A chemical energy approach of avascular tumor growth: multiscale modeling and qualitative results
 Pantelis Ampatzoglou^{1},
 George Dassios^{2},
 Maria Hadjinicolaou^{1}Email author,
 Helen P. Kourea^{3} and
 Michael N. Vrahatis^{4}
Received: 10 September 2015
Accepted: 9 October 2015
Published: 2 November 2015
Abstract
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.
Keywords
Mathematical Subject Classification
Background
In 2014 in the US, there is an estimation of 1,665,540 new diagnosed cancer cases and 585,720 deaths. Cancer maintains its ranking as the second most common cause of death in the developed and developing countries, accounting for nearly a quarter of deaths (Siegel et al. 2015). Among others, these statistics make cancer the most urgently investigated disease in today’s research efforts.
Cancer description
Neoplasm or tumor is an independently growing mass of abnormal cells. The tumors that remain localized, cannot metastasize to other organs and are therefore innocent called benign tumors. Malignant tumors, also called cancers, are locally destructive neoplasms with the potential for distant spread, thus causing death (Goldschmidt and Chief 2013). Cancers are classified by the tissue from which they arise and by the type of cells involved. For example, leukemia is a cancer of circulating white blood cells, sarcoma is a type of cancer arising from tissues of mesenchymal derivation such as adipose or connective tissues, and carcinoma is a cancer originating from epithelial cells. The epithelial cells, are cells closely interconnected in such a way as to form acinar or glandular structures, or to line cavities or tubular organs. Such examples are the cells forming the respiratory or gastrointestinal tract or solid organs such as the breast, the pancreas and the prostate gland, among several others. Until recently, the classical model of carcinogenesis stated that cancer starts when a cell inside a tissue is subjected to DNA mutations that change its phenotype to cancerous. A primary tumor, which is a tumor in its original site of occurrence, is usually traced to a single mutated cell, from which over a period of time, a colony of cells is formed. Progeny of cancer cells reproduce at a faster rate than normal cells forming a colony. In cancers of epithelial origin, namely carcinomas, this colony is delimited by a basement membrane that isolates the neoplastic cells. At this phase, the tumor is called carcinoma in situ and has no metastatic potential. Once the neoplastic cells acquire additional abilities to break out of the basement membrane, than they invade the surrounding extracellular matrix (ECM) and the tumor is called invasive carcinoma and can find access to vessels and produce metastases. Cancers can grow up to 1–2 mm\(^3\) by obtaining oxygen and nutrients through the existing vasculature (Folkman 1990). During this stage the tumor is said to be avascular. Invasive carcinoma cells can find access to old and newly formed blood vessels. Once the tumor cells enter the blood vessels they can be transported via the circulation to other organs, where they can exit from the circulation, engraft in the new environment and start to grow again to produce a metastatic or secondary tumor. In this manuscript we consider only the early stages of cancer development, while the tumor remains in an avascular state. Traditionally the cause of cancer is considered being the fact that DNA replication and repair is not a 100% accurate process, and cancers emerge as a result of the many gene mutations accumulating in the human body over a person's lifetime. There is evidence that a single abnormal cell, which gives rise to a tumor, has risen through a number of genetic alterations, or epigenetic mutations; the latter means a change of gene expression as a result of blocking of gene promoters. The two main ways by which genes can become oncogenic are: (1) a stimulating gene becomes hyperactive, or upregulated; such an abnormal gene is called oncogene; and (2) an inhibitory gene becomes inactive, or downregulated; it is called a tumor suppressor gene, an example being the p53 gene which controls the progression of the cell cycle. In order to continue to grow, the tumor requires new sources of nutrients. It does it by secreting chemicals called tumor growth factors, which stimulate the formation of new blood vessels, attracting them into the tumor. This is the process of angiogenesis; a tumor which has developed beyond this stage is said to be vascularized. Further on, the progeny of cancer cells have a higher probability of mutations that lead to increased cancer aggressiveness over time. Breakthroughs in medical science led to the discovery that cancer is initiated not only by the mutation of a cell’s DNA but rather by the mutation of a stem cell’s DNA (Reya et al. 2001; Jordan et al. 2006). This mutated stem cell now named cancer stem cell travels inside the tissue like normal stem cells, producing cancer cells.
Modeling
Eykhoff in 1974 defined a mathematical model as a ‘representation of the essential aspects of a system which presents knowledge of that system in usable form’ (Åström and Eykhoff 1971). Considering cancer as a multiparametric complex dynamical system, mathematical modeling aims to provide a better understanding of the mechanisms causing the tumor growth and contribute to the diagnosis, therapy and prevention. Through mathematical methods, the empirical and qualitative observations and experimental data can be explained and integrated to realistic models that may also serve as a indispensable ‘noninvasive tools’, for either assessing a treatment (therapy) or for understanding the biological rules of growth. Numerous mathematical models have been developed in the last 50 years for the study of tumor growth, in different phases (Lowengrub et al. 2010). Most of these mathematical approaches fall initially into two general categories, the continuum and the discrete; based on the assumptions made for the tumor tissue. Continuum models represent the main category of models employed in this field, (see for example Murray 2003; Chaplain and Lolas 2005), where the tumor is considered to be homogeneous and continuous averaging out the effects of individual cells. Its growth is described through the evolution of its boundary. Exemplary efforts of continuum models for cancer growth are those provided by Byrne and