BACKGROUND: ONE OF THE MAIN LESSONS OF THE COVID-19 PANDEMIC IS THAT WE MUST PREPARE TO FACE ANOTHER PANDEMIC LIKE IT. CONSEQUENTLY, THIS ARTICLE AIMS TO DEVELOP A GENERAL FRAMEWORK CONSISTING OF EPIDEMIOLOGICAL MODELING AND A PRACTICAL IDENTIFIABILITY APPROACH TO ASSESS COMBINED VACCINATION AND NON-PHARMACEUTICAL INTERVENTION (NPI) STRATEGIES FOR THE DYNAMICS OF ANY TRANSMISSIBLE DISEASE. MATERIALS AND METHODS: EPIDEMIOLOGICAL MODELING OF THE PRESENT WORK RELIES ON DELAY DIFERENTIAL EQUATIONS DESCRIBING TIME VARIATION AND TRANSITIONS BETWEEN SUITABLE COMPARTMENTS. THE PRACTICAL IDENTIFIABILITY APPROACH RELIES ON PARAMETER OPTIMIZATION, A PARAMETRIC BOOTSTRAP TECHNIQUE, AND DATA PROCESSING. WE IMPLEMENTED A CAREFUL PARAMETER OPTIMIZATION ALGORITHM BY SEARCHING FOR SUITABLE INITIALIZATION ACCORDING TO EACH PROCESSED DATASET. IN ADDITION, WE IMPLEMENTED A PARAMETRIC BOOTSTRAP TECHNIQUE TO ACCURATELY PREDICT THE ICU CURVE TREND IN THE MEDIUM TERM AND ASSESS VACCINATION. RESULTS: WE SHOW THE FRAMEWORK?S CALIBRATION CAPABILITIES FOR SEVERAL PROCESSED COVID-19 DATASETS OF DI ERENT REGIONS OF CHILE. WE FOUND A UNIQUE RANGE OF PARAMETERS THAT WORKS WELL FOR EVERY DATASET AND PROVIDES OVERALL NUMERICAL STABILITY AND CONVERGENCE FOR PARAMETER OPTIMIZATION. CONSEQUENTLY, THE FRAMEWORK PRODUCES OUTSTANDING RESULTS CONCERNING QUANTITATIVE TRACKING OF COVID-19 DYNAMICS. IN ADDITION, IT ALLOWS US TO ACCURATELY PREDICT THE ICU CURVE TREND IN THE MEDIUM TERM AND ASSESS VACCINATION. FINALLY, IT IS REPRODUCIBLE SINCE WE PROVIDE OPEN-SOURCE CODES THAT CONSIDER PARAMETER INITIALIZATION STANDARDIZED FOR EVERY DATASET. CONCLUSION: THIS WORK ATTEMPTS TO IMPLEMENT A HOLISTIC AND GENERAL MODELING FRAMEWORK FOR QUANTITATIVE TRACKING OF THE DYNAMICS OF ANY TRANSMISSIBLE DISEASE, FOCUSING ON ACCURATELY PREDICTING THE ICU CURVE TREND IN THE MEDIUM TERM AND ASSESSING VACCINATION. THE SCIENTIFIC COMMUNITY COULD ADAPT IT TO EVALUATE THE IMPACT OF COMBINED VACCINATION AND NPIS STRATEGIES FOR COVID

