Dynamics between Cancer Cell Subpopulations Reveals a Model Coordinating with Both Hierarchical and Stochastic Concepts

Tumors are often heterogeneous in which tumor cells of different phenotypes have distinct properties. For scientific and clinical interests, it is of fundamental importance to understand their properties and the dynamic variations among different phenotypes, specifically under radio- and/or chemo-therapy. Currently there are two controversial models describing tumor heterogeneity, the cancer stem cell (CSC) model and the stochastic model. To clarify the controversy, we measured probabilities of different division types and transitions of cells via in situ immunofluorescence. Based on the experiment data, we constructed a model that combines the CSC with the stochastic concepts, showing the existence of both distinctive CSC subpopulations and the stochastic transitions from NSCCs to CSCs. The results showed that the dynamic variations between CSCs and non-stem cancer cells (NSCCs) can be simulated with the model. Further studies also showed that the model can be used to describe the dynamics of the two subpopulations after radiation treatment. More importantly, analysis demonstrated that the experimental detectable equilibrium CSC proportion can be achieved only when the stochastic transitions from NSCCs to CSCs occur, indicating that tumor heterogeneity may exist in a model coordinating with both the CSC and the stochastic concepts. The mathematic model based on experimental parameters may contribute to a better understanding of the tumor heterogeneity, and provide references on the dynamics of CSC subpopulation during radiotherapy.     

Three‐dimensional cell culture model utilization in cancer stem cell research

Three‐dimensional (3D) cell culture models are becoming increasingly popular in contemporary cancer research and drug resistance studies. Recently, scientists have begun incorporating cancer stem cells (CSCs) into 3D models and modifying culture components in order to mimic in vivo conditions better. Currently, the global cell culture market is primarily focused on either 3D cancer cell cultures or stem cell cultures, with less focus on CSCs. This is evident in the low product availability officially indicated for 3D CSC model research. This review discusses the currently available commercial products for CSC 3D culture model research. Additionally, we discuss different culture media and components that result in higher levels of stem cell subpopulations while better recreating the tumor microenvironment. In summary, although progress has been made applying 3D technology to CSC research, this technology could be further utilized and a greater number of 3D kits dedicated specifically to CSCs should be implemented.     

Cancer stem cells: the lessons from pre-cancerous stem cells

How a cancer is initiated and established remains elusive despite all the advances in decades of cancer research. Recently the cancer stem cell (CSC) hypothesis has been revived, challenging the long-standing model of "clonal evolution" for cancer development and implicating the dawning of a potential cure for cancer [1]. The recent identification of precancerous stem cells (pCSCs) in cancer, an early stage of CSC development, however, implicates that the "clonal evolution" is not contradictory to the CSC hypothesis, but is rather an aspect of the process of CSC development [2]. The discovery of pCSC has revealed and will continue to reveal the volatile properties of CSC with respects to their phenotype, differentiation and tumorigenic capacity during initiation and progression. Both pCSC and CSC might also serve as precursors of tumor stromal components such as tumor vasculogenic stem/progenitor cells (TVPCs). Thus, the CSC hypothesis covers the developing process of tumor-initiating cells (TIC) --> pCSC --> CSC --> cancer, a cellular process that should parallel the histological process of hyperplasia/metaplasia (TIC) --> precancerous lesions (pCSC) --> malignant lesions (CSC --> cancer). The embryonic stem (ES) cell and germline stem (GS) cell genes are subverted in pCSCs. Especially the GS cell protein piwil2 may play an important role during the development of TIC --> pCSC --> CSC, and this protein may be used as a common biomarker for early detection, prevention, and treatment of cancer. As cancer stem cell research is yet in its infancy, definitive conclusions regarding the role of pCSC can not be made at this time. However this review will discuss what we have learned from pCSC and how this has led to innovative ideas that may eventually have major impacts on the understanding and treatment of cancer.     

