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Unformatted text preview: ODE - Medical Akzo Nobel problem II-4-1 4 Medical Akzo Nobel problem 4.1 General information The problem consists of 2 partial di erential equations. Semi-discretization of this system yields a sti ODE. The parallel-IVP-algorithm group of CWI contributed this problem to the test set in collaboration with R. van der Hout from Akzo Nobel Central Research. The software part of the problem is in the le medakzo.f available at [ MI03 ]. 4.2 Mathematical description of the problem The problem is of the form d y d t = f ( t; y ) ; y (0) = g; (II.4.1) with y 2 IR 2 N ; t 20 : Here, the integer N is a user-supplied parameter. The function f is given by f 2 j 1 = j y 2 j +1 y 2 j 3 2¡ + j y 2 j 3 2 y 2 j 1 + y 2 j +1 (¡ ) 2 k y 2 j 1 y 2 j ; f 2 j = k y 2 j y 2 j 1 ; where j = 2( j ¡ 1) 3 c 2 ; j = ( j ¡ 1) 4 c 2 : Here, j ranges from 1 to N , ¡ = 1 N , y 1 ( t ) = ( t ), y 2 N +1 = y 2 N 1 and g 2 IR 2 N is given by g = (0 ; v ; ; v ; : : : ; ; v ) T : The function is given by ( t ) = 2 for t 2 (0 ; 5] ; for t 2 (5 ; 20] : which means that f undergoes a discontinuity in time at t = 5. Suitable values for the parameters k , v and c are 100, 1 and 4, respectively. 4.3 Origin of the problem The Akzo Nobel research laboratories formulated this problem in their study of the penetration of radio-labeled antibodies into a tissue that has been infected by a tumor [ Hou94 ]. This study was carried out for diagnostic as well as therapeutic purposes. Let us consider a reaction di usion system in one spatial dimension: @u @t = @ 2 u @x 2 kuv; (II.4.2) @v @t = kuv; (II.4.3) II-4-2 ODE - Medical Akzo Nobel problem which originates from the chemical reaction A + B k ! C: Here A , the radio-labeled antibody, reacts with substrate B , the tissue with the tumor, and k denotes the rate constant.the rate constant....
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- Spring '09
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