Some mathematical aspects of the medical diagnostic process. I: a general mathematical model

Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.     

Some mathematical aspects of the diagnostic process. II: A mathematical model of electrocardiographic diagnosis

In a previous paper (Bartholomay, 1971), a general mathematical model of the medical diagnostic process was described. The present paper amounts to a relization of that process in terms of conventional 12-lead electrocardiographic diagnosis as enunciated by Dr. Harold D. Levine (1966) in the course of a collaborative study by Dr. Levine and the present author at the Peter Bent Brigham Hospital of the Harvard Medical School between 1963 and 1966. The main details of the cognitive component of that model are described in detail here. The model has been programmed onto a computer system consisting of an analog-digital converter and general purpose digital computer and amounts to a simulation of Dr. Levine’s electrocardiographic analysis procedure.     

Some mathematical aspects of mapping DNA cosmids

A number of experimental and mathematical problems must be solved before high resolution physical maps of mammalian chromosomes can be reliably determined. Such a map might consist of an ordered set of nonsequenced, overlapping DNA fragments 20,000-40,000 bases long, produced by digestion of a chromosome, using two restriction enzymes. Map construction requires assigning a signature to each fragment that differentiates it unambiguously from every other fragment, and then devising a computationally efficient algorithm that will provide a unique ordering of the fragments. In the first part of this paper we present a polynomial time algorithm that yields a unique map, and is largely independent of the method for assigning signatures. In the next section we analyze the distribution of lengths of restriction digest fragments and discuss the implications for the algorithm, including the expected number of map gaps. Finally, we discuss a specific method for assigning signatures proposed by Hans Lehrach, based on which of a panel of probes binds to a given fragment. In particular we examine the effects of fragment length heterogeneity on the theoretical optimum length and number of probes, and the extent to which false signatures might be obtained by nonspecific binding. We conclude that the Lehrach strategy is effective provided the number of probes is >-150, but that each fragment will need testing with at most 25 probes.     

Some aspects of laboratory cometary models

In this paper, we discuss some aspects of laboratory experiments proposed in the literature, associated with the chemical reactions that may occur in the comets. These comments are made in the light of the use of energetic particles or radiation as energy sources, relevant chemical reactions and compounds detected in those experiments.     

Some bio-sociological aspects of the mathematical theory of communication

In the first part of the paper a general discussion of the transmission of information through neural chains is given in terms of the Shannon-Weaver theory. It is pointed out that with the all-or-none law a single chain of neurons connected in series transmits one bit of information per signal. A set ofN independent parallel chains transmitsN bits per signal. If, however, the chains are interconnected, the amount of information is reduced. At the same time, however, the degree of coordination of the final neuromuscular reaction is increased. A relation between the maximum possible speed of a reaction and its degree of coordination is derived, and possible applications to spoken language are suggested. A general quantitative discussion of the relation between amount of information and amount of knowledge which an individual may obtain when confronted with the external world is made and a possible connection with new trends in logical thinking is pointed out. In the second part transmission of information through “social chains” is discussed under certain special assumptions. An expression for the “social channel noise” in terms of the length of the channel is derived. Finally an expression is given for the amount of information transmitted from one individual to another in a social group of uniform density as a function of the physical distance between the two individuals.     

Some remarks on the mathematical aspects of automobile driving

Some aspects which involve the interaction of the human element and of the machine element in driving are discussed. As an example a simple equation is derived for the maximum safe speed of a car on an empty road. The parameters of the equation are partly of physiological nature, partly of mechanical nature. Another example treats in a similar manner the problem of a car passing another car moving in the same direction.     

The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling

In this paper simple models for tumour growth incorporating age-structured cell cycle dynamics are considered in the presence of two non-cross-resistant S-phase specific chemotherapeutic drugs. According to the seminal work of Goldie and Coldman, if one cannot deliver two cell cycle phase non-specific, non-cross-resistant drugs simultaneously, for example due to toxicity, and both drugs are identical apart from resistance, one should alternate their delivery as rapidly as possible. However consider S-phase specific drugs. One might speculate that, for example, alternating the two drugs at intervals of T, where T is the mean cell cycle time, is better than alternating the drugs at intervals of T/2, as the latter strategy allows the possibility of a cell cycle sanctuary. Such speculation implicitly requires a sufficiently low variance of the cell cycle time, and hence it is not clear if such reasoning prevents a generalisation of the results of Goldie and Coldman. This question is addressed in this paper via a detailed modelling investigation, as motivated by suggestions for future colorectal adjuvant chemotherapy trials and developments in hepatic arterial infusion technology. It is shown that the cell cycle distribution of the resistant cell populations is strongly influenced by the chemotherapy schedule. The consequences of this can be dramatic, and can lead to chemotherapy failure at resonant chemotherapy timings, especially for a small standard deviation of the cell cycle time. The novel aspects of this observation are highlighted compared to other models in the literature exhibiting resonant behaviour in the timing of a periodic chemotherapy protocol. The above investigation also results in the principal prediction of this paper that reducing the drug alternation time to approximately a few hours, if possible, can result in substantial improvements in predicted chemotherapy outcomes. Critically, such improvements are not predicted by the Goldie Coldman model or other chemotherapy scheduling models in the literature.     

