2.1 Ethics statement

The study of publicly available data and retrospective analysis of City of Hope patient data were approved with a waiver of consent by the City of Hope Institutional Review Board, Protocol 15286.

2.2 Theory of nested DCE-MRI transport models

Compartmental models of contrast transport for DCE-MRI are constructed assuming that the total tissue CA concentration C_t(t) can be modeled as the sum of the CA concentration in the plasma space (PS) (C_p(t)) and in the extracellular extravascular space (EES) (C_e(t)), i.e. (1) where v_e and v_p are the fractional EES and plasma volumes, respectively. The explicit description of the CA concentration in each of these compartments depends on the assumptions made to estimate both C_e(t) and C_p(t) and, thus, on the chosen model, as illustrated in Fig 1A.

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Fig 1. Models and data.

(A) Schematic illustrations of the four nested transport models from simple (Patlak) to complex (Leaky Tofts–Kety): the contrast agent concentration C_t(t) is evaluated using the functions C_p(t), the CA concentration in the plasma compartment, which is assumed to be given by the arterial input function, and C_e(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 K^trans, v_e, v_p, 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.

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The CA concentration for the plasma compartment, C_p(t), is assumed to be not affected by local transport and it is given by the empirical vascular input function (VIF): (2) The dynamics of the VIF are characterized by a sharp uptake, followed by a peak value, and then a washout dynamic (Fig 1B). Several techniques have been proposed to measure the VIF [10, 24]. Here we consider two different estimation methods. First, we consider a population-based analytical expression of the VIF, commonly referred to as the Parker VIF [25], and second we estimate the VIF directly from the MRI signal [26]. The Parker VIF uses a mixture of 2 Gaussians plus an exponential modulated by a sigmoid function: (3) Here A_n, T_n, and σ_n are the scaling constants, centers, and widths of the n^th Gaussian, α and β are the amplitude and decay constant of the exponential, and l and τ are the width and center of the sigmoid, respectively. For the second method, VIF estimation is performed by drawing a region of interest (ROI) on a major vessel in the tissue and directly quantifying the contrast agent concentration, assuming the CA bolus arrives simultaneously to the entire tissue of interest. Population-based VIFs are widely used in DCE-MRI due to their simplicity and the fact that they do not require additional measurements, though they may ignore inter-subject variability. In contrast, the accuracy of individual-based VIFs derived from estimations of the CA concentration in large vessels depends on MRI characteristics, which may contribute to identifiability issues, as will become apparent through our analysis.

For the evolution of the CA concentration in the EES compartment, and, thus, to derive the expression for C_t(t), we analyze a class of nested compartmental models consisting of the Patlak model (PM), Tofts–Kety (TK) and its extended version (eTK), and the Leaky Tofts–Kety (LTK) model. The Patlak model, firstly introduced in [27], represents the simplest tissue uptake model and it assumes that CA diffuses from the PS to the EES at a rate governed by the forward transfer constant K^trans. It neglects the reflux from the EES back into the plasma space due to the assumption on low permeability and short measuring time. Thus, the concentration into the EES compartment varies according to (4) and, assuming that the initial concentration in the EES is zero (C_e(0) = 0), from Eq (1), the tissue CA concentration is given by (5) To overcome the low permeability hypothesis of the Patlak model the Tofts–Kety model was introduced in [28, 29]. The TK model accounts for bidirectional transfer of the contrast agent from the plasma to the EES compartment at rates governed by the forward transfer constant K^trans and the reverse constant , respectively. However, it ignores the intravascular (plasma) compartment contribution (i.e., v_p ≈ 0). The dynamics of the CA concentration in the EES compartment are described by (6) The transfer constant K^trans now has a different physiologic interpretation, depending on the balance between capillary permeability and blood flow: in high-permeability situations, K^trans is equal to the blood plasma flow per unit volume of tissue; in low permeability scenarios, K^trans is equal to the permeability surface area product between blood plasma and the EES, per unit volume of tissue [29]. Under the assumption that v_p ≈ 0 and setting the initial concentration in the EES to zero (C_e(0) = 0), we can derive from Eq (1) the expression of the tissue CA concentration as (7) The hypothesis of tissue being weakly vascularized (v_p ≈ 0) may be invalid in some situations, especially in the case of highly vascular tumors. Thus, a modification of the TK model was proposed in [30]. Known as the extended Tofts–Kety model, it includes the vascular contribution v_pC_p(t) to the overall CA tissue concentration, maintaining the same dynamics shown in Eq (6) for the EES concentration C_e(t): (8) The TK and eTK models have experienced some problems to obtain reliable estimation for the involved parameters when the data depict a persistent uptake curve (Type I) [31]. The PM model, with its assumption on unidirectional flux from the plasma to the EES compartment, is able to overcome this issue, but assuming a negligible reflux from the EES back into the plasma space is not realistic, especially in the high permeability regime [32]. The Leaky Tofts–Kety (LTK) model was introduced to address this issue [31]. The LTK model, also known as Leaky Tracer Kinetic model (LTKM), adds an additional compartment, the leakage compartment, to the already described plasma and permeable compartments. The permeable space and the leakage space are considered subcompartments of the EES region. The former assumes a bidirectional exchange between plasma and EES, while the latter considers a unidirectional flow from which the contrast, at a concentration C_L(t), does not flow back into the vasculature. Thus, for the LTK model, the overall CA tissue concentration is given by The contrast agent in the permeable compartment C_e(t) evolves following Eq (6), while the rate of contrast change in the leakage space can be read as (9) Here, λ is the volume transfer constant from the plasma compartment to the leakage compartment. Assuming that the initial concentration in the EES and leakage compartments are zero (C_L(0) = C_e(0) = 0), then the CA tissue evolution is given by (10)