IN THIS WORK WE PROPOSE A NEW MODEL TO SIMULATE BIOFILM STRUCTURES (?FINGER-LIKE?, AS WELL AS, COMPACT STRUCTURES) AS A RESULT OF MICROBIAL GROWTH IN DIFFERENT ENVIRONMENTAL CONDITIONS. AT THE SAME TIME, THE NUMERICAL METHOD THAT WE USE IN ORDER TO CARRY OUT THE COMPUTATIONAL SIMULATIONS IS NEW TO THE BIOLOGICAL COMMUNITY, AS FAR AS WE KNOW. THE USE OF OUR MODEL SHEDS LIGHT ON THE BIOLOGICAL PROCESS OF BIOFILM FORMATION SINCE IT SIMULATES SOME CENTRAL ISSUES OF BIOFILM GROWTH: THE PATTERN FORMATION OF HETEROGENEOUS STRUCTURES, SUCH AS FINGER-LIKE STRUCTURES, IN A SUBSTRATE-TRANSPORT-LIMITED REGIME, AND THE FORMATION OF MORE COMPACT STRUCTURES, IN A GROWTH-LIMITED-REGIME. THE MAIN ADVANTAGE OF OUR APPROACH IS THAT WE CONSIDER SEVERAL OF THE MOST RELEVANT ASPECTS OF BIOFILM MODELING, PARTICULARLY, THE EXISTENCE AND EVOLUTION OF A BIOFILM?LIQUID INTERFACE. AT THE SAME TIME, IN ORDER TO PERFORM NUMERICAL SIMULATIONS, WE HAVE USED SOPHISTICATED NUMERICAL TECHNIQUES BASED ON MIXING THE IMMERSED INTERFACE METHOD AND THE LEVEL-SET METHOD, WHICH ARE WELL DESCRIBED IN THE PRESENT WORK.

MICROCIRCULATION ANALYSIS OF THE BRAIN CORTEX IS CHALLENGING BECAUSE SURFACE PERFUSION VARIES RAPIDLY IN SMALL SPACE-TIME REGIONS AND IS BONE PROTECTED. THE LASER SPECKLE CONTRAST IMAGING (LSCI) TECHNIQUE ALLOWS ANALYZING IN VIVO BRAIN VASCULAR PERFUSION GENERATING A LARGE AMOUNT OF DATA THAT REQUIRES SOPHISTICATED DATA ANALYTICS, MAKING RESEARCHERS INVEST MUCH EFFORT IN PROCESSING. OUR RESEARCH QUESTION WAS WHETHER THE REDUCED PLACENTAL PERFUSION MODEL (RUPP) OF PREECLAMPSIA (PE) WAS ASSOCIATED WITH IMPAIRED BLOOD PERFUSION IN THE OFFSPRING'S BRAINS. WE AIMED TO DEVELOP A ROBUST NUMERICAL APPROACH THAT MAINLY CONSISTED OF APPLYING A SIGNAL-PROCESSING TOOL FOR CALCULATING OPTIMAL SEGMENTATION AND PIECE-WISE FITS OF THE OFFSPRING'S BRAIN PERFUSION SIGNALS OBTAINED FROM THE LSCI TECHNIQUE. WE COMBINED THIS TOOL WITH THE USUAL STATISTICAL ANALYSIS, IMPLEMENTING BOTH IN MATLAB SOFTWARE. WE PERFORMED BRAIN PERFUSION MEASUREMENTS FROM OFFSPRING (FIVE DAYS POSTNATAL, P5) OF CONTROL PREGNANT DAMS (SHAM, N = 13) AND OF RUPP DAMS (RUPP, N = 7) USING THE PERICAM® PSI-HR SYSTEM AT A BASAL CONDITION AND AFTER THERMAL STIMULI (WARM AND COLD). WE FOUND THAT PUPS OF RUPP MICE EXHIBITED SIGNIFICANT DIFFERENCES IN PERFUSION AND VASCULAR RESPONSE TO THERMAL STIMULI COMPARED TO THE SHAM MICE. THESE DIFFERENCES WERE ASSOCIATED WITH HIGH DATA VARIABILITY IN THE SHAM GROUP, WHILE IN THE RUPP GROUP, PERFUSION LOOKS ?STIFFER.? DATA ALSO SUGGEST SEXDIMORPHISM IN THE VASCULAR RESPONSE SINCE FEMALE PUPS IN THE SHAM GROUP BUT NOT MALE PUPS SHOWED STATISTICALLY SIGNIFICANT DIFFERENCES IN RESPONSE TO THE WARM STIMULUS. AGAIN, THIS SEX-RELATED DIFFERENCE WAS ABSENT IN PUPS OF RUPP MICE. IN CONCLUSION, WE PRESENT A ROBUST QUANTITATIVE APPROACH FOR LSCI MEASUREMENTS THAT REVEALED ANOMALIES IN THE BRAIN BLOOD FLOW IN OFFSPRING OF THE RUPP MODEL OF PE.

