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Evaluation of some mathematical models for tumor growth.

A number of mathematical models for tumor growth have been proposed in the past to increase the understanding of the tumor growth process. This study evaluates the exponential, the Gompertz, the Bertalanffy and the logistic models. The data used for the evaluation of the models consists of: the untreated primary carcinoma of the human lung, and induced sarcoma in mice. The non-linear regression method was used for the analysis of the data. The logistic equation gave the best fit in the cases of all seven patients. However, the Bertalanffy equation was the best in seven out of 10 cases of mice. The models were also judged by comparing the percentage error in predicting the volume of a tumor.

Animals↗

Construction, mathematical study and numerical simulation of a calcium turnover model during skeletal muscle contraction.

Starting from recent physiological data we establish a first order non-linear differential system which models the calcium turnover during skeletal muscular contraction. We describe the methods allowing one to calculate with a computer the non-experimentally directly-measured initial conditions. In order to ensure the consistency of the system with physiology we prove the positivity, a priori inequalities and both the existence and the uniqueness of the solutions. To obtain a complete knowledge of the system during a twitch, the aim is to identify, using the experimental data the calcium, two re-uptake constants and the function describing its ejection rate. We demonstrate the uniqueness of the identification problem, we establish an original algorithm for the identification of the two constants, and we show that the function to be identified is the solution of an integral equation for the resolution of which we give an algorithm whose convergence is proved. According to this mathematical study the model is numerically simulated on a computer; this allows us thus to find again experimental data in a consistent frame, and to predict the values of some non-measured constants and functions.

Actins↗

A mathematical model for analyzing beat-to-beat difference in the human fetal heart rate during gestation and labor.

We devised a model suitable for mathematical analysis of beat-to-beat differences (BBDs) in fetal heart rates (FHRs). Factor analysis was applied, using a computer system, on a specially devised model of an FHR matrix. This matrix was arranged with FHRs and BBDs at 1-beat/min intervals by rows and columns, respectively. After obtaining the BBD by subtracting the antecedent FHR from the following FHR in a given pair of two consecutive FHRs, both variables of the antecedent FHR and BBD were crossed and the number of one was recorded at the corresponding element of the matrix. This procedure was done for all pairs of FHR and BBD yielding a matrix containing the cumulative incidences. Investigated was a total of 225,282 FHRs obtained from 20 fetuses between 37 and 41 weeks of gestation, by means of a scalp-lead electrocardiograph taken during labor. As shown by clusters of BBDs in units of beats/min, three different factors become evident: fluctuation around zero bpm, plus deviations and minus deviations. The first is considered to play a role in maintaining so-called baseline FHRs, and the second and the third indicate accelerating and decelerating actions on FHRs, respectively. This analytical model is discussed with reference to the findings obtained.

Computer Simulation↗

Population dynamics of Trichostrongylus colubriformis in sheep: mathematical model of worm fecundity.

A mathematical model was constructed to predict the egg production of Trichostrongylus colubriformis worms as a function of worm age and host experience of infection. The model set egg production at zero until the worm was 14 days old, when a linear increase to maximum egg production levels occurred over 7 days. It was assumed that egg production remained at maximum levels until a threshold total worm burden was exceeded, when an exponential decline in egg production occurred. The rate of decline was assumed independent of worm age or worm burden. The estimated parameters (maximum egg production, threshold, lag and rate of decline) were optimized by fitting values predicted from the model to faecal egg counts observed in continuously infected sheep, giving R2 = 0.80. The model was validated against faecal egg counts obtained in two other continuous infection experiments, one performed at the same laboratory and the other in Britain.

Animals↗

A mathematical model of the flow in the posterior communicating arteries.

This paper reports on a mathematical model designed to study the hemodynamics of one posterior communicating artery and its afferent and efferent vessels. The variables in the model are the diameter of the posterior communicating artery, the resistance in the vertebral artery and the ratio of the two peripheral resistances. In the model, the "posterior communicating artery" exhibits a compensatory capacity, as defined in the introduction, which appears to be independent of its diameter. The fluxes in the efferent vessels are dominated by the peripheral resistances.