In summary, the main differences between these models are the directional flows for the CA exchange between the plasma space, the extravascular extracellular space, and eventually the leakage space. The Patlak model is a unidirectional model which assumes CA transfer from EES to PS is negligible due to low permeability and short measurement time [27]. The Tofts–Kety model assumes that the CA diffuses from, and returns to the PS, and the vascular (plasma) compartment contribution is negligible [28]. This assumption of tissue being weakly vascularized is overcome with the extended Tofts model, which includes the vascular contribution to the tissue concentration [30]. Finally, the Leaky Tofts–Kety model extends the eTK, adding an additional leakage compartment with unidirectional flow for disease applications such as gliomas, where slow accumulation of CA from neighboring voxels is common [31]. In all models where the vascular component is considered, it is assumed that the concentration of the contrast agent within the vasculature is not affected by local transport, (i.e. it is an empirical forcing function), and that the difference between the empirical vascular input function and the local tissue concentration is the primary driver of transport between tissue and vasculature.

2.3 Data

Two DCE-MRI scans were used in this study. A brain scan was obtained from a patient with pathology-confirmed diagnosis of glioblastoma who underwent MRI at City of Hope National Medical Center. The T1-weighted brain DCE-MRI scan was acquired as follows: TR = 5.1ms, TE = 2.1ms, variable flip angle, with field of view 240mm x 240mm, matrix size 128x128 and image size 256 x 256 with 12 slices with slice thickness 6mm, and a temporal resolution of 6.03 seconds with 32 dynamic phases and 3 baseline images prior to contrast administration. A breast DCE-MRI scan was obtained from the Quantitative Imaging Network (QIN) BREAST-02 study, UPN-01. The image acquisition details for the breast MRI can be found in the documentation for the BREAST-02 study [33, 34]. In both cases, a variable flip angle (VFA) scan was acquired for direct contrast agent quantification [35]. The VFA scan was used to calculate the baseline T1 relaxation, T₁₀, of the tissue, which is used for calculating the local contrast agent concentration [36].

2.4 Identifiability analysis

Identifiability analysis is a fundamental mathematical tool used to assess a model’s capability to describe data with unique determined parameters. It focuses on whether it is possible to identify a unique vector of parameter values for a given model structure, or whether multiple parameter values will fit the data equally well. It is important to distinguish between structural identifiability, which concerns how the model structure itself affects the possible unique identification of the involved parameters, and practical identifiability, which is based on the analysis of the model’s ability to estimate parameters from the data with adequate precision.