IN THIS STUDY, WE AIMED TO DESCRIBE A MODERN, EFFICIENT, AND REPRODUCIBLE METHODOLOGY TO OPTIMIZE RESPIRATION RATE PARAMETERS COUPLED WITH TRANSPORT PROPERTIES IN MASS BALANCES DESCRIBING MODIFIED ATMOSPHERE PACKAGING (MAP) SYSTEMS. WE CONSIDERED MASS BALANCES FOR THREE DIFFERENT RESPIRATION RATE J FILM (EXPONENTIAL, COMPETITIVE AND UNCOMPETITIVE MICHAELIS?MENTEN KINETICS) COUPLED WITH TRANSPORT PROPERTIES FOR TWO DIFFERENT PACKAGING FILMS. EXPERIMENTS WERE CONDUCTED TO VALIDATE THE METHODOLOGY USING GRAPES PLACED IN A POLYPROPYLENE CONTAINER OPENED ON THE TOP AND SEALED WITH PACKAGING FILMS. THE METHODOLOGY RELIES ON A NUMERICAL OPTIMIZATION PROCEDURE CALLED THE TRUST-REGION-REFLECTIVE ALGORITHM. WE DETERMINED THE PREDICTIVE CAPABILITY OF MODELS USING GOODNESS-OF-FIT CRITERIA AND ASSESSED PARAMETER UNCERTAINTY THROUGH STANDARD ERRORS. WE ALSO CALCULATED THE FIRST-ORDER OPTIMALITY MEASURE AND THE RELATIVE CHANGE IN THE SUM OF SQUARES TO VERIFY THE CONVERGENCE OF THE IMPLEMENTED ALGORITHM. RESULTS SHOWED THAT THE RESPIRATION RATE PARAMETERS OBTAINED WITH THIS METHODOLOGY FOR THE EXPONENTIAL MODEL PROVIDED A BETTER FIT THAN FOR THE OTHER TWO MODELS. THE FITTING FOR THE KINETIC MODELS IS NOT VERY SUITABLE SINCE WE FOUND THAT THE NORMALIZED STANDARD ERRORS WERE RATHER HIGH. IN CONCLUSION, THE METHODOLOGY IS ROBUST, AND WE EXPECT THAT IT SERVES AS A TOOL FOR ASSESSING MAP TECHNOLOGY DESIGN.

THE MATHEMATICAL WORKING SPACES (MWS) THEORETICAL FRAMEWORK HAS DEVELOPED A GROWING INTEREST IN HOW MATHEMATICAL MODELLING IS RECOGNISED, ANALYSED, AND ARTICULATED FROM ITS THEORETICAL AND EMPIRICAL SCOPES. GIVEN THE ARTICULATIONS IDENTIFIED IN THE LITERATURE, IT IS INTERESTING TO DETERMINE WHETHER NEW NETWORKS WITH THE MWS PROVIDE POWERFUL STRATEGIES FOR ANALYSING MATHEMATICAL MODELLING TASK RESOLUTION. OBJECTIVE: TO CHARACTERISE THE MODELLING ACTIVITY FROM THE NETWORK COMPOSED BY THE BLOMHØJ MODELLING CYCLE AND THE MWS IN ENGINEERING STUDENTS. DESIGN: THIS WORK IS A CASE STUDY WITH A QUALITATIVE APPROACH, WHICH ANALYSES THE RESOLUTION OF A MODELLING TASK THROUGH THE PROPOSED NETWORK. SETTING AND PARTICIPANTS: THE EXPERIMENTATION WAS CARRIED OUT IN AN INTEGRAL CALCULUS COURSE FOR A COMPUTER CIVIL ENGINEERING CAREER AT A CHILEAN UNIVERSITY. MODELLING PRACTICES ARE NOT USUAL IN THIS SECOND-YEAR SUBJECT, ALTHOUGH ENHANCING THEIR USE IN PROFESSIONAL EDUCATION IS NECESSARY. DATA COLLECTION AND ANALYSIS: WE COLLECTED THE WRITTEN RECORDS OF THE STUDENTS, SELECTING ONE TO PERFORM THE IN-DEPTH ANALYSIS DUE TO ITS HIGH REPRESENTATIVENESS AND CLARITY, AS EVIDENCED IN THE DOCUMENTS. THREE STEPS WERE FOLLOWED TO CHARACTERISE THE MODELLING ACTIVITY IN THE WRITTEN RECORD: DESCRIPTION, ANALYSIS, AND INTERPRETATION. RESULTS: THE BLOMHØJ MODELLING CYCLE MIGHT PRESENT CONNECTIONS WITH THE MWS AT THE FORMULATION OF THE PROBLEM AND NOT ONLY AT THE SYSTEMATISATION, WHICH IS A NOVELTY IN THIS FIELD OF RESEARCH. CONCLUSIONS: A NOVEL

