A new wavelet-based method for the solution of the population balance equation

A new wavelet-based method for solving population balance equations with simultaneous nucleation, growth and agglomeration is proposed, which uses wavelets to express the functions. The technique is very general, powerful and overcomes the crucial problems of numerical diffusion and stability that often characterize previous techniques in this area. It is also applicable to an arbitrary grid to control resolution and computational efficiency. The proposed technique has been tested for pure agglomeration, simultaneous nucleation and growth, and simultaneous growth and agglomeration. In all cases, the predicted and analytical particle size distributions are in excellent agreement. The presence of moving sharp fronts can be addressed without the prior investigation of the characteristics of the processes. (C) 2001 Published by Elsevier Science Ltd.

The Dubinin-Radushkevich equation and the underlying microscopic adsorption description

The Dubinin-Radushkevich (DR) equation is widely used for description of adsorption in microporous materials, especially those of a carbonaceous origin. The equation has a semi-empirical origin and is based on the assumptions of a change in the potential energy between the gas and adsorbed phases and a characteristic energy of a given solid. This equation yields a macroscopic behaviour of adsorption loading for a given pressure. In this paper, we apply a theory developed in our group to investigate the underlying mechanism of adsorption as an alternative to the macroscopic description using the DR equation. Using this approach, we are able to establish a detailed picture of the adsorption in the whole range of the micropore system. This is different from the DR equation, which provides an overall description of the process. (C) 2001 Elsevier Science Ltd. All rights reserved.

Clinical aspects of gestational trophoblastic disease: A review based partly on 25-year experience of a statewide registry

Background: Gestational trophoblastic disease is a fascinating group of pregnancy disorders characterised by abnormal proliferation of trophoblast, ranging from benign to malignant. Because the disease is uncommon, there is a need to formulate management with the assistance of collective information. Methodology: A review of available information from English written literature was undertaken especially data reported by registries around the world (Charing Cross Hospital in England, the North-western University and the New England area in the USA as well as our own experience in Queensland, Australia). Where possible, collated data from relevant studies were analysed to answer some of the questions posed in clinical practice, with reference to metastatic disease to liver and brain, twinning of molar gestation and coexisting fetus, and placental-site tumour. Results: We found that molar gestation can be classified according to its clinical presentation which influences the time taken to reach human chorionic gonadotropin (HCG) 'negativity' and the risk of persisting disease. Categorisation of risk is the basis for choice of chemotherapy to achieve good outcomes. Metastases to liver and brain remain problems in management; the development of 'new' metastases during chemotherapy is a very poor prognostic factor. In the variant of twinning with molar gestation and coexisting fetus, it is important to elucidate the fetal karyotype in planning management: a 69XXX fetus is not salvageable but a normal 46XX or 46XY fetus faces the prospect of early preterm delivery. The placental-site tumour is very rare; localised disease is curable by surgery; chemotherapy is less effective in disseminated disease. From collated worldwide data, the recurrence rate after one mole is 1.3% and after two or more is 20%. Reproductive outcome in subsequent pregnancies, even after multidrug chemotherapy, is not different from the general population. Because of the increased risk long-term of second tumours after multidrug chemotherapy a closer surveillance of these patients is necessary Conclusion: In general, the disease in its persisting or malignant form is 'a cancer model par excellence' because of an identifiable precursor condition, a reliable HCG marker, and sensitivity of the disease to cytotoxic drugs. With current management, retention of fertility is possible and normal reproductive outcome assured.

South-South relations and the English School of International Relations: Chinese and Brazilian ideas and involvement in Sub-Saharan Africa

The rise of large developing countries has led to considerable discussions of re-balancing global relations and giving greater priority to understanding South-South relations. This paper, in exploring the central ideas of Chinese and Brazilian foreign policy and the behavior of these two rising Southern countries toward Sub-Saharan Africa, argues that the English School of International Relations is well suited to understanding the intentions and actions that characterize South-South relations.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Finanças comportamentais: sobre-reação e sub-reação no mercado bolsista português

Dissertação para obtenção do Grau de Mestre em Contabilidade e Finanças Orientador: Mestre Luis Pereira Gomes

An application of structural equation modeling of test dispositional optimism as mediator or moderator in quality of life in patients with chronic disease

