Outline Motivation Matrix Calculus Review Sensitivity Analysis Complete State Variable Sensitivity Quantity of Interest Sensitivity Adjoints Summary of Sensitivity Analysis Worked Examples Conclusions…..

Malignant Tumors In experiments[6][5], radioactively marked tumor cells added to bloodstream of laboratory mice and levels of cancer cells measured overtime. Proposed model has two compartments (accounts for fact that rate of decay did not follow a simple exponential decay model). Let x 1 and x 2 denote number of cancer cells in capillaries and lung tissue, respectively. Loss of cancer cells from capillaries by being dislodged and carried away by the blood and transfer from capillaries to lung tissue modelled by linear functions * 1 x 1 and * 2 x 1 , respectively. Loss of cancer cells in the lung tissue modelled by * 3 x 2 .
Lecture1: Calculus Based Approach to Sensitivity Analysis Motivation Malignant Tumors Linear System of Differential Equations * ú x 1 = (* 1 +* 2 ) x 1 ú x 2 =* 2 x 1 * 3 x 2 qofi In experiments, state variables could not be measured individually, but the total amount of radioactivity, x 1 +x 2 was measurable, so theqofiinthis case is qofi=x 1 +x 2 . We can analyze theoretical sensitivities of x 1 and x 2 w.r.t parameters, but can only experimentally verify sensitivity of qofi w.r.t parameters.
Lecture1: Calculus Based Approach to Sensitivity Analysis Motivation Epidemics Description of model[5] This is the classical SIR class-structured model . We let K=S+I+R denote the total population. I There is a constant birthrate/natural death rate µ (all births placed inS class) I Infection rate is * I Recovery rate is I Fatality rate of disease is * …

Download Lecture1: Calculus Based Approach to Sensitivity Analysis.Pdf