Jerry Aroesty

Tumor growth and chemotherapy: Mathematical methods, computer simulations, and experimental foundations

Abstract Mathematical methods and computer simulation techniques for describing tumor growth in the presence or absence of chemotherapy are reviewed. The experimental and clinical background is discussed briefly, and several major current issues are identified. These include the development of appropriate descriptions for variable tumor growth rates and kinetic parameters over the entire range of tumor size or maturity, the effects of cell-cycle specific therapy on tumor growth, and the possible implications during treatment of cell-cycle time distributions and degree of correlation between succeeding generations of tumor cells.

Convection and diffusion in the microcirculation.

Abstract It has been conjectured that the motion of the plasma in the microcirculation can play a significant role in enhancing the transport of diffusing species, such as oxygen or carbon dioxide, between capillary endothelium and erythrocyte. A detailed theroretical and numerical investigation of the bolus model of capillary flow has been performed, and it is shown that for this highly idealized situation, the convective motions of the plasma, and the enhanced mixing due to these motions, do not appreciably augment diffusional species transport rates for dissolved gases. The result of our investigation of the equations of motion, and the equations of species transport, indicate that the plasma mixing is important only in the transfer of materials such as macromolecules, which may diffuse more slowly than dissolved gases.

The Mathematics of Pulsatile Flow in Small Vessels: I. Casson Theory.

Abstract Recent experiments in vivo confirm the existence of pulsatile flow in the microcirculation. This paper is primarily a mathematical analysis of a particular model of blood rheology, the Casson fluid, which possesses finite yield stress and shear-dependent viscosity when it is subjected to a periodic pressure gradient in a long rigid tube. The coupled nonlinear implicit equations of motion and constitutive relations are nondimensionalized, and approximate solutions valid for small values of the Womersley frequency parameter are derived. It is shown that the inertial or unsteady terms in the solutions for pressure, flow, and wall shear are negligible for conditions of physiological relevance. Thus, the flow behavior in arterioles and venules can be accurately approximated by the quasi-steady solution.

A preliminary model of doable-minate-mediated gene amplification

A mathematical model of double minute (dm) population dynamics has been developed based upon current concepts of the saltatory replication, random partitioning, nuclear exclusion and loss, and cellular growth inhibition of these extrachromosomal elements. A highly accurate approximate analytical solution has been obtained for the dm frequency distribution at steady state and preliminary analysis of transient states has been performed. The steady state solution has been fit to experimental frequency data of the SW527N carcinoma line, the excellent goodness of fit (X2 = 2.6, d.f. = 29) providing preliminary evidence for the consistency of this set of mechanisms. Two special cases are examined in which extrareplicative dms are produced on both the chromosome and existing dms at equal rates or on the chromosome alone. The model predicts that the population--average rate of extrareplicative dm production is 0.039 +/- S.E. 0.009 dms/hr/cell in the first case and is tenfold higher than when such replication occurs on the chromosome alone (0.0043 +/- S.E. 0.0004 dms/hr/cell). Allowable ranges of the extent of dm-related growth inhibition and dm loss are determined for the SW527N cell line. It is found that dm-related growth inhibition can be nearly as high as that observed for the S180 sarcoma lines (on the order of 0.5% per dm lengthening of the doubling time) or as low as zero.

An analysis of predictor variables for adjuvant treatment of breast cancer.

SummaryModern computer-based statistical analysis of a well-documented data base can facilitate the selection of breast cancer patients for adjuvant chemotherapy. Traditional selection criteria are dominated by the number of positive axillary lymph nodes found by pathologic examination. However, patients of the same nodal status (0+, 1–3+, 4–7+, >7+) are still heterogeneous with regard to risk of metastatic recurrence and benefits of adjuvant treatment. A pilot study of 103 patients of similar nodal status (1–3+) followed for up to 10 years was undertaken to determine whether data already in the patients record at the time of pathologic examination and derived from the patients history as well as from clinical, surgical and pathologic examination could supplement nodal status in predicting disease recurrence. Preliminary processing and screening for promising variables were performed with CLINFO; maximum-likelihood procedures were then used to relate the probability of disease recurrence to those variables that appeared to be significant. Parametric models of hazard rate for the individual patient were employed corresponding to both exponential and Wiebull distributions of disease-free interval. The hazard rate was related log-linearly to a set of prognostic variables, and model parameters were determined by fitting to the data. Factors that favor longer disease-free intervals (in quantitative order of importance) are: (1) Nipple involved clinically at presentation; (2) Lesion had soft or rubbery consistency on palpation; (3) Disease discovered by physician; (4) Homolateral lymph nodes not involved clinically; (5) Margin from tumor to fascia >1 cm; (6) No maternal history of breast cancer; (7) Increasing age of patient; (8) Presence of specialized histology.Based on the findings of this pilot study, a quantitative summary of personal (SK) and institutional experience is developed in which the probability of recurrence for the individual patient and the associated confidence intervals are used to classify patients with regard to risk of recurrence.

Simple relations for the stability of heated-water laminar boundary layers

Dunn-Lin theory (1955) is shown to be applicable in the estimation of minimum critical Reynolds number for heated boundary layers. A parameter is defined to reflect the curvature of the velocity profile between wall and critical layer, and includes the variable kinematic viscosity. The value of the parameter is determined by assuming a linear viscosity profile, a parabolic velocity profile, and by calculating a dominant viscous term in the asymptotic solution of the Orr-Sommerfeld equation. The inclusion of the curvature permits accurate results for the laminar velocity profiles in the wall region. Comparisons are made between the obtained values and those derived from a numerical computation using the Orr-Sommerfeld equation for exact boundary layer profiles. Agreement is noted in predicting the location of the maximum attainable critical Reynolds number in certain cases, and the method is considered reliable within experimentally available pressure gradients and surface overheats.

The Importance of Plasma Mixing in Bolus Flow.

The fluid-mechanical and biochemical processes which occur during the movement of a red blood cell through a capillary are exceedingly complex. The small scale and complexity of the capillary flow system makes quantitative experimental description very difficult and prevents a direct empirical modeling of the system at the present time. A theoretical approach directed at certain basic features of the fluid mechanics and transport processes in the microcirculation can indicate those features which must be considered in the development of a capillary flow model.