “spikey” and square signal drops that approach zero. For this reason we visually inspected the candidate arterial and venous curves and rejected those showing these effects. PVEs can also cause reductions in signal drops. However, our method is partially resistant to such errors for the following reason. The calibration is based both on a comparison of bolus area, A, and κ which is the ratio of post bolus concentration to A. Essentially κ is a ratio of concentrations since A is the integral of concentrations. If C is reduced by a factor α due to partial volume effects then the estimate of C is

$$\widehat{C}={\mathrm{\Lambda }}^{-1}\left(\mathrm{\Lambda }(\alpha C)\right)$$

where Λ⁻¹ is the inverse of Λ, i.e., the calibration function that we are seeking, and we have assumed T_E = 1 for simplicity. If Λ⁻¹ is incorrect then

$$\widehat{C}=\alpha C+\epsilon (C)$$

where ε is an error term, which is dependent on C because Λ is non-linear. Because of this

$$\frac{{\widehat{C}}_1}{{\widehat{C}}_2}\ne \frac{C_1}{C_2}.$$

Thus, if partial volume effects are substantial and if our calibration function is incorrect it will introduce large errors between measured and fitted curves in Figure 1.

There a number of limitations to this study: First, we took no account of T₁ saturation effects. However, with our sequence parameters saturation should be only 13% and we saw no evidence of any effects such as an initial or post bolus increases in signal intensity. Second, hematocrit and OEF will influence relaxivity (15) so that large variations in these parameters will reduce the accuracy of our calibration values. However, a previous study by Akbudak et al. demonstrates that provided plasma concentrations are compared hematocrits between 30% and 40% do not produce large differences in R₂^* within the range of concentration in which we are interested (0 – 5 mM) (7). Since hematocrit is equal in arteries and veins any effects will tend to cancel out. Thirdly, the in vitro experiments used ProHance as a gadolinium chelate, not Gd-DTPA. This may explain some of the discrepancies though there appears to be little evidence in the literature to suggest major relaxivity differences between most contrast agents. Lastly, the in vitro relaxivity curve was determined over a smaller range of concentrations than the in vivo so that the latter may be more reliable at high concentrations.