The aim of the present study was to test a hypothetical model to examine if dispositional optimism exerts a moderating or a mediating effect between personality traits and quality of life, in Portuguese patients with chronic diseases. A sample of 540 patients was recruited from central hospitals in various districts of Portugal. All patients completed self-reported questionnaires assessing socio-demographic and clinical variables, personality, dispositional optimism, and quality of life. Structural equation modeling (SEM) was used to analyze the moderating and mediating effects. Results suggest that dispositional optimism exerts a mediator rather than a moderator role between personality traits and quality of life, suggesting that “the expectation that good things will happen” contributes to a better general well-being and better mental functioning.

Automatic learning for the classification of chemical reactions and in statistical thermodynamics

This Thesis describes the application of automatic learning methods for a) the classification of organic and metabolic reactions, and b) the mapping of Potential Energy Surfaces(PES). The classification of reactions was approached with two distinct methodologies: a representation of chemical reactions based on NMR data, and a representation of chemical reactions from the reaction equation based on the physico-chemical and topological features of chemical bonds. NMR-based classification of photochemical and enzymatic reactions. Photochemical and metabolic reactions were classified by Kohonen Self-Organizing Maps (Kohonen SOMs) and Random Forests (RFs) taking as input the difference between the 1H NMR spectra of the products and the reactants. The development of such a representation can be applied in automatic analysis of changes in the 1H NMR spectrum of a mixture and their interpretation in terms of the chemical reactions taking place. Examples of possible applications are the monitoring of reaction processes, evaluation of the stability of chemicals, or even the interpretation of metabonomic data. A Kohonen SOM trained with a data set of metabolic reactions catalysed by transferases was able to correctly classify 75% of an independent test set in terms of the EC number subclass. Random Forests improved the correct predictions to 79%. With photochemical reactions classified into 7 groups, an independent test set was classified with 86-93% accuracy. The data set of photochemical reactions was also used to simulate mixtures with two reactions occurring simultaneously. Kohonen SOMs and Feed-Forward Neural Networks (FFNNs) were trained to classify the reactions occurring in a mixture based on the 1H NMR spectra of the products and reactants. Kohonen SOMs allowed the correct assignment of 53-63% of the mixtures (in a test set). Counter-Propagation Neural Networks (CPNNs) gave origin to similar results. The use of supervised learning techniques allowed an improvement in the results. They were improved to 77% of correct assignments when an ensemble of ten FFNNs were used and to 80% when Random Forests were used. This study was performed with NMR data simulated from the molecular structure by the SPINUS program. In the design of one test set, simulated data was combined with experimental data. The results support the proposal of linking databases of chemical reactions to experimental or simulated NMR data for automatic classification of reactions and mixtures of reactions. Genome-scale classification of enzymatic reactions from their reaction equation. The MOLMAP descriptor relies on a Kohonen SOM that defines types of bonds on the basis of their physico-chemical and topological properties. The MOLMAP descriptor of a molecule represents the types of bonds available in that molecule. The MOLMAP descriptor of a reaction is defined as the difference between the MOLMAPs of the products and the reactants, and numerically encodes the pattern of bonds that are broken, changed, and made during a chemical reaction. The automatic perception of chemical similarities between metabolic reactions is required for a variety of applications ranging from the computer validation of classification systems, genome-scale reconstruction (or comparison) of metabolic pathways, to the classification of enzymatic mechanisms. Catalytic functions of proteins are generally described by the EC numbers that are simultaneously employed as identifiers of reactions, enzymes, and enzyme genes, thus linking metabolic and genomic information. Different methods should be available to automatically compare metabolic reactions and for the automatic assignment of EC numbers to reactions still not officially classified. In this study, the genome-scale data set of enzymatic reactions available in the KEGG database was encoded by the MOLMAP descriptors, and was submitted to Kohonen SOMs to compare the resulting map with the official EC number classification, to explore the possibility of predicting EC numbers from the reaction equation, and to assess the internal consistency of the EC classification at the class level. A general agreement with the EC classification was observed, i.e. a relationship between the similarity of MOLMAPs and the similarity of EC numbers. At the same time, MOLMAPs were able to discriminate between EC sub-subclasses. EC numbers could be assigned at the class, subclass, and sub-subclass levels with accuracies up to 92%, 80%, and 70% for independent test sets. The correspondence between chemical similarity of metabolic reactions and their MOLMAP descriptors was applied to the identification of a number of reactions mapped into the same neuron but belonging to different EC classes, which demonstrated the ability of the MOLMAP/SOM approach to verify the internal consistency of classifications in databases of metabolic reactions. RFs were also used to assign the four levels of the EC hierarchy from the reaction equation. EC numbers were correctly assigned in 95%, 90%, 85% and 86% of the cases (for independent test sets) at the class, subclass, sub-subclass and full EC number level,respectively. Experiments for the classification of reactions from the main reactants and products were performed with RFs - EC numbers were assigned at the class, subclass and sub-subclass level with accuracies of 78%, 74% and 63%, respectively. In the course of the experiments with metabolic reactions we suggested that the MOLMAP / SOM concept could be extended to the representation of other levels of metabolic information such as metabolic pathways. Following the MOLMAP idea, the pattern of neurons activated by the reactions of a metabolic pathway is a representation of the reactions involved in that pathway - a descriptor of the metabolic pathway. This reasoning enabled the comparison of different pathways, the automatic classification of pathways, and a classification of organisms based on their biochemical machinery. The three levels of classification (from bonds to metabolic pathways) allowed to map and perceive chemical similarities between metabolic pathways even for pathways of different types of metabolism and pathways that do not share similarities in terms of EC numbers. Mapping of PES by neural networks (NNs). In a first series of experiments, ensembles of Feed-Forward NNs (EnsFFNNs) and Associative Neural Networks (ASNNs) were trained to reproduce PES represented by the Lennard-Jones (LJ) analytical potential function. The accuracy of the method was assessed by comparing the results of molecular dynamics simulations (thermal, structural, and dynamic properties) obtained from the NNs-PES and from the LJ function. The results indicated that for LJ-type potentials, NNs can be trained to generate accurate PES to be used in molecular simulations. EnsFFNNs and ASNNs gave better results than single FFNNs. A remarkable ability of the NNs models to interpolate between distant curves and accurately reproduce potentials to be used in molecular simulations is shown. The purpose of the first study was to systematically analyse the accuracy of different NNs. Our main motivation, however, is reflected in the next study: the mapping of multidimensional PES by NNs to simulate, by Molecular Dynamics or Monte Carlo, the adsorption and self-assembly of solvated organic molecules on noble-metal electrodes. Indeed, for such complex and heterogeneous systems the development of suitable analytical functions that fit quantum mechanical interaction energies is a non-trivial or even impossible task. The data consisted of energy values, from Density Functional Theory (DFT) calculations, at different distances, for several molecular orientations and three electrode adsorption sites. The results indicate that NNs require a data set large enough to cover well the diversity of possible interaction sites, distances, and orientations. NNs trained with such data sets can perform equally well or even better than analytical functions. Therefore, they can be used in molecular simulations, particularly for the ethanol/Au (111) interface which is the case studied in the present Thesis. Once properly trained, the networks are able to produce, as output, any required number of energy points for accurate interpolations.

