Random paths in random arrays of cylinders.

Distributions associated with random tracks in bodies are important to understanding the effects of radiation on objects and their response to it. Results of computer simulations are presented for the distribution densities, average lengths, and relative frequencies of the various types of random tracks resulting from the traversal of random arrays of cylinders by random lines. We consider arrays consisting of freely overlapping long circular cylinders distributed randomly with their axes parallel to a line or a plane, or oriented randomly in the three-dimensional space. The problem of random tracks lying outside the cylinders (external random tracks) is treated analytically, while the distributions of random tracks found within the cylinders (internal random tracks) are determined using a simulation procedure based on a discrete step-by-step random walk mechanism applied in finite samples of the arrays. The numerical results are used to validate various general analytical results for the distribution densities and average lengths of random tracks in structures of arbitrary shape. It is found that the distributions of internal random tracks exhibit strong dependence on the cylinder volume fraction, while the external random tracks are distributed independently of the cylinder volume fraction for all three cases of orientation distribution. At the limit of high cylinder volume fraction, the internal random track distributions approach the distributions of the external random tracks, while at the other extreme, they become, as expected, identical to those determined analytically for a single infinite cylinder.

Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH.

Tumors and multicellular tumor spheroids can develop gradients in oxygen concentration, glucose concentration, and extracellular pH as they grow. In order to calculate these gradients and assess their impact on tumor growth, it is necessary to quantify the effect of these variables on tumor cell metabolism and growth. In this work, the oxygen consumption rates, glucose consumption rates, and growth rates of EMT6/Ro mouse mammary tumor cells were measured at a variety of oxygen concentrations, glucose concentrations, and extracellular pH levels. At an extracellular pH of 7.25, the oxygen consumption rate of EMT6/Ro cells increased by nearly a factor of 2 as the glucose concentration was decreased from 5.5 mM to 0.4 mM. This effect of glucose concentration on oxygen consumption rate, however, was slight at an extracellular pH of 6.95 and disappeared completely at an extracellular pH of 6.60. The glucose consumption rate of EMT6/Ro cells increased by roughly 40% when the oxygen concentration was reduced from 0.21 mM to 0.023 mM and decreased by roughly 60% when the extracellular pH was decreased from 7.25 to 6.95. The growth rate of EMT6/Ro cells decreased with decreasing oxygen concentration and extracellular pH; however, severe conditions were required to stop cell growth (0.0082 mM oxygen and an extracellular pH of 6.60). Empirical correlations were developed from these data to express EMT6/Ro cell growth rates, oxygen consumption rates, and glucose consumption rates, as functions of oxygen concentration, glucose concentration, and extracellular pH. These empirical correlations make it possible to mathematically model the gradients in oxygen concentration, glucose concentration, and extracellular pH in EMT6/Ro multicellular spheroids by solution of the diffusion/reaction equations. Computations such as these, along with oxygen and pH microelectrode measurements in EMT6/Ro multicellular spheroids, indicated that nutrient concentration and pH levels in the inner regions of spheroids were low enough to cause significant changes in nutrient consumption rates and cell growth rates. However, pH and oxygen concentrations measured or calculated in EMT6/Ro spheroids where quiescent cells have been observed were not low enough to cause the cessation of cell growth, indicating that the observed quiescence must have been due to factors other than acidic pH, oxygen depletion, or glucose depletion.

Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids.

In order to determine the role of micromilieu in tumour spheroid growth, a mathematical model was developed to predict EMT6/Ro spheroid growth and microenvironment based upon numerical solution of the diffusion/reaction equation for oxygen, glucose, lactate ion, carbon dioxide, bicarbonate ion, chlorine ion and hydrogen ion along with the equation of electroneutrality. This model takes into account the effects of oxygen concentration, glucose concentration and extracellular pH on cell growth and metabolism. Since independent measurements of EMT6/Ro single cell growth and metabolic rates, spheroid diffusion constants, and spinner flask mass transfer coefficients are available, model predictions using these parameters were compared with published data on EMT6/Ro spheroid growth and micro-environment. The model predictions of reduced spheroid growth due to reduced cell growth rates and cell shedding fit experimental spheroid growth data below 700 microns, but overestimated the spheroid growth rate at larger diameters. Predicted viable rim thicknesses based on predicted near zero glucose concentrations fit published viable rim thickness data for 1000 microns spheroids grown at medium glucose concentrations of 5.5 mM or less. However, the model did not accurately predict the onset of necrosis. Moreover, the model could not predict the observed decreases in oxygen and glucose metabolism seen in spheroids with time, nor could it predict the observed growth plateau. This suggests that other unknown factors, such as inhibitors or cell-cell contact effects, must also be important in affecting spheroid growth and cellular metabolism.

Correction of cellular autofluorescence in flow cytometry by mathematical modeling of cellular fluorescence.

A method for the correction of background fluorescence in flow cytometry with special relevance to the quantitation of low levels of cellular surface membrane antigens is presented. The method is based on the mathematical modeling of cellular fluorescence distributions of background fluorescence (autofluorescence control or irrelevant antibody control) and total fluorescence (positively stained cells). Algorithms based on two models and utilizing only the routinely available background and total fluorescence histograms are developed and implemented in computer programs. These allow estimation of the fluorescence histogram corresponding exclusively to immunofluorescence staining of the cell surface antigen of interest. Thus, the correction of background fluorescence is effected solely with software processing of routinely available data; no additional hardware or parameter determinations are necessary. Two models were chosen to be physically plausible and to represent extremes in correlation between background and probe fluorescence. Extremes were chosen to assess the solution dependence on model and to provide bounds to the actual solution when no information on correlation is available. Results are presented for both computer simulations and for an actual assay of the CR1 complement receptor on human erythrocytes to test and illustrate the technique. Alternatively, data can be tested assuming a particular model to explore the relationship, if any, between specific and nonspecific fluorescence.

Glucose diffusivity in multicellular tumor spheroids.

In order to understand the role of glucose limitations in controlling multicellular tumor spheroid growth, knowledge of the glucose diffusion coefficient is essential. The effective diffusivity of glucose in spheroids of rodent and human tumor cell lines has been determined by measuring the efflux of tritium labeled L-glucose from spheroids with time. When the rapid and irreversible binding of L-glucose in spheroids is properly taken into account, measurements of the efflux of this diffusion tracer from spheroids into label-free medium can be correlated to the diffusion equation in order to obtain the effective glucose diffusivity in spheroids. Such measurements have been made in EMT6/Ro mouse mammary tumor spheroids as well as in spheroids derived from human colon carcinoma cells (HT29, CO112, and WiDr) and from human squamous carcinoma cells (CaSki and A431). EMT6/Ro spheroids have a glucose diffusivity of 1.1 x 10(-6) cm2/s, while glucose diffusion coefficients in the human cell spheroids studied vary from 5.5 x 10(-7) cm2/s to 2.3 x 10(-7) cm2/s. These values are low enough to suggest that significant gradients in glucose concentration may exist in spheroids and tumors. It is thus believed that these glucose diffusivities, as well as their variation with cell line, may have important implications for the role played by glucose in the growth and cellular heterogeneity of spheroids and tumors.