Analysis of a Mathematical Model Describing Necrotic Tumor Growth

Escher, Joachim; Matioc, Anca–Voichita; Matioc, Bogdan–Vasile

doi:10.1007/978-3-642-20490-6_10

Cited by 8 publications

(7 citation statements)

References 18 publications

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“…Thus necrosis formation is an important factor to be accounted for in serial images to obtain accurate parameters such as growthrate. Necrosis effects have been modeled theoretically by moving boundary methods modeling nutrient concentration (mainly by oxygen diffusion) (13)(14)(15)(16)(17), but on the other hand, in these models, typically, tumor cell density or volume effects (6)(7)18) are not explicitly considered. Another model that incorporates necrosis is the theoretical ode-compartment volume model proposed by Wallace and Guo (18).…”

Section: Introductionmentioning

confidence: 99%

Experimental method and statistical analysis to fit tumor growth model using SPECT/CT imaging: a preclinical study

Hidrovo¹,

Dey²,

Chesal³

et al. 2017

Quant. Imaging Med. Surg.

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Background: Over the last decade, several theoretical tumor-models have been developed to describe tumor growth. Oncology imaging is performed using various modalities including computed tomography (CT), magnetic resonance imaging (MRI), single-photon emission computed tomography (SPECT) and fluorodeoxyglucose-positron emission tomography (FDG-PET). Our goal is to extract useful, otherwise hidden, quantitative biophysical parameters (such as growth-rate, tumor-necrotic-factor, etc.) from these serial images of tumors by fitting mathematical models to images. These biophysical features are intrinsic to the tumor types and specific to the study-subject, and expected to add valuable information on the tumor containment or spread and help treatment plans. Thus, fitting realistic but practical models and assessing parameter-errors and degree of fit is important. Methods:We implemented an existing theoretical ode-compartment model and variants and applied them for the first time, in vivo. We developed an inversion algorithm to fit the models for tumor growth for simulated as well as in vivo experimental data. Serial SPECT/CT scans of mice breast-tumors were acquired, and SPECT data was used to segment the proliferating-layers of tumors.Results: Results of noisy data simulation and inversion show that 5 out of 7 parameters were recovered to within 4.3% error. In particular, tumor "growth-rate" parameter was recovered to 0.07% error. For model fitting to in vivo mice-tumors, regression analysis on the P-layer volume showed R 2 of 0.99 for logistic andGompertzian while surface area model yielded R weights of the models (giving their relative probability of being the best Kullback-Leibler (K-L) model among the set of candidate models) were ~0, 0.43 and 0.57 for surface-area, logistic and Gompertzian models.Conclusions: Model-fitting to mice tumor studies demonstrates feasibility of applying the models to in vivo imaging data to extract features. Akaike information criterion (AIC) evaluations show Gompertzian or logistic growth model fits in vivo breast-tumors better than surface-area based growth model. IntroductionOncology imaging is performed using various modalities including computed tomography (CT), magnetic resonance imaging (MRI), single-photon emission computed tomography (SPECT) and fluorodeoxyglucose-positron emission tomography (FDG-PET).Quantification of cancer from images is important for assessment of disease progress and treatment. Tumors grow in a specific way and theoretical tumor-models of increasing complexity have been developed that may be potentially used to find (otherwise concealed) biophysical information from serial images of tumors. If accurate, tumor growthrate, cell-motility (diffusion), tumor-necrotic-factor, can be consistently and reliably extracted from oncological images, they may add valuable information on the tumor containment/spread and help treatment plan. For example, assessing the growth-rate accurately will help in treatment planning whether in surgery or in determining the numbe...

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Section: Introductionmentioning

confidence: 99%

Experimental method and statistical analysis to fit tumor growth model using SPECT/CT imaging: a preclinical study

Hidrovo¹,

Dey²,

Chesal³

et al. 2017

Quant. Imaging Med. Surg.

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“…The moving boundary problems associated to the models presented in [1,4] have been investigated by many authors (see [5][6][7][8][9]11,12,[19][20][21] and the references therein). Unlike the models studied in [10,13,23,24] where existence of a necrotic core consisting of death cells is taken into account, the model derived in [3] and studied herein does not possess this feature. However, system (1.2) has two phases, and the nutrient concentrations σ T and σ H , inside and outside the solid tumor, are connected through a diffraction problem, a feature which relates system (1.2) to the Muskat problem, cf.…”

Section: Introductionmentioning

confidence: 86%

Analysis of a two-phase model describing the growth of solid tumors

Escher¹,

Matioc²

2012

Eur. J. Appl. Math

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In this paper we consider a two-phase model describing the growth of avascular solid tumors when taking into account the effects of cell-to-cell adhesion and taxis due to nutrient. The tumor is surrounded by healthy tissue which is the source of nutrient for tumor cells. In a three-dimensional context, we prove that the mathematical formulation corresponds to a well-posed problem, and find radially symmetric steady-state solutions of the problem. They appear in the regime where the rate of cell apoptosis to cell proliferation is less than the far field nutrient concentration. Furthermore, we study the stability properties of those radially symmetric equilibria and find, depending on the biophysical parameters involved in the problem, both stable and unstable regimes for tumor growth.

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“…The strict positivity of D (in the sense of Assumption 3.1) yields 18) i.e., Θ 0 is independent of z. Moreover, integrating (3.18) with respect to z and using the matching condition (3.6) applied to Θ 0 gives…”

Section: Expansions To Leading Ordermentioning

confidence: 98%

“…Different models have been suggested to describe free boundary problems involving multiphase tumour growth. In particular, the effect of a necrotic core has been studied in [13,18]. In this section, we will perform a formally matched asymptotic analysis for the phase field model (2.35) in order to derive new free boundary problems for tumour growth.…”

Section: Sharp Interface Asymptoticsmentioning

confidence: 99%

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

Garcke

Lam

Nürnberg

et al. 2018

Math. Models Methods Appl. Sci.

107

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We derive a Cahn-Hilliard-Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. Via a coupling of the Cahn-Hilliard-Darcy equations to a system of reaction-diffusion equations a multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, can be included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth is investigated numerically.

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Analysis of a Mathematical Model Describing Necrotic Tumor Growth

Cited by 8 publications

References 18 publications

Experimental method and statistical analysis to fit tumor growth model using SPECT/CT imaging: a preclinical study

Experimental method and statistical analysis to fit tumor growth model using SPECT/CT imaging: a preclinical study

Analysis of a two-phase model describing the growth of solid tumors

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

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