Structural non-identifiability implies practical non-identifiability: in fact, if a model is not structurally identifiable, then the quality of the data collected does not matter; it will not be possible to uniquely estimate parameters in practice. The converse, however, is not true. Structural identifiability implies practical identifiability only when there is availability of infinitely resolved data, with zero measurement noise. However, even if the model structure theoretically allows parameters to be estimated, one still needs to have the appropriate data to achieve practically identifiable parameters.

2.4.2 Theory of practical identifiability.

As already briefly explained, a parameter that is structurally identifiable may still be practically non-identifiable. This issue arises frequently if the quantity and/or quality of the experimental data is insufficient. Assessing the practical identifiability of a model (or a parameter) is fundamental in obtaining reliable parameter estimations and, thus, for ensuring good model prediction capabilities. While the notion of practical identifiability is not uniquely defined in the literature, we consider a definition based on the concept of profile likelihoods and confidence intervals [48]. Using this framework, we evaluate the agreement between the experimental data and the observable predicted by our set of nested models choosing as Maximum Likelihood Estimator (MLE) the objective (or cost) function defined as: Here, N is the number of time points t_i available from the data, are the CA data values at time t_i, is the CA value at time t_i predicted by the model with the estimation θ of the parameter vector, and σ is the standard deviation of the noise of the data. In particular, this standard deviation is estimated as the inverse of the Signal-To-Noise (SNR) ratio of the data [21]. The state variable x(t) represents the CA concentration C_t(t) and u(t) is the vascular input function. The parameter vector used to evaluate the practical identifiability of the model is given by To determine the confidence interval of the estimated set of parameter , we consider the profile likelihood (PL) method. To build the profile likelihood plot, the idea is to choose one parameter θ_i, vary its value around the maximum likelihood estimate , and re-optimize the remaining parameters. Thus, the profile likelihood is given by Then, the confidence intervals (CI) of estimated parameters are defined as where Δ_α represents the α-th quantile of the chi squared (χ²) distribution with k = 1 degrees of freedom for a point-wise confidence interval [37, 49]. Thus, given the optimal value of one parameter, defining the confidence interval to a confidence level α implies that the true value of the parameter is located within this interval with probability α. When this confidence interval is finite, then a parameter θ_i is practically identifiable. Otherwise, if the confidence region is infinitely extended, although the numerical algorithm finds a unique minimum , then the parameter is practically non-identifiable. Precisely, we analyze the profiles of the versus each parameter θ_i. The choice of a least squares function for the MLE is equivalent to working with an additive Gaussian noise measurement model with constant variance. We point out the reader that the assumption of additive Gaussian noise may not be the more appropriate under certain conditions, especially for data close to zero. We highlight that other choices of measurement error model may be considered in order to relax this assumption of additive Gaussian noise (see [50] for a recent review on the topic). In any case, the method of the profile likelihood analysis here implemented holds when different measurement models are used.

A structurally non-identifiable parameter is characterized by a flat profile likelihood, whereas the profile likelihood of a practically non-identifiable parameter may have a minimum, but does not have a well-defined confidence level α for increasing and/or decreasing values of θ_i. Instead, for an identifiable parameter the profile likelihood exceeds such threshold in both directions of θ_i. Fig 2 shows an illustrative example of likelihood profiles. Practical identifiability as determined by profile likelihoods methods are standard methods for determining the unique parametrization of a real-world model-data pairing. We refer the interested reader to [37] for a detailed primer on profile likelihoods and practical identifiability. Profile likelihood-based confidence intervals are often used in biological applications to ensure that there is sufficient data and data quality to assume the validity of the model [14, 51–54]. With respect to other methods for determining confidence intervals, this particular method allows for asymmetric intervals that are invariant under re-parametrizations of the model. The analysis performed here refers to univariate likelihood profiles, where we are considering only one target parameter. For the reader convenience, it is worthy to mention that other options are available when pairs (or more) target parameters are considered. In particular, in this cases bivariate or higher-order profiles can be generated and studied [53].

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Fig 2. Illustrative example of profile likelihood for an identifiable parameter and structurally and practically non-identifiable parameters.

