Dual‐input tracer kinetic modeling of dynamic contrast‐enhanced MRI in thoracic malignancies

Abstract Pulmonary perfusion with dynamic contrast‐enhanced (DCE‐) MRI is typically assessed using a single‐input tracer kinetic model. Preliminary studies based on perfusion CT are indicating that dual‐input perfusion modeling of lung tumors may be clinically valuable as lung tumors have a dual blood supply from the pulmonary and aortic system. This study aimed to investigate the feasibility of fitting dual‐input tracer kinetic models to DCE‐MRI datasets of thoracic malignancies, including malignant pleural mesothelioma (MPM) and nonsmall cell lung cancer (NSCLC), by comparing them to single‐input (pulmonary or systemic arterial input) tracer kinetic models for the voxel‐level analysis within the tumor with respect to goodness‐of‐fit statistics. Fifteen patients (five MPM, ten NSCLC) underwent DCE‐MRI prior to radiotherapy. DCE‐MRI data were analyzed using five different single‐ or dual‐input tracer kinetic models: Tofts‐Kety (TK), extended TK (ETK), two compartment exchange (2CX), adiabatic approximation to the tissue homogeneity (AATH) and distributed parameter (DP) models. The pulmonary blood flow (BF), blood volume (BV), mean transit time (MTT), permeability‐surface area product (PS), fractional interstitial volume (v I), and volume transfer constant (K Trans) were calculated for both single‐ and dual‐input models. The pulmonary arterial flow fraction (γ), pulmonary arterial blood flow (BFPA) and systemic arterial blood flow (BFA) were additionally calculated for only dual‐input models. The competing models were ranked and their Akaike weights were calculated for each voxel according to corrected Akaike information criterion (cAIC). The optimal model was chosen based on the lowest cAIC value. In both types of tumors, all five dual‐input models yielded lower cAIC values than their corresponding single‐input models. The 2CX model was the best‐fitted model and most optimal in describing tracer kinetic behavior to assess microvascular properties in both MPM and NSCLC. The dual‐input 2CX‐model‐derived BFA was the most significant parameter in differentiating adenocarcinoma from squamous cell carcinoma histology for NSCLC patients.


1
(1) with 1 1 4 and , where and are the relative volumes of the plasma and interstitial spaces in the considered tissue region with volume , is the CA-specific permeability surface area product (in mL/min), (in mL) is the plasma volume, and is the unit step function that explains is valid only at 0.
The AATH model describes a closed-form solution of the Johnson and Wilson tissue homogeneity model in the time domain by considering adiabatic (slow) changes of CA concentration in the interstitial compartment relative to that in the plasma compartment 1 , whereby this model accounts for the concentration gradient of CA between the arterial and venous ends of the capillary, but CA concentration in the interstitial space is assumed to be well mixed. The TRF for the AATH model is given by: where 1 ⁄ is the first-pass extraction fraction from the plasma space to the interstitial space, and ⁄ (in min) is the capillary transit time.
The DP model also describes the concentration gradient of CA along the axial length of a capillary tube like the AATH model, but unlike the AATH model, the interstitial compartment is modeled as a series of infinitesimal compartments that exchange CA only with nearby locations in the capillary bed. Thus, the CA concentrations in the plasma and interstitial compartments both depend on the position of the capillary tube. The TRF for the DP model is given by: where denotes the modified Bessel function of the first kind. The integral term, including the modified Bessel function, cannot be solved into a fully analytic form. To simplify the formulation of the integral term in , an alternative derivation can be considered by evaluating a Taylor series expansion 2 . The , can be simplified as: The ETK model assumes negligible capillary transit time in comparison to the data sampling interval, resulting in a situation where CA plasma concentration is equal to the AIF. The TRF for the ETK model is given by where is the Dirac delta function that denotes the idealized impulse excitation of a unit mass source at t = 0.
The TK model provides a further simplification of the two-compartment situation, whereby it is assumed that ≪ and thus the contribution of CA in the plasma compartment to is ignored. The TRF for the TK model is given by The symbols and definitions for kinetic parameters used in the current study are summarized in supplementary document 2.

B. Modeling of the dual-input function
Because the dual arterial inputs join in the capillary bed, they can be effectively replaced by a single net input function with their mixed contributions, i.e., a weighted sum of the pulmonary and systemic AIFs: are the body transfer function (BTF) that models leakage into the whole-body interstitial space during the recirculation phase. Thus, and are modeled as a full-pass AIF that describes superposition of the bolus shape (first-pass) and its shape after modification by the BTF (recirculation). An AIF model that represents a sums-of-exponentials function can be adopted for modeling the dual-input arterial curves as follows 3 : , , , where was expressed with its convolution components for providing abbreviated reference, while with its analytic solution. Note that the analytic solution of is of the same form as that of , though the scaling constants and the onset times should be distinguished between them.

C. Analytic solutions of the five different models
For convenience in expressing analytic solutions of for each model, it was defined that , , ⨂ 1 ⁄ , and , , ⨂ 1 ⁄ . Once the dual AIF is modeled as a continuous-time parametric functional form, a subsequent analytic solution of can be derived by incorporating the scaling constants and the onset times of the dual AIF into the convolution integral with the TRF for each tracer kinetic model. By assuming that tracer transport within the capillary-tissue system can be modeled as a linear time-invariant (stationary) system, the analytic form of can be given by where log ln 2 1 is the maximized log likelihood with the sample size (i.e., the number of temporal data points) and ⁄ the normalized residual sum of squares 4 . The number of the estimated model parameters (including ) is . It is advocated to use the cAIC when the ratio ⁄ is small (<40). Based on the cAIC values, for each voxel, the optimal model was chosen by selecting the cAIC min . To assess the relative likelihood of a model, the cAIC differences (∆ ) were calculated between models as with being the cAIC value of candidate model . The model estimated to be the best has ∆ 0. For each model out of the set of alternative models, the AW, , was calculated from ∆ :