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A mathematical model of a heterogenous tumor as a system of interrelating cell populations is described, including a pool of quiescent cells, cell-to-cell variability in maturation rates, and cell migration from growth area to necrotic one. Computer simulation results are given, model labeled mitoses and labeled index curves for the Lewis carcinoma are compared with experimental data.
A mathematical model for heating and cooling during hyperthermia has been developed from an appropriate solution of a bioheat transfer equation. Predicted cooling rates obtained from the model have been compared with cooling rates obtained from experiments performed on both perfused and non-perfused normal canine brain tissue. The agreement between the predicted and observed cooling rates in non-perfused tissue is satisfactory (within 6-11 per cent) and provides confidence that the conduction process is being accurately represented. The model is then used to estimate the relative contribution of conductive and convective (blood flow) heat loss during cooling for the in vivo experiments. Estimates of blood flow dynamics are made from cooling data taken early and late in a heating course using the model to correct for conductive heat loss. Simplified forms of the bioheat transfer equation are examined. An adequate model for the observed cooling data is one that treats heat loss (both conduction and blood flow) as a heat sink (i.e. an effective perfusion model) rather than an effective thermal conductivity model.
A mathematical model that predicts the thermal response of tissue during and after the application of localized pressure is described. The model predictions compare favorably with previously obtained experimental results. It is inferred that the principal cause of the transient temperature rise occurring after pressure application is a reactive hyperemic effect, while changes in metabolism are of little importance. The values of the tissue blood flow rates during reactive hyperemia are estimated.
A mathematical model of the isolated intracranial system including autoregulation of cerebral blood flow with the aid of a variable cerebrovascular resistance is described. The rate of formation of cerebrospinal fluid is assumed to depend on the regional blood flow through the choroid plexuses. This model is extended by cardiovascular components including the left ventricle of the heart, the aorta and the peripheral resistance. Additionally the model contains control circuits to simulate the short-time behaviour of the blood pressure regulation with the aid of the baroreceptor reflex. Disturbances of central regulation of blood pressure are simulated depending on changes of the regional blood flow through the brain stem. The application of the model is demonstrated by the analysis of the influence of arterial blood pressure upon the intracranial pulse pressure relationship (PPR) and upon the pressure response to a volume pressure test. Theoretical considerations and simulations reveal an opposite effect of arterial blood pressure (ABP) and its amplitude upon PPR. The ICP amplitude rises with decreasing ABP or increasing ABP amplitude. Breakpoints and other deviations from a linear PPR over the whole ICP range are studied by the analysis of the transfer function. The application of the model concerning parameter estimation methods is demonstrated and discussed. Simulations of rhythmic phenomena with the aid of the extended model point out possible approaches to quantitative descriptions of disturbances of central regulation.
A mathematical model of lactate kinetics after exercise has been constructed from the application of the mass conservation law and the following assumptions: 1. The total lactate distribution space is composed of two compartments, i.e., (M) the previously working muscles and (S) the remaining lactate space; 2. The rates of lactate release and utilization in (M) and (S) are proportional to the lactate contents of these compartments; 3. The post-exercise lactate production rates in (M) and (S) are constants; 4. Arterial lactate concentration can represent the average lactate concentration in (S). Consideration of experimental facts reported in the literature shows these assumptions to be reasonable. The relationships obtained express the compatibility of parameters and time functions concerning lactate concentrations, as well as rates of production, uptake, release, and utilization. They open the way to various applications, especially those involving numerical fits to observed time courses of lactate concentrations.
A mathematical model was developed to analyze the elimination kinetics of drug interactions in the rat. The model is based on physiological blood flow rates and organ weights and includes Michaelis-Menten equations for enzymatic processes which are involved in the elimination of the drug; competitive inhibition interactions are computed for shared pathways. Using data from the single drugs, the model can simulate the results of experiments of the acute warfarin-BSP interactions in rats.
A mathematical model has been developed to optimize tumor targeting with labeled antibodies. The model is compartmental and nonlinear, incorporating saturable binding. Published parameter values have been used in the model, and the resulting stiff differential equations have been solved using FACSIMILE, a computer package that can simulate very stiff differential systems. Results show that successful tumor targeting depends on an optimal combination of antibody dose, affinity, and molecular size. The model has allowed an assessment to be made of the complicated and interrelated dynamic relationships that these factors have on tumor targeting. It has also offered an explanation for previously unsatisfactory results from tumor targeting with labeled antibodies. The structural identifiability of the model parameters is also analyzed and it is shown that, with the prior knowledge of some parameters which is likely in practice, the remaining model parameters are uniquely identifiable.
