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The authors present methodical ways of diagnosis of the type of the epidemic process in dysentery with the aid of computers base on the application of the theory of recognition of imapes.
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The goal of cancer immunotherapy is to boost a patient's immune response to a tumour. Yet, the design of an effective immunotherapy is complicated by various factors, including a potentially immunosuppressive tumour microenvironment, immune-modulating effects of conventional treatments and therapy-related toxicities. These complexities can be incorporated into mathematical and computational models of cancer immunotherapy that can then be used to aid in rational therapy design. In this review, we survey modelling approaches under the umbrella of the major challenges facing immunotherapy development, which encompass tumour classification, optimal treatment scheduling and combination therapy design. Although overlapping, each challenge has presented unique opportunities for modellers to make contributions using analytical and numerical analysis of model outcomes, as well as optimization algorithms. We discuss several examples of models that have grown in complexity as more biological information has become available, showcasing how model development is a dynamic process interlinked with the rapid advances in tumour-immune biology. We conclude the review with recommendations for modellers both with respect to methodology and biological direction that might help keep modellers at the forefront of cancer immunotherapy development.
A mathematical formalism is presented for use with digital computers to permit the routine fitting of data to physical and mathematical models. Given a set of data, the mathematical equations describing a model, initial conditions for an experiment, and initial estimates for the values of model parameters, the computer program automatically proceeds to obtain a least squares fit of the data by an iterative adjustment of the values of the parameters. When the experimental measures are linear combinations of functions, the linear coefficients for a least squares fit may also be calculated. The values of both the parameters of the model and the coefficients for the sum of functions may be unknown independent variables, unknown dependent variables, or known constants. In the case of dependence, only linear dependencies are provided for in routine use. The computer program includes a number of subroutines, each one of which performs a special task. This permits flexibility in choosing various types of solutions and procedures. One subroutine, for example, handles linear differential equations, another, special non-linear functions, etc. The use of analytic or numerical solutions of equations is possible.
Analysis of large-scale gene expression studies usually begins with gene clustering. A ubiquitous problem is that different algorithms applied to the same data inevitably give different results, and the differences are often substantial, involving a quarter or more of the genes analyzed. This raises a series of important but nettlesome questions: How are different clustering results related to each other and to the underlying data structure? Is one clustering objectively superior to another? Which differences, if any, are likely candidates to be biologically important? A systematic and quantitative way to address these questions is needed, together with an effective way to integrate and leverage expression results with other kinds of large-scale data and annotations. We developed a mathematical and computational framework to help quantify, compare, visualize and interactively mine clusterings. We show that by coupling confusion matrices with appropriate metrics (linear assignment and normalized mutual information scores), one can quantify and map differences between clusterings. A version of receiver operator characteristic analysis proved effective for quantifying and visualizing cluster quality and overlap. These methods, plus a flexible library of clustering algorithms, can be called from a new expandable set of software tools called CompClust 1.0 (http://woldlab.caltech.edu/compClust/). CompClust also makes it possible to relate expression clustering patterns to DNA sequence motif occurrences, protein-DNA interaction measurements and various kinds of functional annotations. Test analyses used yeast cell cycle data and revealed data structure not obvious under all algorithms. These results were then integrated with transcription motif and global protein-DNA interaction data to identify G1 regulatory modules.
The proper evaluation of the accuracy of a new computer program for radiation-therapy dosimetry requires consideration of both the mathematical algorithm used in the program and the performance required in clinical applications. As an example, our evaluation of the irregular-fields dosimetry program currently marketed by SHM for their Rad-8 system is described. The evaluation begins with an explanation of the mathematical computation described. The evaluation begins with an explanation of the mathematical computation method, with emphasis on the points where the calculation differs from previous methods. Next, the procedure for setting up the beam data file is discussed. Finally, a step-by-step procedure is described in which calculated doses are compared with measured doses, using a Varian Clinac-4 with lead flattening filter, and the limits within which a +/- 5% accuracy is attainable are estimated. Some sources of error and areas for possible improvements are mentioned.
BACKGROUND: Advocates of acute normovolemic hemodilution (ANH) frequently neglect to consider the decreasing hematocrit of the patient during both hemodilution and the subsequent operative procedure and the need to begin transfusion at some minimal hematocrit. STUDY DESIGN AND METHODS: For more accurate prediction of the efficacy of ANH, equations were derived and a computer model developed that allowed accounting for the decreasing hematocrit due to blood loss in an isovolemic patient and calculating the red cell volume on a minute-by-minute basis; the model also began the transfusion of ANH blood on a mL-for-mL basis when the minimal hematocrit was reached and transfused any remaining blood following completion of the case. The red cell volume saved by performing ANH for a given estimated blood volume (EBV) was expressed as either the fraction of the red cell volume of a routinely banked unit of blood (red cells stored in additive solution: volume 350 mL, hematocrit 0.65) or the number of units saved. RESULTS: The number of units saved in a typical example--EBV, 5000 mL; pre-ANH hematocrit, 0.40; minimal hematocrit at which transfusion was begun, 0.25 over a range of estimated blood losses (500-2500 mL); and 1 to 5 ANH units drawn--never exceeded 0.6. Even with extensive hemodilution, as in a child (EBV, 1500 mL; pre-ANH hematocrit, 0.40; minimal hematocrit at which transfusion was begun, 0.15; 5 units drawn; and estimated blood losses, 2500, 1500, and 1000 mL) with a postdilution hematocrit of 0.16, the savings would have been only 0.29, 0.44, and 0.49 units, respectively. CONCLUSION: Because of the decreasing hematocrit in a bleeding isovolemic patient and the need to begin transfusion at some minimal hematocrit, the theoretic savings in red cell volume attributable to ANH is less than had previously been appreciated, and additional ANH does not necessarily result in additional patient benefit.