Chaplain (1996) and Byrne et al. (2003) where the proposed models describe the evolution of the boundary of an avascular solid tumor driven by nutrient supplied by the extracellular matrix. Cellular movement has been studied in (Macklin and Lowengrub 2007; Greenspan 1976; Friedman 2007; Byrne and Matthews 2002; Chaplain and Sleeman 1993). Continuum cancer models are focusing on the evolution of the densities of cells (abnormal, normal, or dead), and the evolution of boundaries of the tumor regions which are due mainly to changes of the concentrations of biochemical species, are described by differential equations. Their approach is based on the principles of continuum mechanics. Some of these models use only ordinary differential equations (ODEs) (Greenspan 1972; Andersen et al. 2005), overpassing the spatial heterogeneity of tumor growth. Otherwise Partial Differential Equations (PDEs) of reaction diffusion type take into account spatial effects and also may describe the time evolution of the tumor region. Since a deterministic approach neglects random influences on the growth process, stochastic differential equations can be regarded as more adequate models for the development of a population (Rosenkranz 1985). Additionally one of the methods that is applied to describe tumor growth is the dynamic scaling of interfaces based on the work given by Brú et al. (2003). There is strong experimental evidence that the fluctuations of tumor region boundaries show temporospatial behavior, that contain the characteristics of selfaffinity. The interface fluctuations seem to evolve according to power laws. This approach is used as a starting point to describe the evolution of tumor boundaries through the use of stochastic partial differential equations.
Discrete models on the other hand consist a separate category, where the behavior of the tumor is determined by the interaction between individual cells and the microenvironment both. Recently, discretenumerical models (Swanson et al. 2002; Clatz et al. 2005) are given much attention due to the increase in computer power that is available nowadays. In these numerical models the biological processes are described using mathematical tools, available from numerical analysis methods. Special reference is due to the Cellular Potts Model (CPM) also known as extended largeq Potts model or Glazier and Graner model. The CPM is a latticebased computational modeling method to simulate the collective behavior of cellular structures with notable applications on cancer modeling (Graner and Glazier 1992; Ghaemi and Shahrokhi 2006; Turner and Sherratt 2002; Stott et al. 1999). In the CPM the process of the simulation progresses by updating the cell lattice one pixel at a time based on a set of stochastic rules. CPM can be thought of as an agent based generalized cellular automaton method in which cell agents interact through precise methods. Hybrid models employ differed methods from both continuum and discrete mathematics in order to archive a better modeling approach of the studied phenomenon. In the case of hybrid tumor modeling, continuum methods are being used mainly to model the tumor on a macroscopic scale while discrete functions are applied mainly to the microscopic scale. Multiphase modeling is used in modeling physical phenomena that are described by twoormore liquids that flow on different phases. Each of the phases is considered to have a separately defined volume fraction and velocity field. In the case of cancer multiphase modeling, the tumor is described as two or three separate liquids, depending on the assumptions of each approach. These liquids are correlated with corresponding types of tumor cells and are presumed to flow with different velocity fields, thus simulating the evolution of the various tumor regions in time (Sciumè et al. 2013). Multiscale approach is employed in order to solve problems which have important features at multiple scales. Multiscale models may serve as theoretical tools but also allow for a deeper understanding of the underlined biological system. These models incorporate biological mechanisms that refer to intracellular, cellular or extracellular scale. While employing multiscale modeling several restrictions have to be taken under consideration, the most important of which is scale linking (de Pablo 2011; Steinhauser 2008). A detailed review of the various proposed mathematical models for tumor growth can be found in the work of Lowengrub et al. (2010).
Results and discussion
In the present manuscript we propose a hybrid multiscale model for avascular tumor growth. At the tissue scale, the concentrations of the encountered biochemical species are described through diffusion equations, while at the cellular scale cellular life is modeled as an cellular automaton. At this scale we introduce the concept of a function that encounters the chemical energy level of each cell in ATPs. The existence of this function in the model affects the behavior of the cells and consequentially determines the morphology of the tumor. At the intracellular scale, stochastic methods describe the behavior of the cellular organs.
Results
Based on the configuration and methodology explicitly presented in the following sections, we demonstrate our early results (of 30 simulations) that describe the behavior of a tumor. The qualitative characteristics of the proposed model are in accordance to other mathematical models of avascular tumor growth, while in addition, qualitative information of the tumor morphology is gained, which is the formation of necrotic islands in the center region of the tumor. The obtained results coincide with observed clinical data. In more detail the following figures depict a series of cross sections of a simulated three dimensional tumor at different time iterations, where the simulated cells are shown as circles.
After the 52 time iteration, we notice that the amount of nutrient doesn’t suffice to allow the entire population of cancer cells to remain proliferating and so we observe a quiescent region appearing in the center of the tumor. This results in reduction of the growth rate of the cancer cell population. Such behavior is depicted in Fig. 1c. As simulation time progresses, we observe that the cancer stem cell produced a new proliferating cancer cell that in turn, starts an exponentially growing second tumor. Such behavior is depicted in Fig. 1d. As time progresses, we notice that the growth of the population of the second tumor, even though it remains exponential, it is of a lower rate than the one of the original tumor. This behavior is due to the fact that the total amount of nutrient inflow, through the tissue remains constant. Therefore the second tumor behaves as if it grows in a poorer biochemical environment. This is shown in Fig. 1e.