Sea urchin embryo, DNA-damaged cell cycle checkpoint and the mechanisms initiating cancer development

Cell division is an essential process for heredity, maintenance and evolution of the whole living kingdom. Sea urchin early development represents an excellent experimental model for the analysis of cell cycle checkpoint mechanisms since embryonic cells contain a functional DNA-damage checkpoint and since the whole sea urchin genome is sequenced. The DNA-damaged checkpoint is responsible for an arrest in the cell cycle when DNA is damaged or incorrectly replicated, for activation of the DNA repair mechanism, and for commitment to cell death by apoptosis in the case of failure to repair. New insights in cancer biology lead to two fundamental concepts about the very first origin of cancerogenesis. Cancers result from dysfunction of DNA-damaged checkpoints and cancers appear as a result of normal stem cell (NCS) transformation into a cancer stem cell (CSC). The second aspect suggests a new definition of "cancer", since CSC can be detected well before any clinical evidence. Since early development starts from the zygote, which is a primary stem cell, sea urchin early development allows analysis of the early steps of the cancerization process. Although sea urchins do not develop cancers, the model is alternative and complementary to stem cells which are not easy to isolate, do not divide in a short time and do not divide synchronously. In the field of toxicology and incidence on human health, the sea urchin experimental model allows assessment of cancer risk from single or combined molecules long before any epidemiologic evidence is available. Sea urchin embryos were used to test the worldwide used pesticide Roundup that contains glyphosate as the active herbicide agent; it was shown to activate the DNA-damage checkpoint of the first cell cycle of development. The model therefore allows considerable increase in risk evaluation of new products in the field of cancer and offers a tool for the discovery of molecular markers for early diagnostic in cancer biology. Prevention and early diagnosis are two decisive elements of human cancer therapy.     

Cancer stem cells: A guide for skeptics

Cancer stem cells (CSC) were postulated to exist many years ago as cells within a tumor that regenerate the tumor following treatment. A stochastic clonal evolution model was used to explain observed tumor heterogeneity. Recently, xenotransplantation studies have demonstrated that prospectively identifiable subpopulations from human cancers can initiate tumors in immune deficient mice, and these results along with recent advances in stem cell biology have generated much excitement in the cancer field. The modern CSC theory posits a hierarchy of cells analogous to normal stem cell development. Some controversy remains, however, as to whether these tumor initiating cells truly represent CSC, and whether the modern CSC field can live up to the promise of providing improved cancer treatments based on a novel model of cancer biology. Recent data from CSC investigators are discussed critically. J. Cell. Biochem. 106: 745–749, 2009. © 2009 Wiley‐Liss, Inc.     

Spatial processes and evolutionary models: a critical review

Evolution is a fundamentally population level process in which variation, drift and selection produce both temporal and spatial patterns of change. Statistical model fitting is now commonly used to estimate which kind of evolutionary process best explains patterns of change through time using models like Brownian motion, stabilizing selection (Ornstein–Uhlenbeck) and directional selection on traits measured from stratigraphic sequences or on phylogenetic trees. But these models assume that the traits possessed by a species are homogeneous. Spatial processes such as dispersal, gene flow and geographical range changes can produce patterns of trait evolution that do not fit the expectations of standard models, even when evolution at the local‐population level is governed by drift or a typical OU model of selection. The basic properties of population level processes (variation, drift, selection and population size) are reviewed and the relationship between their spatial and temporal dynamics is discussed. Typical evolutionary models used in palaeontology incorporate the temporal component of these dynamics, but not the spatial. Range expansions and contractions introduce rate variability into drift processes, range expansion under a drift model can drive directional change in trait evolution, and spatial selection gradients can create spatial variation in traits that can produce long‐term directional trends and punctuation events depending on the balance between selection strength, gene flow, extirpation probability and model of speciation. Using computational modelling that spatial processes can create evolutionary outcomes that depart from basic population‐level notions from these standard macroevolutionary models.     