Applying Escudero's formula to Bayes' theorem of conditional probabilities, we have developed a mathematical model to diagnose type-specific brain tumours with great accuracy. We have transformed a multidimensional system of n characteristics on computed tomography into a linear system, making this model very easy to use and, thus, accessible to the clinician and radiologist. We tested this model in 177 brain tumours. In 86% of the cases we obtained a type-specific correct diagnosis, achieving better results than those previously reported. Suggestions are given as to how to improve this method further.
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A mathematical model of ventricular volume regulation based on fluid mechanical principles has been constructed using a systems engineering approach. The parameters used in the model are based on clinical observation, laboratory investigation, and presumptions that will be tested later. The model was constructed to be the basis of a computer simulation. Using the computer simulation, information obtained from the literature and laboratory hypotheses regarding pathophysiology, several enigmatic conditions were tested. The model predicted that over-production of cerebrospinal fluid, as in the case of choroid plexus papilloma, could by itself lead to distention of the ventricular system. In simulating pseudotumor cerebri, if cerebrospinal fluid absorption at the arachnoid villi is impaired and the brain itself is rendered incompressible by swelling, intracranial pressure rises and ventricular volume diminishes. Conversely, in normal-pressure hydrocephalus, if cerebrospinal fluid flow is restricted between the spinal and cortical subarachnoid spaces and the brain is made more compressible, the ventricular volume increases with minimal increases in intracranial pressure. This mathematical model and its associated computer simulation is useful in predicting the behavior of the volume of the cerebral ventricles to a variety of pathological phenomena.
The relationship between verbal and visual-spatial working memory and mathematical computation skill was examined in children and adults with and without disabilities in mathematics. A hierarchical regression analysis showed that, when partialing for the influence of reading ability, age, and gender, mathematical computation was better predicted by verbal than by visual-spatial working memory. Furthermore, the results showed that the relationship between mathematics ability and working memory were not significantly moderated by age but were stable across a broad age span. We concluded that, regardless of age, deficits in mathematics are mediated by both a domain-general and a domain-specific working memory system.
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A mathematical model of electrophoretic separation processes has been developed and adapted for computer simulations. The model is used to predict the characteristic behavior of a variety of electrophoretic techniques from a knowledge of chemical equilibria and physical transport phenomena. The model provides a unifying basis for a rational classification of all electrophoretic processes.
Following a pretest, 11 participants who were naive with regard to various algebraic and trigonometric transformations received an introductory lecture regarding the fundamentals of the rectangular coordinate system. Following the lecture, they took part in a computer-interactive matching-to-sample procedure in which they received training on particular formula-to-formula and formula-to-graph relations as these formulas pertain to reflections and vertical and horizontal shifts. In training A-B, standard formulas served as samples and factored formulas served as comparisons. In training B-C, factored formulas served as samples and graphs served as comparisons. Subsequently, the program assessed for mutually entailed B-A and C-B relations as well as combinatorially entailed C-A and A-C relations. After all participants demonstrated mutual entailment and combinatorial entailment, we employed a test of novel relations to assess 40 different and complex variations of the original training formulas and their respective graphs. Six of 10 participants who completed training demonstrated perfect or near-perfect performance in identifying novel formula-to-graph relations. Three of the 4 participants who made more than three incorrect responses during the assessment of novel relations showed some commonality among their error patterns. Derived transfer of stimulus control using mathematical relations is discussed.
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A mathematical model was developed to describe a dialysis process for the continuous fermentation of whey lactose to lactic acid, with neutralization to a constant pH by ammonia. In the process, whey of a relatively high concentration is fed into the fermentor circuit at a relatively low rate so that the residual concentration of lactose is low. The fermentor effluent contains ammonium lactate, bacterial cells, and residual whey solids and could be used as a nitrogen-enriched feedstuff for ruminant animals. Only water is fed into the dialysate circuit at a relatively high rate. The dialysate effluent contains purified ammonium lactate and could be converted to lactic acid and ammonium sulfate for industry. The fermentation was specifically modeled as a set of equations representing material balances and rate relationships in the two circuits. Dialysis continuous fermentations, in general, were modeled by combining these equations and by using dimensionless parameters. The generalized model was then solved for the steady state and used to simulate the specific fermentation on a digital computer. The results showed the effects of various material and operational and kinetic parameters on the process and predicted that it could be operated efficiently.