In Fig. 1f we observe the existence of a necrotic region, which appears at the center of the first tumor. This is due to the lack of sufficient amount of nutrients in the simulated area. Thus the tumor can not sustain the entire cell population and cells that exist in the domain with the poorest biochemical environment undergo necrosis.
Model validity and generalization
Reduction to known results
Discussion
Improving our previous work on the proposed model (Ampatzoglou and Hadjinicolaou 2013) we extended the model by implementing a mechanism for the induction of cancer cells from cancer stem cells. In medical literature, these CSC are considered to travel inside the tissue and spore at times new cancer cells. This expansion is included in the proposed model by a mechanism that allows for a CSC to travel freely inside the simulated area and randomly produce daughters that are cancer cells which can produce new tumor ‘islands’. This expansion of the model derives simulation results that are consistent with the previously proposed model and are in accordance with the observations of invivo cancer tumors that usually show a nonwellformed and consistent cancer tumor, but rather multiple and fluctuated tumors that appear in the form of cancer agglomerations within the tissue. Moreover, given the finite rate of inflow of biochemical factors inside the tumor, we observe a competition for nourishment between the different tumor islands. Simulations show that the new tumor islands that are introduced to the model from the cancer stem cell deprive already existing tumors from nutrients, thus forcing them to reduce the number of cells.
It is well documented both invino and invitro, that avascular carcinomas can show complex structures that deviate from the standard spheroidal patterns (see for example BredelGeissler et al. 1992; Byrne and Matthews 2002). Similar morphologic characteristics are evident in the proposed model mainly in the development of the necrotic region where the necrotic region is not a spherical or a symmetric continuous domain, but rather is divided in two subregions. One in the center that is spherical and is occupied solely from necrotic cells and a second area that is occupied from both quiescent and necrotic cells with the later forming complex clusters and agglomerations. Similar formations documented appear in many types of human tumors such as the case of human prostate cancer (Hedlund et al. 1999) and seems to be in accordance with real data obtained in the case of the Ductal Carcinoma InSitu of the breast published from Fonseca et al. (1997).
Conclusions
We propose a lattice free multiscale model, that describes avascular tumor growth through a chemical energy vantage point, using the ATP molecules as a quantification approach to reveal cellular dynamics. The proposed health function offers greater resolution and insights to cellular dynamics with respect to small time intervals; in contrast to other tumor models where such effects are averaged. Tumor cells are persevered as incompressible bodies that react to the cellular environment both biochemicaly and mechanicaly. The biochemical environment is described by the concentrations of biochemical species, that propagate through the studied area through diffusion. The values of the concentrations of these species are calculated using finite element methodology. Cellular movement is implemented as a result of both chemotaxis and a spring based cellular adhesion hypothesis. Estimations made for various parameters of the model are explained. The model requites calibration in order to produce results that are better approaches to observed tumor behavior.
The model predicts (1) avascular tumors that are growing within a circular or spherical extracellular environment are likely to reach and oscillate around equilibrium. (2) The population of tumor cells depends on the amount of nutrition that it is provided to the tumor by the host tissue through the ECM. This is a result of the implemented chemical energy approach that restricts the population of cells that can be sustained from the nutrients that are offered to the tumor by the ECM. (3) The model demonstrates complex formations of necrotic regions scattered around the center of the tumor as shown in Fig. 3.
Region \(\varOmega _{A}\) is an innovation when compared with other mathematical models of cancer tumor development for the following reasons: such a region is not predicted by other models; or is observed under a forced behavior as a direct result or border fluctuations. In the proposed model no such rule or assumption is made that forces the creation of such artifacts inside the tumor region. Rather, they are produced as a result of the health level function that allows for greater insights to cellular behavior.
Methods
The proposed model is a multiscale model for tumor growth. The tumor is presumed to consist of cancer cells that develop and evolve freely inside the studied area. These cells in accordance with other cancer models belong to three categories that are explained in the following section. The behavior of the tumor as a whole as well as the behavior of each participating cell individually, is an implicit result of the biochemical environment in which the cells exist. The participating biochemical species are: oxygen, glucose, waste, growth factors and growth inhibitors. The later three are abstract biochemicals that include a number of different factors and proteins of cellular life affecting the tumor similarly. For example the generic term waste envelops the total of metabolic end products that cells produce such as pyruvate, lactate, alanine, proline, aspartate, and citrate (Lanks and Li 1988). Further, we presume that inside the studied area, cancer stem cells (CSC) exist, that are able to randomly produce new cancer cells inside the ECM every 70 time intervals.
Formulation of the problem
Tissue scale
Diffusion constants for biochemical species (Jiang et al. 2005)
Oxygen  Glucose  Waste  Growth  Inhibitor  