THIS ARTICLE IS CONCERNED WITH THE CONVERGENCE OF THE LEVEL-SET ALGORITHM INTRODUCED BY ASLAM (J COMPUT PHYS 167 (2001), 413?438) FOR TRACKING THE DISCONTINUITIES IN SCALAR CONSERVATION LAWS IN THE CASE OF LINEAR OR STRICTLY CONVEX FLUX FUNCTION. THE NUMERICAL METHOD IS DEDUCED BY THE LEVEL-SET REPRESENTATION OF THE ENTROPY SOLUTION: THE ZERO OF A LEVEL-SET FUNCTION IS USED AS AN INDICATOR OF THE DISCONTINUITY CURVES AND TWO AUXILIARY STATES, WHICH ARE ASSUMED CONTINUOUS THROUGH THE DISCONTINUITIES, ARE INTRODUCED. WE REWRITE THE NUMERICAL LEVEL-SET ALGORITHM AS A PROCEDURE CONSISTING OF THREE BIG STEPS: (A) INITIALIZATION, (B) EVOLUTION, AND (C) RECONSTRUCTION. IN (A), WE CHOOSE AN ENTROPY ADMISSIBLE LEVEL-SET REPRESENTATION OF THE INITIAL CONDITION. IN (B), FOR EACH ITERATION STEP, WE SOLVE AN UNCOUPLED SYSTEM OF THREE EQUATIONS AND SELECT THE ENTROPY ADMISSIBLE LEVEL-SET REPRESENTATION OF THE SOLUTION PROFILE AT THE END OF THE TIME ITERATION. IN (C), WE RECONSTRUCT THE ENTROPY SOLUTION USING THE LEVEL-SET REPRESENTATION. WE PROVE THE CONVERGENCE OF THE NUMERICAL SOLUTION TO THE ENTROPY SOLUTION IN FOR EVERY , USING -WEAK BOUNDED VARIATION (BV) ESTIMATES AND A CELL ENTROPY INEQUALITY. IN ADDITION, SOME NUMERICAL EXAMPLES FOCUSED ON THE ELEMENTARY WAVE INTERACTION ARE PRESENTED. © 2014 WILEY PERIODICALS, INC. NUMER METHODS PARTIAL DIFFERENTIAL EQ, 2014

WE INVESTIGATE THE HALLEY METHOD OF EXPONENTIAL MAPS. OUR MAIN RESULT IS THAT, UNLIKE NEWTONS METHOD, THE JULIA SET OF HALLEYS METHOD MAY BE DISCONNECTED WHEN APPLIED TO ENTIRE MAPS OF FORM F=PEQ WHERE P AND Q ARE POLYNOMIALS AND Q IS NON-CONSTANT. WE ALSO DESCRIBE THE NATURE OF THE FIXED POINTS AND CLASSIFY RATIONAL HALLEYS MAPS OF ENTIRE FUNCTIONS.

