From passive diffusion to active cellular migration in mathematical models of tumour invasion

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction-diffusion models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface.     

Cellular automaton simulation of tumour growth -- equivocal relationships between simulation parameters and morphologic pattern features.

OBJECTIVE: To develop an interpretation procedure which estimates simulation parameters (tumour cell motility, tumour cell adhesion, autocrine and paracrine growth control, stroma destruction) of simulated patterns solely based on morphometric features of the morphologic pattern. METHODS: A cellular automaton computer simulation program was developed which produces morphologic patterns by growth of a seed of tumour cells. At the beginning of each simulation run certain simulation parameters are assigned to the tumour cells. After the run has been completed, the resulting pattern is evaluated by a set of morphometric features. Simulation parameters and resulting morphometric features of 27,800 simulations were stored in a database and were used for the evaluation of potential relationships. RESULTS: Correlation analysis showed highly significant correlations between morphometric features on the one hand and the preset simulation parameters (tumour cell motility, tumour cell adhesion, autocrine and paracrine growth control, stroma destruction) on the other. Correlation coefficients, however, varied from 0.72 to 0.99. When only one simulation parameter varied while all others were kept constant, morphometric features yielded a highly reliable estimate of the particular simulation parameter. When variability was extended to 4 simulation parameters, morphometric features were less effective in estimating the setting of the parameters. Though in all patterns tested several possible simulation parameter constellations could be ruled out, morphometric features were usually compatible with more than one set of simulation parameters thus preventing a straightforward interpretation. CONCLUSIONS: Though simulation parameters significantly and reproducibly influence the resulting morphologic pattern as characterized by morphometric features, estimates of the simulation parameters based on morphometric features yield equivocal results.     

A cellular automaton model for the proliferation of migrating contact-inhibited cells.

A cellular automaton is used to develop a model describing the proliferation dynamics of populations of migrating, contact-inhibited cells. Simulations are carried out on two-dimensional networks of computational sites that are finite-state automata. The discrete model incorporates all the essential features of the cell locomotion and division processes, including the complicated dynamic phenomena occurring when cells collide. In addition, model parameters can be evaluated by using data from long-term tracking and analysis of cell locomotion. Simulation results are analyzed to determine how the competing processes of contact inhibition and cell migration affect the proliferation rates. The relation between cell density and contact inhibition is probed by following the temporal evolution of the population-average speed of locomotion. Our results show that the seeding cell density, the population-average speed of locomotion, and the spatial distribution of the seed cells are crucial parameters in determining the temporal evolution of cell proliferation rates. The model successfully predicts the effect of cell motility on the growth of isolated megacolonies of keratinocytes, and simulation results agree very well with experimental data. Model predictions also agree well with experimentally measured proliferation rates of bovine pulmonary artery endothelial cells (BPAE) cultured in the presence of a growth factor (bFGF) that up-regulates cell motility.     

A phenotypic switch in the dispersal strategy of breast cancer cells selected for metastatic colonization

An important question in cancer evolution concerns which traits make a cell likely to successfully metastasize. Cell motility phenotypes, mediated by cell shape change, are strong candidates. We experimentally evolved breast cancer cells in vitro for metastatic capability, using selective regimes designed to simulate stages of metastasis, then quantified their motility behaviours using computer vision. All evolved lines showed changes to motility phenotypes, and we have identified a previously unknown density-dependent motility phenotype only seen in cells selected for colonization of decellularized lung tissue. These cells increase their rate of morphological change with an increase in migration speed when local cell density is high. However, when the local cell density is low, we find the opposite relationship: the rate of morphological change decreases with an increase in migration speed. Neither the ancestral population, nor cells selected for their ability to escape or invade extracellular matrix-like environments, displays this dynamic behavioural switch. Our results suggest that cells capable of distant-site colonization may be characterized by dynamic morphological phenotypes and the capacity to respond to the local social environment.     

