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Purpose The aim of this study was to assess the uncertainty in estimation of MR tracer kinetic parameters and water exchange rates in T1-weighted dynamic contrast enhanced (DCE)-MRI. Methods Simulated DCE-MRI data were used to assess four kinetic models; general kinetic model with a vascular compartment (GKM2), GKM2 combined with the 3S2X model (SSM2), adiabatic approximation of the tissue homogeneity model (ATH), and ATH combined 3S2X model (ATHX). Results In GKM2 and SSM2, increase in transfer constant (Ktrans) led to underestimation of vascular volume fraction (vb), and increase in vb led to overestimation of Ktrans. Such coupling between Ktrans and vb was not observed in ATH and ATHX. The precision of estimated intracellular water lifetime (τi) was substantially improved in both SSM2 and ATHX when Ktrans > 0.3 min−1. Ktrans and vb from ATHX model had significantly smaller errors than those from ATH model (p<0.05). Conclusion The results of this study demonstrated the feasibility of measuring τi from DCE-MRI data albeit low precision. While the inclusion of the water exchange model improved the accuracy of Ktrans, vb, and the interstitial volume fraction estimation (ve), it lowered the precision of other kinetic model parameters within the conditions investigated in this study.

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In this paper we study the mathematical program with geometric constraints such that the image of a mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplund space, the optimality condition obtained can be expressed in terms of the limiting subdifferential, while in the general case it can be expressed in terms of the Clarke subdifferential. One of the technical difficulties in obtaining such a result in an infinite dimensional space is that no compactness result can be used to show the existence of local minimizers of a perturbed problem. In this paper we employ the celebrated Ekeland's variational principle to obtain the results instead. The enhanced Fritz John condition allows us to obtain the enhanced Karush-Kuhn-Tucker condition under the pseudo-normality and the quasi-normality conditions which are weaker than the classical normality conditions. We then prove that the quasi-normality is a sufficient condition for the existence of local error bounds of the constraint system. Finally we obtain a tighter upper estimate for the subdifferentials of the value function of the perturbed problem in terms of the enhanced multipliers.

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In this paper, we perform sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints (MPEC). We show that the value function is directionally differentiable in every direction under the MPEC relaxed constant rank regularity condition, the MPEC no nonzero abnormal multiplier constraint qualification, and the restricted inf-compactness condition. This result is new even in the setting of nonlinear programs in which case it means that under the relaxed constant rank regularity condition, the Mangasarian-Fromovitz constraint qualification, and the restricted inf-compactness condition, the value function for parametric nonlinear programs is directionally differentiable in every direction. Enhanced Mordukhovich (M-) and Clarke (C-) stationarity conditions are M-and C-stationarity conditions with certain enhanced properties and the sets of enhanced M-and C-multipliers are usually smaller than their associated sets of M-and C-multipliers. In this paper, we give upper estimates for the subdifferential of the value function in terms of the enhanced M-and C-multipliers, respectively. Such estimates give sharper results than their M-and C-counterparts.