Carotid Arteries↗

Mathematical principles of cell division.

The report given here is an attempt of the mathematical formulations to solve a problem concerning cell division and the control of mass and volume of the nucleus and cytoplasm. The study of the interaction forces on the nucleus and the vibrational frequencies during cell division are analyzed in detail. Lastly, the vibrations of a cell and the coalescence of the vacuoles inside a living plant cell are discussed theoretically.

Animals↗

A mathematical model of aortic valve vibration.

It has been shown experimentally that the second heart sound is produced by diastolic vibrations of the closed aortic valve. In the present paper a mathematical model of this vibration is developed from first principles. The model assumes a one dimensional but non-linear fluid behavior. The problem is coupled through a non-linear, planar valve. A solution is obtained using the method of characteristics developed in finite difference form. The resulting valve frequency and amplitude are in good agreement with patient data. The model predicts a strong dependency of response on the valve forcing function and valve stiffness; and a weaker dependency of response on valve mass.

Aortic Valve↗

A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes.

Following a survey of existing mathematical models of running, a new analysis is developed, based on the first law of thermodynamics. The method properly accounts for each term in the energy balance, and avoids the use of mechanical efficiency factors. A relationship is derived between race distance and the time taken to run that distance. An excellent correlation of results from recent Olympic Games is established for events over distances from 100 m to 10,000 m. The velocity-time relationship for a sprinter running 100 m at maximum available power is obtained by numerical integration of the power equation. It is shown that the peak velocity is achieved in the middle stages of the race, a result which is consistent with practice, but which previous calculations based on Newton's laws have failed to predict. Further applications of the analysis are indicated.

Biomechanical Phenomena↗

A mathematical analysis of the influence of adverse and favourable winds on sprinting.

A mathematical analysis of running performance, based on the first law of thermodynamics and originally derived for still air conditions, is extended to account for the effects of favourable and adverse winds. Solutions to the full theory have been obtained by numerical integration of the governing equations. Simplifications to the full calculation procedure have also been investigated. Calculations for races over a distance of 100 m show that the advantage to an athlete of a following wind increases progressively with increasing wind speed. A favourable wind of 2 ms-1 provides a benefit of about 0.18 s on running time.

Air Movements↗

A mathematical model of the flow in the circle of Willis.

A mathematical model of the flow in the circle of Willis has been designed and the effects of (a) the large anatomical variation of the communicating arteries and (b) physiological changes of the resistances of the vertebral arteries have been studied. The influence of the posterior perforating arteries on the flow in the posterior communicating arteries has been investigated as well, with special attention being paid to the possible occurrence of a 'dead point'. In the model, the influence of diameters of the communicating arteries on the flow in the afferent vessels and the segments of the circle turns out to be considerable, especially in the range of the anatomical variation of the diameters. Within this range flow reductions due to an increased resistance of the vertebral artery will be compensated for by the system. Assuming that the values and ratios of the peripheral resistances are within the physiological range, a dead point is not to be expected in the flow in the posterior communicating arteries.

Cerebrovascular Circulation↗

A mathematical model of the patellofemoral joint.

A mathematical model of the patellofemoral joint taking into account movements and forces in the sagittal plane is described. The system parameters of the model are the locations of the attachments of the quadriceps muscle and the patellar ligament, the length of the patellar ligament, the dimensions of the patella and the geometry of the articulating surfaces. They were obtained from ten autopsy knees. The model enables calculation of the relative position of the patella, patellar ligament and quadriceps tendon, the location of the patellofemoral contact point and the magnitude of the patellofemoral compression force and the force in the patellar ligament as a function of the location of the tibial tuberosity at different flexion-extension angles of the knee. The model is validated by comparing model data with experimentally determined data.

Biomechanical Phenomena↗

Mathematical modelling of trabecular bone structure: the evaluation of analytical and quantified surface to volume relationships in the femoral head and iliac crest.

