Fig 1.
(A) Schematic illustrations of the four nested transport models from simple (Patlak) to complex (Leaky Tofts–Kety): the contrast agent concentration Ct(t) is evaluated using the functions Cp(t), the CA concentration in the plasma compartment, which is assumed to be given by the arterial input function, and Ce(t), for the CA concentration in the EES space. The rate of forward and backward volume transfer and the fractional EES and plasma volumes are the quantities Ktrans, ve, vp, and λ. For each model, the involved parameters are listed in brackets. (B) Different enhancement patterns of CA signal: Type I—persistent curve—is a progressive increasing intensity signal; Type II—plateau curve—is characterized by an initial peak followed by a relatively constant enhancement; Type III—wash-out curve—refers to a sharp uptake followed by an enhancement decrease over time. (C) Examples of Type I (right plot) and Type III (left plot) enhancement curves obtained from breast and brain tumors, respectively.
Fig 2.
Illustrative example of profile likelihood for an identifiable parameter and structurally and practically non-identifiable parameters.
Fig 3.
Best fitting of CA evolution signal obtained with the LTK model (10).
The three types of CA time-enhancement curves (columns) for the three cases of study (AA), (RA), and (RR) (rows). The last row (RR) is CA time course data from the GBM DCE-MRI. Similar results for the breast cancer dataset are provided in Fig C in S3.1 Text.
Table 1.
Parameter values used for the synthetic data set in the (AA) case.
Table 2.
Parameter values used for the synthetic data set in the (RA) case of study.
Fig 4.
Ktrans practical identifiability for LTK model and Type I enhancement curve.
Top row: profile likelihood and confidence levels at 68%, 80%, and 95% for the parameter Ktrans in the (AA), (RA), and (RR) case for the Type I enhancement curve. Inset in the first subplot shows a zoom of the region around the best-fitted value (red marker). Bottom row: compensating profiles of the parameters ve, vp, and λ with respect to variation of Ktrans around its best-fitted value. Variation of ±50% around the optimal value of Ktrans are considered. Two colors are for two different y-axis: black curves refer the left y-axis and red curves to the right y-axis. Different line styles are used to distinguish curves referring to the same y-axis. For each curve, the name of the corresponding parameter is indicated above the line in the same color.
Fig 5.
Leakage (λ) practical identifiability for LTK model and Type I enhancement curve.
Top row: profile likelihood and confidence levels at 68%, 80%, and 95% for the parameter Kλ in the (AA), (RA), and (RR) case for the Type I enhancement curve. Inset in the first subplot shows a zoom of the region around the best-fitted value (red marker). Bottom row: compensating profiles of the parameters Ktrans, ve, and vp with respect to variation of λ around its best-fitted value. Variation of ±50% around the optimal value of λ are considered. Two colors are for two different y-axis: black curves refer the left y-axis and red curves to the right y-axis. Different line styles are used to distinguish curves referring to the same y-axis. For each curve, the name of the corresponding parameter is indicated above the line in the same color.
Fig 6.
Study on the noise effect on parameter practical identifiability in the LTK model.
Top row: Artificial VIF obtained from the analytical expression (Eq (3)) by adding a noisy signal with increased amplitudes. No noise VIF (first subplot) is the artificial VIF used in the (AA) case, while 5%, 10%, and 15% noise VIF (second to fourth subplots) are obtained by adding noise to the noise-free VIF. Bottom row: profile likelihood and confidence levels at 68%, 80%, and 95% for the parameter Ktrans for a Type I enhancement curve obtained by repeating the study on Ktrans with the increasing noisy VIF illustrated in the top row. Red markers indicate the best-fitted value .
Fig 7.
Study on the smoothing effect on parameter practical identifiability for the LTK model.
Top row: CA profile obtained from the GBM data in a Type I enhancement curve by smoothing the data using a moving average method with different values of the smoothing factor γ, i.e., γ = 0.05 (second column), γ = 0.1 (third column), and γ = 0.15 (fourth column). The original GBM data (first column) are the same used in the (RR) case. Bottom row: profile likelihood and confidence levels at 68%, 80%, and 95% for the parameter Ktrans obtained by repeating the study on Ktrans with the smoothed data illustrated in the top row.