STOCHASTIC & DETERMINISTIC SOLVERS

STOCHASTIC & DETERMINISTIC SOLVERS

Outline Spatial Scales of Optical Technologies & Mathematical Models Prototype RTE Problems 1-D Transport in Slab Geometry: Exact Solution Stochastic Models: Monte Carlo Simulation Fidelity of Stochastic Model: Model Equivalence P N & Standard Diffusion Approximation (SDA) Fluence Comparisons: RTE vs SDA

Spatial Scales of Current Imaging Techniques

Need for Model-Based Computation Optical technologies rely upon the precise delivery, detection, and analysis of light signals for (a) accurate dosimetry planning (forward RTE problems) (b) extraction of optical and physiological properties from measurements (inverse RTE problems) (c) guidance in instrument measurement and design (involves both forward and inverse RTE solvers However, model-based computation has lagged behind development of optical technologies Virtual Photonics Core Technology in LAMMP

Spatial Scales of Mathematical Models

Resolution vs. Depth linked to Physics

With Different Optical Regimes

Core LAMMP Optical Technologies & Mathematical Models SDA RTE Maxwell s eqs

Exact RTE Solutions A small number of RTE equations can be solved in closed form. An important family of such problems describes transport in one dimension These provide interesting and useful benchmarks for approximate RTE methods Surprisingly perhaps, they capture certain features of 3D RTE solutions quite well We introduce this 1D family next

RTE in 1-D Slab Geometry A.J. Welch and M.J.C. van Gemert (eds.), Optical-Thermal Response of Laser-Irradiated Tissue, Plenum, 1995 Input: μ a,μ s,d (slab thickness), source Q 0 = 1 p(μ ) = p f δ(μ-1) + p b δ(μ+1) p f + p b = 1, p f p b = g d L x c x c x ( ) 1exp( ) 2exp( ) L x d x d x ( ) 1exp( ) 2exp( ) where c 1, c 2, d 1, d 2 depend on input data and a( a ' s) atr This characteristic decay rate for RTE is different than the characteristic decay rate for SDA, as we will see

RTE Solutions We focus on the steady state RTE L( r, ) ( r) L( r, ) ( r) L( r, ') p( r, ' ) d' Q( r, ) s S 2 t Solution L(r, Ω) determines light field everywhere Add detection f(r,ω) to indicate location(s) and angular span(s) for measurement(s) Add target(s): region(s) in phase space needing maximal interrogation These ideas are schematically described on next slide

Depictions of RTE Problem Classes Radiance (blue) produced by collimated point source (left) Photon banana (red) indicates paths that reach detector from source (middle) Information density function (red) reveals whether targeted region is receiving adequate light that reaches detector (right)

Practical Utility of RTE Problem Types In Lecture 1 it was noted that measurements relevant to therapy & fluorescence excitation rely on virtual detection within the tissue, while diagnostics and imaging are detected by measurements at the boundaries of the tissue. In fact, capturing light signals anywhere can be expressed as weighted integrals of the radiance f ( r,, t) L( r,, t) drddt where the function f ( r,, t), the detector function, restricts the light collected to specific locations r, orientations Ω and times t, either inside the tissue (therapeutics) or on the surface (diagnostics)

Forward/Inverse Problems Given problem input: I= {μ t, μ a,p, n, Q, Γ, BC, IC, f₁, f₂,, f D } Determine output O= {I₁, I₂,, I D }, D=# of measurements (detectors) The forward problem map: F: I O The inverse problem map: F -1 :O I In general, F <1 (F is contractive) while O >1 (inverse problem is ill-conditioned). F depends on choice of model and implementation strategy

Forward Light Propagation Models Problem Definition layer of VTS establishes the problem input Solver choice (EM, RTE, SDA, P n ) determines mathematical model that produces the RTE solution and (weighted) integrals of describe the problem output Lr (, ) Lr (, ) that VTS platform In this lecture we discuss stochastic (Monte Carlo) & diffusionbased RTE solvers (including SDA)

