Case Studies in cancer and its treatment by radiotherapy

Mathematical modelling has played a major role in certain aspects of the wide area of study of cancer and its treatment. Relevant background material is followed by mathematical modelling case studies in tumour growth and the response of tumour cells to irradiation. These components are synthesized in Case Study 4 where the problem of the derivation of optimal radiotherapy treatment schedules is discussed. The case studies are presented in a form suitable for classroom development.     

Preliminary Investigation on the Effect of Production Parameters on the Strength of Large Tablets

A case study of large tablet manufacturing in the chemical industry is presented. Since the tablets are much larger than the usual scale in the pharmaceutical industry, common practice that was developed mainly for pharmaceutical tablets could not be applied directly. Customer complaints concerning damaged tablets were the reason for this comprehensive study, which was aimed first at evaluating a testing method for process control, then to provide some hints for stabilizing the manufacturing process and for producing stronger tablets. Three test methods were evaluated: measuring compression and tension crush strengths and wear tests in a rotating drum. It was shown that the wear test provides sufficient results faster and easier than the other methods. By testing hundreds of tablets, produced in various batches, production parameters such as granule aging, surrounding humidity, tableting pressure and tablet aging were evaluated in order to provide stronger tablets. The strongest tablets were achieved by using fresh granules at very high humidity and compacting pressure and by allowing the tablets to age for at least 2 weeks.     

Equilibrium,kinetics, and mechanism of the malonic acid–iodine reaction

The equilibrium constant for the reaction CH₂(COOH)₂ + I₃^? ? CHI(COOH)₂ + 2I^? + H⁺, measured spectrophotometrically at 25°C and ionic strength 1.00M (NaClO₄), is (2.79 ± 0.48) × 10^?4M². Stopped-flow kinetic measurements at 25°C and ionic strength 1.00M with [H⁺] = (2.09-95.0) × 10^?3M and [I^?] = (1.23-26.1) × 10^?3M indicate that the rate of the forward reaction is given by (k₁[I₂] + k₃[I₃^?]) [HOOCCH₂COO^?] + (k₂[I₂] + k₄[I₃^?]) [CH(COOH)₂] + k₅[H⁺] [I₃^?] [CH₂(COOH)₂]. The values of the rate constants k₁-k₅ are (1.21 ± 0.31) × 10², (2.41 ± 0.15) × 10¹, (1.16 ± 0.33) × 10¹, (8.7 ± 4.5) × 10^?1M^?1·sec^?1, and (3.20 ± 0.56) × 10¹M^?2·sec^?1, respectively. The rate of enolization of malonic acid, measured by the bromine scavenging technique, is given by k_en[CH₂(COOH)₂], with k_en = 2.0 × 10^?3 + 1.0 × 10^?2 [CH₂(COOH)₂]. An intramolecular mechanism, featuring a six-member cyclic transition state, is postulated to account for the results on the enolization of malonic acid. The reactions of the enol, enolate ion, and protonated enol with iodine and/or triodide ion are proposed to account for the various rate terms.     

A generalization of Malho’s method for obtaining large cardinal numbers

Mahlo used a method by which fixed points of an enumeration of regular cardinals were employed to get a hierarchy of “large cardinals.” He also employed a second method which, in a certain sense, is much stronger than the first. Here the methods are investigated and generalized and the relations between them are clarified. This stronger method turns out to be a kind of “least upper bound” to all “fixed-points operations.” Possibilities of strengthening these processes in a natural way are pointed out.