Images for high grade gliomas were extracted from different collections in The Cancer Imaging Archive (Chang et al., 2011), (Clark et al., 2013); The Cancer Genome Atlas Low Grade Glioma (TCGA-LGG) data collection (Pedano et al., 2020), the Repository of Molecular Brain Neoplasia Data (REMBRANDT) (Scarpace et al., 2019) for astrocytomas and oligodendrogliomas of grades 2 and 3, and The Cancer Genome Atlas Glioblastoma multiforme [TCGA-GBM] collection (Scarpace et al., 2016) for glioblastoma multiforme. Also, data coming from the RSNA-ASNR-MICCAI Brain Tumor Segmentation (BraTS) Challenge 2021 (Baid et al., 2021) (Menze et al., 2015) (Bakas et al., 2017). For other brain neoplasias, such as meningiomas and acoustic schwannomas, local image datasets were used. Among these collections, only contrast enhanced T1-weighted images, with tumor lesions clearly identified as such and separated from anatomical structures, were selected for analysis.

Image digital levels were clustered using the Dynamic Quantum Clustering algorithm (DQC) (Weinstein and Horn, 2009) (Horn and Gottlieb, 2001) which assumes that data are described by a set of points, each one defined with some uncertainty, σ and the distribution for all points in data space is given by a Parzen estimator, φ, which satisfies the time independent Schrödinger equation for its ground state, − 1 2 σ 2 ∇ 2 φ + V X ⃗ φ = E φ = 0 (1) which leads to the evaluation of an energy potential V X ⃗ = 1 2 σ 2 ∇ 2 φ φ (2) The number of potential minima (Horn and Gottlieb, 2001) was previously used (Martín-Landrove et al., 2020) to determine the number of classes in a K-means algorithm alone. In the present work, the full dynamic quantum clustering algorithm is used and by Ehrenfest theorem, digital levels evolve in time toward the potential minima according to the following equation of motion (Lafata et al., 2018), σ 2 d 2 X k ⃗ d t 2 = − ∇ V X k ⃗ − γ d X k ⃗ d t (3) which is a second order Langevin equation with a dissipative term γ. The clustering process is performed with a suitable selection of parameters such as σ ², which is an equivalent to a “mass”, and determines the number of classes among the digital levels (Horn and Gottlieb, 2001), dissipation, γ and time interval. These parameters are estimated according to the magnitude of the digital levels in the image and assess the evolution of all the data to the potential minima. An example of clustering of digital levels is shown in Figure 1. For image segmentation the dynamics is performed as a two step process (Sánchez and Martín-Landrove, 2021):

Dynamic A: First application of the dynamics upon the original image using the potential calculated from the original image histogram. At the end of this step, a Parzen estimator is evaluated using the clustered image histogram, allowing for the calculation of a “trap” potential, which defines the number of classes if a further classification algorithm, such as K-means, is to be used. It has been proved that the dynamic quantum clustering algorithm provides the same set of centroids as the K-means algorithm (Sánchez and Martín-Landrove, 2021).

Dynamic B: Second application of the dynamics upon the original image using the “trap” potential.

In Figure 2 it is shown how the selection of the appropriate σ ² determines the number of classes and therefore the image segmentation. It is important to note that the “trap” potential compared to the original Schrödinger potential, exhibit a well defined minima, which allows for an unsupervised definition of the number of clusters and an improvement in the assessment of the tumor interface.