Stochastic Solver Ideas Assumption: RTE describes average behavior of photons; i.e., RTE describes the average behavior of a stochastically varying ensemble of particles RTE coefficients μ t, μ s, p and source Q determine the radiance, L generate photon biographies by A. initiating biography from Q, applying Beer s Law (using μ t, μ s ) to determine intercollision distances, B. using p(μ 0 ) to determine changes in direction; μ 0 = Ω Ω C. alternating sampling between distance & direction until photon leaves tissue or is absorbed

Analog Monte Carlo Simulation Each photon represents a unit of radiant energy that is unvarying throughout the biography Energy is conserved: each photon is either absorbed, specularly or diffusely reflected or transmitted Recall Beer s Law: In homogeneous media: L(r, Ω) = L 0 exp(- µ t s) Distances between successive collisions ~ µ t exp(- µ t s) (produces correct total mean free path, 1/ µ t ) Changes in direction ~ p(ω Ω ) = p(μ 0 = cos θ) Photon biographies each contribute tally of 1 either to absorbed energy, reflectance or transmittance

Weighted Monte Carlo Simulation Each photon represents a beam of continuously (or discretely) varying radiant intensity, with unit initial intensity Energy is conserved, as before Launch each photon with weight W = 1 Distances between successive scattering collisions ~ µ s exp(- µ s s) (produces correct scattering mean free path, 1/ µ s ) Distribute a weight W [1- exp(- µ a s)] of absorbed energy density along track of length s, and scatter the complementary weight W exp(- µ a s) making use of the phase function p; i.e., changes in direction determined from p(ω Ω ) = p(cos θ) Photon biographies deposit correct weights for absorption, reflectance, transmittance

Illustration: Continuous Absorption Weighting This method (CAW) performs weighted MC simulation in tissue in which only scattering is modeled, but absorption is nevertheless accounted for using weights For any µ a > 0, absorbed and scattered energy densities are estimated as on previous slide, using weighted estimators Method applies much more generally, with appropriate modifications A single set of photon biographies can be assigned a set of weights to mimic tissue with arbitrary optical properties, layer thicknesses, etc. This idea is crucial to the use of perturbation Monte Carlo (pmc) in studying sensitivities and solving inverse problems

Flow Charts: Weighted (CAW) & Analog MC Weighted MC Analog MC

Fidelity of Stochastic Model: Model Equivalence There are (infinitely) many equivalent stochastic solution models for every RTE. E.g, Discrete Absorption Weighting (DAW) de-weights) only at discrete collision events by μ s /μ t. How can we be certain the one we ve chosen models RTE exactly? Each model produces AN N If one can show that N as that is, N 0 A N that, then AN is an unbiased estimator. Intuitively plausible estimators may not possess this property. Conversely, many estimators that are developed strictly from mathematical considerations are not only unbiased but give rise to larger FOMs than more intuitive choices. A

Fidelity of Stochastic Model: Model Equivalence Physical/Analytic Model: {Γ, RTE, BCs, f} Stochastic/Probabilistic Model: {B, ξ, M} B = sample space of all possible photon biographies ξ(b) = weight associated with biography b The RTE induces an analog probability measure M on B E[ξ] = (M(B) = 1) B ξ( b) dm ( b) = exact mean value of ξ with respect to M In what sense are these models equivalent? It has been rigorously established J. Spanier and E.M. Gelbard, Monte Carlo Principles and Neutron Transport Problems, Dover, 2008 E[ξ] = B ( b) dm ( b) f ( r, ) L( r, ) drd S 2

Spatially Resolved Diffuse Reflectance R( ) L( x, y, z 0, ) d where (ρ 2 = x 2 + y 2 ) and S 2- designates the full hemisphere of exiting directions at the tissue surface, z = 0 (matched n s) The variables/parameters of interest to vary: Distance S 2 between source(s) & detector(s), Number and locations of detectors In practice, light is collected over reduced numerical apertures and small surface areas that include the desired values of ρ; that is, R(, ) L( x, y, z 0, ) dd

Time-Resolved Diffuse Reflectance R(,, t) L( x, y, z 0,, t) ddt t Time-resolved measurements depend on introduction of a pulsed light source and collection of light that arrives at each detector at different times (equivalently, distances traversed) For both spatially- and time-resolved measurements, avoiding redundancy requires intelligent placement of detectors

