1.  Graded Mesh  Change of variable in the vessel to allow for resolution of the boundary layer  Alternating Directions Implicit (ADI)  Discretize in time:  Split into operators on C n+1 and C n and factor into differential operators in x and y  Solve the resulting scheme using Finite Difference methods  On-and-Off Fluid Freezing Methodology  Evolve algorithm until concentration in vessel reaches steady state  “Freeze” concentration in vessel by not applying solver in vessel region  Iterate twice using large time steps  Reduce time step, unfreeze vessel, and iterate until concentration in vessel reaches steady state  Repeat  Change of Unknown  Use the following change of unknown to transform the differential operator in y in the membrane and tissue regions into the Helmholtz equation to make use of a previously known fast time-stepping method A Fast, Accurate Algorithm Enabling Efficient Solution of a Drug Delivery Problem Catherine E. Beni, Oscar P. Bruno Applied and Computational Mathematics California Institute of Technology Introduction Algorithm Conclusions Framework References Numerical Results  The VMT convection-diffusion problem:  The parameters D, v x, and v y vary in each layer  VMT geometry:  The VMT solver is based on a combination of  Use of a graded mesh to adequately resolve boundary layers  The Alternating Directions Implicit (ADI) method to overcome the overwhelmingly restrictive CFL condition imposed by the fine spatial discretization mentioned above  An on-and-off fluid-freezing methodology that allows for efficient treatment of the multiple time-scales that coexist in the problem (whose equilibria arise through a complex balance of fluid-flow, magnetic-pull and diffusion effects)  A change of unknown that enables evaluation of steady states in tissue and membrane layers through a highly accelerated time-stepping procedure “The Behaviors of Ferro-Magnetic Nano-Particles in Blood Vessels under Applied Magnetic Fields”, A. Nacev, C.E. Beni, O.P. Bruno, B. Shapiro (to be submitted)  Developed a fast, efficient solver for a drug delivery problem  432 times faster than commercial package COMSOL Multiphysics  1000 times reduced memory requirements  Allows for solution of previously intractable problems  The goal of magnetic drug delivery is to use magnetic fields to direct and confine magnetically-responsive particles bound to therapeutic agents to specific regions in a patient’s body-- thus allowing for focused treatment in an area of interest.  To design a method leading to confinement of the magnetically-responsive particles to a particular region of the body, a predictive capability must be used to evaluate the effects of external magnetic forces on the convection and diffusion of magnetic particles through the bloodstream and in membranes and tissue.  The numerical solution of the Vessel-Membrane-Tissue (VMT) convection diffusion problem proposed by Grief and Richardson is highly challenging:  Greatly disparate time-scales  Extremely steep boundary layers  Occurrence of very small diffusion coefficients Tissue MembraneVessel D =.0001, v y =.0001, Ren =.01 COMSOLVMT Solver Speed36 hours< 5 minutes Memory Requirements32 GB32.7 MB → 432 times faster. → 1000 times reduction in memory requirements. D =.00001, v y =.00001, Ren =.001 COMSOLVMT Solver SpeedN/A< 8 minutes Memory Requirements > Available 32Gb 98.3 Mb Future work  Finite Difference methods restrict us to a rectangular geometry  Room for accuracy improvement  These two problems will be fixed by solving the ODEs present in the ADI method with the new Fourier Continuation-Alternating Directions (FC-AD) methodology  See the talk by O.P. Bruno for more details

2.  Approximate the Radon transform with its Fourier series: where a k and b k are the Fourier coefficients  Compute derivative of Hilbert transform:  Approximate C T and S T, the Hilbert transforms of cosine and sine respectively, as follows:  Combine with the derivative of the Hilbert transform and integrate to obtain the modified-Filtered Back Projection (mFBP) algorithm A Noise-tolerant Fejér-based modified-FBP Reconstruction Algorithm (Fejér-mFBP) for Positron Emission Tomography C.E. Beni, O.P. Bruno Applied and Computational Mathematics California Institute of Technology Reconstructed Images Framework Introduction Modified-FBP  Images can be reconstructed from Positron Emission Tomography (PET) scanners via two methods: Iterative methods and Direct methods  Direct methods, such as the well-known Filtered Back Projection (FBP) algorithm are fast, but reconstruct images that are low resolution.  Iterative methods, such as ML-EM (Maximum Likelihood-Expectation Maximization) and OSEM (Ordered Subset Expectation Maximization), are much slower (each iteration takes the same amount of time as a full reconstruction using a direct method and approximately 20-30 iterations are required), but provide high quality reconstructions.  Both methods suffer in the presence of noise:  Direct methods amplify noise and show a dramatic loss of information in the reconstructed images  Iterative algorithms are not guaranteed to converge  Goal: to design a fast, accurate reconstruction algorithm that does not degrade substantially in the presence of noise  Radon Transform:  Geometry  Inverse Radon Transform:  h( ½, µ ) is the Hilbert transform of the Radon transform  Reconstructions using 711 values of ½, 200 values of µ, and 200 Fourier modes → Unrealistic noise, unrealistically sensitive device Original FBP  Compute the Hilbert transform and its derivative as follows:  Integrate to obtain the inverse Radon transform Fejér-mFBP Numerical Results  Both algorithms were implemented in C++  Each reconstruction requires ~3.6 seconds, the same amount of time used by MATLAB’s built-in ‘iradon’ function  All reconstructions shown here are of the well-known Shepp-Logan phantom generated using MATLAB’s built-in ‘phantom’ command MATLAB’s ‘iradon’mFBPFejér-mFBP MATLAB’s ‘iradon’mFBPFejér-mFBP MATLAB’s ‘iradon’mFBPFejér-mFBP MATLAB’s ‘iradon’mFBPFejér-mFBP Conclusions References  Developed a new reconstruction algorithm that, in presence of noise, yields iterative-solver-like quality at FBP computational costs.  “A Noise-Tolerant Fejér-based modified-FBP Reconstruction Algorithm (Fejér-mFBP) for Positron Emission Tomography”, C.E. Beni, O.P. Bruno (to be submitted)  Reconstructions using 100 values of ½, 200 values of µ, and 200 Fourier modes → Unrealistic noise, realistically sensitive device  Reconstructions using 711 values of ½, 200 values of µ, and 200 Fourier modes with 18.5% noise present → Realistic noise, unrealistically sensitive device  Reconstructions using 100 values of ½, 200 values of µ, and 200 Fourier modes with 18.5% noise present → Realistic noise, realistically sensitive device  Fejér series of a given function:  By approximating the Radon transform with a Fejér series instead, we obtain the Fejér-mFBP algorithm:  Shepp-Logan phantom