Cell-surface interaction in the cellular Potts model is regulated by the surface energy coefficients.

<p>Three cells with cell indices 1, 2 and 3, respectively, each one covering several lattice sites, interact with each other at the cell surfaces. The cells 1 and 3 are of type A, depicted in dark grey, the cell 3 is of type B, depicted in light grey. The strength of the interaction depends on the cell types. There are also interactions between the cells and the medium (white, cell index 0). Possible boundary interactions are not shown.</p

A mechanistic multiscale framework is characterized by the coupling of multiple spatial and temporal scales on the basis of abstracted rules.

<p>The assumed intercellular interaction may depend on an interplay with cellular characteristics and intercellular details. By determining the distinctive characteristics at the tissue level and their comparison with experimental observation, it can be tested wether a specific mechanism explains the behavior of an experimentally studied cell system.</p

Cell surface fluctuations are the central device in the realization of the multiscale concept in CPMs.

<p>Both the rules of intercellular interaction and the considered cellular characteristics are eventually coded, via the Hamiltonian or directly for extended models, into an expression that regulates the intensity of CPM-cells' surface fluctuations. Additional technical parameters are integrated into the Hamiltonian to be able to suppress phenomenologically unrealistic behavior. The actual impact of the Hamiltonian on the intensity of CPM cells' surface fluctuations is attenuated by a voter-like portion in the transition rates. The surface fluctuations drive simultaneously the actual behavior of a CPM at the cellular scale, the specifics of intercellular interaction and the emerging behavior at the tissue scale. Single aspects of the cellular properties in the model, for instance the cell shape flexibility, the magnitude of random cell displacements or the cells' surface roughness, and of the intercellular interaction, like the strength of intercellular adhesion, cannot be controlled individually but are interlinked with each other. Likewise, purely model-technical control parameters such as the cellular integrity, that is the property of CPM cells to span over connected, essentially convex lattice domains, are coupled indirectly with biologically interpretable cellular and intercellular properties. The emerging tissue scale behavior is solely rooted in the specified characteristics of the CPM cells' surface fluctuations and not linked directly to cellular and intracellular specifics.</p

Taylor coefficients and coefficient multipliers of Hardy and Bergman-type spaces

This book provides a systematic overview of the theory of Taylor coefficients of functions in some classical spaces of analytic functions and especially of the coefficient multipliers between spaces of Hardy type. Offering a comprehensive reference guide to the subject, it is the first of its kind in this area. After several introductory chapters covering the basic material, a large variety of results obtained over the past 80 years, including the most recent ones, are treated in detail. Several chapters end with discussions of practical applications and related topics that graduate students and experts in other subjects may find useful for their own purposes. Thus, a further aim of the book is to communicate to non-specialists some concrete facts that may be of value in their own work. The book can also be used as a textbook or a supplementary reference for an advanced graduate course. It is primarily intended for specialists in complex and functional analysis, graduate students, and experts in other related fields

Parameter estimation from epidemiological data and derivation of the PA-regression-function.

<p><b>A</b>. In order to estimate the clinically observed fraction of PA-I, we utilize data from the literature. <b>B</b>. The estimated fraction of PA-I cases <math><mi>p</mi><mo>^</mo></math> is interpreted as absorption probability in state <i>N</i> in our model. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004662#pcbi.1004662.e003" target="_blank">Eq (3)</a> allows to estimate the corresponding risk coefficient <math><mi>γ</mi><mo>^</mo></math>. <b>C</b>. Substituting the estimated risk coefficient <math><mrow><mi>γ</mi><mo>^</mo><mo>=</mo><mn>0</mn><mo>.</mo><mn>152</mn></mrow></math> in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004662#pcbi.1004662.e016" target="_blank">Eq (5)</a> determines the PA-regression-function <i>β</i>_0.152(<i>ρ</i>) which is plotted in purple. Furthermore, the blue plots indicate the corresponding regression functions for values of <i>γ</i> which are obtained within the standard deviation of <math><mi>p</mi><mo>^</mo></math>.</p