Measuring Proliferation In Routine Fine Needle Aspirates. Immunocytochemical Detection of Bromodeoxyuridine Incorporation and Ki-67 Expression In Breast Aspirates

Two simple quantitative means of measuring tumour proliferation which can be applied to cytological material are described. One method involves immunocytochemical staining of cytological smears prepared from breast aspirates with the monoclonal antibody Ki-67. The other method involves incubation of aspirated material with 5-Bromo-2-deoxyuridine (BrdU). Direct measurement of the S phase of the cell cycle is feasible in breast fine needle aspirates by Bromodeoxyuridine incorporation and subsequent immunocytochemical detection. The proliferation indices obtained correlate with those derived from Ki-67 staining. This technique is suitable for routine use in the assessment of tumour proliferation.     

Probabilistic model of the spatial distribution of muscle fibres in human muscles

An analytical model and computer simulation model for measuring fibre density in motor units of human skeletal muscles have been described. The model was developed for Gaussian distribution of the fibres in the motor unit territory. It has been shown that fibre density measurement using a triggering fibre was a biased estimate of the actual density of the fibres in the territory. The effects of varying the standard deviation of the spatial distribution on the estimate of fibre density has been investigated, and it has been shown that for high values of standard deviation a uniform distribution of the fibres in the territory was a good first order approximation.     

Embryonic Tissue Interactions as the Basis for Morphological Change in Evolution

Phenotypic changes over geological time must result from alterationsin the morphogenetic mechanisms associated with organogenesis.Recent advances in our understanding of such mechanisms suggestthat there are certain patterns of metabolic activity and organellesynthesis which are present in many different cell lineagesat different times during embryonic development, and that thisubiquity of genomic potential is available for expression atany time. Microevolutionary sequences of morphological changeprobably result from modulations of quantitative aspects oforganogenesis, e.g., rates of cell proliferation, or cell density.It is also possible that certain macroevolutionary steps ("neomorphs")may result from qualitative changes in the developmental programs,and such events underly "ultimate refinement" of morphologicaladaptation. The selection pressures involved in these differentsequences of morphological change "see" embryonic processesin different ways. This thesis is illustrated with referenceto the evolutionary history of the tetrapod limb.     

Undirected motility of filamentous cyanobacteria produces reticulate mats

The roles of biology in the morphogenesis of microbial mats and stromatolites remain enigmatic due to the vast array of physical and chemical influences on morphology. However, certain microbial behaviors produce complex morphological features that can be directly attributed to motility patterns. Specifically, laboratory experiments with a strain of the cyanobacteria Pseudanabaena demonstrate that distinctive morphologies arise from the undirected gliding and colliding of filaments. When filamentous cells collide, they align and clump, producing intersecting ridges surrounding areas with low cell density, i.e. reticulate structures. Cell motility is essential for the development of reticulates and associated structures: filaments organize into reticulates faster than cell division and growth, and conditions that inhibit motility also inhibit reticulate formation. Cell density of the inoculum affects the frequency of cell–cell collisions, and thus the time required for biofilm organization into reticulate structures. This also affects the specific geometry of the reticulates. These patterns are propagated into larger structures as cyanobacterial cell numbers increase and cells remain motile. Thus, cell motility is important for templating and maintaining the morphology of these microbial communities, demonstrating a direct link between a microbial behavior and a community morphology. Reticulate geometries have been identified in natural microbial mats as well as in the fossil record, and these structures can be attributed to the motility of filamentous bacteria.     

Gompertzian growth pattern correlated with phenotypic organization of colon carcinoma,malignant glioma and non-small cell lung carcinoma cell lines

Abstract. In the current study we present a Gompertzian model for cell growth as a function of cell phenotype using six human tumour cell lines (A-549, NCI-H596, NCI-H520, HT-29, SW-620 and U-251). Monolayer cells in exponential growth at various densities were quantified over a week by sulforhodamine B staining assay to produce cell-growth curves. A Gompertz equation was fitted to experimental data to obtain, for each cell line, three empirical growth parameters (initial cell density, cell-growth rate and carrying capacity – the maximal cell density). A cell-shape parameter named deformation coefficient D (a morphological relationship among spreading and confluent cells) was established and compared by regression analysis with the relative growth rate parameter K described by the Gompertz equation. We have found that coefficient D is directly proportional to the growth parameter K . The fit curve significantly matches the empirical data ( P  < 0.05), with a correlation coefficient of 0.9152. Therefore, a transformed Gompertzian growth function was obtained accordingly to D . The degree of correlation between the Gompertzian growth parameter and the coefficient D allows a new interpretation of the growth parameter K on the basis of morphological measurements of a set of tumour cell types, supporting the idea that cell-growth kinetics can be modulated by phenotypic organization of attached cells.     

