Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory

Modeling tumor growth using fractal calculus: Insights into tumor dynamics

Important concepts like fractal calculus and fractal analysis, the sum of squared residuals, and Aikaike's information criterion must be thoroughly understood in order to correctly fit cancer-related data using the proposed models. The fractal growth models employed in this work are classified in three main categories: Sigmoidal growth models (Logistic, Gompertz, and Richards models), Power Law growth model, and Exponential growth models (Exponential and Exponential-Lineal models)". We fitted the data, computed the sum of squared residuals, and determined Aikaike's information criteria using Matlab and the web tool WebPlotDigitizer. In addition, the research investigates "double-size cancer" in the fractal temporal dimension with respect to various mathematical models.

Revolutionizing anti-cancer drug discovery against breast cancer and lung cancer by modification of natural genistein: an advanced computational and drug design approach

Breast and lung cancer are two of the most lethal forms of cancer, responsible for a disproportionately high number of deaths worldwide. Both doctors and cancer patients express alarm about the rising incidence of the disease globally. Although targeted treatment has achieved enormous advancements, it is not without its drawbacks. Numerous medicines and chemotherapeutic drugs have been authorized by the FDA; nevertheless, they can be quite costly and often fall short of completely curing the condition. Therefore, this investigation has been conducted to identify a potential medication against breast and lung cancer through structural modification of genistein. Genistein is the active compound in Glycyrrhiza glabra (licorice), and it exhibits solid anticancer efficiency against various cancers, including breast cancer, lung cancer, and brain cancer. Hence, the design of its analogs with the interchange of five functional groups-COOH, NH2 and OCH3, Benzene, and NH-CH2-CH2-OH-have been employed to enhance affinities compared to primary genistein. Additionally, advanced computational studies such as PASS prediction, molecular docking, ADMET, and molecular dynamics simulation were conducted. Firstly, the PASS prediction spectrum was analyzed, revealing that the designed genistein analogs exhibit improved antineoplastic activity. In the prediction data, breast and lung cancer were selected as primary targets. Subsequently, other computational investigations were gradually conducted. The mentioned compounds have shown acceptable results for in silico ADME, AMES toxicity, and hepatotoxicity estimations, which are fundamental for their oral medication. It is noteworthy that the initial binding affinity was only -8.7 kcal/mol against the breast cancer targeted protein (PDB ID: 3HB5). However, after the modification of the functional group, when calculating the binding affinities, it becomes apparent that the binding affinities increase gradually, reaching a maximum of -11.0 and -10.0 kcal/mol. Similarly, the initial binding affinity was only -8.0 kcal/mol against lung cancer (PDB ID: 2P85), but after the addition of binding affinity, it reached -9.5 kcal/mol. Finally, a molecular dynamics simulation was conducted to study the molecular models over 100 ns and examine the stability of the docked complexes. The results indicate that the selected complexes remain highly stable throughout the 100-ns molecular dynamics simulation runs, displaying strong correlations with the binding of targeted ligands within the active site of the selected protein. It is important to further investigate and proceed to clinical or wet lab experiments to determine the practical value of the proposed compounds.

Estimation of Risk Factors Affecting Screening Outcomes of Prostate Cancer Using the Bayesian Ordinal Logistic Model

Prostate cancer occurs when cells in the prostate gland grow out of control. Almost all prostate cancers are adenocarcinomas. The survival rate for prostate cancer patients depends on the screening outcome, which can be either no prostate cancer, early detection, and late detection or advanced stage detection. The main objective of this study was to estimate the risk factors affecting the screening outcome of prostate cancer. With ordinal outcomes, a generalized Bayesian ordinal logistic model was considered in the analysis. The generalized Bayesian ordinal logistic model helped in estimation of coefficient parameters of the risk factors affecting each level of prostate cancer-screening outcomes. In the study, positive coefficients, that is, ${\beta }_k>0$ , indicated that the higher values on the explanatory variable increased the chances of the respondent being in a higher category of the dependent variable than the current one, while the negative coefficients, that is, ${\beta }_k<0$ , signified that the higher values on the explanatory variable increased the likelihood of being in the current or lower category of prostate cancer. For instance, from the analysis, positive or negative outcomes of prostate cancer showed that an increase in weight lowered the chances of an individual having the disease.