Diffusion constant  \(5.94\times 10^{2}\)  \(1.52\times 10^{3}\)  \(2.124\times 10^{3}\)  \(10^{6}\)  \(10^{6}\) 
Unit  (\(\text{cm}^{2}/\text{h}\)) 

\(C_{i}\): concentration of biochemical specie ‘i’

t: time.

\(D_{i}\): diffusion coefficient of biochemical factor ‘i’

\(Q_{i,j}\): rate of productionconsumption of biochemical factor ‘i’ dependent on the state of cell ‘j’ the values of which given in Table 2.
Metabolic rates for the biochemical species for proliferating, quiescent and necrotic cancer cells (Jiang et al. 2005)
State  Oxygen  Glucose  Waste  Growth  Inhibitor 

Proliferating  108  162  240  1  0 
Quiescent  50  80  110  0.5  1 
Necrotic  0  0  0  0  2 
Units  mM/h/cm^{3}  %/h/cm^{3} 
Cellular scale
On the microscale we employ a discrete cellbased model where each cell is represented by a set of variables that uniquely characterize it (Harjanto 2010). These are: the location, the state of each cell (proliferating, quiescent and necrotic) and the number of mitoses that separate it from the original cancer cell that gave rise to the tumor. At this scale our proposed model extends to the analogous models that are found in literature, by introducing a health level function h. This function represents the overall condition of each cell and is correlated with/to the equivalent amount of Adenosine 5triphosphate (ATP) molecules of that each cell has got (Knowles 1980).
This is a fundamental concept of the proposed model which is explicitly justified as follows. Adenosine triphosphate is a nucleoside triphosphate used in cells as a coenzyme. According to Knowles (1980) ATP is the ‘molecular unit of currency’ of intracellular energy transfer. While Campbell states that ‘ATP transports chemical energy within cells for metabolism. It is one of the end products of photophosphorylation, cellular respiration, and fermentation and is used by enzymes and structural proteins in many cellular processes, such as biosynthetic reactions, motility, and cell division’ (Neil et al. 2004). Specifically, a molecule of ATP contains three phosphate groups, and it is produced by a wide variety of enzymes, e.g. ATP synthase, from adenosine diphosphate (ADP) or adenosine monophosphate (AMP) along with various phosphate group donors. Substrate level phosphorylation, oxidative phosphorylation in cellular respiration, and photophosphorylation in photosynthesis are the three major mechanisms of ATP biosynthesis. Metabolic processes that use ATP as energy source convert it back to its precursors. Therefore ATP is continuously recycled in organisms. ATP is used by cells ‘as a substrate in signal transduction pathways by kinases that phosphorylate proteins and lipids, as well as by adenylate cyclase, which uses ATP to produce the second messenger molecule cyclic AMP. The ratio of ATP over AMP used as a way for a cell to sense how much energy is available, and control the metabolic pathways that produce and consume ATP’ (Hardie and Hawley 2001). According to Warburg et al. (1956; Hardie and Hawley 2001; Moses 1962) the rate of respiration in cancer cells is within an order of accuracy, identical to that of normal cells, while the glucose uptake is approximately 10 times higher than that of a normal cell. Furthermore, the complete combustion of glucose through the citric acid cycle and the electrontransport chain is at about 200 times decreased against the high rates of the anaerobic glycolysis process with lactic acid as an end product. All glucose molecules that are taken up, are primarily oxidized via respiration, while the remaining glucose