CELL PROLIFERATION IN EMT6 TUMOURS TREATED WITH SINGLE DOSES OF X-RAYS OR HYDROXYUREA II. COMPUTER SIMULATIONS

A computer simulation technique was used to analyse data on the proliferation of clonogenic cells in EMT6 tumours treated with 5 mg/mouse of hydroxyurea (HU) or 3·0 Gy (300 rads) X-rays. This simulation technique is able to determine the respective roles of selective killing, blocks in cell progression and recruitment of the treated population. When the technique was applied to tumours treated with HU, it was possible to prove that both a G₁/S block and recruitment occurred. These phenomena could not have been demonstrated quantitatively, or even qualitatively, without the use of the simulation. After irradiation, blocks in cell progression and differences in the proliferative patterns of the surviving clonogenic cells and the total tumour cell population were found.     

Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows

Amoeboid cell motility is essential for a wide range of biological processes including wound healing, embryonic morphogenesis, and cancer metastasis. It relies on complex dynamical patterns of cell shape changes that pose long-standing challenges to mathematical modeling and raise a need for automated and reproducible approaches to extract quantitative morphological features from image sequences. Here, we introduce a theoretical framework and a computational method for obtaining smooth representations of the spatiotemporal contour dynamics from stacks of segmented microscopy images. Based on a Gaussian process regression we propose a one-parameter family of regularized contour flows that allows us to continuously track reference points (virtual markers) between successive cell contours. We use this approach to define a coordinate system on the moving cell boundary and to represent different local geometric quantities in this frame of reference. In particular, we introduce the local marker dispersion as a measure to identify localized membrane expansions and provide a fully automated way to extract the properties of such expansions, including their area and growth time. The methods are available as an open-source software package called AmoePy, a Python-based toolbox for analyzing amoeboid cell motility (based on time-lapse microscopy data), including a graphical user interface and detailed documentation. Due to the mathematical rigor of our framework, we envision it to be of use for the development of novel cell motility models. We mainly use experimental data of the social amoeba Dictyostelium discoideum to illustrate and validate our approach.     

Image cytometry of aneuploidy, growth fraction (MoAb Ki-67) and hormone receptors (ER, PR) immunocytochemical assays in breast carcinomas

DNA nuclear content was assessed in human breast carcinomas (n = 132) using image cytometry. Optical density histograms of Feulgen stained cell imprints from fresh tissue samples, subsequently frozen for immunocytochemical assays, were determined by the SAMBA system and used for the DNA index, the ploidy balance (PB) and the proliferation index (PI) computation. The three parameters were correlated to (i) histological data (tumour grade, vascular and/or lymph node invasion) and to (ii) growth fraction (Ki67), hormone receptor antigenic sites (ER, PR) and intramedullar (bone marrow) biopsies and anti-KL1-positive epithelial cells. It was shown that 57% of breast carcinomas were aneuploid. Aneuploidy PI significantly correlated to the criteria of poor prognosis such as high tumour grade, vascular and lymphatic invasion and to increased Ki67-positive cells, and the absence of or low ER and PR. Since image cytometry is easy to handle and perfectly suitable for current diagnostic practice in pathology departments, particularly for tumour cell ploidy assessment and standardized analysis of immunostaining procedures with morphological control of the preparation, we conclude that image cytometry, as performed with the SAMBA, must be regarded as a relevant tool for prognosis evaluation and therapy guidance in individual patients.     