Identification of significant genes signatures and prognostic biomarkers in cervical squamous carcinoma via bioinformatic data

Background

Cervical squamous cancer (CESC) is an intractable gynecological malignancy because of its high mortality rate and difficulty in early diagnosis. Several biomarkers have been found to predict the prognose of CESC using bioinformatics methods, but they still lack clinical effectiveness. Most of the existing bioinformatic studies only focus on the changes of oncogenes but neglect the differences on the protein level and molecular biology validation are rarely conducted.

Methods

Gene set data from the NCBI-GEO database were used in this study to compare the differences of gene and protein levels between normal and cancer tissues through significant pathway selection and core gene signature analysis to screen potential clinical biomarkers of CESC. Subsequently, the molecular and protein levels of clinical samples were verified by quantitative transcription PCR, western blot and immunohistochemistry.

Results

Three differentially expressed genes (RFC4, MCM2, TOP2A) were found to have a significant survival (P < 0.05) and highly expressed in CESC tissues. Molecular biological verification using quantitative reverse transcribed PCR, western blotting and immunohistochemistry assays exhibited significant differences in the expression of RFC4 between CESC and para-cancerous tissues (P < 0.05).

Conclusion

This study identified three potential biomarkers (RFC4, MCM2, TOP2A) of CESC which may be useful to clarify the underlying mechanisms of CESC and predict the prognosis of CESC patients.

A three dimensional model of multicellular aggregate compression

Multicellular aggregates are an excellent model system to explore the role of tissue biomechanics, which has been demonstrated to play a crucial role in many physiological and pathological processes. In this paper, we propose a three-dimensional mechanical model and apply it to the uniaxial compression of a multicellular aggregate in a realistic biological setting. In particular, we consider an aggregate of initially spherical shape and describe both its elastic deformations and the reorganisation of the cells forming the spheroid. The latter phenomenon, understood as remodelling, is accounted for by assuming that the aggregate undergoes plastic-like distortions. The study of the compression of the spheroid, achieved by means of two parallel, compressive plates, needs the formulation of a contact problem between the living spheroid itself and the plates, and is solved with the aid of the augmented Lagrangian method. The results of the performed numerical simulations are in qualitative agreement with the biological observations reported in the literature and can also be used to estimate quantitatively some fundamental aggregate mechanical parameters.

Patented therapeutic approaches targeting LRP/LR for cancer treatment

Introduction: The ubiquitously expressed 37 kDa/67 kDa high-affinity laminin receptor (laminin receptor precursor/laminin receptor, LRP/LR) is a protein found to play several roles within cells. The receptor is located in the nucleus, cytosol and the cell surface. LRP/LR mediates cell proliferation, cell adhesion and cell differentiation. As a result, it is seen to enhance tumor angiogenesis as well as invasion and adhesion, key steps in the metastatic cascade of cancer. Recent findings have shown that LRP/LR is involved in the maintenance of cell viability through apoptotic evasion, allowing for tumor progression. Thus, several patented therapeutic approaches targeting the receptor for the prevention and treatment of cancer have emerged.Areas covered: The several roles that LRP/LR plays in cancer progression as well as an overview of the current therapeutic patented strategies targeting LRP/LR and cancer to date.Expert opinion: Small molecule inhibitors, monoclonal antibodies and small interfering RNAs might act used as powerful tools in preventing tumor angiogenesis and metastasis through the induction of apoptosis and telomere erosion in several cancers. This review offers an overview of the roles played by LRP/LR in cancer progression, while providing novel patented approaches targeting the receptor as potential therapeutic routes for the treatment of cancer as well as various other diseases.