Some considerations on vibrational mathematical models of the ulna

An approximate mechanical model of the ulna is proposed. It has realistic complicating factors: variable cross-sectional area and moment of inertia, and non-uniform mechanical properties.     

Growth and reproduction in higher plants I. Theoretical analysis by mathematical models

To analyse the whole life of higher plants, an attempt was made to describe their growth and reproduction by mathematical models based on the elements determining matter production and economy of the matter. A plant body was regarded as a compound system of two parts; “productive part” and “reproductive part”. A parameter (reproductive index) was introduced to connect these two parts, and a set of the mathematical models describing the quantitative growth of these two parts were established. Two basic patterns of reproduction in higher plants were distinguished into “D-reproduction” and “I-reproduction”. The state of matter production of the mother plant determined an initial size of the daughter plant in theD-reproduction, while, in theI-reproduction, it did not determine the initial size of the daughter, but determined the number of propagules. The model of each reproduction pattern was also constructed. A formula determining the initial size of a plant in a given generation was constructed as the model of theD-reproduction. The model for theI-reproduction described the number of propagules produced in a given generation. Some aspects of the plant life, e.g. the optimum reproductive index, the switch-over time from the vegetative to the reproductive growth phase, the seed number, types of expansive reproduction, were theoretically analysed and discussed under these mathematical models.     

Optimization of batch fermentation processes. I. Development of mathematical models for batch penicillin fermentations

Two kinds of mathematical models have been developed for batch penicillin fermentations: (1) general models, based on averaged, nondimensionalized cell and penicillin synthesis curves from plant, scale fermentors and (2) particular models developed from specific sets of experimental data from two sources. Parameter-temperature functions used with the general models were assumed to have general shapes which could apply to many fermentations, i.e., they were based on the familiar temperature response of enzyme-catalyzed reactions. Parameter-temperature functions for the particular models were determined from experimental data for batch runs at various temperatures.     

A mathematical model of brain tumour response to radiotherapy and chemotherapy considering radiobiological aspects

Glioblastoma is the most frequent and malignant brain tumour. For many years, the conventional treatment has been maximal surgical resection followed by radiotherapy (RT), with a median survival time of less than 10 months. Previously, the use of adjuvant chemotherapy (given after RT) has failed to demonstrate a statistically significant survival advantage. Recently, a randomized phase III trial has confirmed the benefit of temozolomide (TMZ) and has defined a new standard of care for the treatment of patients with high-grade brain tumours. The results showed an increase of 2.5 months in median survival, and 16.1% in 2 year survival, for patients receiving RT with TMZ compared with RT alone. It is not clear whether the major benefit of TMZ comes from either concomitant administration of TMZ with RT, or from six cycles of adjuvant TMZ, or both.The objectives were to develop our original model, which addressed survival after RT, to construct a new module to assess the potential role of TMZ from clinical data, and to explore its synergistic contribution in addition to radiation. The model has been extended to include radiobiological parameters. The addition of the linear quadratic equation to describe cellular response to treatment has enabled us to quantify the effects of radiation and TMZ in radiobiological terms.The results indicate that the model achieves an excellent fit to the clinical data, with the assumption that TMZ given concomitantly with RT synergistically increases radiosensitivity. The alternative, that the effect of TMZ is due only to direct cell killing, does not fit the clinical data so well. The addition of concomitant TMZ appears to change the radiobiological parameters. This aspect of our results suggests possible treatment developments.Our observations need further evaluations in real clinical trials, may suggest treatment strategies for new trials, and inform their design.     

Some population genetic models combining artificial and natural selection pressures: I. One-locus theory

A series of one-locus genetic models involving the combined effects of artificial and natural selection forces are analyzed. Other factors incorporated in this work include the influence of the imposed or inherent mating system, the relevance of timing in application of the two types of selection forces, the importance of multiallelism and dominance relations.