A mathematical model of tumor growth including autocrine and paracrine control has been developed. The model starts with the logistic equation of Verhulst: dV/dt = rV (1-V/K). Autocrine controls are described as modifiers of the Malthusian growth rate (r), while paracrine controls modify the carrying capacity (K) of the system. The control mechanisms are expressed in terms of "candidate" functions, which are based upon the dynamic distribution of TGF-alpha TGF-beta in the local tumor environment. Three paradigms of tissue growth have been modeled: normal tissue wound repair, unrestricted, unperturbed tumor growth, and tumor growth in a (radiation) damaged environment (the Tumor Bed Effect, TBE). These scenarios were used to test the dynamics of the system against known phenomena. Computer simulations are presented for each case. The mode is being extended to include the description of heterogeneous tumors, within which subpopulations can express differential degrees of growth activity. Heterogeneous tumor models, with and without emergent subpopulations, and models of terminal differentiation are also discussed.
A mathematical model was used to estimate the cost-effectiveness of colorectal cancer screening strategies for people who are at high risk because of a first-degree relative with colorectal cancer. The model uses indirect evidence about such factors as cancer incidence, sensitivity and specificity of different tests, and treatment effectiveness. The analysis indicates that for screening people over 40 yr old an annual fecal occult blood test may reduce colorectal cancer mortality by about one-third, either colonoscopy or barium enema may reduce mortality by approximately 85%, a 3-5-yr frequency for endoscopies or barium enemas preserves 70%-90% of the effectiveness of an annual frequency, and beginning screening at age 50 reduces effectiveness by 5%-10%. Although both barium enemas and colonoscopies appear to be effective in reducing mortality, the lower cost of the barium enema makes it a more cost-effective strategy. All of these estimates depend on the baseline estimates of each of the factors incorporated in the model; the conclusions are most sensitive to assumptions about the natural history of adenomatous polyps, the bleeding of adenomas and presymptomatic cancers, and the sensitivity of the fecal occult blood test. Recommendations about colorectal cancer screening must also consider factors such as discomfort, inconvenience, and the availability of various technologies.
A mathematical model simulating the formation of gas bubbles in biological tissues under decompression is presented. It is written as a system of partial differential equations solved on a computer. For the nitrogen-oxygen gas mixture, used for respiration in deep-water immersions, the effects of the physico-chemical properties of the gases, the magnitude of pressure differentials and the density of bubble-formation centres on the bubble size and rate of growth were studied. It is shown that in the case of drastic pressure differentials the formation of bubbles capable of producing microcirculatory disturbances is accomplished within a few seconds.
A mathematical model of the umbilical placental circulation was used to examine the effect of different physiological variables on the pulsatility index (PI) of the umbilical artery Doppler waveform. The variables include the umbilical and placental resistances, the volume flow rate and the pressure. In the model the branching structure of the placental villous tree is considered in detail, while each arterial branch is itself represented simply using a resistor and a capacitor. Placental vascular disease is modelled as obliteration of a fraction of the terminal branches of the tree. The model umbilical artery PI depends on the ratio of the placental resistance to the umbilical artery resistance. The PI increases with vascular disease, but the rate of increase is not uniform. Initially, the placental resistance and the PI increase very slowly with vessel obliteration. Once the level of vessel obliteration has reached a large enough value--typically between 60% and 90% obliteration--the PI begins to rise sharply. A larger placental vascular bed can accommodate a greater level of vessel obliteration before this rapid PI rise begins. The umbilical artery PI also depends on the pulsatility of the input (aortic bifurcation) pressure waveform, but blood pressure variations in the physically attainable range cannot account for the very high PI values associated with fetal compromise. Physically attainable pressure waveform changes would, however, enable the fetus with substantial placental vascular disease to maintain umbilical volume flow rate, and at the same time exhibit a raised umbilical artery PI value.
A mathematical model is presented which comprises the reactions of glycolysis, the hexose monophosphate shunt (HMS) and the glutathione system in erythrocytes. The model is used to calculate stationary and time-dependent metabolic states of the cell in vitro and in vivo. The model properly accounts for the following metabolic features observed in vitro: (a) stimulation of the oxidative pentose pathway after addition of pyruvate due to a NADP-dependent lactate dehydrogenase as coupling enzyme between glycolysis and the oxidative pentose pathway, (b) relative share of the oxidative pentose pathway in the total consumption of glucose amounting to approximately 10% in the normal case and to approximately 90% under conditions of oxidative stress excreted by methylene blue. From the application of the model to in vivo conditions it is predicted that (c) under normal conditions glycolysis and the HMS are independently regulated by the energetic and oxidative load, respectively, (d) under conditions of enhanced energetic or oxidative load both glycolysis and the HMS are mainly controlled by the hexokinase; in this situation the highest possible values of the energetic and oxidative load which are compatible with cell integrity are strongly coupled and considerably restricted in comparison with the normal case, (e) the stationary states possess bifurcation points at high and low values of the energetic load.