Identification of Significant Gene Signatures and Prognostic Biomarkers for Patients With Cervical Cancer by Integrated Bioinformatic Methods

Cervical cancer is the leading cause of death with gynecological malignancies. We aimed to explore the molecular mechanism of carcinogenesis and biomarkers for cervical cancer by integrated bioinformatic analysis. We employed RNA-sequencing details of 254 cervical squamous cell carcinomas and 3 normal samples from The Cancer Genome Atlas. To explore the distinct pathways, messenger RNA expression was submitted to a Gene Set Enrichment Analysis. Kyoto Encyclopedia of Genes and Genomes and protein–protein interaction network analysis of differentially expressed genes were performed. Then, we conducted pathway enrichment analysis for modules acquired in protein–protein interaction analysis and obtained a list of pathways in every module. After intersecting the results from the 3 approaches, we evaluated the survival rates of both mutual pathways and genes in the pathway, and 5 survival-related genes were obtained. Finally, Cox hazards ratio analysis of these 5 genes was performed. DNA replication pathway (P < .001; 12 genes included) was suggested to have the strongest association with the prognosis of cervical squamous cancer. In total, 5 of the 12 genes, namely, minichromosome maintenance 2, minichromosome maintenance 4, minichromosome maintenance 5, proliferating cell nuclear antigen, and ribonuclease H2 subunit A were significantly correlated with survival. Minichromosome maintenance 5 was shown as an independent prognostic biomarker for patients with cervical cancer. This study identified a distinct pathway (DNA replication). Five genes which may be prognostic biomarkers and minichromosome maintenance 5 were identified as independent prognostic biomarkers for patients with cervical cancer.

A chemical energy approach of avascular tumor growth: multiscale modeling and qualitative results

In the present manuscript we propose a lattice free multiscale model for avascular tumor growth that takes into account the biochemical environment, mitosis, necrosis, cellular signaling and cellular mechanics. This model extends analogous approaches by assuming a function that incorporates the biochemical energy level of the tumor cells and a mechanism that simulates the behavior of cancer stem cells. Numerical simulations of the model are used to investigate the morphology of the tumor at the avascular phase. The obtained results show similar characteristics with those observed in clinical data in the case of the Ductal Carcinoma In Situ (DCIS) of the breast.

On the stability of a spherical tumour

The mathematical analysis of the tumour growth attracted a lot of interest in the last two decades. However, as of today no generally accepted model for tumour growth exists. This is due partially to the incomplete understanding of the related pathology as well as the extremely complicated procedure that guides the evolution of a tumour. In the present work, we analyse the stability of a spherical tumour for four continuous models of an avascular tumour. Conditions for the stability are stated and the results are implemented numerically. It is observed that the steady-state radii that secure the stability of the tumour are different for each of the four models, although the differences are not very pronounced.

Elastic free energy drives the shape of prevascular solid tumors

It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an in vitro tumor model, which isolates mechanical interactions between cancer tumor cells and a hydrogel, we find that tumors grow as ellipsoids, resembling the same, oft-reported observation of in vivo tumors. Specifically, an oblate ellipsoidal tumor shape robustly occurs when the tumors grow in hydrogels that are stiffer than the tumors, but when they grow in more compliant hydrogels they remain closer to spherical in shape. Using large scale, nonlinear elasticity computations we show that the oblate ellipsoidal shape minimizes the elastic free energy of the tumor-hydrogel system. Having eliminated a number of other candidate explanations, we hypothesize that minimization of the elastic free energy is the reason for predominance of the experimentally observed ellipsoidal shape. This result may hold significance for explaining the shape progression of early solid tumors in vivo and is an important step in understanding the processes underlying solid tumor growth.

Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth

The idea that one can possibly develop computational models that predict the emergence, growth, or decline of tumors in living tissue is enormously intriguing as such predictions could revolutionize medicine and bring a new paradigm into the treatment and prevention of a class of the deadliest maladies affecting humankind. But at the heart of this subject is the notion of predictability itself, the ambiguity involved in selecting and implementing effective models, and the acquisition of relevant data, all factors that contribute to the difficulty of predicting such complex events as tumor growth with quantifiable uncertainty. In this work, we attempt to lay out a framework, based on Bayesian probability, for systematically addressing the questions of Validation, the process of investigating the accuracy with which a mathematical model is able to reproduce particular physical events, and Uncertainty quantification, developing measures of the degree of confidence with which a computer model predicts particular quantities of interest. For illustrative purposes, we exercise the process using virtual data for models of tumor growth based on diffuse-interface theories of mixtures utilizing virtual data.

