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BENG 449 Problem 9 5

# BENG 449 Problem 9 5 - 4:22 PM MATLAB Command Window 1 of 5...

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4/23/08 4:22 PM MATLAB Command Window 1 of 5 >> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Alex Lemon % % BENG 449 -- Biomedical Data Analysis % % Problem Set 9, Question 5 % % Due April 22, 2008 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The Logan plot method estimates the total distribution volume VT from % % PET data by fitting the linear portion of the graph (t > t*) to a % % straight line. The goal of this problem is to evaluate the effect of % % your selection of t* on bias and noise of your estimate. Assume 10% % % noise, perform 25 replicates and for each replicate, compute the Logan % % slope using different numbers of points from the end of the curve. % % Graph the mean and standard deviation of the VT estimates as a % % function of t*. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Generate the data for t and CP to be used as the input to the system % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t_CP = [0:0.01:10, 11:60, 65:5:120]; CP = CP_function(t_CP); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Initialize the problem parameters % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K1 = 0.30; k2 = 0.15; k3 = 0.05; k4 = 0.01; params = [K1 k2 k3, k4]; VT = K1 / k2 * (1 + k3 / k4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Solve for the ideal C1(t), C2(t) and CT(t) numerically using ode45. % % First, integrate the differential equations using ode45 to obtain a % % set of solution points. Then, use deval to evaluate the solution at % % the same time points used for CP(t). Finally, compute CT(t) as the % % sum of C1(t) and C2(t). % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dur_ct = horzcat(ones(1,6), 2, 2, ones(1,22) * 5); t_CT = cumsum(dur_ct) - dur_ct / 2.; % mid time of scan C0 = [0 0]; tspan = [0 max(t_CT)]; sol = ode45(@dCdT, tspan, C0, [], CP, t_CP, params); C = deval(sol, t_CT); CT = C(1,:) + C(2,:);

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4/23/08 4:22 PM MATLAB Command Window 2 of 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integrate CP(t), so that it can be used in the Logan plot without % % being recalculated during every iteration of the trial loop. The % % plot is based on the time scale for CT(t), so it is also necessary to % % use interpolation to convert the time scale of the resulting integral. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INT_CP = cumtrapz(t_CP, CP); INT_CP = interp1(t_CP, INT_CP, t_CT); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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