molecules split to form lactic acid (Tiedemann 1952). Thus Warburg stated that although the anaerobic glycolysis yields only a fraction of the necessary cell energy (2 ATP), compared to the complete combustion of glucose (32 ATP). This relatively inefficient metabolic pathway is extremely preferable by the rapidly growing tumor cells even in the presence of oxygen. Recent studies, based on a more accurate estimation of ATP yields, during the oxidative phosphorylation steps, show that the complete oxidation of glucose rarely produces the full potential of 32 ATPs while a value of 30 ATPs is a more accurate estimation (Hinkle et al. 1991). In cancer cells this process is taking place reversely, where the anaerobic glycolysis acts as the mainstream metabolic pathway of glucose, whereas the Krebs cycle and oxidative phosphorylation have a supportive role (Gatenby and Gillies 2004).
Evaluation of parameters
In order to perform the following simulations, some more estimations and assumptions have to be made. These approximations are necessary since most of the medical and biological data that are available in the literature are qualitative ones. Model parameter extraction from biological data and estimations are presented in the following paragraphs.
Cell size
In the following simulations we assume that the tumor consists of a number of eukaryotic cells. Literature states that eukaryotic cell size varies between 10 and 30 μm, thus the approximation of 20 μm was used for the diameter of the cells.
Metabolic rates
In accordance to the principles of Warburg et al. we propose that the glucose absorbed by each cell, will follow both metabolic paths. The amount of glucose that can be aerobically metabolized will do so in the mitochondria abiding to the stoichiometric laws using six parts of oxygen and one part of glucose to produce 30 parts of ATPs and the remaining glucose will be converted to 2 parts of ATP per glucose. Combining the work of Warburg et al. to the cell’s metabolic rates experimentally produced by Jiang et al. based on EMT6/Ro mouse mammary tumor cell line (Jiang et al. 2005), specific rates at which the proposed biochemical species produced or consumed from the tumor cells are provided on Table 2. Including the aforementioned methodology we are able to derive a new set of metabolic rates for each state of cancer cells as shown in the following examples:
Example of proliferating cell
18 mM/h/cm\(^{3}\) glucose will aerobically metabolize using the 108 mM/h/cm\(^{3}\) of oxygen and will produce 540 mM/h/cm\(^{3}\) ATP. The remaining 144 mM/h/cm\(^{3}\) of glucose will produce 252 mM/h/cm\(^{3}\) ATPs. Thus the total amount of ATP produced in the proliferating cells is 828 mM/h/cm\(^{3}\). In what follows, this value will be indicated as \(H_{P}\).
Example of quiescent cells
8.3 mM/h/cm\(^{3}\) of glucose will aerobically metabolize with the 50 mM/h/cm\(^{3}\) of oxygen and produce 250 mM/h/cm\(^{3}\) ATPs. The remaining 71.7 mM/h/cm\(^{3}\) of glucose will produce 143.4 mM/h/cm\(^{3}\) ATP. Thus the total amount of ATP produced in quiescent cells is 393.4 mM/h/cm\(^{3}\). In what follows of the present manuscript his value will be indicated as \(H_{Q}\).
Health function
Oxygen

\(\left[ HB\right]\) is the tetra haemoglobin concentration in blood that is equal to \(0.0048\, \text{mM/cm}^{3}\)

\(P_{O_{2}}\) is the oxyphoric power of tetra haemoglobin and is equal to 4

h is the Hill coefficient equal to 2.73 (Hill 1910)

\(P_{50}\) the value of \(P_{O2}\) at which haemoglobin is 50% saturated equal to 26 mmHg

