\fFDG travels through the arterial system. where we measure its concentration {CPLand crosses the blood brain barrier. It's metabolized in the brain according to the compartment model displayed in the diagram. The governing equations for this are given along with the rate constants for this model. (Note: I have taken care of unit normalization and all numerical values should he used as given.) At the time I was working with Rodney Brooks. a physicist who had built a PET scanner for NIH. He published a paper for the model given in this paper {you will have it available for this project). In it he solved the set of coupled differential equations you see below {not using Laplace transforms}. However the easiest way to solve this problem is by the method of Laplace transforms. That is what I am asking you to do! And more! {a} In the model shown in Figure l the rates of change of ("It {free}. FLEDG in brain tissue and Cm {trapped} , FLEDGP in brain tissue are equal to the net transport of PEG and FDGoP into their compartments. That is ac [1-Er=ngng+kgepugem dC dy=hgkm [2] Figure l where initial concentrations are assumed to be zero. Note mat (IF the concentration of FDG in the bland, as a function of time, is an experimentally acquired function. Then the total brain tissue concentration. CL is given by [3} C; = [it + Cm Solve for Ci using the method of Laplace transforms starting with the coupled differential equations given above. Show every detail of the calculation and reproduce formula {4} in the associated paper by Brooks [Answer #1]. However instead of presenting this in the form given in the paper present it in terms of convolutions {those integrals in the paper are in fact convolutions do you recognize that its also a hint}. A few other hints follow: 1. CPU) is not given as an analytic function it is in fact tabulated data below. 1|When solving this system of equations simple carry it as this symbol or its Laplace transform quantity Ep(s} when appropriate. 2. I would use Cramer's rule to solve for CE [5], Em[s} 3. You may also want to use the quadratic formula to get some factors for a quadratic expression you get. 4. Remember how the convolution arises when applying the Laplace transform rnedrodology. that is when you ltave a product of functions in sspacc