The Stability of Compressible Mixing Layers in Binary Gases

We present the results of a study of the inviscid two-dimensional spatial stability of a parallel compressible mixing layer in a binary gas. The parameters of this study are the Mach number of the fast stream, the ratio of the velocity of the slow stream to that of the fast stream, the ratio of the temperatures, the composition of the gas in the slow stream and in the fast stream, and the frequency of the disturbance wave. The ratio of the molecular weight of the slow stream to that of the fast stream is found to be an important quantity and is used as an independent variable in presenting the stability characteristics of the flow. It is shown that differing molecular weights have a significant effect on the neutral-mode phase speeds, the phase speeds of the unstable modes, the maximum growth rates, and the unstable frequency range of the disturbances. The molecular weight ratio is a reasonable predictor of the stability trends. We have further demonstrated that the normalized growth rate as a function of the convective Mach number is relatively insensitive (≈25%) to changes in the composition of the mixing layer. Thus, the normalized growth rate is a key element when considering the stability of compressible mixing layers, since once the basic stability characteristics for a particular combination of gases is known at zero Mach number, the decrease in growth rates due to compressibility effects at the larger convective Mach numbers is somewhat predictable. (C) 1996 American Institute of Physics

The stability of compressible mixing layers in binary gases

We present the results of a study of the inviscid two-dimensional spatial stability of a parallel compressible mixing layer in a binary gas. The parameters of this study are the Mach number of the fast stream, the ratio of the velocity of the slow stream to that of the fast stream, the ratio of the temperatures, the composition of the gas in the slow stream and in the fast stream, and the frequency of the disturbance wave. The ratio of the molecular weight of the slow stream to that of the fast stream is found to be an important quantity and is used as an independent variable in presenting the stability characteristics of the flow. It is shown that differing molecular weights have a significant effect on the neutral-mode phase speeds, the phase speeds of the unstable modes, the maximum growth rates and the unstable frequency range of the disturbances. The molecular weight ratio is a reasonable predictor of the stability trends. We have further demonstrated that the normalized growth rate as a function of the convective Mach number is relatively insensitive (Approx. 25%) to changes in the composition of the mixing layer. Thus, the normalized growth rate is a key element when considering the stability of compressible mixing layers, since once the basic stability characteristics for a particular combination of gases is known at zero Mach number, the decrease in growth rates due to compressibility effects at the larger convective Mach numbers is somewhat predictable

The structure of variable property, compressible mixing layers in binary gas mixtures

We present the results of a study of the structure of a parallel compressible mixing layer in a binary mixture of gases. The gases included in this study are hydrogen (H2), helium (He), nitrogen (N2), oxygen (02), neon (Ne) and argon (Ar). Profiles of the variation of the Lewis and Prandtl numbers across the mixing layer for all thirty combinations of gases are given. It is shown that the Lewis number can vary by as much as a factor of eight and the Prandtl number by a factor of two across the mixing layer. Thus assuming constant values for the Lewis and Prandtl numbers of a binary gas mixture in the shear layer, as is done in many theoretical studies, is a poor approximation. We also present profiles of the velocity, mass fraction, temperature and density for representative binary gas mixtures at zero and supersonic Mach numbers. We show that the shape of these profiles is strongly dependent on which gases are in the mixture as well as on whether the denser gas is in the fast stream or the slow stream

Modelling of Tirapazamine effects on solid tumour morphology

Bioreductive drugs are in clinical practice to exploit the resistance from tumour microenvironments especially in the hypoxic region of tumour. We pre-sented a tumour treatment model to capture the pharmacology of one of the most prominent bioreductive drugs, Tirapazamine (TPZ) which is in clinical trials I and II. We calculated solid tumour mass in our previous work and then integrated that model with TPZ infusion. We calculated TPZ cytotoxicity, concentration, penetra-tion with increasing distance from blood vessel and offered resistance from micro-environments for drug penetration inside the tumour while considering each cell as an individual entity. The impact of these factors on tumour morphology is also showed to see the drug behaviour inside animals/humans tumours. We maintained the heterogeneity factors in presented model as observed in real tumour mass es-pecially in terms of cells proliferation, cell movement, extracellular matrix (ECM) interaction, and the gradients of partial oxygen pressure (pO2) inside tumour cells during the whole growth and treatment activity. The results suggest that TPZ high concentration in combination with chemotherapy should be given to get maximum abnormal cell killing. This model can be a good choice for oncologists and re-searchers to explore more about TPZ action inside solid tumour