Toward an Ising model of cancer and beyond

The holy grail of tumor modeling is to formulate theoretical and computational tools that can be utilized in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies to control cancer growth. In order to develop such a predictive model, one must account for the numerous complex mechanisms involved in tumor growth. Here we review the research work that we have done toward the development of an 'Ising model' of cancer. The Ising model is an idealized statistical-mechanical model of ferromagnetism that is based on simple local-interaction rules, but nonetheless leads to basic insights and features of real magnets, such as phase transitions with a critical point. The review begins with a description of a minimalist four-dimensional (three dimensions in space and one in time) cellular automaton (CA) model of cancer in which cells transition between states (proliferative, hypoxic and necrotic) according to simple local rules and their present states, which can viewed as a stripped-down Ising model of cancer. This model is applied to study the growth of glioblastoma multiforme, the most malignant of brain cancers. This is followed by a discussion of the extension of the model to study the effect on the tumor dynamics and geometry of a mutated subpopulation. A discussion of how tumor growth is affected by chemotherapeutic treatment, including induced resistance, is then described. We then describe how to incorporate angiogenesis as well as the heterogeneous and confined environment in which a tumor grows in the CA model. The characterization of the level of organization of the invasive network around a solid tumor using spanning trees is subsequently discussed. Then, we describe open problems and future promising avenues for future research, including the need to develop better molecular-based models that incorporate the true heterogeneous environment over wide range of length and time scales (via imaging data), cell motility, oncogenes, tumor suppressor genes and cell-cell communication. A discussion about the need to bring to bear the powerful machinery of the theory of heterogeneous media to better understand the behavior of cancer in its microenvironment is presented. Finally, we propose the possibility of using optimization techniques, which have been used profitably to understand physical phenomena, in order to devise therapeutic (chemotherapy/radiation) strategies and to understand tumorigenesis itself.

Front instabilities and invasiveness of simulated 3D avascular tumors

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

In silico cancer modeling: is it ready for prime time?

At the dawn of the era of personalized, systems-driven medicine, computational or in silico modeling and the simulation of disease processes is becoming increasingly important for hypothesis generation and data integration in both experiments and clinics alike. Arguably, the use of these techniques is nowhere more visible than in oncology. To illustrate the field's vast potential, as well as its current limitations, we briefly review selected studies on modeling malignant brain tumors. Implications for clinical practice, and for clinical trial design and outcome prediction, are also discussed.

Morphological instability and cancer invasion: a 'splashing water drop' analogy

Background

Tissue invasion, one of the hallmarks of cancer, is a major clinical problem. Recent studies suggest that the process of invasion is driven at least in part by a set of physical forces that may be susceptible to mathematical modelling which could have practical clinical value.

Model and conclusion

We present an analogy between two unrelated instabilities. One is caused by the impact of a drop of water on a solid surface while the other concerns a tumor that develops invasive cellular branches into the surrounding host tissue. In spite of the apparent abstractness of the idea, it yields a very practical result, i.e. an index that predicts tumor invasion based on a few measurable parameters. We discuss its application in the context of experimental data and suggest potential clinical implications.

The role of mechanical host-tumour interactions in the collapse of tumour blood vessels and tumour growth dynamics

A mathematical model of residual stress evolution in a growing vascular tumour is presented, in an attempt to elucidate the poorly understood phenomenon of vascular collapse. Whereas earlier studies in this area have neglected the effects of mechanical interactions between the tumour and the surrounding host tissue, the significance of these interactions for the long-term development of a tumour is now considered. The model predicts tumour stress distributions which reflect the distinctive patterns of vascular collapse reported in experimental studies. Moreover, while neglecting mechanical host/tumour interactions results in the eventual complete regression of the tumour to its avascular dormant size in the event of vascular collapse, this new model points to the possibility of oscillations in the tumour's size in the long term.