Supplier's selection model based on an empirical study

The problem of selecting suppliers/partners is a crucial and important part in the process of decision making for companies that intend to perform competitively in their area of activity. The selection of supplier/partner is a time and resource-consuming task that involves data collection and a careful analysis of the factors that can positively or negatively influence the choice. Nevertheless it is a critical process that affects significantly the operational performance of each company. In this work, there were identified five broad selection criteria: Quality, Financial, Synergies, Cost, and Production System. Within these criteria, it was also included five sub-criteria. After the identification criteria, a survey was elaborated and companies were contacted in order to understand which factors have more weight in their decisions to choose the partners. Interpreted the results and processed the data, it was adopted a model of linear weighting to reflect the importance of each factor. The model has a hierarchical structure and can be applied with the Analytic Hierarchy Process (AHP) method or Value Analysis. The goal of the paper it's to supply a selection reference model that can represent an orientation/pattern for a decision making on the suppliers/partners selection process

Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems

Dissertação para obtenção do Grau de Doutor em Matemática

Energy efficiency in energy policy: Empirical evidence of direct rebound effect in Portuguese households' demand for electricity

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Economics from the NOVA – School of Business and Economics

Caracterização da aterosclerose sub-clínica e doença coronária em doentes com síndrome metabólica e diabetes mellitus