IN THIS WORK WE STUDY THE NUMERICAL RESOLUTION OF SYSTEMS OF FUZZY NONLINEAR EQUATIONS. MORE PRECISELY, WE DESCRIBE, ANALYZE AND SIMULATE NUMERICAL METHODS, SUCH AS NEWTON METHOD, IN ORDER TO APPROXIMATE EFFICIENTLY THE SOLUTIONS TO SUCH PROBLEMS. ONE OF THE MAIN ISSUES OF THIS TYPE OF PROBLEMS IS THAT THE STANDARD ANALYTICAL TECHNIQUES FOR FINDING SOLUTIONS, ARE NOT APPROPRIATE TO RESOLVE THEM. FOR THIS REASON, IN THIS PAPER WE FOCUS IN THE STUDY OF KNOWN RESULTS FOR THE CLASSICAL METHODS AND THEIR ADAPTATION TO THE RESOLUTION OF FUZZY PROBLEMS.

THE COVID-19 DISEASE HAS FORCED COUNTRIES TO MAKE A CONSIDERABLE COLLABORATIVE EFFORT BETWEEN SCIENTISTS AND GOVERNMENTS TO PROVIDE INDICATORS TO SUITABLE FOLLOW-UP THE PANDEMIC?S CONSEQUENCES. MATHEMATICAL MODELING PLAYS A CRUCIAL ROLE IN QUANTIFYING INDICATORS DESCRIBING DIVERSE ASPECTS OF THE PANDEMIC. CONSEQUENTLY, THIS WORK AIMS TO DEVELOP A CLEAR, EFFICIENT, AND REPRODUCIBLE METHODOLOGY FOR PARAMETER OPTIMIZATION, WHOSE IMPLEMENTATION IS ILLUSTRATED USING DATA FROM THREE REPRESENTATIVE REGIONS FROM CHILE AND A SUITABLE GENERALIZED SIR MODEL TOGETHER WITH A FITTED POSITIVITY RATE. OUR RESULTS REPRODUCE THE GENERAL TREND OF THE INFECTED?S CURVE, DISTINGUISHING THE REPORTED AND REAL CASES. FINALLY, OUR METHODOLOGY IS ROBUST, AND IT ALLOWS US TO FORECAST A SECOND OUTBREAK OF COVID-19 AND THE INFECTION FATALITY RATE OF COVID-19 QUALITATIVELY ACCORDING TO THE REPORTED DEAD CASES.

THE FUTURE CHALLENGES IN ONCOLOGY IMAGING ARE TO ASSESS THE RESPONSE TO TREATMENT EVEN EARLIER. AS AN ADDITION TO FUNCTIONAL IMAGING, MATHEMATICAL MODELING BASED ON THE IMAGING IS AN ALTERNATIVE, CROSS-DISCIPLINARY AREA OF DEVELOPMENT. MODELING WAS DEVELOPED IN ONCOLOGY NOT ONLY IN ORDER TO UNDERSTAND AND PREDICT TUMOR GROWTH, BUT ALSO TO ANTICIPATE THE EFFECTS OF TARGETED AND UNTARGETED THERAPIES. A VERY WIDE RANGE OF THESE MODELS EXIST, INVOLVING MANY STAGES IN THE PROGRESSION OF TUMORS. FEW MODELS, HOWEVER, HAVE BEEN PROPOSED TO REPRODUCE IN VIVO TUMOR GROWTH BECAUSE OF THE COMPLEXITY OF THE MECHANISMS INVOLVED. MORPHOLOGICAL IMAGING COMBINED WITH """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""SPATIAL"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" MODELS APPEARS TO PERFORM WELL ALTHOUGH FUNCTIONING IMAGING COULD STILL PROVIDE FURTHER INFORMATION ON METABOLISM AND THE MICRO-ARCHITECTURE. THE COMBINATION OF IMAGING AND MODELING CAN RESOLVE COMPLEX PROBLEMS AND DESCRIBE MANY FACETS OF TUMOR GROWTH OR RESPONSE TO TREATMENT. IT IS NOW POSSIBLE TO CONSIDER ITS CLINICAL USE IN THE MEDIUM TERM. THIS REVIEW DESCRIBES THE BASIC PRINCIPLES OF MATHEMATICAL MODELING AND DESCRIBES THE ADVANTAGES, LIMITATIONS AND FUTURE PROSPECTS FOR THIS IN VIVO APPROACH BASED ON IMAGING DATA.

