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2015 3-D in vivo brain tumor geometry study by scaling analysis (Estudio in vivo en 3d de la geometría de tumores cerebrales por análisis de escalamiento)

Universidad Cooperativa de ColombiaSala de Conocimiento20153-D in vivo brain tumor geometry study by scaling analysis (Estudio in vivo en 3d de la geometría de tumores cerebrales por análisis de escalamiento)

3-D in vivo brain tumor geometry study by scaling analysis (Estudio in vivo en 3d de la geometría de tumores cerebrales por análisis de escalamiento)


3-D in vivo brain tumor geometry study by scaling analysis (Estudio in vivo en 3d de la geometría de tumores cerebrales por análisis de escalamiento)​​


Francisco José Torres Hoyos 
Doctor en Física - Universidad Simón Bolívar - Venezuela
Magíster en Ciencias Físicas - Universidad Nacional de Colombia - Colombia
Especialista en Ciencias Físicas - Universidad Nacional de Colombia - Colombia
Licenciado en Matemáticas y Física - Universidad de Córdoba - Colombia

The analysis and understanding of tumor growth have been the object of multidisciplinary research. In the last decade an increased number of complex physical–mathematical models have been published, with different approaches due to its multiscale nature. The simplest theoretical models involve only the total number of cells in the tumor, when they have been applied to this problem, growth is usually assumed to be exponential, gompertzian, or logistic [1,2]. Such models do not consider the spatial arrangement of the cells at a specific anatomical location, or the spatial spread of the cancerous cells. These spatial aspects are essential in estimating tumor growth since they determine the invasiveness of the tumor.

 Tomada de:

Dynamic scaling is a technique that exploits the geometric properties of the growth fronts using different concepts of the theory of stochastic processes and fractal geometry [3,4]. There have been studies of the tumor-host interface, which indicate that the behavior of the fluctuations leads to stochastic evolution equations, which are analyzed under radial symmetry [5].

Tumor growth is a complex process that depends on the corresponding cell proliferation and then spread in the host tissue. The approach used in the present work is based on the fractality and scale invariance of the boundary of cell colony. Tumor cells form colonies that can be characterized by a fractal dimension, which is a measure of the degree of complexity, allowing for the use of scaling analysis [6–10].

Recent studies of in vitro tumor cell colonies and resected tumor sections [8,9] show that some kind of universality in tumor growth dynamics can be determined by evaluating their boundary geometrical properties through critical exponents.

The present work attempts to perform a detailed study of tumor growth scaling behavior from in vivo brain tumors by the evaluation of the local roughness exponent [11,12] of the tumor-host interface. To our knowledge, this kind of study based on scaling analysis has not yet been realized.

1. Image processing
Figure 1. High contrast three-dimensional image for a glioblastoma.

2. Results and discussion 
Scaling analysis was applied to a total of 62 tumor lesions in the brain. The tumors were classified according to its histological characteristics into four groups: astrocytomas (10 cases), glioblastomas (7 cases), metastases (18 cases) and benign brain tumors (27 cases), such as meningiomas and acoustic schwannomas. In all cases the tumor-host interface exhibits fractal geometry characterized by a non-integer dimension in the range 2.1–2.32, which is consistent with what is expected for fractal interfaces embedded in a three dimensional Euclidean space [4,17,18]. The local roughness exponent αloc was calculated and a power-law behavior was obtained in all cases, with small variations within each histological group, for the cases of glioblastomas multiformes, astrocytomas, metastases and benign tumors, respectively. Analysis  reveals not only a diminution of the roughness exponent, αloc , as we proceed from high grade malignant tumors, such as glioblastomas, through astrocytomas and metastases, but also an increase of the dispersion of this exponent. From the roughness exponent point of view, high grade tumors, i.e., grade IV, become more similar than for example, grades I through III. In the case of metastases, an increase in the dispersion is expected due to the diverse histological origin and development in brain.

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