RESUMO- Introdução: A obesidade e a Síndrome Metabólica (SM) são atualmente um importante problema de saúde pública, com prevalências crescentes, que se acompanham também por aumento da prevalência de Diabetes Mellitus (DM).Estudos prévios demonstram associação destas entidades com o aumento de risco de eventos cardiovasculares, em particular a DM. A SM tem sido uma entidade muito debatida nos últimos anos, com aparecimento de diversas definições, contribuindo para resultados díspares no que diz respeito à influência da SM nas doenças cardiovasculares. Também têm sido descritas variações étnicas e regionais. Para além de alguns estudos epidemiológicos na população geral, a informação relativamente à sua influência na presença de doença cardiovascular é desconhecida em Portugal, em particular em populações com suspeita de doença coronária. Objetivos - Esclarecimento de questões relacionadas com a prevalência de SM e a sua influência na evolução de doença ateroclerótica arterial por avaliação de uma população com suspeita de doença coronária. População e Métodos - Estudo observacional, transversal, com inclusão prospetiva de indivíduos admitidos letivamente para realização de angiografia coronária por suspeita de doença coronária, tendo sido também efetuadas análises laboratoriais e ecografia carotidea para avaliação da espessura intima-média carotidea (EIMc) e da presença de placas carotídeas. Efetuou-se avaliação dos parâmetros demográficos, antropométricos, determinação do perfil lipídico, glicémia e insulinémia. Os exames angiográficos foram analisados por análise quantitativa semi-automática. Foram excluídos indivíduos com antecedentes conhecidos de doença cardíaca. Resultados - Incluíram-se 300 doentes, com idade média de 64 ± 9 anos, 59% do género masculino. A prevalência de SM de acordo com a definição da AHA/NHLBI foi 48,4% (ajustada para idade e género da população portuguesa) e a prevalência de DM foi 14,8% (ajustada). A concordância global das três definições mais recentes de SM foi de apenas 43%. A prevalência de SM aumenta com a idade e é também mais elevada no género feminino. O componente mais frequente foi a hipertensão arterial, seguido pela obesidade abdominal, a elevação da glicémia e por fim as alterações dos triglicéridos e do colesterol HDL. Por outro lado, a presença de doença coronária significativa (lesões ≥50%) ocorreu em apenas 51,3% dos doentes, sendo ainda mais baixa no género feminino. Demonstrou-se também uma baixa capacidade preditiva para doença coronária dos testes não invasivos clássicos, em particular no género feminino. A prevalência de doença coronária significativa foi idêntica nos indivíduos com SM comparativamente com indivíduos sem alterações metabólicas (46,3% vs. 48,2%, respectivamente), sendo mais elevado nos diabéticos (65,2%). Os fatores predizentes independentes de doença coronária significativa foram a idade, o género masculino, a elevação da glicémia e dos triglicéridos. Pelo contrário, o Índice de Massa Corporal (IMC) mostrou uma associação protetora relativamente à presença de doença coronária. A SM não é fator predizente de doença coronária. Relativamente às dimensões dos vasos coronários, o IMC correlaciona-se positivamente e a glicémia / DM correlacionam-se negativamente. A EIMc aumenta com o aumento da idade e no género masculino. A EIMc foi intermédia nos doentes com SM (0,88 ± 0,31 mm) comparativamente com os doentes diabéticos (0,97 ± 0,34 mm) e os indivíduos “Normais” (0,85 ± 0,34 mm). Os fatores predizentes independentes de EIMc foram a idade, o género masculino, o colesterol HDL e a insulinémia. A EIMc permite predizer com uma acuidade moderada a presença de doença coronária significativa (AUC 0,638), em particular no género feminino, sendo um fator predizente independente de presença de doença coronária (OR 2,35, IC 95% 1,04-5,33. p=0,04). Apesar de não se correlacionar com o número de vasos coronários com doença, correlacionou-se com a gravidade da doença (pelo score de Gensini). A insulinémia e o índice HOMA aumentam diretamente com a idade e com o IMC, sendo contudo sobreponíveis em ambos os géneros. Os fatores predizentes de índice HOMA (resistência à insulina) foram o IMC, bem como os restantes componentes de SM, estando o índice HOMA relacionado com a presença de SM e o número dos seus componentes presentes. O limiar para resistência à insulina foi de 2,66 e para SM foi 2,41. Ao contrário das restantes definições de SM, a definição da AHA/NHLBI não é predizente da presença de DM no género masculino. A associação da resistência à insulina com doença coronária foi limiar (OR 1,13, IC 95% 1,00-1,28, p=0,045). Conclusões - Numa população com suspeita de doença coronária, a prevalência de SM é muito elevada (superior a 50%), sendo a prevalência de DM de 23%. Também a obesidade e o excesso de peso foram extremamente prevalentes nesta população. A concordância entre definições de SM é baixa. A hipertensão arterial e a obesidade abdominal são os componentes mais frequentes de SM, sendo menos prevalentes as alterações lipídicas. Pelo contrário, a presença de doença coronária significativa foi muito baixa, em particular nas mulheres. A SM não se associou à presença de doença coronária significativa, estando esta mais dependente das alterações do metabolismo glicídico e dos triglicéridos, bem como de outros fatores de risco não modificáveis, nomeadamente a idade e o género. A EIMc da carótida comum e a presença de placas carotídeas é mais elevada nos indivíduos diabéticos, estando também ligeiramente aumentada nos doentes com SM, sendo os fatores predizentes de EIMc apenas a idade, o género, a hiperinsulinémia bem como os níveis baixos de colesterol HDL. A utilização da avaliação da EIMc na estratificação de risco