Model-Based Evaluation of Spontaneous Tumor Regression in Pilocytic Astrocytoma

<div><p>Pilocytic astrocytoma (PA) is the most common brain tumor in children. This tumor is usually benign and has a good prognosis. Total resection is the treatment of choice and will cure the majority of patients. However, often only partial resection is possible due to the location of the tumor. In that case, spontaneous regression, regrowth, or progression to a more aggressive form have been observed. The dependency between the residual tumor size and spontaneous regression is not understood yet. Therefore, the prognosis is largely unpredictable and there is controversy regarding the management of patients for whom complete resection cannot be achieved. Strategies span from pure observation (wait and see) to combinations of surgery, adjuvant chemotherapy, and radiotherapy. Here, we introduce a mathematical model to investigate the growth and progression behavior of PA. In particular, we propose a Markov chain model incorporating cell proliferation and death as well as mutations. Our model analysis shows that the tumor behavior after partial resection is essentially determined by a risk coefficient <i>γ</i>, which can be deduced from epidemiological data about PA. Our results quantitatively predict the regression probability of a partially resected benign PA given the residual tumor size and lead to the hypothesis that this dependency is linear, implying that removing any amount of tumor mass will improve prognosis. This finding stands in contrast to diffuse malignant glioma where an extent of resection threshold has been experimentally shown, below which no benefit for survival is expected. These results have important implications for future therapeutic studies in PA that should include residual tumor volume as a prognostic factor.</p></div

Non-existence of an EOR threshold in PA.

<p>The derived PA-regression-function (purple line) allows to quantitatively predict the regression probability based on the critical tumor size estimated as 9 cm³, see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004662#pcbi.1004662.t001" target="_blank">Table 1</a>. Roughly, one cm³ of resected tumor mass will elevate the chance of regression by 10%. The direct consequence is the non-existence of an EOR threshold implying that any proportion of resected tumor mass will improve prognosis. This stands in contrast to the behavior of the regression function for a fictive high value of the risk coefficient of e.g. <i>γ</i> = 50 (black line).</p

Additional file 11 of Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model

Sensitivity to the cell dissemination threshold distanc. Sensitivity to the cell dissemination threshold distance for γ=0.25. (a) shows the disseminated cell ratio rdiss over time for different threshold distances. (b) shows rdiss over time for different threshold distances in a higher resolution for low values. From both (a) and (b) one can see that rdiss decreases with the cell dissemination threshold distance as would be expected. There is a striking difference for the cell dissemination threshold distance of 5 where rdiss increases drastically until it saturates at high ratios. (c) shows the mean equilibrium adhesive state of disseminated cells ā D $\bar {a}^{D}$ for different threshold distances. (d) shows the difference in adhesion phenotypes d a between the two subpopulations for different threshold distances. As would be expected, the adhesion phenotype does not strongly depend on the distance threshold except for a very low distance threshold of 5. In the latter case more than half of the cells are considered disseminated so that the differentiation between the adhesion phenotypes is blurred. This is not surprising as within such short distance cells are likely to disseminate and re-join the cell population due to stochasticity. Accordingly, the effect is rather a model artefact than a biological phenomenon. (PNG 8294 kb

Tumor regression in the TGP model.

<p><b>A</b>. Partial resection of a PA-I tumor reduces the number of tumor cells to size <i>k</i> which is assumed to be below the critical tumor size <i>N</i>. The residual tumor can regrow, progress or regress based on the same dynamics that led to the primary tumor. Hence, the TGP dynamics with relevant cell number <i>N</i> is utilized to describe the further development of the residual tumor. Regression is achieved if state 0 is reached, i.e. no tumor cells are present anymore. <b>B</b>. The red area indicates the resected part of the diagnosed PA-I tumor. This resection leads to removal of both tumor and wild-type cells. Subsequently, the residual number of tumor cells <i>k</i> competes with other wild-type cells which can lead to regrowth, regression or progression of the residual tumor. We assume that <i>N</i> is the relevant cell number for this competition as in the formation of the primary tumor. This relevant cell number is indicated by the blue circle.</p