The three-dimensional architecture of trabecular bone has structural trends related to physical function as described by Wolff's law. Mathematical modelling provides a means of analysing these structures through the use of simplified representations. A single measure of mineralized bone volume per unit volume of structure (Vv) and the surface area of mineralized bone per unit volume of structure (Sv) does not identify a particular architecture in any detail; the way in which Sv changes in relation to Vv does provide this information as the structure remodels. A series of structures using the elements of plates and rods were created. The rates of change of Sv with respect to Vv for trabecular structures give insight into differences in such models. Structures in the femoral head and iliac crest were analysed by power curve regression. In the principal compressive region, just above the medial cortex, advanced osteoarthritis was associated with a preferential loss of rods from the normal trabecular structure, resulting in a more plate-like architecture. The iliac crest remodelling that takes place in the osteoporotic appears to be the result of a generalised bone loss with some of the thinner elements of the structure being removed completely, resulting in an increase in unit cell dimension. The consequence of changing unit cell size has a major impact on surface availability for osteoblastic and osteoclastic activity. The simple plate model as a basis for the stereological analysis of trabecular structures is therefore limited because of the mixed plate and rod nature of trabecular architecture.

Anthropometry↗

Mathematical models of the flow in the basilar artery.

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. This study deals with the question whether a parabolic (Poiseuille) profile will have been established before the basilar artery divides into both posterior cerebral arteries. The inlet length (that is, the downstream distance needed for the flow to become approximately equal to the limiting Poiseuille flow) and velocity profiles have been computed from two- and three-dimensional mathematical models in which flow pulsatility and vessel wall distensibility have been neglected and the complex geometry of the junction has been taken into account in a simplified form. The results show that the flow at the end of the basilar artery is far from being parabolic and that an asymmetry in the entrance flow will be carried along towards the end of the basilar artery, thus affecting flows in the circle of Willis.

Basilar Artery↗

The simulation of aerial movement--II. A mathematical inertia model of the human body.

A mathematical inertia model which permits the determination of personalized segmental inertia parameter values from anthropometric measurements is described. The human body is modelled using 40 geometric solids which are specified by 95 anthropometric measurements. A 'stadium' solid is introduced for modelling the torso segments using perimeter and width measurements. This procedure is more accurate than the use of elliptical discs of given width and depth and permits a smaller number of such solids to be used. Inertia parameter values may be obtained for body models of up to 20 segments. Errors in total body mass estimates from this and other models are discussed with reference to the unknown lung volumes.

Anthropometry↗

Counting knowledge and skill in cognitive addition: a comparison of normal and mathematically disabled children.

The relationship between counting knowledge and computational skills (i.e., skill at counting to solve addition problems) was assessed for groups of first-grade normal and mathematically disabled (MD) children. Twenty-four normal and 13 MD children were administered a series of counting tasks and solved 40 computer-administered addition problems. For the addition task, problem-solving strategies were recorded on a trial-by-trial basis. Performance on the counting tasks suggested that the MD children were developmentally delayed in the understanding of essential and unessential features of counting and were relatively unskilled in the detection of certain forms of counting error. On the addition task, the MD children committed many more computational errors and tended to use developmentally immature counting procedures. The immature counting knowledge of the MD children, combined with their relatively poor skills at detecting counting errors, appeared to underlie their poor computational skills on the addition task. Suggestions for future research are presented.

Child↗

Quantitative analysis of interleukin-2-induced proliferation in the presence of inhibitors using a mathematical model.

The proliferative response of CTLL-2 cells to human recombinant interleukin-2 (IL-2) can be modeled mathematically using enzyme kinetic equations. This approach has been used to analyze dose-response curves (IL-2 concentration vs. level of proliferation) measured by MTT and [3H]TdR assays. The values of functional dissociation constants, equivalent to IL-2 concentrations giving 50% of the maximal response, depended on the cell concentration and increased from 4 to 60 pM for the [3H]TdR assay and from 40 to 140 pM for the MTT assay when the cell concentration was increased from 2 x 10(3) to 4 x 10(4) cells/well. The types of inhibition and dissociation constants for various inhibitors of IL-2-dependent proliferation such as mAbs against IL-2 receptor (7D4 and AMT13) and normal mouse serum (NMS) were also analyzed. Both mAbs exhibited competitive mechanisms of inhibition whereas NMS inhibited IL-2-driven proliferation in a mixed manner. Two gel-filtration fractions of NMS with inhibitory activity manifested different types of inhibition: purely competitive type of inhibition in the case of a 10-15 kDa fraction and a mixed type of inhibition for a 100-150 kDa fraction. The proposed model can also be used for quantitative analysis of the influence of various factors (pH, temperature, cultivation condition) on the level of proliferation.