pré-angiografia coronária, poderá ser útil no género feminino. A hiperinsulinémia e o índice HOMA (índice de resistência à insulina), estão relacionados com o IMC e consequentemente com a presença de obesidade, embora também se correlacione de forma independente com os outros componentes de SM. A resistência à insulina associou-se à presença de SM. Relativamente à capacidade preditiva da coexistência com DM, verificou-se associação com a definição da NCEP-ATP III e da IDF, contudo, a definição da AHA/NHLBI só foi predizente de DMnas mulheres. -------------ABSTRACT - Introduction: Obesity and Metabolic Syndrome (MS) are a major public health problem, with increasing prevalence, that follows the increase in diabetes prevalence. Previous studies showed an association of both entities with increased cardiovascular risk, particularly diabetes. MS has been debated in the last few years, with several definitions and different results when analysed the influence of MS on cardiovascular diseases. There are also some regional and ethnical variations. Beyond general population epidemiological studies, information about the influence on cardiovascular disease in Portugal is unknown, particularly in patients with suspected coronary disease. Objectives- To clarify several questions regarding the prevalence of MS and the influence in arterial atherosclerotic disease by evaluation of a population with suspected coronary artery disease. Population and Methods- Observational, cross-sectional study with prospective inclusion of individuals admitted electively for coronary angiography with suspicion of coronary artery disease. All individuals also performed laboratorial evaluation and carotid ultrasound to evaluate carotid intima-media thickness (cIMT) and carotid plaques. We also evaluated demographic, anthropometric parameters, lipid profile, blood glucose and blood insulin. Angiographic data was obtained by semi-automated quantitation. Individuals with previously known cardiac history were excluded from the study. Results- We included 300 individuals with a mean age of 64 ± 9 years, 59% males. MS prevalence according to AHA/NHLBI definition was 48.4% (adjusted for age and gender of the Portuguese population) and the adjusted prevalence of diabetes was 14.8%. Global agreement between the more recent three definitions of MS was only 43%. MS prevalence increases with age and is also higher in women. The most frequent components were hypertension and abdominal obesity, followed by elevated glucose and triglicerides and low HDL-cholesterol. Significant coronary artery disease (stenosis ≥50%) was present in only 51.3% of patients, being lower in females. Non-invasive tests also had a low predictive capacity, particularly in females. The prevalence of significant coronary disease was identical in patients with MS compared with normal metabolism individuals (46.3% vs. 48.2%, respectively), being higher in diabetics (65.2%). Independent predictive factors for coronary disease were age, male gender, high blood glucose and triglycerides. On the contrary, Body Mass Index (BMI) was a protective factor for coronary disease. MS wasn’t a predictor of coronary disease. BMI showed a positive correlation with coronary vessel diameter and glucose /diabetes had a negative correlation. CIMT increased with age and was higher in males. CIMT was intermediate in patients with MS (0.88 ± 0.31 mm) when compared to diabetic patients (0.97 ± 0.34 mm) and “Normal” individuals (0.85 ± 0.34 mm). Independent predictors for cIMT were age, male gender, HDL-cholesterol and insulin. CIMT had a moderate predictive accuracy for coronary disease (AUC 0,638), particularly in females and is an independent predictor of the presence of significant coronary disease (OR 2.35, 95% CI 1.04-5.33. p=0.04). Although it did not correlate with the number of diseased coronary arteries, it correlated with coronary disease severity by the Gensini score. Insulin and HOMA index increase directly with age and BMI, but were identical in both genders. Predictive factors for HOMA index (insulin resistance) were BMI as well as the other MS components. HOMA index is related to MS and the number of its components. The cut-off for insulin resistance was 2.66 and for MS 2.41. Unlike other MS definitions, AHA/NHLBI definition is not a predictor of diabetes in males. There was a borderline association between insulin resistance and coronary disease (OR 1.13, 95% CI 1.00-1.28, p=0.045). Conclusions - In a population of patients with suspected coronary disease, MS prevalence is extremely high (above 50%) with a diabetes prevalence of 23%. Also obesity and overweight are very prevalent in this population. Global agreement between MS definitions is however low. Hypertension and abdominal obesity are the most frequent components, with a lower prevalence of lipid abnormalities. Coronary disease prevalence was low, particularly in women. MS wasn’t associated with coronary disease. Coronary disease was related to glucose and triglycerides, as well as with other non-modifiable factors such as age and gender. CIMT and carotid plaques are increased in diabetic patients, and also slightly elevated in patients with MS, but cIMT independent predictors were age, male gender, insulin and HDLcholesterol. CIMT can be useful in risk stratification before coronary angiography particularly in women. Elevated insulin and HOMA index (an insulin resistance index) are related with BMI and consequently with obesity, and it was also correlated with other MS components. Insulin resistance was associated with MS. The presence of diabetes was associated with the presence of MS by NCEP-ATP III and IDF definitions; however, AHA/NHLBI definition was only predictive of diabetes in females.

