Modelling acidosis and the cell cycle in multicellular tumour spheroids

A partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells and dead cell material (necrotic and apopotic) within a multicellular tumour spheroid. The rates of cell quiescence and necrosis depend upon the local nutrient and acid concentrations and quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Rather, it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis.

Dipole modes on open periodic structures — an experimental study

An experimental study is made of the lower pass-band of waveguides built as necklaces of oblate spheroids. Short lengths of guide are tested in open resonators. The dominant mode is found to be a glow-wave dipole type having no low-frequency cut off. High Q factors indicate low attenuations. Perturbation measurements demonstrate this energy to be concentrated in the vicinity of the guide.

Modelling the formation of necrotic regions in avascular tumours

The mechanisms underlying the formation of necrotic regions within avascular tumours are not well understood. In this paper, we examine the relative roles of nutrient deprivation and of cell death, from both the proliferating phase of the cell cycle via apoptosis and from the quiescent phase via necrosis, in changing the structure within multicellular tumour spheroids and particularly the accumulation of dead cell material in the centre. A mathematical model is presented and studied that accounts for nutrient diffusion, changes in cell cycling rates, the two different routes to cell death as well as active motion of cells and passive motion of the dead cell material. In studying the accumulation of dead cell matter we do not distinguish between the route by which each was formed. The resulting mathematical model is examined for a number of scenarios. Results show that in many cases the size of the necrotic core is closely correlated with low levels in nutrient concentration. However, in certain cases, particularly where the rate of necrosis is large, the resulting necrotic core can lead to regions of non-negligible nutrient concentration-dependent upon the mode of cell death.

A comparison among four different retrieval methods for ice-cloud properties using data from CloudSat, CALIPSO, and MODIS

The A-Train constellation of satellites provides a new capability to measure vertical cloud profiles that leads to more detailed information on ice-cloud microphysical properties than has been possible up to now. A variational radar–lidar ice-cloud retrieval algorithm (VarCloud) takes advantage of the complementary nature of the CloudSat radar and Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) lidar to provide a seamless retrieval of ice water content, effective radius, and extinction coefficient from the thinnest cirrus (seen only by the lidar) to the thickest ice cloud (penetrated only by the radar). In this paper, several versions of the VarCloud retrieval are compared with the CloudSat standard ice-only retrieval of ice water content, two empirical formulas that derive ice water content from radar reflectivity and temperature, and retrievals of vertically integrated properties from the Moderate Resolution Imaging Spectroradiometer (MODIS) radiometer. The retrieved variables typically agree to within a factor of 2, on average, and most of the differences can be explained by the different microphysical assumptions. For example, the ice water content comparison illustrates the sensitivity of the retrievals to assumed ice particle shape. If ice particles are modeled as oblate spheroids rather than spheres for radar scattering then the retrieved ice water content is reduced by on average 50% in clouds with a reflectivity factor larger than 0 dBZ. VarCloud retrieves optical depths that are on average a factor-of-2 lower than those from MODIS, which can be explained by the different assumptions on particle mass and area; if VarCloud mimics the MODIS assumptions then better agreement is found in effective radius and optical depth is overestimated. MODIS predicts the mean vertically integrated ice water content to be around a factor-of-3 lower than that from VarCloud for the same retrievals, however, because the MODIS algorithm assumes that its retrieved effective radius (which is mostly representative of cloud top) is constant throughout the depth of the cloud. These comparisons highlight the need to refine microphysical assumptions in all retrieval algorithms and also for future studies to compare not only the mean values but also the full probability density function.

Radar scattering from ice aggregates using the horizontally aligned oblate spheroid approximation

The assumed relationship between ice particle mass and size is profoundly important in radar retrievals of ice clouds, but, for millimeter-wave radars, shape and preferred orientation are important as well. In this paper the authors first examine the consequences of the fact that the widely used ‘‘Brown and Francis’’ mass–size relationship has often been applied to maximumparticle dimension observed by aircraftDmax rather than to the mean of the particle dimensions in two orthogonal directions Dmean, which was originally used by Brown and Francis. Analysis of particle images reveals that Dmax ’ 1.25Dmean, and therefore, for clouds for which this mass–size relationship holds, the consequences are overestimates of ice water content by around 53% and of Rayleigh-scattering radar reflectivity factor by 3.7 dB. Simultaneous radar and aircraft measurements demonstrate that much better agreement in reflectivity factor is provided by using this mass–size relationship with Dmean. The authors then examine the importance of particle shape and fall orientation for millimeter-wave radars. Simultaneous radar measurements and aircraft calculations of differential reflectivity and dual-wavelength ratio are presented to demonstrate that ice particles may usually be treated as horizontally aligned oblate spheroids with an axial ratio of 0.6, consistent with them being aggregates. An accurate formula is presented for the backscatter cross section apparent to a vertically pointing millimeter-wave radar on the basis of a modified version of Rayleigh–Gans theory. It is then shown that the consequence of treating ice particles as Mie-scattering spheres is to substantially underestimate millimeter-wave reflectivity factor when millimeter-sized particles are present, which can lead to retrieved ice water content being overestimated by a factor of 4.h

Rayleigh scattering by hexagonal ice crystals and the interpretation of dual-polarisation radar measurements

Dual-polarisation radar measurements provide valuable information about the shapes and orientations of atmospheric ice particles. For quantitative interpretation of these data in the Rayleigh regime, common practice is to approximate the true ice crystal shape with that of a spheroid. Calculations using the discrete dipole approximation for a wide range of crystal aspect ratios demonstrate that approximating hexagonal plates as spheroids leads to significant errors in the predicted differential reflectivity, by as much as 1.5 dB. An empirical modification of the shape factors in Gans's spheroid theory was made using the numerical data. The resulting simple expressions, like Gans's theory, can be applied to crystals in any desired orientation, illuminated by an arbitrarily polarised wave, but are much more accurate for hexagonal particles. Calculations of the scattering from more complex branched and dendritic crystals indicate that these may be accurately modelled using the new expression, but with a reduced permittivity dependent on the volume of ice relative to an enclosing hexagonal prism.