\(\alpha\) is the solubility coefficient equal to \(1.39\times 10\times 10^{3}\,\text{mmol.L}^{1}.\text{(mmHg)}^{1}\)
Glucose
In humans, the normal glucose concentration of extracellular fluid that is regulated by homeostasis is presumed to have a value of 0.005 mM/cm^{3}.
Waste–growth and inhibitor factors
For these biochemical species we assume that all the factors that are produced by the tumor cells are nullified when they are diluted through blood circulation inside the human body. Thus an assumption is made that the Dirichlet data for these species at the ECM are set equal to zero.
Mitosis

\(h_{t}\) is the health value of the mother cell at time t

M corresponds to the mean probability of mitosis and it’s a function of \(r, \mu , C_{GF}, C_{GI}\)

r corresponds to the average mitosis probability of a normal tissue cell with value of 0.0315 as described by RamisConde et al. (2008)

\(\mu\) corresponds to the effect on mitosis of the number of phenotype mutations

\(\sigma\) is the standard deviation of mitosis probability

\(C_{GF}\) and \(C_{GI}\) the concentrations for, growth factors and growth inhibitor factors respectively

N corresponds to the number of generations that separate each cell from its original cancer cell

\(\epsilon (t)\) is the Gaussian noise term
Cellular adhesion
Cellular adhesion is employed in the proposed model by implementing the differential adhesion hypothesis (DAH) as postulated by Malcolm Steinberg (2007), to model the mechanism of adhesion of the cells in the same state. The theory of cell adhesion advanced, to explain the mechanism by which heterotypic cells in mixed aggregates sort out into isotypic territories. The DAH postulates that tissues are viscoelastic liquids, and as such possess measurable tissue surface tensions. These surface tensions have been determined for a variety of tissues, including embryonic tissues. The surface tensions correspond to the mutual sorting behavior: the tissue type with the higher surface tension will occupy an internal position with regard to a tissue with a lower surface tension. Differences in homo and heterotypic adhesion presumed to be adequate to account for the phenomenon without requiring cell type specific adhesion systems. According to DAH, cellular movement and assortment is governed by the spontaneous rearrangement of cells, in much the same way as a liquid to obtain a more thermodynamically stable equilibrium. This is achieved by maximizing the amount of energy that is utilized in adhering the cells together, which decreases the free energy available in the system.
As cells with similar strengths of surface adhesion bond to each other, bonding energy in the overall system increases, and interfacial free energy decreases causing the arrangement to be thermodynamically more stable. Liquids behave in a similar manner, but with molecules moving due to their kinetic energy instead of motile cells moving around due to a combination of their kinetics and active movement. This allows examples of tissue spatial relation to correspond to the behavior of liquids, such as one tissue spreading across another corresponding to oil spreading across water; the oil spreads crossways the water to minimize weak oilwater interactions and maximize stronger waterwater and oiloil interactions, the cells similarly sort themselves to be near other cells of similar adhesive strength and bond with them. Other tissue interactions that DAH offers an explanation for included tissue hierarchy, are tissues with weaker surface adhesion that surround tissue with stronger surface adhesion, the rounding of irregular cell masses to become spherical, and the cell sorting and construction of anatomical structures that occurs independent of the path taken. This DAH mechanism forces the cells of the tumor, especially the proliferating cells, to remain connected to the tumor and not roam freely inside the simulated area while reinforcing the creation of agglomerations of necrotic cells.
Chemotaxis

all cells are considered to be incompressible bodies

between two adjacent cells an adhesion force is assumed in the form of a spring. Each of these springs forces attraction between the cells equal to \(F = kd\), as defined by Hooke’s law, where k is a constant factor characteristic of the spring, and d is the distance between the two neighboring cells.

for proliferating cells subjected to chemotaxis, an extra force is assumed moving the cell on a vector of the nutrientwaste function, give in (19)
Declarations
Authors' contributions
PA conceived the proposed model, carried out the implementation and the simulations, drafted the manuscript. MH conceived and designed the research, accomplished overall supervision and coordination, drafted the manuscript. GD participated in the mathematical formulation, drafted the manuscript HPK accomplished supervision of the formulation of the biological/medical aspects of the model, drafted the manuscript MNV assisted in verification and model’s reduction to known results, drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
All the authors are grateful to A.Sgourou and F.Kariotou for the fruitful discussions, during the preparation of the present manuscript. PA and MH acknowledge: This research has been cofinanced by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)  Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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