Energy efficiency in energy policy: Empirical evidence of direct rebound effect in Portuguese household's demand for electricity

This project the direct rebound effect for the electricity demand in Portugal. While we find evidence of such an effect, the estimations also reflect the institutional arrangement that has characterized the electricity market in the country. Also, issues related to energy efficiency promotion are addressed in general putting into context the case study developed.

Implementation of a probabilistic based framework for safety assessment of existing timber elements

The assessment of existing timber structures is often limited to information obtained from non or semi destructive testing, as mechanical testing is in many cases not possible due to its destructive nature. Therefore, the available data provides only an indirect measurement of the reference mechanical properties of timber elements, often obtained through empirical based correlations. Moreover, the data must result from the combination of different tests, as to provide a reliable source of information for a structural analysis. Even if general guidelines are available for each typology of testing, there is still a need for a global methodology allowing to combine information from different sources and infer upon that information in a decision process. In this scope, the present work presents the implementation of a probabilistic based framework for safety assessment of existing timber elements. This methodology combines information gathered in different scales and follows a probabilistic framework allowing for the structural assessment of existing timber elements with possibility of inference and updating of its mechanical properties, through Bayesian methods. The probabilistic based framework is based in four main steps: (i) scale of information; (ii) measurement data; (iii) probability assignment; and (iv) structural analysis. In this work, the proposed methodology is implemented in a case study. Data was obtained through a multi-scale experimental campaign made to old chestnut timber beams accounting correlations of non and semi-destructive tests with mechanical properties. Finally, different inference scenarios are discussed aiming at the characterization of the safety level of the elements.

Fractional pennes' bioheat equation: theoretical and numerical studies

In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bioheat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.