Approximating RTE Solutions: Recapturing Some Angular Variation (SDA & P N ) We revisit the steady state RTE L( r, ) ( r) L( r, ) ( r) L( r, ') p( r, ' ) d' Q( r, ) s S 2 t to sketch the ideas that lead to the standard diffusion approximation (SDA) and higher order (P N ) methods. For simplicity we specialize to an infinite plane geometry, reducing L to a function of the spatial variable, x, and the cosine, μ, of arccos (Ω Ω x ). With μ 0 = cos (Ω Ω ), the RTE reduces to dl L ( x, ) t ( x ) L ( x, ) dx 1 s ( x) (, ') (, 0 L x p x ) d ' Q( x, ) 1

Derivation of P N Equations The general approach is to expand all functions in Legendre polynomials, P j : 2j 1 L( x, ) Lj( x) Pj( ) 2j 1 Q( x, ) j0 2 Qj( x) Pj( ) j0 2 2j 1 p( 0) pjpj( 0) j0 2 Because the P j are orthogonal 1 and normalized, Lj( x) (, ) j L x P ( x) d and since P 0 =1 and P 1 = μ we find that 1 L0( x) L( x, ) d ( x) is the fluence and N =1 1 1 1 L1 x L x d J x 1 ( ) (, ) ( ) 1 3 L x L x L x 2 2 (, ) 0( ) 1( ) is the net energy flux

The Standard Diffusion Approximation (SDA) Substituting into the RTE we find dl1( x) dl0( x) ( t sp0) L0( x) Q0( x) dx dx 1 3 L( x, ) L0( x) L1( x) 2 2 1 K1(1 3 D)exp( x) K 2(1 3 D)3 L1( x)exp( x) 2 where 1 D and SDA a / D 3 a( a s ') 3atr 3( t sg) which compares with RTE a tr 3( t sp1) L1( x) 3 Q1( x) found for RTE

The Significance of μ s in SDA Modeling From the SDA equations dl1 x ( ) dx ( t sp0) L0( x) Q0( x) dl0 x ( ) dx 3( t sp1) L1( x) 3 Q1( x) Since p 0 = 1 and p 1 = g, we note that L 0 depends only on μ a + μ s (1-g) = μ a + μ s = μ tr and L 1 only depends on μ a and L 0 For fixed μ a, all (μ s, g) pairs with identical μ s values give rise to identical SDA solutions. this feature can be used advantageously in both SDA and RTE modeling through choices of pairs (μ s, g) that offer computational advantages however, it also means that one can expect difficulty in predicting the optical property distribution (inverse problem solution) from measurements of the RTE solution

Model Transport Problem: RTE & SDA S.A. Dupree & S.K. Fraley, A Monte Carlo Primer, Plenum, 2002 It is instructive to examine fluence distributions produced by an isotropic point source in an infinite medium: We compare RTE and SDA solutions in this spherical geometry for different ratios of scattering to absorption and different anisotropies g ( r) ( r) ( r) ( r r ') 2 2 eft 1 ( r) exp( eft r r ' ), eft 3atr 4 r r'

Fluences with Varying μ s /μ a, g l* = 1, g = 0, μ s /μ a = 1, 10, 100

Fluences with Varying μ s /μ a, g l* = 1, g = 0.8, μ s /μ a = 1, 10, 100

Summary and Take Home Messages SDA and RTE are basic and important VTS solvers Monte Carlo simulation is a stochastic model for solving the RTE that offers great generality: FOM measures computational efficiency of Monte Carlo simulations and is easy to estimate There are many stochastically equivalent MC simulations based on weighted Monte Carlo with different FOMs Exact SDA solvers are formula-based and fast (for simple geometries and boundary conditions) Choosing the right solver requires understanding the range and limitations of each

The VTS Computational Platform After lunch today, Lisa will present an overview of the VTS computational platform This platform provides access to both command line and GUI-enabled learning & research capabilities We ll use VTS throughout the Short Course and in the hands-on computational labs each afternoon