细胞信号转导网络的计算机网络模型构建初探

根据细胞信号转导的特点,利用计算机通信技术,提出了一个类似Internet的细胞信号转导计算机网络模型,对该模型的信息编码提出了自己的规则。并利用此模型对Grill等提出的ABA对气孔运动调节的信号转导模型进行实例化。利用wpa—Pack软件包,用VC＋＋6．0开发了封装、发送数据包的程序。仿真结果表明：细胞信号可以根据所定义的规则编码、封装后在构建的网络上进行数据传输。     

A viscoelastic model of blood capillary extension and regression: derivation, analysis, and simulation

This work studies a fundamental problem in blood capillary growth: how the cell proliferation or death induces the stress response and the capillary extension or regression. We develop a one-dimensional viscoelastic model of blood capillary extension/regression under nonlinear friction with surroundings, analyze its solution properties, and simulate various growth patterns in angiogenesis. The mathematical model treats the cell density as the growth pressure eliciting a viscoelastic response from the cells, which again induces extension or regression of the capillary. Nonlinear analysis captures two cases when the biologically meaningful solution exists: (1) the cell density decreases from root to tip, which may occur in vessel regression; (2) the cell density is time-independent and is of small variation along the capillary, which may occur in capillary extension without proliferation. The linear analysis with perturbation in cell density due to proliferation or death predicts the global biological solution exists provided the change in cell density is sufficiently slow in time. Examples with blow-ups are captured by numerical approximations and the global solutions are recovered by slow growth processes, which validate the linear analysis theory. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.     

Expression of CD40, CD44, bcl-2 antigens and rate of cell proliferation on fine needle aspirates from metastatic melanoma

The clinical behaviour of melanoma is often unpredictable using clinical and histological criteria. Tumour cell markers related to cell cycle regulation, apoptosis, cell-cell interactions and cell proliferation might improve the possibility of predicting the clinical course of melanoma. The aim of the present study was to refine prognostic criteria by an immunocytochemical investigation of CD44, CD40, bcl-2 antigens and cell proliferation in tumour cells aspirated from metastases of malignant melanoma. CD40 is a cell surface receptor shown to be expressed by lymphomas as well as carcinomas, and is thought to play a central role in the process of tumour progression. CD44 is a transmembrane glycoprotein, which is involved in growth signal transmission of importance in the binding of tumour cells to endothelium, cell migration and enhancement of cell motility, which makes it of interest to study in relation to the metastasizing capacity of tumours. The bcl-2 protein is active in the process of programmed cell death (apoptosis) as an antiapoptotic agent and its expression may reflect tumour progression. Mean/median percentages of tumour cell positivity were 8.5/3.0 for CD40, 76.1/86.3 for CD44 and 7.4/3.3 for bcl-2. A significant correlation was observed between expression of apoptosis-associated bcl-2 antigen and overall survival (r = 0.33). The CD44 positive cell fraction was higher in patients with short overall survival than those with long survival but this difference was not statistically significant. The expression of CD40 did not correlate with overall survival. The mean/median proliferation fraction assessed by MIB-1 monoclonal antibody was 25.8/23.9 and showed a significant correlation with survival after diagnosis of melanoma metastasis (r = 0.32). Lack of bcl-2 expression and a high proportion of tumour cells expressing Ki-67 antigen are predictors of poor prognosis that are independent of the traditionally accepted Breslow's thickness of the primary melanomas.     

Quantitative Stain-Free and Continuous Multimodal Monitoring of Wound Healing In Vitro with Digital Holographic Microscopy

Impaired epithelial wound healing has significant pathophysiological implications in several conditions including gastrointestinal ulcers, anastomotic leakage and venous or diabetic skin ulcers. Promising drug candidates for accelerating wound closure are commonly evaluated in in vitro wound assays. However, staining procedures and discontinuous monitoring are major drawbacks hampering accurate assessment of wound assays. We therefore investigated digital holographic microscopy (DHM) to appropriately monitor wound healing in vitro and secondly, to provide multimodal quantitative information on morphological and functional cell alterations as well as on motility changes upon cytokine stimulation. Wound closure as reflected by proliferation and migration of Caco-2 cells in wound healing assays was studied and assessed in time-lapse series for 40 h in the presence of stimulating epidermal growth factor (EGF) and inhibiting mitomycin c. Therefore, digital holograms were recorded continuously every thirty minutes. Morphological changes including cell thickness, dry mass and tissue density were analyzed by data from quantitative digital holographic phase microscopy. Stimulation of Caco-2 cells with EGF or mitomycin c resulted in significant morphological changes during wound healing compared to control cells. In conclusion, DHM allows accurate, stain-free and continuous multimodal quantitative monitoring of wound healing in vitro and could be a promising new technique for assessment of wound healing.     

