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 Cn+1 and Cn

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 EfficientSolution of a Drug Delivery Problem

Catherine E. Beni, Oscar P. BrunoApplied and Computational Mathematics

California Institute of Technology

Introduction Algorithm

Conclusions

Framework

References

Numerical Results

The VMT convection-diffusion problem:

The parameters D, vx, and vy 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 Membrane Vessel

D = .0001, vy = .0001, Ren = .01

COMSOL VMT Solver

Speed 36 hours < 5 minutes

Memory Requirements 32 GB 32.7 MB

→ 432 times faster.→ 1000 times reduction in memory requirements.

D = .00001, vy = .00001, Ren = .001

COMSOL VMT Solver

Speed N/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

Approximate the Radon transform with its Fourier series:

where ak and bk are the Fourier coefficients

Compute derivative of Hilbert transform:

Approximate CT and ST, 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. BrunoApplied 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’ mFBP Fejér-mFBP

MATLAB’s ‘iradon’ mFBP Fejér-mFBP

MATLAB’s ‘iradon’ mFBP Fejér-mFBP

MATLAB’s ‘iradon’ mFBP Fejér-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