Kinetic Modeling of Contrast-Enhanced MRI: An Automated Technique for Assessing Inflammation in the Rheumatoid Arthritis Wrist

Abstract

In recent years, development of rheumatoid arthritis (RA) drug therapy has been more directly targeted to counteract specific mechanisms of inflammation, and it is now believed that early aggressive treatment with disease modifying drugs is important to inhibit future structural joint damage. The development of these new treatments has increased the need for methodologies to assess disease activity in RA and monitor the effectiveness of drug therapy. Unlike X-ray, which shows only structural bone damage, magnetic resonance imaging (MRI) can depict soft tissue damage and synovitis, the primary pathology of RA. Recent studies have also indicated that MRI is sensitive to pathophysiologic changes that may predate radiographic erosions and may predict future joint damage. In this study, we have developed a computer automated analysis technique for MR wrist images that provides an objective measure of RA synovitis. This method applies a two-compartment pharmacokinetic model to every voxel of a dynamic contrast-enhanced MRI (DCE-MRI) dataset and outputs resulting parametric images. The aim of this technique is to not only objectively quantify the severity of rheumatoid synovitis, but to also locally determine where areas of serious disease activity are situated through kinetic modeling of blood-tissue exchange. Preliminary results show good correlation to early enhancement rate, which has previously been shown to be a useful clinical marker of RA activity. However, the use of tracer kinetic modeling methods potentially provides more specific information regarding underlying RA physiology. This approach could provide a useful new tool in RA patient management and could substantially improve RA therapeutic studies by calculating objective biomarkers of the disease state.

Appendix

Appendix I. Blood-Tissue Exchange Model Description and Early Enhancement Rate

In this simplified version of our special case of the Patlak method, EER is calculated as the early enhancement slope of the dynamic measurement in a simplified blood-tissue exchange model (see Fig. A1). The dynamic measurement, s(t), is calculated as the sum of the interstitial space (tissue), q ₃, and the scaled vascular region, f _BV· q ₁, and is also represented as the change in signal intensity from pre- to post-Gd-DTPA-BMA enhancement. Specifically:

Figure A1.

Simplified blood-tissue exchange model. Similar to Figs. 1 and 2, in this simplified model the state variable q ₁ (t) represents Gd concentration (mol l⁻¹) in the vasculature. The rate constant k ₀₁ (s⁻¹) describes the fractional flux of Gd leaving of the vasculature. A further simplification is that there is a bolus Gd injection instead of a 30-s infusion. The state variable q ₃(t) represents Gd concentration (mol l⁻¹) in the interstitial space. The fractional rate constant k ₃₁ (s⁻¹) describes the flux of Gd from vasculature into interstitial space. The amount of blood present in the tissue is represented by f _BV (ml blood 100 cc⁻¹ tissue), or fractional blood volume (fraction of q ₁). The arbitrary unit, m, represents the measured value of Gd concentration in both compartments (f _BV · q ₁  + q ₃  ( ΔSI). It should be apparent that this model is simpler than, but similar to, the model used for our analyses.

$$ s(t) = f_{{\hbox{BV}}} \cdot q_1 (t) + q_3 (t) = m(t) - m(t_0 ) $$

(7)

where m(t) is the post-Gd signal intensity at time t, and m(t ₀) is the pre-Gd signal intensity.

The simplified blood-tissue exchange model can be represented as a set of ordinary differential equations, whose solutions can be approximated using the first order Taylor series:

$$ \begin{aligned}{} & \frac{{{\hbox{d}}q_1 (t)}} {{{\hbox{d}}t}} = - k_{01} \cdot q_1 (t) \\ & \frac{{{\hbox{d}}q_3 (t)}} {{{\hbox{d}}t}} = k_{31} \cdot \left( {f_{{\hbox{BV}}} \cdot q_1 (t)} \right) = K^{{\hbox{PS}}} \cdot q_1 (t) \\ & q_1 (t) = q_1 (t_0 )\exp ( - k_{01} t) \\ & q_1 (t) \approx q_1 (t_0 )(1 - k_{01} t) \\ & q_3 (t) = K^{{\hbox{PS}}} \cdot \int_{t_0 }^t {q_1 (\tau )\,{\hbox{d}}\tau } = q_1 (t_0 )\frac{{K^{{\hbox{PS}}} }} {{k_{01} }}(1 - {\hbox{exp}}( - k_{01} t)) \\ & q_3 (t) \approx q_1 (t_0 )\frac{{K^{{\hbox{PS}}} }} {{k_{01} }}(1 - (1 - k_{01} t)) = q_1 (t_0 )K^{{\hbox{PS}}} t \\ \end{aligned} $$

(8)

The dynamic measurement is thus approximated by the first order Taylor series according to the following equation:

$$ s(t) \approx q_1 (t_0 )(f_{{\hbox{BV}}} (1 - k_{01} t) + K^{{\hbox{PS}}} t) $$

(9)

where q ₁(t ₀) and k ₀₁ are constant for every voxel (defined by the input function). This approximation is valid for small values of t. The initial rate of Gd enhancement can be estimated as the spatially normalized constant slope of Eq. 9:

$$ {\hbox{EER}} = \frac{{s(t)}} {{\bar m(t_0 ) \cdot t}} \approx \frac{{q_1 (t_0 )}} {{\bar m(t_0 )}}\left( {K^{{\hbox{PS}}} + \left( {\frac{1} {t} - k_{01} } \right)f_{{\hbox{BV}}} } \right) $$

(10)

where $\bar m(t_0 )$ is the spatial average of the baseline image and t is approximately 55 s. It is important to note that the spatially normalizing factor, $\bar m(t_0 )$, is constant for all voxels in our case, much like the approach in previously published ROI analyses.30,3 Therefore, the only spatially varying parameters are K ^PS and f _BV. Moreover, Eq. 10 should correlate strongly with a linear combination of the two parameters we have estimated with the Patlak method: K ^PS and f _BV. That is,

$$ {\hbox{EER}}\tilde \propto K^{{\hbox{PS}}} + A \cdot f_{{\hbox{BV}}} ,\quad \quad A = \left( {\frac{1} {t} - k_{01} } \right) $$

(11)

for small values of t, where A is a proportionality constant and does not vary spatially. We think that this general conclusion based on the simplified blood-tissue exchange model may also be applicable to the more complex model used in our analyses.

Appendix II. Automated Selection of SFE Masking Threshold

To select the optimum SFE masking threshold we analyze the number of voxels included in the SFE mask for all unit SFE thresholds, from 1 to 100. The SFE threshold that causes the first slope decrease (first point of inflection) in the number of voxels is selected as the optimum SFE threshold. This point occurs where the number of voxels in the mask is most sensitive to the chosen threshold value. Figure A2 shows the sigmoid relationship between the natural logarithm of the SFE threshold and the number of voxels in the mask.

Figure A2.

Optimal SFE masking threshold. The vertical axis shows the number of voxels included in the SFE mask. The horizontal axis shows the natural logarithm of the SFE threshold. Searching from low to high SFE thresholds, the software determines the first SFE threshold value where the slope of the curve decreases (first point of inflection). The SFE threshold value at the first point of inflection on this plot is considered the optimal SFE masking threshold, as it marks the point of maximum sensitivity of the number of voxels to the choice of the SFE threshold.