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In addition to the profile likelihood, we also analyze the compensating profiles of the model parameters. These are obtained by perturbing one parameter θ_i around its maximum likelihood estimate and plotting the variation of the re-optimized parameters versus the perturbed θ_i value. This analysis demonstrates how a non-identifiable parameter (whether structural or practical), may be compensated for by variations of other parameters. The small dimensionality of our systems allows us to use the described method for the analysis proposed in Section 3.2. We analyze the practical identifiability for the family of nested models described in Section 2.2 using a self-implemented code in Matlab based on the Particle Swarm algorithm for global optimization [55, 56]. The codes used for the analysis proposed in these notes are available in the corresponding GitHub repository: https://github.com/mconte93/Practical_Identifiability.git.

3.1 Structural identifiability of the LTK model

Following the formalism of the differential algebra approach described in Section 2.4.1, model (10) can be rewritten as (11) where y is the observable concentration of the contrast agent in the tissue C_t(t), u(t) is the external input given by the concentration of contrast agent in the plasma compartment (i.e., C_p(t) in (2)), x₁ and x₂ are the state variables for the CA concentration into the EES and leakage compartment, respectively. Differentiating the last equation in system (11) and plugging inside the first two equations of (11), we obtain the differential equation for y(t) of the form (12) where v is defined as and the coefficients a_i are given by (13) We observe that a₂, a₃, a₄ < 0, while a₁ > 0. Solving system (13) for the four parameters of the LTK model provides the following expressions: (14) For v_e to be well-defined we need , which always holds as the four parameters of the LTK model are positive. From this analysis, we can conclude that LTK model is structurally identifiable. The structural identifiability of the nested PM, TK, and eTK models has also been analyzed and details are provided in S1 Text. Moreover, for the reader’s convenience, in S2 Text we provide an example of a structurally non-identifiable model obtained by modifying the LTK model structure and changing the connections among the compartments.

3.2 Practical identifiability of the LTK model

Practical identifiability is performed using the method described in Section 2.4.2 and results are compared with the analytical results obtained in Section 3.1 regarding parameter structural identifiability. Precisely, we study the effect of varying the noise characteristics of the DCE-MRI data used for estimating the VIF and CA concentration on the accuracy of the identification of the model parameters. Each analysis is performed on the three characteristic enhancement patterns observed in CA evolution [10] (Fig 3). As shown in Fig 1B, CA time-enhancement curves are classified into three types: Type I—persistent curve—a progressive signal intensity increase (Fig 1C, right plot); 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 (Fig 1C, left plot). We consider two different tissues to study the role of the tissue characteristics on model identifiability. In both tissues, we replicate the analysis for the three different CA time-enhancement curves. We report results on the Glioblastoma multiforme (GBM) brain tumor data in the main text. Results for the breast cancer data are presented in S3.1 Text. For practical identifiability analysis we distinguish between the following three cases of study (referring to the rows of Fig 3).

1. (AA). The first case is referred to as Artificial-Artificial. Here, we consider an artificial VIF given analytically by Eq (3) and three artificial data sets. These are generated by running forward simulations of the LTK model, given by Eq (10), with the parameter values listed in Table 1, and replicate the three CA time-enhancement curves. In particular, these artificial data sets represents noise-free experimental data. Thus, in this case, there is no noise in the VIF or in the data which could affect the parameter identifiability. For the VIF we set A₁ = 48.54 mM × s, A₂ = 18.64 mM × s, σ₁ = 3.378 s, σ₂ = 7.92 s, T₁ = 10.2276 s, T₂ = 21.9 s, α = 1.05 mM, β = 0.0028 s⁻¹, l = 0.6346 s⁻¹, and τ = 28.98 s.
2. (RA). The second case is referred to as Real-Artificial. Here, we consider a real estimation of the VIF directly from DCE-MRI data by drawing a ROI around a major vessel, and three artificial data sets. These are obtained by running forward simulations of the LTK model, given by Eq (10), with the parameter values listed in Table 2 and the real VIF. This real VIF includes noise, as compared to the analytic expression for the artificial VIF as in the (AA) case.
3. (RR). The last case is referred to as Real-Real. Here, we consider both a real estimation of the VIF and a real evolution of the contrast agent in the tissue. We analyze data from a GBM brain tumor and a breast cancer MRI dataset.