A NUMERICAL METHOD BASED ON ARTIFICIAL NEURAL NETWORKS IS USED TO SOLVE THE INVERSE SCHRÖDINGER EQUATION FOR A MULTI-PARAMETER CLASS OF POTENTIALS. FIRST, THE FINITE ELEMENT METHOD WAS USED TO SOLVE REPEATEDLY THE DIRECT PROBLEM FOR DIFFERENT PARAMETRIZATIONS OF THE CHOSEN POTENTIAL FUNCTION. THEN, USING THE ATTAINABLE EIGENVALUES AS A TRAINING SET OF THE DIRECT RADIAL BASIS NEURAL NETWORK A MAP OF NEW EIGENVALUES WAS OBTAINED. THIS RELATIONSHIP WAS LATER INVERTED AND REFINED BY TRAINING AN INVERSE RADIAL BASIS NEURAL NETWORK, ALLOWING THE CALCULATION OF THE UNKNOWN PARAMETERS AND THEREFORE ESTIMATING THE POTENTIAL FUNCTION. THREE NUMERICAL EXAMPLES ARE PRESENTED IN ORDER TO PROVE THE EFFECTIVENESS OF THE METHOD. THE RESULTS SHOW THAT THE METHOD PROPOSED HAS THE ADVANTAGE TO USE LESS COMPUTATIONAL RESOURCES WITHOUT A SIGNIFICANT ACCURACY LOSS.

IN THIS WORK WE SIMULATE BIOFILM STRUCTURES (?FINGER-LIKE?, AS WELL AS, COMPACT STRUCTURES) AS A RESULT OF MICROBIAL GROWTH IN DIFFERENT ENVIRONMENTAL CONDITIONS. AT THE SAME TIME, THE NUMERICAL METHOD THAT WE USE IN ORDER TO CARRY OUT THE COMPUTATIONAL SIMULATIONS IS NEW TO THE BIOLOGICAL COMMUNITY, AS FAR AS WE KNOW. THE USE OF OUR MODEL SHEDS LIGHT ON THE BIOLOGICAL PROCESS OF BIOFILM FORMATION SINCE IT SIMULATES SOME CENTRAL ISSUES OF BIOFILM GROWTH : THE PATTERN FORMATION OF HETEROGENEOUS STRUCTURES, SUCH AS FINGER-LIKE STRUCTURES, IN A SUBSTRATE-TRANSPORT-LIMITED REGIME, AND THE FORMATION OF MORE COMPACT STRUCTURES, IN A GROWTH-LIMITED-REGIME.