Cancer stem cell plasticity and tumor hierarchy

The origins of the complex process of intratumoral heterogeneity have been highly debated and different cellular mechanisms have been hypothesized to account for the diversity within a tumor. The clonal evolution and cancer stem cell(CSC) models have been proposed as drivers of this heterogeneity. However, the concept of cancer stem cell plasticity and bidirectional conversion between stem and non-stem cells has added additional complexity to these highly studied paradigms and may help explain the tumor heterogeneity observed in solid tumors. The process of cancer stem cell plasticity in which cancer cel s harbor the dynamic ability of shifting from a non-CSC state to a CSC state and vice versa may be modulated by specific microenvironmental signals and cellular interactions arising in the tumor niche. In addition to promoting CSC plasticity, these interactions may contribute to the cellular transformation of tumor cells and affect response to chemotherapeutic and radiation treatments by providing CSCs protection from these agents. Herein, we review the literature in support of this dynamic CSC state, discuss the effectors of plasticity, and examine their role in the development and treatment of cancer.     

The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed “tumor growth paradox”—accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.     

肿瘤干细胞及其模型的研究现状及临床治疗最新进展

癌症发生最有名的模型之一肿瘤CSCcancer stem cell(CSC)模型已经被确立为一种细胞机制,有助于体现不同类型的肿瘤表型和功能的异质性。然而,最新研究突出显示了肿瘤CSC模型的复杂性和挑战性:CSC表型在不同的病人间有本质上的不同,肿瘤可能含有多种表型或遗传上不同的CSC,转移的肿瘤CSC可能由原始肿瘤CSC进展而来,肿瘤细胞可进行可逆的表型的改变。虽然在特定情况下,肿瘤CSC的概念具有临床相关性,但越来越多的证据表明,定位、鉴定出肿瘤灶中所有亚型的肿瘤CSC将对临床上预防肿瘤复发具有特殊意义。有以下几个因素可以促使异质性的发生:基因突变,后生的变化,与微环境间的相互作用以及存在或缺失细胞分级等。肿瘤异质性可以用不同的细胞机制来解释。虽然CSC具有CSC自我更新和分化的特性,但是它们不一定是正常组织CSC转化而形成的。这种CSC模型引起了科研工作者广泛的关注,我们结合新近的相关文献综述如下。     

Revisiting the role of lysophosphatidic acid in stem cell biology

Stem cells possess unique biological characteristics such as the ability to self-renew and to undergo multilineage differentiation into specialized cells. Whereas embryonic stem cells (ESC) can differentiate into all cell types of the body, somatic stem cells (SSC) are a population of stem cells located in distinct niches throughout the body that differentiate into the specific cell types of the tissue in which they reside in. SSC function mainly to restore cells as part of normal tissue homeostasis or to replenish cells that are damaged due to injury. Cancer stem-like cells (CSC) are said to be analogous to SSC in this manner where tumor growth and progression as well as metastasis are fueled by a small population of CSC that reside within the corresponding tumor. Moreover, emerging evidence indicates that CSC are inherently resistant to chemo- and radiotherapy that are often the cause of cancer relapse. Hence, major research efforts have been directed at identifying CSC populations in different cancer types and understanding their biology. Many factors are thought to regulate and maintain cell stemness, including bioactive lysophospholipids such as lysophosphatidic acid (LPA). In this review, we discuss some of the newly discovered functions of LPA not only in the regulation of CSC but also normal SSC, the similarities in these regulatory functions, and how these discoveries can pave way to the development of novel therapies in cancer and regenerative medicine.     

Modeling evolutionary dynamics of epigenetic mutations in hierarchically organized tumors

The cancer stem cell (CSC) concept is a highly debated topic in cancer research. While experimental evidence in favor of the cancer stem cell theory is apparently abundant, the results are often criticized as being difficult to interpret. An important reason for this is that most experimental data that support this model rely on transplantation studies. In this study we use a novel cellular Potts model to elucidate the dynamics of established malignancies that are driven by a small subset of CSCs. Our results demonstrate that epigenetic mutations that occur during mitosis display highly altered dynamics in CSC-driven malignancies compared to a classical, non-hierarchical model of growth. In particular, the heterogeneity observed in CSC-driven tumors is considerably higher. We speculate that this feature could be used in combination with epigenetic (methylation) sequencing studies of human malignancies to prove or refute the CSC hypothesis in established tumors without the need for transplantation. Moreover our tumor growth simulations indicate that CSC-driven tumors display evolutionary features that can be considered beneficial during tumor progression. Besides an increased heterogeneity they also exhibit properties that allow the escape of clones from local fitness peaks. This leads to more aggressive phenotypes in the long run and makes the neoplasm more adaptable to stringent selective forces such as cancer treatment. Indeed when therapy is applied the clone landscape of the regrown tumor is more aggressive with respect to the primary tumor, whereas the classical model demonstrated similar patterns before and after therapy. Understanding these often counter-intuitive fundamental properties of (non-)hierarchically organized malignancies is a crucial step in validating the CSC concept as well as providing insight into the therapeutical consequences of this model.     