Measuring morphological complexity of segmented animals: centipedes as model systems

In segmented animals it is possible to define morphological complexity as the degree of morphological differentiation of segments. A quantitative method for measuring morphological complexity of segmental patterns was devised by McShea in 1992, who introduced three geometrical indices. Here, we introduce a new index of morphological complexity and emphasize the possible decoupling between segmentation and segment differentiation and illustrate different patterns of variation within segmental series and how these could affect morphological evolution and evolvability. Concepts are illustrated by contrasting the segmental models of two groups of centipedes (Chilopoda): the elongate Geophilomorpha and the short‐bodied Lithobiomorpha. A preliminary application of the new metric provides no evidence of macroevolutionary increase in morphological complexity of centipede segmental organization.     

Curve of labeled mitoses in different kinetic states of cell proliferation. III. Experiments with a mathematical model]

By means of computer simulation, the dynamic structure of labeled mitoses curve (LMC) has been investigated. Experiments using computer demonstrated the dependence of LMC form not only on the values of temporal parameters of the mitotic cycle, but also on the state of cell proliferation kinetics before and after pulse labeling. Of particular interest was a case of a temporal block of cell entry into the S period which has been studied in a greater detail. The results of modelling experiments have shown that the universally accepted graphic procedures for the evaluation of mean durations of S- and G2-periods (using 50%-level of LMC) yield the satisfactory estimates only for systems with stationary age distribution for cells in the mitotic cycle (steady, in the strict sense, and exponential states of cell proliferation kinetics). The paper presents also the kinetic interpretation of certain abnormalities in LMC form arising from some peculiarities of cell population kinetics in the real biological experiment.     

Analysis of endothelial cell locomotion: Differential effects of motility and contact inhibition

Video microscopy and digital time-lapse recording were used to monitor locomotion and proliferation of bovine pulmonary artery endothelial (BPAE) cells cultured with varying concentrations of basic fibroblast growth factor (bFGF). Cell trajectories were reconstructed using a generalized nearest-neighbor algorithm and analyzed to determine how cell motility is affected by cell-cell collisions, cell divisions, and increasing cell density. The temporal evolution patterns of the average speed of locomotion for all cells in a culture were computed and the effects of varying bFGF concentrations were analyzed. Intermediate concentrations of bFGF (30 and 50 ng/mL) significantly increased the speed of locomotion above the levels we observed with 0 and 100 ng/mL concentrations of bFGF. Increases in cell density due to proliferation were immediately accompanied by a decrease in the average speed of locomotion of the cell population. Finally, the effect of bFGF concentration on the overall cell proliferation rates was assessed. With the addition of 30 or 50 ng/mL of bFGF to the culture media, the observed cell proliferation rates increased significantly. The proliferation rates decreased when the bFGF concentration increased to 100 ng/mL. These results show that bFGF concentrations that increase the motility of BPAE cells also increase the observed cell proliferation rates. (c) 1994 John Wiley & Sons, Inc.     

Changes in cellularity induced by radiation in a solid tumour.

The growth, and cellular responses of Morris hepatoma 3924 A to a locally-administered dose of 3750 R X-rays were studied using the following parameters; (1) relative tumour volume changes; (2) tritiated thymidine (3H-TdR) incorporation into DNA; (3) tumour DNA content and (4) cellular analysis, including 3H-TdR labelling index, mitotic index, aberrant mitotic frequency and relative cell density. Before depression of tumour growth, cell proliferation is temporarily interuppted. As proliferation is reinitiated, a short-lived synhcrony and prolongation of cell-cycle traverse are reflected in (a) the labelling index and mitotic index, (b) the relative cell density, and (c) the rate of incorporation of 3H-TdR into DNA. Within 4 days after radiation, cell proliferation and 3H-TdR incorporation are significantly depressed. Simultaneously there are reductions in both the relative cell density and tumour DNA contents, and these remain depressed as the tumours initiate regression. From these studies, it is apparent that the cellular responses to radiation insult occur well in advance of measurable volume changes and are observed both in tumours that continue to regress and in those that initiate regrowth.