MATHEMATICAL MODELLING: GENERAL CONCEPTS AND INTRODUCTION

1. MATHEMATICAL MODELLING: GENERAL CONCEPTS AND INTRODUCTION Mario Grassi (mariog@dicamp.univ.trieste.it) Department of Chemical Engineering (DICAMP) UINVERSITY OF TRIESTE

2. VISUALIZE SOMETHING THAT CAN NOT BE DIRECTLY OBSERVED (MOLECULE) IMITATION OR EMULATION OF SOMETHING WHAT IS A MODEL? [1] AN ARTIFICIAL REPRESENTATTION OF SOMETHING NEEDED FOR

3. KNOWLEDGE CONSEQUENTLY, MODELLING IS A COGNITIVE ACTIVITY AIMED TO THE DESCRIPTION OF HOW PHENOMENA OCCUR OR OBJECTS BEHAVE

4. 4 3 2 1 SKETCHES PHYSICAL MODELS DRAWINGS Leonardo da Vinci MATHEMATICS MODELLING CAN BE PERFORMED ACCORDING TO:

5. MATHEMATICAL MODEL IS [2] “... niuna umana investigazione si può dimandare vera scienzia, s’essa non passa per le matematiche dimostrazioni” Leonardo da Vinci, Trattato della Pittura1651 A MATHEMATICAL METAPHOR OF SOME ASPECTS OF REALITY (DRUG RELEASE AND ABSORPTION)

6. 4 5 6 7 PHARMACY 1 ENGINEERING BIOLOGY 2 PHYSIC MEDICINE CHEMISTRY 3 PSYCHOLOGY MANY DIFFERENT FIELDS CAN TAKE ADVANTAGE OF MATHEMATICAL MODEL [3-5]

7. CULTURAL REASONS PRACTICAL REASONS HUMAN MIND CATEGORISATION OF KNOWLEDGE INTRINSIC COMLEXITY OF PHARMACEUTICAL BIOLOGICAL MEDICAL PSYCHOLOGICALPHENOMENA Field 1 Field 2 Field 3 no no no no no no Impermeable barriers hindering transversal tools diffusion WHY NOT TYPICAL IN BLUEFIELDS (4-7)[1]

8. Reliable Model Phenomenon analysis Physical hypotheses Identification of the mechanisms ruling the phenomenon Mathematical hypotheses OK Mechanisms mathematical form and assembling to get the MODEL Definition of proper experiments aimed to check the hypotheses: NO OK MODEL comparison with EXPERIMENTAL DATA: FITTING MODEL comparison with EXPERIMENTAL DATA: PREDICTION NO

9. MATHEMATICAL MODEL EXPRESSION Y = f(Z1, Z2, ….Zn, A1, A2, ….Am) dependent variable Independent variables Model parameters: Z1, Z2, … Zn independent EMPIRICAL A1, A2, ..Am do not posses a physical meaning THEORETICAL A1, A2, ..Am do posses a physical meaning C0 receiver D Linear relation C=0 donor h

10. MODEL FITTING PARAMETERS SOME MODEL PARAMETERS ARE KNOWN … A1, A2, A4 in the previous example receiver donor …THE REMAINING ARE UNKOWN AND ARE CALLED FITTING PARAMETERS AND NEED TO BE DETERMINED C0 A3 in the previous example D C=0 h

11. EXPERIMENTAL DATA DIFFERENCE [Cr experim – Cr model] Cr(t) MODEL t t1 t2 t3 t4 t5 t6 Standard deviation MODEL DATA FITTING

12. Mean Square Error degrees of freedom n2 = N-M Mean Square Regression degrees of freedom n1 = M-1 M = model fitting parameters number N = experimental data number Mean Statistic tells us that the most probable set of model parameters (A1, A2, … Am) is that yielding the lowest c2 value IS THE FITTING SATISFACTORY?

14. MODEL COMPARISON: AKAIKE CRITERION1 The most probable model is that characterised by the smallest AIC The probability of being right is defined by: EXAMPLE: IF D = 2 =====> pAIC = 0.73 N = number of experimental data M = number of fitting parameters

15. UNKNOWN MODEL PARAMETERS (FITTING P.) FITTING Y = f(Z1, Z2, ….Zn, A1, A2, ….Am) Cr MODEL PREDICTION EXPERIMENTAL DATA t MODEL EVALUATION: PREDICTION

16. MODEL PREDICTION YES E.D. STD. DEV. Cr EXP. DATA MODEL STD. DEV. t MODEL STD DEV IS CALCULATED ASSUMING FITTING PARAMETERS SETS THAT MAXIMISE AND MINIMISE PREDICTION: Ai = value ± S.D. IS MODEL PREDICTION SATISFACTORY?

17. EXP. DATA d6 Traditional Fitting d4 Cr(t) d5 MODEL EXP. DATA d2 MODEL d3 d1 t1 t2 t3 t4 t5 t6 t1 t2 t3 t4 t5 t6 ROBUST FITTING

18. Y FRICTION (FR = b*M*v) FM FR FM X t = 0 HEAT due to friction (WRINKLED SURFACE) EXAMPLE 1: SLIDING SLAB

19. SLAB MOTION CAN BE SIMULATED POSITION, VELOCITY, ACCELERATION, TEMPERATURE MODEL: NEWTON EQUATION

20. POSITION VELOCITY TEMPERATURE PROFILE ACCELERATION

22. EXAMPLE 2: HUMAN MEMORY [6] Performance Sensory buffer STS kr(d) kj+1(d) r Lost information kj(d) j+1 Buffer retrieval j kj-1(d) j-1 k1=d 1 t*/n mij LTS f(r,d,t,q) Model fitting parameters Stimulus vp

23. FREE VERBAL RECALL TEST [7] 48 young subjects (19-28 years old) are asked to remember a list containing twelve items (word) each (8 repetitions) Vp = 0.5 item/s

24. 0.25 item/s 0.5 item/s 1 item/s 2 item/s r = 3;  = 1.75; t = 0.76; d = 0.27 Fitting parameters

25. REFERENCES • Grassi, M., Grassi, G., Lapasin, R., Colombo, I. Understanding drug release and absorption mechanisms: a physical and mathematical approach. CRC, Boca Raton, Florida, USA, 2006. • Israel, G. Balthazar van der Pol e il primo modello matematico del battito cardiaco, in Modelli matematici nelle scienze biologiche, Freguglia, P. Ed., Edizioni Quattro Venti, 1998. • Dym, C. L. Principles of Mathematical Modeling, 2nd edition, Elsevier Academic Press, San Diego, London 2004. • Bourne, D. W. A. Mathematical Modeling of Pharmacokinetic Data, Technomic publishing CO., INC., Lancaster, Basel, 1995. • Cartensen, J. T. Modeling and data treatment in the pharmaceutical sciences, Technomic publishing CO., INC., Lancaster, Basel, 1996. • Atkinson, R. C. and Shiffrin, R. M., Human memory: A proposed system and its control processes, in The Psychology of Learning and Motivation: Advances in Research and Theory, Spence, K. W. and Spence J.T. Eds., Academic Press, 1968

26. Peirano, F. Rievocazione libera e interpretazione della curva di posizione seriale negli anziani, Tesi di Laurea, Università di Padova, Dipartimento di Psicologia, Giugno 2000. • Grassi, M., Voinovich, D., Moneghini, M., Franceschinis, Perissutti, B., Filipovic-Grcic, J. Preparation and evaluation of a melt pelletised paracetamol stearic acid sustained release delivery system. Journal of Controlled Release 88 (2003) 381–391.