Tumor initiating cells in pancreatic cancer: A critical view

Emerging evidence points to the existence of pancreatic cancer stem cells (CSC) as the culprit in the initiation, maintenance, metastasis, and treatment resistance of pancreatic cancer. The existence of such a cell population would have an important impact on the design of novel therapies against this devastating disease. However, no in vivo validation or rebuttal of the pancreatic CSC hypothesis exists. Major backlashes in the discussion on CSC are firstly, the confusion between the terms CSC and cell of origin of pancreatic ductal adenocarcinoma (PDAC), secondly the ambiguity of the cell of origin itself and thirdly, the fact that the CSC hypothesis is based on cell sorting and xenografting experiments; the latter of which often precludes solid conclusions because of the lack of a natural microenvironment and differences in drug delivery. Nonetheless, recent studies in other cancers partially support the CSC hypothesis by demonstrating a link between epithelial-to-mesenchymal transdifferentiation/transition (EMT) and CSC properties. Such a link is again open to dispute as EMT is a reversible process which is highly dependent on major oncogenic pathways in PDAC [e.g. K-Ras, transforming growth factor-β (TGF-β)] rather than on presumed cancer stem cell pathways. Hence, the available evidence does not robustly support the CSC concept in PDAC and a thorough validation of this hypothesis in well-defined genetically engineered mouse models of pancreatic cancer is required.     

Identification and targeting of cancer stem cells

Cancer stem cells (CSC) represent malignant cell subsets in hierarchically organized tumors, which are selectively capable of tumor initiation and self‐renewal and give rise to bulk populations of non‐tumorigenic cancer cell progeny through differentiation. Robust evidence for the existence of prospectively identifiable CSC among cancer bulk populations has been generated using marker‐specific genetic lineage tracking of molecularly defined cancer subpopulations in competitive tumor development models. Moreover, novel mechanisms and relationships have been discovered that link CSC to cancer therapeutic resistance and clinical tumor progression. Importantly, proof‐of‐principle for the potential therapeutic utility of the CSC concept has recently been provided by demonstrating that selective killing of CSC through a prospective molecular marker can inhibit tumor growth. Herein, we review these novel and translationally relevant research developments and discuss potential strategies for CSC‐targeted therapy in the context of resistance mechanisms and molecular pathways preferentially operative in CSC.     

Phylogenetic mixture models can reduce node-density artifacts

We investigate the performance of phylogenetic mixture models in reducing a well-known and pervasive artifact of phylogenetic inference known as the node-density effect, comparing them to partitioned analyses of the same data. The node-density effect refers to the tendency for the amount of evolutionary change in longer branches of phylogenies to be underestimated compared to that in regions of the tree where there are more nodes and thus branches are typically shorter. Mixture models allow more than one model of sequence evolution to describe the sites in an alignment without prior knowledge of the evolutionary processes that characterize the data or how they correspond to different sites. If multiple evolutionary patterns are common in sequence evolution, mixture models may be capable of reducing node-density effects by characterizing the evolutionary processes more accurately. In gene-sequence alignments simulated to have heterogeneous patterns of evolution, we find that mixture models can reduce node-density effects to negligible levels or remove them altogether, performing as well as partitioned analyses based on the known simulated patterns. The mixture models achieve this without knowledge of the patterns that generated the data and even in some cases without specifying the full or true model of sequence evolution known to underlie the data. The latter result is especially important in real applications, as the true model of evolution is seldom known. We find the same patterns of results for two real data sets with evidence of complex patterns of sequence evolution: mixture models substantially reduced node-density effects and returned better likelihoods compared to partitioning models specifically fitted to these data. We suggest that the presence of more than one pattern of evolution in the data is a common source of error in phylogenetic inference and that mixture models can often detect these patterns even without prior knowledge of their presence in the data. Routine use of mixture models alongside other approaches to phylogenetic inference may often reveal hidden or unexpected patterns of sequence evolution and can improve phylogenetic inference.     