IN THIS WORK WE DEVELOP A GENERAL MATHEMATICAL MODEL AND DEVISE A PRACTICAL IDENTIFIABILITY APPROACH FOR GASTROINTESTINAL STROMAL TUMOR (GIST) METASTASIS TO THE LIVER, WITH THE AIM OF QUANTITATIVELY DESCRIBING THERAPY FAILURE DUE TO DRUG RESISTANCE. TO THIS END, WE HAVE MODELED METASTATIC GROWTH AND THERAPY FAILURE PRODUCED BY RESISTANCE TO TWO STANDARD TREATMENTS BASED ON TYROSINE KINASE INHIBITORS (IMATINIB AND SUNITINIB) THAT HAVE BEEN OBSERVED CLINICALLY IN PATIENTS WITH GIST METASTASIS TO THE LIVER. THE PARAMETER IDENTIFICATION PROBLEM IS DIFFICULT TO SOLVE, SINCE THERE ARE NO GENERAL RESULTS ON THIS ISSUE FOR MODELS BASED ON ORDINARY DIFFERENTIAL EQUATIONS (ODE) LIKE THE ONES STUDIED HERE. WE PROPOSE A GENERAL MODELING FRAMEWORK BASED ON ODE FOR GIST METASTATIC GROWTH AND THERAPY FAILURE DUE TO DRUG RESISTANCE AND ANALYZED FIVE DIFFERENT MODEL VARIANTS, USING MEDICAL IMAGE OBSERVATIONS (CT SCANS) FROM PATIENTS THAT EXHIBIT DRUG RESISTANCE. THE ASSOCIATED PARAMETER ESTIMATION PROBLEM WAS SOLVED USING THE NELDER-MEAD SIMPLEX ALGORITHM, BY ADDING A REGULARIZATION TERM TO THE OBJECTIVE FUNCTION TO ADDRESS MODEL INSTABILITY, AND ASSESSING THE AGREEMENT OF EITHER AN ABSOLUTE OR PROPORTIONAL ERROR IN THE OBJECTIVE FUNCTION. WE COMPARED THE GOODNESS OF FIT TO DATA FOR THE PROPOSED MODEL VARIANTS, AS WELL AS EVALUATED BOTH ERROR FORMS IN ORDER TO IMPROVE PARAMETER ESTIMATION RESULTS. FROM THE MODEL VARIANTS ANALYZED, WE IDENTIFIED THE ONE THAT PROVIDES THE BEST FIT TO ALL THE AVAILABLE PATIENT DATA SETS, AS WELL AS THE BEST ASSUMPTION IN COMPUTING THE OBJECTIVE FUNCTION (ABSOLUTE OR PROPORTIONAL ERROR). THIS IS THE FIRST WORK THAT REPORTS MATHEMATICAL MODELS CAPABLE OF CAPTURING AND QUANTITATIVELY DESCRIBING THERAPY FAILURE DUE TO DRUG RESISTANCE BASED ON CLINICAL IMAGES IN A PATIENT-SPECIFIC MANNER.

THE POSSIBILITY THAT FUNDAMENTAL DISCRETENESS IMPLICIT IN A QUANTUM GRAVITY THEORY MAY ACT AS A NATURAL REGULATOR FOR ULTRAVIOLET SINGULARITIES ARISING IN QUANTUM FIELD THEORY HAS BEEN INTENSIVELY STUDIED. HERE, ALONG THE SAME EXPECTATIONS, WE INVESTIGATE WHETHER A NONSTANDARD REPRESENTATION CALLED POLYMER REPRESENTATION CAN SMOOTH AWAY THE LARGE AMOUNT OF NEGATIVE ENERGY THAT AFFLICTS THE HAMILTONIANS OF HIGHER-ORDER TIME DERIVATIVE THEORIES, RENDERING THE THEORY UNSTABLE WHEN INTERACTIONS COME INTO PLAY. WE FOCUS ON THE FOURTH-ORDER PAIS-UHLENBECK MODEL WHICH CAN BE REEXPRESSED AS THE SUM OF TWO DECOUPLED HARMONIC OSCILLATORS ONE PRODUCING POSITIVE ENERGY AND THE OTHER NEGATIVE ENERGY. AS EXPECTED, THE SCHRODINGER QUANTIZATION OF SUCH MODEL LEADS TO THE STABILITY PROBLEM OR TO NEGATIVE NORM STATES CALLED GHOSTS. WITHIN THE FRAMEWORK OF POLYMER QUANTIZATION WE SHOW THE EXISTENCE OF NEW REGIONS WHERE THE HAMILTONIAN CAN BE DEFINED WELL BOUNDED FROM BELOW.

