On Some Probabilistic Aspects of Diffusion Models for Tissue Growth

Kiseľák, Jozef; Pardasani, Kamal Raj; Adlakha, Neeru; Stehlík, Milan; Agrawal, Mithlesh

doi:10.2174/1876527001305010014

Cited by 3 publications

(4 citation statements)

References 28 publications

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“…Discrimination between mammary cancer and mastopathy tissues plays a crucial role in a clinical practice; see [9]. Noninvasive techniques may produce generally an inverse problems, e.g., estimating a Hausdorff fractal dimension from boundary of examined tissue; see [10]. Main problem here can be formulated as follows: ''How can be cancer tissue discriminated from healthy tissue?''…”

Section: Spocu With C = ¥mentioning

confidence: 99%

“SPOCU”: scaled polynomial constant unit activation function

Kiseľák

Lü

Švihra

et al. 2020

Neural Comput & Applic

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We address the following problem: given a set of complex images or a large database, the numerical and computational complexity and quality of approximation for neural network may drastically differ from one activation function to another. A general novel methodology, scaled polynomial constant unit activation function ''SPOCU,'' is introduced and shown to work satisfactorily on a variety of problems. Moreover, we show that SPOCU can overcome already introduced activation functions with good properties, e.g., SELU and ReLU, on generic problems. In order to explain the good properties of SPOCU, we provide several theoretical and practical motivations, including tissue growth model and memristive cellular nonlinear networks. We also provide estimation strategy for SPOCU parameters and its relation to generation of random type of Sierpinski carpet, related to the [pppq] model. One of the attractive properties of SPOCU is its genuine normalization of the output of layers. We illustrate SPOCU methodology on cancer discrimination, including mammary and prostate cancer and data from Wisconsin Diagnostic Breast Cancer dataset. Moreover, we compared SPOCU with SELU and ReLU on large dataset MNIST, which justifies usefulness of SPOCU by its very good performance.

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Section: Spocu With C = ¥mentioning

confidence: 99%

“SPOCU”: scaled polynomial constant unit activation function

Kiseľák

Lü

Švihra

et al. 2020

Neural Comput & Applic

Self Cite

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“…The predicting or tracing of the tissue growth can be carried out by using the fractal analysis as well, see e.g. [7]. The expansion of the mathematical and computational modelling in recent decades facilitated the biomechanical research inter alia, enabled to simulate, observe and predict some processes of living organisms.…”

Section: Introductionmentioning

confidence: 99%

Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation

Minárová

Sumec

2016

Open Physics

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Abstract:The paper deals with the biomechanical investigation on the human lumbar intervertebral disc under the static load. The disc is regarded as a two -phased ambient consisting of a fibrous outer part called annulus fibrosis and a liquid inner part nucleus pulposus. Due to the fibrous structure, the annulus fibrosis can be treated by using a special case of anisotropy -transversal isotropy. In the paper the corresponding tensor of material constants is derived. The tensor consequently incomes to the constitutive equations determining the stress -strain relation in the material. In order to study the mechanical behaviour the disc is observed within the motion segment, the basic unit for motion tracing. The motion segment involves two neighbouring vertebrae and the intervertebral disc between them that connect them both. When constitutive equations are accomplished, they can be incorporated in the finite element analysis. The illustrative example of the intervertebral disc L2/L3, the disc between the second and the third lumbar vertebrae the lumbar part of spine, with its computer implementation is performed. Finally the comparison of the results of using anisotropic and homogenized approach is provided. The comparison illustrates the eligibility of such a kind of approach.

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“…Kiseľák et al . discuss the relation between tissue growth and the mathematical model of diffusion. It is clear that a deterministic fractal is not an appropriate model for cancer growth (see also ).…”

Section: Introductionmentioning

confidence: 99%

“…Stochastic fractals (in our area of applications, relevant references are, e.g., , which relate to fractional Brownian motions; for more details see Section 6) are more appropriate; however, a validation measure should be developed for a better comparability with advanced stochastic geometry models, for example, Quermass‐interaction processes. A relation between temperature and geometry of the tissue is studied in . Therein, partial results on nonstandard diffusions for benign alterations and malignant tumors originating from glandular tissues (e.g., mammary, prostatic, or pancreatic) are given.…”

Section: Introductionmentioning

confidence: 99%

Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process

et al. 2015

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Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

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On Some Probabilistic Aspects of Diffusion Models for Tissue Growth

Cited by 3 publications

References 28 publications

“SPOCU”: scaled polynomial constant unit activation function

“SPOCU”: scaled polynomial constant unit activation function

Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation

Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process

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