Cancer stem cell, niche and EGFR decide tumor development and treatment response: A bio-computational simulation study

Recent research in cancer biology has suggested the hypothesis that tumors are initiated and driven by a small group of cancer stem cells (CSCs). Furthermore, cancer stem cell niches have been found to be essential in determining fates of CSCs, and several signaling pathways have been proven to play a crucial role in cellular behavior, which could be two important factors in cancer development. To better understand the progression, heterogeneity and treatment response of breast cancer, especially in the context of CSCs, we propose a mathematical model based on the cell compartment method. In this model, three compartments of cellular subpopulations are constructed: CSCs, progenitor cells (PCs), and terminal differentiated cells (TCs). Moreover, (1) the cancer stem cell niche is, considered by modeling its effect on division patterns (symmetric or asymmetric) of CSCs, and (2) the EGFR signaling pathway is integrated by modeling its role in cell proliferation, apoptosis. Our simulation results indicate that (1) a higher probability for symmetric division of CSC may result in a faster expansion of tumor population, and for a larger number of niches, the tumor grows at a slower rate, but the final tumor volume is larger; (2) higher EGFR expression correlates to tumors with larger volumes while a saturation function is observed, and (3) treatments that inhibit tyrosine kinase activity of EGFR may not only repress the tumor volume, but also decrease the CSCs percentages by shifting CSCs from symmetric divisions to asymmetric divisions. These findings suggest that therapies should be designed to effectively control or eliminate the symmetric division of CSCs and to reduce or destroy the CSC niches.     

Cell-to-cell signaling influences the fate of prostate cancer stem cells and their potential to generate more aggressive tumors

An increasing number of malignancies has been shown to be initiated and propelled by small subpopulations of cancer stem cells (CSC). However, whether tumor aggressiveness is driven by CSC and by what extent this property may be relevant within the tumor mass is still unsettled. To address this issue, we isolated a rare tumor cell population on the basis of its CD44(+)CD24(-) phenotype from the human androgen-independent prostate carcinoma cell line DU145 and established its CSC properties. The behavior of selected CSC was investigated with respect to the bulk DU145 cells. The injection of CSC in nude mice generated highly vascularized tumors infiltrating the adjacent tissues, showing high density of neuroendocrine cells and expressing low levels of E-cadherin and β-catenin as well as high levels of vimentin. On the contrary, when a comparable number of unsorted DU145 cells were injected the resulting tumors were less aggressive. To investigate the different features of tumors in vivo, the influence of differentiated tumor cells on CSC was examined in vitro by growing CSC in the absence or presence of conditioned medium from DU145 cells. CSC grown in permissive conditions differentiated into cell populations with features similar to those of cells held in aggressive tumors generated from CSC injection. Differently, conditioned medium induced CSC to differentiate into a cell phenotype comparable to cells of scarcely aggressive tumors originated from bulk DU145 cell injection. These findings show for the first time that CSC are able to generate differentiated cells expressing either highly or scarcely aggressive phenotype, thus influencing prostate cancer progression. The fate of CSC was determined by signals released from tumor environment. Moreover, using microarray analysis we selected some molecules which could be involved in this cell-to-cell signaling, hypothesizing their potential value for prognostic or therapeutic applications.     

