Parameter estimation using weighted total least squares in the two‐compartment exchange model

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In dynamic contrast‐enhanced magnetic resonance imaging, high temporal resolution T1‐weighted images are acquired serially before, during and after the administration of a contrast agent (CA) 1. Quantitative analysis of the T1‐weighted images using tracer‐kinetic modeling produces estimates of physiological parameters such as compartment volumes, blood flow, and capillary permeability 2. Applications can be found in e.g., assessment of head and neck cancer 3, studies of blood‐brain barrier disruptions in a variety of pathologies 4, and in the assessment of liver function 5.
The parameters are typically estimated by fitting the model curves to the data by nonlinear least squares (NLLS) 6. This process can be quite time‐consuming, in particular when the parameters are computed on the pixel level. Further, the NLLS method may converge to a local minimum and can therefore be biased by the starting guess used by the algorithm (6).
Due to these limitations of the NLLS method, linear least squares (LLS) estimation methods are becoming increasingly popular in dynamic contrast‐enhanced magnetic resonance imaging. The LLS methods are attractive since they have a single global minimum that easily can be found by solving a linear system of equations followed by a nonlinear transform. For the popular Tofts and extended Tofts models 7, an LLS estimator has been known for over a decade 8. Similar developments for more complex models such as the two‐compartment exchange model (2CXM), the two‐compartment filtration model (2CFM) 9, and the compartmental tissue uptake model 10 are, on the other hand, more recent. In the positron emission tomography (PET) literature, the LLS 11, the generalized LLS 12, and the total least squares 13 methods have been used for a much longer time.
In the work of Flouri et al. 9, where the LLS estimators for the 2CXM and the 2CFM were developed, it was found that a better fit could be obtained by using weighted LLS. The weights were selected in a heuristic manner and it was noted that finding optimal weights required further studies.
Due to the limitations of the LLS method described above, the purpose of this work was to develop an estimator that accounts for the full noise covariance in the linearized 2CXM.

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