Let X(t) be the number of tumor cells at time t, and Pr{X(t) = n} = pn(t) is the density of X. A “birth”, i.e., an increase of one of the total population of cancer cells, can occur either by mutation of a normal cell caused by the action of the carcinogen, consisting of randomly (Poisson) distributed hits, or by reproduction of existing cancer cells. A death of a tumor cell occurs as an additive combination of non-immunological and immunological elements. Once a tumor is initiated by carcinogenic action, it undergoes a birth and death process with infinitesimal birth rate linear and infinitesimal death rate composed of a linear and a nonlinear term, the former due to non-immunological deaths, the latter to immunological feedback. The death rate per tumor cell due to immunological response is proportional to the total number of antigen-producing (tumor) cells; thus, the total death rate is quadratic. Although this assumes a very simple mechanism for the action of immunological feedback, it is nevertheless a first step.

The Mathematical Model

Evaporation of pure liquid sessile and spherical suspended drops: a review.

We review the evolution of the nonparametric regression modeling in imaging from the local Nadaraya-Watson kernel estimate to the nonlocal means and further to transform-domain filtering based on nonlocal block-matching. The considered methods are classified mainly according to two main features: local/nonlocal and pointwise/multipoint. Here nonlocal is an alternative to local, and multipoint is an alternative to pointwise. These alternatives, though obvious simplifications, allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques. Within this framework, we introduce a novel single- and multiple-model transform domain nonlocal approach. The Block Matching and 3-D Filtering (BM3D) algorithm, which is currently one of the best performing denoising algorithms, is treated as a special case of the latter approach.

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From Local Kernel to Nonlocal Multiple-Model Image Denoising

The literature on drying sessile droplets and deposition of suspended material is reviewed including the simple explanation of the “coffee ring” deposit given by Deegan et al.1 Analytical and numerical solutions for the flow are given, including the effect of Marangoni stresses, pinning or movement of the contact line, and viscous, thermal, gravitational, and other effects. The solution space is explored using dimensionless groups governing mass, momentum, and heat transfer effects in the droplet, external gas, and substrate. The most common types of deposition patterns are summarized, including those produced by pinned contact lines, sticking-and-slipping contact lines, and Marangoni effects. The influence of contact-line deposits is also reviewed, and the effects of colloidal, polymeric, and other depositing materials. Advanced applications from ink-jet printing to disease diagnosis are discussed as well. The review helps readers take stock of what has been learned and what remains incompletely explained. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1538–1571, 2014

Transport and deposition patterns in drying sessile droplets

This review is aimed at presenting the evaporation of macroscopic sessile droplets on inert substrates in normal atmosphere in simple cases, as a basis for more complex analyses.

Evaporation of macroscopic sessile droplets

We report the results of physical experiments that demonstrate the strong influence of the thermal conductivity of the substrate on the evaporation of a pinned droplet. We show that this behaviour can be captured by a mathematical model including the variation of the saturation concentration with temperature, and hence coupling the problems for the vapour concentration in the atmosphere and the temperature in the liquid and the substrate. Furthermore, we show that including two ad hoc improvements to the model, namely a Newton's law of cooling on the unwetted surface of the substrate and the buoyancy of water vapour in the atmosphere, give excellent quantitative agreement for all of the combinations of liquid and substrate considered.

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The strong influence of substrate conductivity on droplet evaporation

An experimental and theoretical study of the effect of the atmosphere on the evaporation of pinned sessile droplets of water is described. The experimental work investigated the evaporation rates of sessile droplets in atmospheres of three different ambient gases (namely, helium, nitrogen, and carbon dioxide) at reduced pressure (from 40 to 1000 mbars) using four different substrates (namely, aluminum, titanium, Macor, and polytetrafluoroethylene) with a wide range of thermal conductivities. Reducing the atmospheric pressure increases the diffusion coefficient of water vapor in the atmosphere and hence increases the evaporation rate. Changing the ambient gas also alters the diffusion coefficient and hence also affects the evaporation rate. A mathematical model that takes into account the effect of the atmospheric pressure and the nature of the ambient gas on the diffusion of water vapor in the atmosphere and the thermal conductivity of the substrate is developed, and its predictions are found to be in encouraging agreement with the experimental results.

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On the effect of the atmosphere on the evaporation of sessile droplets of water

/pdf/a-mathematical-model-for-the-evaporation-of-a-thin-sessile-w7gvr05my3.pdf

A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory

A mathematical model for the quasisteady evaporation of a thin liquid droplet on a thin substrate that incorporates the dependence of the saturation concentration of vapor at the free surface of the droplet on temperature is used to examine an atypical situation in which the substrate has a high thermal resistance relative to the droplet (i.e., it is highly insulating and/or is thick relative to the droplet). In this situation diffusion of heat through the substrate is the rate-limiting evaporative process and at leading order the local mass flux is spatially uniform, the total evaporation rate is proportional to the surface area of the droplet, and the droplet is uniformly cooled. In particular, the qualitative differences between the predictions of the present model in this situation and those of the widely used “basic” model in which the saturation concentration is independent of temperature are highlighted.

/pdf/evaporation-of-a-thin-droplet-on-a-thin-substrate-with-a-5ap6ss5kbz.pdf

Evaporation of a thin droplet on a thin substrate with a high thermal resistance

HO: Maths underpins our understanding of almost all ‘industrial’ processes including social science (we regard ‘industry’ as a broad term, using the EU definition: ‘any activity of social or economic value’). The aims of ECMI are to introduce academic mathematicians to real problems and to convince industry that they need mathematics and mathematicians. The main mechanisms used are European Study Groups with Industry (ESGIs) and workshops where industrial scientists can meet informally with mathematicians to brainstorm their problems. We support a number of Special Interest Groups in topics ranging from weather prediction to digital factories. These groups act as a focal point for researchers from all over Europe and can lead to EU funded proposals.

G. Dunn

Papers

The strong influence of substrate conductivity on droplet evaporation

On the effect of the atmosphere on the evaporation of sessile droplets of water

A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory

Evaporation of a thin droplet on a thin substrate with a high thermal resistance

The European Consortium For Mathematics in Industry