ONE OF THE MAIN CHALLENGES IN CANCER MODELLING IS TO IMPROVE THE KNOWLEDGE OF TUMOR PROGRESSION IN AREAS RELATED TO TUMOR GROWTH, TUMOR-INDUCED ANGIOGENESIS AND TARGETED THERAPIES EFFICACY. FOR THIS PURPOSE, INCORPORATE THE EXPERTISE FROM APPLIED MATHEMATICIANS, BIOLOGISTS AND PHYSICIANS IS HIGHLY DESIRABLE. DESPITE THE EXISTENCE OF A VERY WIDE RANGE OF MODELS, INVOLVING MANY STAGES IN CANCER PROGRESSION, FEW MODELS HAVE BEEN PROPOSED TO TAKE INTO ACCOUNT ALL RELEVANT PROCESSES IN TUMOR PROGRESSION, IN PARTICULAR THE EFFECT OF SYSTEMIC TREATMENTS AND ANGIOGENESIS. COMPOSITE BIOLOGICAL EXPERIMENTS, BOTH IN VITRO AND IN VIVO, IN ADDITION WITH MATHEMATICAL MODELLING CAN PROVIDE A BETTER UNDERSTANDING OF THESES ASPECTS. IN THIS WORK WE PROPOSED THAT A RATIONAL EXPERIMENTAL DESIGN ASSOCIATED WITH MATHEMATICAL MODELLING COULD PROVIDE NEW INSIGHTS INTO CANCER PROGRESSION. TO ACCOMPLISH THIS TASK, WE REVIEWED MATHEMATICAL MODELS AND CANCER BIOLOGY LITERATURE, DESCRIBING IN DETAIL THE BASIC PRINCIPLES OF MATHEMATICAL MODELLING. WE ALSO ANALYZE HOW EXPERIMENTAL DATA REGARDING TUMOR CELLS PROLIFERATION AND ANGIOGENESIS IN VITRO MAY FIT WITH MATHEMATICAL MODELLING IN ORDER TO RECONSTRUCT IN VIVO TUMOR EVOLUTION. ADDITIONALLY, WE EXPLAINED THE MATHEMATICAL METHODOLOGY IN A COMPREHENSIBLE WAY IN ORDER TO FACILITATE ITS FUTURE USE BY THE SCIENTIFIC COMMUNITY.

EN ESTA COMUNICACIÓN SE PROPONE UN NUEVO MODELO PARA LA FORMACIÓN DE UNA BIOPELÍCULA Y SE ESTUDIAN LOS ASPECTOS NUMÉRICOS DE SU SIMULACIÓN COMPUTACIONAL. NUESTRO MODELO ESTÁ BASADO EN TRES ASPECTOS FUNDAMENTALES: PRIMERO, SE INCORPORA UN MECANISMO DE TRANSPORTE DE NUTRIENTES HACIA LA BIOPELÍCULA; SEGUNDO, SE INCORPORA UN MECANISMO DE CONSUMO Y DE CRECIMIENTO DE LA BIOPELÍCULA; Y TERCERO, SE INCORPORA UN MECANISMO DE NECROSIS Y DE DECAIMIENTO DE LA BIOPELÍCULA. PARA EL MODELAMIENTO DEL TRANSPORTE DE NUTRIENTES, HEMOS USADO UNA ECUACIÓN DE DIFUSIÓN ESTACIONARIA. PARA EL MODELAMIENTO DEL MECANISMO DE CONSUMO Y CRECIMIENTO, ASÍ COMO TAMBIÉN DEL DE NECROSIS Y DECAIMIENTO, HEMOS MODELADO LA BIOPELÍCULA COMO UN FLUIDO VISCOSO ESCURRIENDO EN UN MEDIO POROSO. EL ASPECTO CLAVE DE ESTE MODELO ES QUE EL FRENTE DE LA BIOPELÍCULA ES UNA INCÓGNITA DEL PROBLEMA, LA CUAL DEPEN-DE DE LA SOLUCIÓN DEL MISMO. ASÍ, CON EL FIN DE RECONSTRUIR EL MOVIMIENTO DEL FRENTE, HEMOS USADO EL MÉTODO DEL CONJUNTO DE NIVEL. ADEMÁS, PARA RESOLVER LA ECUACIÓN DE LA PRESIÓN HEMOS USADO EL ASÍ LLAMADO MÉTODO DE LA INTERFAZ INMERSA, EL CUAL FUE ESPECIALMENTE DESARROLLADO PARA PROBLEMAS CON DISCONTINUIDADES A TRAVÉS DE UNA INTERFAZ DENTRO DE UN DOMINIO.