Animals↗

Mathematical modeling of the effects of the signal recognition particle on translation and translocation of proteins across the endoplasmic reticulum membrane.

The kinetics of the signal recognition particle(SRP)-mediated process of protein translocation across the endoplasmic reticulum membrane was studied by mathematical modeling and complementary experiments. The following results were obtained. (1) A model according to which SRP directs the ribosome, rather than the mRNA, to the membrane is supported by experiments designed to discriminate between the two alternatives. (2) This model describes both steady-state and synchronized translation experiments and makes a number of predictions. (3) The interaction between a nascent protein and SRP may be described by two parameters: (i) a binding constant which can be attributed to the structure of the signal peptide, and (ii) the size of the "SRP-window", i.e. the distance between the first and the last site on the polypeptide chain that can interact with SRP. For preprolactin a binding constant of 1 to 2.5 nmol-1l was estimated. Modeling of the synchronized synthesis of ovalbumin indicates that it has a much weaker binding constant than preprolactin (approximately 0.25 nmol-1l) although we cannot exclude the possibility that the SRP-window may be also smaller. (4) A better understanding of the molecular effects of SRP on translation and translocation through the rough endoplasmic reticulum membrane has been achieved. Inhibition of the steady-state rate of translation by SRP requires a stoichiometric interaction of SRP with ribosomes carrying nascent polypeptide chains and will occur only when ribosomes are piled up back to the initiation site. Translocation, on the other hand, requires only the catalytic action of SRP and is determined by the local concentration of protein-synthesizing ribosomes accumulated at the site(s) of SRP interaction. As a consequence, translational inhibition by SRP may sometimes fail to occur, depending either on the type of protein or on experimental conditions, such as a high mRNA concentration, even if translocation can be demonstrated. (5) A rough extrapolation to the conditions in vivo indicates that all synthesized polypeptide chains destined for translocation across or integration into the endoplasmic reticulum membrane are indeed quantitatively translocated and that no translational inhibition occurs.

Endoplasmic Reticulum↗

A mathematical treatise on the fit of crown castings.

A mathematical analysis was given to calculate the marginal cement thickness, Dm, of a cemented crown casting as a function of the minimum cement film thickness, F; the width, W, of the base of the tapered region; the width, W', at the bevel (if present) of the tooth preparation; the taper angle, theta; the marginal angle, beta; and the compensatory expansion, X, of the metal investment system: Dm = [F/sin theta - 0.5(X - 1) (W cot theta - W' cot beta)] sin beta. The elevation of the cemented casting, delta H, above the tooth preparation may also be obtained from this equation by substituting beta = 90 degrees, in which case Dm = delta H. The effect of the preparation of a chamfer margin or bevel on the marginal film thickness may be calculated from this equation. An additional analysis showed that the minimum marginal cement thickness, Dm = F, can be obtained for a shoulder preparation and X values of common metal-investment systems by applying a spacer of appropriate thickness on the entire occlusal and axial walls of the die. This renders the preparation of bevels superfluous. The thickness of the spacer should allow for the cement film thickness, roughness of the tooth and casting surfaces, dimensional inaccuracies of the die, and distortions of the wax pattern. If the spacer is thicker than required, a minimum value of Dm is still obtained for preparations without bevels. If the horizontal part of the die shoulder is covered as well with the spacer, a greater spacer thickness is required to obtain a minimum value of Dm than if the spacer extends to the end of the axial walls. In this case the spacer thickness should be greater on the occlusal die wall than on other regions of the die for values of X smaller than 1. The spacer should also be thicker in the occlusal region if restriction of the investment setting expansion by the wax pattern occurs mainly in this region.

Cementation↗