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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.

https://doi.org/10.1371/journal.pcbi.1012106.g003

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Table 1. Parameter values used for the synthetic data set in the (AA) case.

https://doi.org/10.1371/journal.pcbi.1012106.t001

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Table 2. Parameter values used for the synthetic data set in the (RA) case of study.

https://doi.org/10.1371/journal.pcbi.1012106.t002

The three CA time-enhancement curves used in (AA), (RA), and (RR) cases are shown in Fig 3 (continuous black lines) together with the corresponding model fits obtained with the LTK model (10) (dashed red lines). The particle swarm (PS) method is used to estimate parameter values for fitting the three curve types and in the three cases of study (AA), (RA), and (RR). Values of fitted parameters are collected in Table A in S3 Text.

Recalling the methodology described in Section 2.4.2, we analyze profile likelihood and confidence levels for each of the parameters involved in the LTK model, in the three cases of study (AA), (RA), and (RR), and for the three types of CA time-enhancement curves. Here, we report the results for the Type I enhancement curve, as the LTK model was introduced with the aim of solving the issues of PM, TK, and eTK models concerning parameter estimation for persistent uptake CA profile. Results concerning Type II and Type III time-enhancement curves are collected in Figs A and B in S3 Text. Moreover, we show the results for K^trans and λ parameters. K^trans is the only parameter appearing in all nested models analyzed in this study, thereby allowing for a direct comparison of the results obtainable with the different models. λ is a novel parameter from the LTK model, thus it is important to analyze its identifiability.

Fig 4 shows the results concerning practical identifiability of the parameter K^trans for the LTK model and Type I time-enhancement curve. The columns of Fig 4 refer to the three cases of study (AA), (RA), and (RR). From Section 3.1, we know that K^trans is a structurally identifiable parameter, i.e., it is possible to uniquely determine its value from the given model structure. When we consider a completely artificial data set, obtained with the parameter values collected in Table 1 and the population-based analytical expression of the VIF given in (3), the complete absence of noise in the data and in the VIF allows us to accurately recover K^trans. In the (AA) case, the profile likelihood shows a well-defined parabola, with a unique minimum in the optimal value . Zooming around this minimum, we observe a finite confidence region around for the 95% confidence level. Analyzing the (RA) case (second column), we notice that represents a local minimum for the profile likelihood and it is not possible to identify a lower bound for the confidence regions for both 95%, 80%, and 68%, i.e., it is not possible to define a finite confidence interval. Thus, K^trans is practically non-identifiable in this case. Moreover, from the compensating profiles in the region where the profile likelihood is almost flat (left side with respect to ), we notice v_e and λ compensating for varying K^trans. In contrast, the optimal value estimated for v_p does not significantly changes with respect to K^trans variation. This is in line with the results in (13), where we can notice how v_p does not have any relation with K^trans. In the (RR) case, K^trans is not practically identifiable, evident from the flat likelihood profile, which makes it not possible to identify a lower or upper bound of a confidence region with respect to the optimal value .

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Fig 4. K^trans 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 K^trans 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 v_e, v_p, and λ with respect to variation of K^trans around its best-fitted value. Variation of ±50% around the optimal value of K^trans 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.

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Similar observations can be made from the results of Fig 5, where practical identifiability of the parameter λ for the LTK model and Type I time-enhancement curve is analyzed. As for Fig 4, columns of Fig 5 refer to the three cases (AA), (RA), and (RR). From the structural identifiability analysis of Section 3.1, we know that λ is a structurally identifiable parameter, thus, it is possible to uniquely identify its value from the model structure. For the completely artificial data set used in (AA), we are able to accurately identify λ. The profile likelihood in the first subplot of Fig 5 shows a well-defined parabola with a unique minimum in the optimal value , around which a finite confidence region for the 95% confidence level can be determined (as shown in the zoomed inset). Analyzing the RA and RR cases (second and third columns), we notice that the profile likelihood is characterized by flat regions for which it is not possible to identify a lower or upper bound for 95%, 80%, and 68% confidence levels. Thus, λ is not practically identifiable in these more realistic cases. Looking at the corresponding compensating profiles, we observe that v_p estimation is not influenced by λ variation, while v_e and K^trans show variability in response to λ. This result is reasonable in relation to the results in (14): the expression for v_p is independent from the other parameters, whereas K^trans, v_e, and λ expressions depends on common coefficients (i.e., a₁, a₂, and a₄).