A mathematical model has been developed which allows estimation of the epidemiological and economic effects of different tetanus vaccination strategies. The model was used to simulate the epidemiology of tetanus in Italy from 1955 to 1982, and then applied to a district of Tuscany by utilizing data obtained from a seroepidemiological survey carried out in the same area. For this district we simulated vaccination programmes designed to reach, within 1 or 10 years, coverages of 60 or 90% of the population aged over 10 years who had not been exposed to the neonatal vaccination programme. The most effective strategy, from both the epidemiological and economic point of view, seems to be 90% coverage reached in 1 year's time. Benefits would be increased by improving the reliability of vaccinal anamnesis.
A mathematical model for noise-induced hearing loss is based on the assumption that hair cells are damaged, temporarily or permanently, by metabolic exhaustion, and that the number of damaged hair cells and the hearing loss are monotonically increasing functions of an energy deficiency. The purpose of the model is to focus on the influence of sound intensity, exposure duration, and temporal pattern of the sound exposure on the noise-induced hearing loss from long-duration exposures. The model is restricted to the range of sound levels where metabolic exhaustion probably is the main reason for the hair cell damage. Only exposures with similar frequency spectra and producing moderate hearing losses are considered; frequency dependence is not discussed.
A mathematical model tests possible mechanisms for the progressive failure of blood volume restitution seen after larger hemorrhages ( > 26%) with increasing changes in plasma osmolality. After 10% hemorrhage, the model requires a decrease in net hydrostatic capillary pressure, the release of solute into the extracellular space, and the release of Na+ and K+ from a bound pool in equilibrium with the interstitium to match the experimental data. The solute and released cations expand the interstitium to drive the restitution of volume and protein from 3 to 24 h. After 30% hemorrhage, the best prediction of the average experimental responses occurs when the Na(+)-K(+)-adenosinetriphosphatase (ATPase) in the cell membrane is inhibited by 38.7% from 0.8 to 3 h, and the proportionality between capillary pressure and blood volume is reduced by 68% from its value for 10% hemorrhage. When the change in plasma osmolality is doubled after 30% hemorrhage, an increase in the inhibition of the ATPase to 85% and extension of its duration to 24 h are necessary to match experimental findings. The associated defect in sodium transport may occur after large hemorrhage so that sodium and water move into cells. This response may oppose osmotically driven expansion of the interstitium and thus account for the failure of restitution.
A mathematical model correlating the spread of analgesia to the dose of local anaesthetic and to age or body weight was found analysing the data of 763 caudal blocks in children from age one day to twelve years. Two graphs have been plotted: (1) spread of analgesia, dose, age and (2) spread of analgesia, dose, weight. Both age and weight can be used as predictors to determine the desired level of analgesia, but weight is more useful in very young patients while age is a better guide in older children.
A mathematical model of microvascular exchange in the rat following a burn injury was developed by extending an existing model of normal microvascular exchange to include perturbations characteristic of burn injuries without fluid resuscitation. The changes anticipated for small (10% body surface area) and large (40% body surface area) burns are incorporated systematically into the model until there is no improvement in the statistical fit of the simulation predictions with the experimental data of Lund and Reed (Circulatory Shock 20:91-104, 1986). The "best fit" perturbations for the small burn include the experimentally measured changes in mean arterial pressure and injured tissue pressure as well as changes to plasma protein and fluid transport coefficients in the injured tissue. The larger burn "best fit" simulation required changes to the plasma protein transport coefficients in the intact tissues as well as all of the changes listed above. The simulation results are compared with the available experimental information on burn injuries as well as with the specific data of Lund and Reed (Circulatory Shock 20:91-104, 1986).
A mathematical model has been constructed to predict the workspaces available to patients with reduced ranges of motion at the joints of the upper limb. The model uses the inverse kinematic method of Benati et al. (1982) together with simplified anatomical data. Comparisons with experimental data (Dempster, 1955) show significant discrepancies which are believe to be due to differences in scapular constraint and to simplified anatomical modelling. It is intended to improve the anatomical model by the use of the published data of Eyclesheimer and Shoemaker (1911). Despite the discrepancies, the model already detects significant differences between normal workspaces and those of patients with restriction at the joints of the upper limb. It is believed that this model could, when incorporated with empirical data, form the basis of an "expert" system to assist the therapist and clinician in the planning of rehabilitation of the upper limb.