Results are shown in Figure 5 which exhibits a high dispersion of the data. Average values are indicated by large circles and correspond from left to right to meningioma, acoustic schwannoma, Grade II and Grade III glioma TCGA-LGG and REMBRANDT databases), glioblastoma multiforme and high grade glioma (BraTS 2021 database) and glioblastoma multiforme (TCGA-GBM database). These average values are summarized in Table 1. There is a clear difference between meningiomas and acoustic schwannomas, top of Table 1, with average parameters d _F = 1.97 ± 0.08 and α _loc = 0.68 ± 0.11 and gliomas, bottom of Table 1, with d _F = 2.11 ± 0.12 and α _loc = 0.80 ± 0.14, a result that is consistent with previous research using segmentation schemes based on k-means classification alone and smaller data sets (Torres Hoyos and Martín-Landrove, 2012) (Martín-Landrove et al., 2020) (Martín-Landrove et al., 2016). This fact is also shown in Figure 6 in the comparison of the corresponding histograms for d _F and α _loc . The values of the local roughness exponent α _loc and the fractal dimension d _F give important information about what proliferative-invasive process is taking place in the dynamics of tumor growth, if the sum of these parameters is close to the Euclidean dimension, d _E , as reflected by Equation 8, the tumor growth dynamics corresponds to a ballistic growth model, characterized by the ansatz of Family-Vicsek (Family and Vicsek, 1991) (Barabasi and Stanley, 1995). Inspection of Table 1 reveals that this is indeed the case for Grade II and III gliomas and glioblastoma multiforme, d _F + α _loc = 2.92 ± 0.22. In the case of meningiomas and acoustic schwannomas, the tumor growth dynamics corresponds to a different growth model since d _F + α _loc = 2.65 ± 0.16. Also, from Table 1, lacunarity values, λ, exhibits a variation that it is in correspondence with fractal dimension, d _F . Another important feature comes from the results shown in Figure 7 which depicts the dependence of W _sat with respect to the average size of the tumor lesion, ⟨R⟩. Trend of the data conforms to a power law according to Equation 9 and the results are summarized in Table 2. There is a clear difference between tumor groups, for gliomas, α = 0.948 ± 0.038, and for meningiomas and acoustic schwannomas, α = 0.730 ± 0.087, which is what has to be expected if the growth dynamics governing the tumor interface corresponds to a more invasive process as it happens to be the case for malignant tumors.

The relationship between S _C and S _R are shown in Figure 8. Data points are dispersed below the diagonal since S _C takes into account both inner and outer surfaces while S _R only takes into account the outer surface (Pérez-Beteta et al., 2018). Data points along the diagonal correspond to tumors that lack the presence of either contrast free or necrotic volumes, a condition that occurs more frequently for meningiomas and acoustic schwannomas than for gliomas as seen in Figure 8. Large circles in Figure 8 correspond to average values that are summarized in Table 3. The average values of S _R for meningiomas and acoustic schwannomas, 0.64 ± 0.16 and gliomas, 0.57 ± 0.23, do not differ significantly. On the other hand, average values of S _C are 0.61 ± 0.17 for meningiomas and acoustic schwannomas, and 0.31 ± 0.17 for gliomas, have a difference that clearly discriminates between these groups. In order to enhance the difference among tumor groups, the ratio S _C /S _R is used, as seen from Table 3. Figure 9 shows frequency distributions for meningiomas and acoustic schwannomas, with an average value ⟨S _C /S _R ⟩ = 0.94 ± 0.11, and gliomas with ⟨S _C /S _R ⟩ = 0.58 ± 0.23. Since regularity measures are related to the fractal properties of the tumor interface (Pérez-Beteta et al., 2018) there must be a certain correlation to the scaling parameters d _F and α _loc . For tumors with S _C /S _R close to 1, fractal dimension, d _F , should be close to 2 and α _loc should have its lowest value, i.e., tumor surface is regular and smooth. As S _C /S _R decreases it is expected an increase in the scaling parameters. Figure 10 shows this trend for the dependence of d _F and α _loc on S _C /S _R . The slopes that characterize the linear trend are summarized in Table 4.

Ordered series were extracted from the tumor interface for those slices that contain the maximum number of interface points, visibility graphs were generated and the degree distribution functions were obtained. Results for the average of P (k) distributions obtained for each of the different tumor types are shown in Figure 11A. For values of the connectivity index k > 20, P (k) decays abruptly due to the fact that the ordered series has a finite size which limits the probability P (k). For values of the connectivity index k < 20 all distributions exhibit a power law behavior as shown in Figure 12A with slopes that are summarized in Table 5 and shown in Figure 12B. If the tumor types are discriminated only into two classes: one including meningiomas and acoustic schwannomas and the other including high grade gliomas and glioblastoma multiforme, the slope distributions are clearly distinct for each tumor class as shown in Figure 11B. This result suggest the possibility of using γ as a possible parameter that characterizes tumor interface dynamics supported by the result shown in Figure 12B. Multifractal analysis results are summarized in Figure 13. As expected, the evaluation of generalized fractal dimensions for the one dimensional sampling of the tumor interface, Figure 13A, does not provide with enough information to discriminate between these two classes as seen in Figure 13B and summarized in Table 6. On the other hand, two dimensional detrended fluctuation analysis of the tumor interface yield some differences in the average values of D (q), as shown in Table 6 and Figure 13D, that possibly could be improved by an adequate sampling of the tumor interface, i.e., higher image longitudinal and transverse resolution. In any case, generalized fractal dimensions do not discriminate appropriately between tumor types.