Evolution and Phenotypic Selection of Cancer Stem Cells

Cells of different organs at different ages have an intrinsic set of kinetics that dictates their behavior. Transformation into cancer cells will inherit these kinetics that determine initial cell and tumor population progression dynamics. Subject to genetic mutation and epigenetic alterations, cancer cell kinetics can change, and favorable alterations that increase cellular fitness will manifest themselves and accelerate tumor progression. We set out to investigate the emerging intratumoral heterogeneity and to determine the evolutionary trajectories of the combination of cell-intrinsic kinetics that yield aggressive tumor growth. We develop a cellular automaton model that tracks the temporal evolution of the malignant subpopulation of so-called cancer stem cells(CSC), as these cells are exclusively able to initiate and sustain tumors. We explore orthogonal cell traits, including cell migration to facilitate invasion, spontaneous cell death due to genetic drift after accumulation of irreversible deleterious mutations, symmetric cancer stem cell division that increases the cancer stem cell pool, and telomere length and erosion as a mitotic counter for inherited non-stem cancer cell proliferation potential. Our study suggests that cell proliferation potential is the strongest modulator of tumor growth. Early increase in proliferation potential yields larger populations of non-stem cancer cells(CC) that compete with CSC and thus inhibit CSC division while a reduction in proliferation potential loosens such inhibition and facilitates frequent CSC division. The sub-population of cancer stem cells in itself becomes highly heterogeneous dictating population level dynamics that vary from long-term dormancy to aggressive progression. Our study suggests that the clonal diversity that is captured in single tumor biopsy samples represents only a small proportion of the total number of phenotypes.     

Cancer stem cells--old concepts, new insights

Cancer has long been viewed as an exclusively genetic disorder. The model of carcinogenesis, postulated by Nowell and Vogelstein, describes the formation of a tumor by the sequential accumulation of mutations in oncogenes and tumor suppressor genes. In this model, tumors are thought to consist of a heterogeneous population of cells that continue to acquire new mutations, resulting in a highly dynamic process, with clones that out compete others due to increased proliferative or survival capacity. However, novel insights in cancer stem cell research suggest another layer of complexity in the process of malignant transformation and preservation. It has been reported that only a small fraction of the cancer cells in a malignancy have the capacity to propagate the tumor upon transplantation into immuno-compromised mice. Those cells are termed 'cancer stem cells' (CSC) and can be selected based on the expression of cell surface markers associated with immature cell types. In this review, we will critically discuss these novel insights in CSC-related research. Where possible we integrate these results within the genetic model of cancer and illustrate that the CSC model can be considered an extension of the classic genetic model rather than a contradictory theory. Finally, we discuss some of the most controversial issues in this field.     

Identifying dynamical persistent biomarker structures for rare events using modern integrative machine learning approach

The evolution of omics and computational competency has accelerated discoveries of the underlying biological processes in an unprecedented way. High throughput methodologies, such as flow cytometry, can reveal deeper insights into cell processes, thereby allowing opportunities for scientific discoveries related to health and diseases. However, working with cytometry data often imposes complex computational challenges due to high-dimensionality, large size, and nonlinearity of the data structure. In addition, cytometry data frequently exhibit diverse patterns across biomarkers and suffer from substantial class imbalances which can further complicate the problem. The existing methods of cytometry data analysis either predict cell population or perform feature selection. Through this study, we propose a “wisdom of the crowd” approach to simultaneously predict rare cell populations and perform feature selection by integrating a pool of modern machine learning (ML) algorithms. Given that our approach integrates superior performing ML models across different normalization techniques based on entropy and rank, our method can detect diverse patterns existing across the model features. Furthermore, the method identifies a dynamic biomarker structure that divides the features into persistently selected, unselected, and fluctuating assemblies indicating the role of each biomarker in rare cell prediction, which can subsequently aid in studies of disease progression.     

Non-local Parabolic and Hyperbolic Models for Cell Polarisation in Heterogeneous Cancer Cell Populations

Tumours consist of heterogeneous populations of cells. The sub-populations can have different features, including cell motility, proliferation and metastatic potential. The interactions between clonal sub-populations are complex, from stable coexistence to dominant behaviours. The cell–cell interactions, i.e. attraction, repulsion and alignment, processes critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this study, we develop a mathematical model describing cancer cell invasion and movement for two polarised cancer cell populations with different levels of mutation. We consider a system of non-local hyperbolic equations that incorporate cell–cell interactions in the speed and the turning behaviour of cancer cells, and take a formal parabolic limit to transform this model into a non-local parabolic model. We then investigate the possibility of aggregations to form, and perform numerical simulations for both hyperbolic and parabolic models, comparing the patterns obtained for these models.