Mechanistic mathematical modelling of mercaptopurine effects on cell cycle of human acute lymphoblastic leukaemia cells

The antimetabolite mercaptopurine (MP) is widely used to treat childhood acute lymphoblastic leukaemia (ALL). To study the dynamics of MP on the cell cycle, we incubated human T-cell leukaemia cell lines (Molt-4 sensitive and resistant subline and P12 resistant) with 10 μM MP and measured total cell count, cell cycle distribution, percent viable, percent apoptotic, and percent dead cells serially over 72 h. We developed a mathematical model of the cell cycle dynamics after treatment with MP and used it to show that the Molt-4 sensitive controls had a significantly higher rate of cells entering apoptosis (2.7-fold, P<0.00001) relative to the resistant cell lines. Additionally, when treated with MP, the sensitive cell line showed a significant increase in the rate at which cells enter apoptosis compared to its controls (2.4-fold, P<0.00001). Of note, the resistant cell lines had a higher rate of antimetabolite incorporation into the DNA of viable cells (>1.4-fold, P<0.01). Lastly, in contrast to the other cell lines, the Molt-4 resistant subline continued to cycle, though at a rate slower relative to its control, rather than proceed to apoptosis. This led to a larger S-phase block in the Molt-4 resistant cell line, but not a higher rate of cell death. Gene expression of apoptosis, cell cycle, and repair genes were consistent with mechanistic dynamics described by the model. In summary, the mathematical model provides a quantitative assessment to compare the cell cycle effects of MP in cells with varying degrees of MP resistance

Cell killing and resistance in pre-operative breast cancer chemotherapy

<p>Abstract</p> <p>Background</p> <p>Despite the recent development of technologies giving detailed images of tumours <it>in vivo</it>, direct or indirect ways to measure how many cells are actually killed by a treatment or are resistant to it are still beyond our reach.</p> <p>Methods</p> <p>We designed a simple model of tumour progression during treatment, based on descriptions of the key phenomena of proliferation, quiescence, cell killing and resistance, and giving as output the macroscopically measurable tumour volume and growth fraction. The model was applied to a database of the time course of volumes of breast cancer in patients undergoing pre-operative chemotherapy, for which the initial estimate of proliferating cells by the measure of the percentage of Ki67-positive cells was available.</p> <p>Results</p> <p>The analysis recognises different patterns of response to treatment. In one subgroup of patients the fitting implied drug resistance. In another subgroup there was a shift to higher sensitivity during the therapy. In the subgroup of patients where killing of cycling cells had the highest score, the drugs showed variable efficacy against quiescent cells.</p> <p>Conclusion</p> <p>The approach was feasible, providing items of information not otherwise available. Additional data, particularly sequential Ki67 measures, could be added to the system, potentially reducing uncertainty in estimates of parameter values.</p

Towards Predicting the Response of a Solid Tumour to Chemotherapy and Radiotherapy Treatments: Clinical Insights from a Computational Model

In this paper we use a hybrid multiscale mathematical model that incorporates both individual cell behaviour through the cell-cycle and the effects of the changing microenvironment through oxygen dynamics to study the multiple effects of radiation therapy. The oxygenation status of the cells is considered as one of the important prognostic markers for determining radiation therapy, as hypoxic cells are less radiosensitive. Another factor that critically affects radiation sensitivity is cell-cycle regulation. The effects of radiation therapy are included in the model using a modified linear quadratic model for the radiation damage, incorporating the effects of hypoxia and cell-cycle in determining the cell-cycle phase-specific radiosensitivity. Furthermore, after irradiation, an individual cell's cell-cycle dynamics are intrinsically modified through the activation of pathways responsible for repair mechanisms, often resulting in a delay/arrest in the cell-cycle. The model is then used to study various combinations of multiple doses of cell-cycle dependent chemotherapies and radiation therapy, as radiation may work better by the partial synchronisation of cells in the most radiosensitive phase of the cell-cycle. Moreover, using this multi-scale model, we investigate the optimum sequencing and scheduling of these multi-modality treatments, and the impact of internal and external heterogeneity on the spatio-temporal patterning of the distribution of tumour cells and their response to different treatment schedules