Emerging patterns in tumor systems: simulating the dynamics of multicellular clusters with an agent-based spatial agglomeration model

Brain cancer cells invade early on surrounding parenchyma, which makes it impossible to surgically remove all tumor cells and thus significantly worsens the prognosis of the patient. Specific structural elements such as multicellular clusters have been seen in experimental settings to emerge within the invasive cell system and are believed to express the systems' guidance toward nutritive sites in a heterogeneous environment. Based on these observations, we developed a novel agent-based model of spatio-temporal search and agglomeration to investigate the dynamics of cell motility and aggregation with the assumption that tumors behave as complex dynamic self-organizing biosystems. In this model, virtual cells migrate because they are attracted by higher nutrient concentrations and to avoid overpopulated areas with high levels of toxic metabolites. A specific feature of our model is the capability of cells to search both globally and locally. This concept is applied to simulate cell-surface receptor-mediated information processing of tumor cells such that a cell searching for a more growth-permissive place "learns" the information content of a brain tissue region within a two-dimensional lattice in two stages, processing first the global and then the local input. In both stages, differences in microenvironment characteristics define distinctions in energy expenditure for a moving cell and thus influence cell migration, proliferation, agglomeration, and cell death. Numerical results of our model show a phase transition leading to the emergence of two distinct spatio-temporal patterns depending on the dominant search mechanism. If global search is dominant, the result is a small number of large clusters exhibiting rapid spatial expansion but shorter lifetime of the tumor system. By contrast, if local search is dominant, the trade-off is many small clusters with longer lifetime but much slower velocity of expansion. Furthermore, in the case of such dominant local search, the model reveals an expansive advantage for tumor cell populations with a lower nutrient-depletion rate. Important implications of these results for cancer research are discussed.

Simulated brain tumor growth dynamics using a three-dimensional cellular automaton

We have developed a novel and versatile three-dimensional cellular automaton model of brain tumor growth. We show that macroscopic tumor behavior can be realistically modeled using microscopic parameters. Using only four parameters, this model simulates Gompertzian growth for a tumor growing over nearly three orders of magnitude in radius. It also predicts the composition and dynamics of the tumor at selected time points in agreement with medical literature. We also demonstrate the flexibility of the model by showing the emergence, and eventual dominance, of a second tumor clone with a different genotype. The model incorporates several important and novel features, both in the rules governing the model and in the underlying structure of the model. Among these are a new definition of how to model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.

A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects

Angiogenesis is defined as the outgrowth and formation of new vessels from a pre-existing vascular network (Rakusan, In: Cardiac Growth and Regeneration. Annals of the New York Academy of Sciences, 1995), and is of fundamental importance in understanding the processes by which a tumour achieves vascularization. Diffusible substances, collectively called tumour angiogenesis factors are released from the tumour to elicit a variety of responses from the surrounding tissues, most importantly the migration of endothelial cells (lining neighbouring vessels) towards the tumour. To facilitate locomotion, the cells exert appreciable traction forces upon the interstitial extracellular matrix which, in turn, influences the resulting direction of their migration. In this paper, we examine the role played by cellular traction during cell migration and the corresponding viscoelastic effects of the extracellular matrix.

A patient-specific in vivo tumor model

A significant body of research, spanning approximately the last 25 years, has focused upon the task of developing a better understanding of tumor growth through the use of in vitro mathematical models. Although such models are useful for simulation, in vivo growth differs in significant ways due to the variety of competing biological, biochemical, and mechanical factors present in a living biological system. An in vivo, macroscopic, primary brain tumor growth model is developed, incorporating previous in vitro growth pattern research as well as scientific investigations into the biological and biochemical factors that affect in vivo neoplastic growth. The tumor growth potential model presents an integrated, universal framework that can be employed to predict the direction and extent of spread of a primary brain tumor with respect to time for a specific patient. This framework may be extended as necessary to include the results of current and future research into parameters affecting neoplastic proliferation. The patient-specific primary brain tumor growth model is expected to have multiple clinical uses, including: predictive modeling, tumor boundary delineation, growth pattern research, improved radiation surgery planning, and expert diagnostic assistance.