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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 K^trans, v_e, and v_p 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.

https://doi.org/10.1371/journal.pcbi.1012106.g005

Observation 1. For reader convenience, we point out that it is reasonable to obtain different optimal estimations for K^trans in the different analyzed cases (AA, RA, RR), as moving from AA to RA and, then, RR case, we are changing the VIF and the DCE data on which the inverse problem is performed.

To specifically analyze the influence of the noise in both the data and the individual-based estimation of the VIF function, derived directly from the DCE-MRI data, we perform two different tests, whose results are shown in Figs 6 and 7. We first analyze the impact of increasing the noise in the VIF function, mimicking the process that leads from the AA to the RA case. In fact, the individual-based estimations of the VIF are characterized by a predefined level of noise that is not present in the analytical VIF estimations. To this aim, we define three different amplitudes for the noise-only signal, corresponding to the 5%, 10%, and 15%, respectively, of the noise-free VIF at every element. These amplitudes allow to control the amount of noise. Then, we define the noise-only signal in the form a standard normal distribution. Finally, we add the noise-only signal adjusted with the corresponding amplitude to your original noise-free VIF. This creates the noisy VIF used in the study. More details can be found in the available code. Fig 6 shows the results of this analysis for the parameter K^trans and Type I time-enhancement curve. Analogous results can be obtained for the other parameters and enhancement profiles. The first column, the no noise case, replicates the same results shown in the first column of Fig 4, i.e., the practical identifiability of K^trans with a confidence level of 95%. Increasing the noise in the VIF, up to 5%, we observe that K^trans is still practically identifiable, but with a lower confidence level (80%). Considering higher levels of noise of the VIF (10% or 15%), the practical identifiability of the parameter is no longer guaranteed, as the confidence region bounds become no longer finite for the considered confidence levels. This supports the conclusion that increasing noise in the VIF decreases the confidence level of parameter identifiability, justifying the differences in the results between (AA) and (RA) cases in Figs 4 and 5.

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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 K^trans for a Type I enhancement curve obtained by repeating the study on K^trans with the increasing noisy VIF illustrated in the top row. Red markers indicate the best-fitted value .

https://doi.org/10.1371/journal.pcbi.1012106.g006

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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 K^trans obtained by repeating the study on K^trans with the smoothed data illustrated in the top row.

https://doi.org/10.1371/journal.pcbi.1012106.g007

To analyze the influence of the noise in CA data on the parameter identifiability and, thus, mimicking the process that leads from the RR to the RA case, we perform a second test based on data smoothing. Starting from the CA time evolution data from the GBM dataset for a Type I time-enhancement curve, we use a moving average method with different values for the smoothing factor γ. Precisely, the method is based on the moving average technique that slides a window along the data, computes the mean of the points inside of each window, and replaces each data point with the average of the neighboring data points defined within the span. The window size is fixed and heuristically determined by the algorithm depending on the input data, while γ ∈ [0, 1] adjusts the level of smoothing by scaling the window size. Values of γ near 0 produce smaller moving window sizes, resulting in less smoothing, while values near 1 produce larger moving window sizes, resulting in more smoothing. Fig 7 summarizes the results of this analysis. The first column, referring to the original GBM data, replicates the same results shown in the last column of Fig 4, i.e., K^trans is not practically identifiable, with an almost flat profile likelihood. However, moving from left to right in Fig 7, the higher the value of the smoothing factor γ, i.e., the smoother the data, the less flat the corresponding profile likelihood. In particular, the last column of Fig 7, referring to γ = 0.15, shows the practical identifiability of K^trans can be recovered for the confidence level of 68%. We would not expect to obtain a higher confidence level for K^trans. These results justify the conclusion that high levels of noise in the data decrease the confidence level of parameter identifiability, in line with the previous shown results for RA and RR case in Figs 4 and 5.