Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation

R. Roy; E. M. Sevick-Muraca

doi:10.1364/OE.4.000353

Optics Express
Vol. 4,
Issue 10,
pp. 353-371
(1999)
•https://doi.org/10.1364/OE.4.000353

Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation

R. Roy and E. M. Sevick-Muraca

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R. Roy and E. M. Sevick-Muraca, "Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation," Opt. Express 4, 353-371 (1999)

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Abstract

The development of non-invasive, biomedical optical imaging from time-dependent measurements of near-infrared (NIR) light propagation in tissues depends upon two crucial advances: (i) the instrumental tools to enable photon “time-of-flight” measurement within rapid and clinically realistic times, and (ii) the computational tools enabling the reconstruction of interior tissue optical property maps from exterior measurements of photon “time-of-flight” or photon migration. In this contribution, the image reconstruction algorithm is formulated as an optimization problem in which an interior map of tissue optical properties of absorption and fluorescence lifetime is reconstructed from synthetically generated exterior measurements of frequency-domain photon migration (FDPM). The inverse solution is accomplished using a truncated Newton’s method with trust region to match synthetic fluorescence FDPM measurements with that predicted by the finite element prediction. The computational overhead and error associated with computing the gradient numerically is minimized upon using modified techniques of reverse automatic differentiation.

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Figure 1

Figure 1 Schematic of fluorescence photon migration.

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- \nabla \cdot [D_{x} (\vec{r}) \nabla Φ_{x} (\vec{r}, ω)] + [\frac{i ω}{c} + μ_{a_{xi}} (\vec{r}) + μ_{a_{xf}} (\vec{r})] Φ_{x} (\vec{r}, ω) = 0 on Ω

- \nabla \cdot [D_{m} (\vec{r}) \nabla Φ_{m} (\vec{r}, ω)] + [\frac{i ω}{c} + μ_{a_{m}} (\vec{r})] Φ_{m} (\vec{r}, ω) = {ϕμ}_{a_{x \to m}} \frac{1}{1 - i ωτ} Φ_{x} (\vec{r}, ω) on Ω

Φ_{x} (\vec{r}, ω) - 2 γ D_{x} (\vec{r}) \frac{\partial Φ_{x} (\vec{r}, ω)}{\partial n} + S δ (\vec{r} - {\vec{r}}_{s}) = 0 on d Ω

γ = \frac{(1 + r_{d})}{(1 - r_{d})}

r_{d} = - 1.44 n_{rel}^{- 2} + 0.72 n_{rel}^{- 1} + 0.668 + 0.063 n_{rel}

\int_{Ω} [- \nabla \cdot (D_{x} \nabla Φ_{x}) + (\frac{i ω}{c} + {μ_{a}}_{xi} + μ_{a_{xf}}) Φ_{x}] w_{j} d Ω = 0 j = 1,2, \dots, N

\int_{Ω} [D_{x} (\nabla Φ_{x}) \cdot (\nabla w_{j}) + (\frac{i ω}{c} + μ_{a_{xi}} + μ_{a_{xf}}) Φ_{x} w_{j}] d Ω - \int_{Γ} D_{x} w_{j} \frac{\partial Φ_{x}}{\partial n} d Γ = 0

\int_{Γ} D_{x} w_{j} \frac{\partial Φ_{x}}{\partial n} d Γ = \frac{1}{γ} \int_{Γ} (Φ_{x} + S) w_{j} d Γ = \frac{1}{γ} \int_{Γ} Φ_{x} w_{j} d Γ + \frac{1}{γ} \int_{Γ} S w_{j} d Γ

\int_{Ω} [D_{x} (\nabla Φ_{x}) (\nabla w_{j}) + (\frac{i ω}{c} + μ_{a_{xi}} + μ_{a_{xf}}) Φ_{x} w_{j}] d Ω - \frac{1}{γ} \int_{Γ} Φ_{x} w_{j} d Γ = \frac{1}{γ} \int_{Γ} S w_{j} d Γ

\sum_{el = 1}^{M} [\int_{Ω^{el}} [D_{x}^{el} (\nabla Φ_{x}^{el}) (\nabla w_{j}) + (\frac{iω}{c} + μ_{a_{xi}} + μ_{a_{xf}}^{el}) Φ_{x}^{el} w_{j}] dΩ + \frac{1}{γ} \int_{Γ^{el}} Φ_{x}^{el} w_{j} dΓ] =

\sum_{el}^{M} [\frac{1}{γ} \int_{Γ^{el}} S w_{j} dΓ]

Φ_{x}^{el} = \sum_{j = 1}^{3} L_{j} {(Φ_{x})}_{j}

w_{j} = L_{j} for j = 1,2, \dots, N

\sum_{el = 1}^{M} [\int_{Ω^{el}} [D_{x}^{el} (\frac{\partial Φ_{x}^{el}}{\partial x} \frac{\partial L_{j}}{\partial x} + \frac{\partial Φ_{x}^{el}}{\partial y} \frac{\partial L_{j}}{\partial y}) + (\frac{iω}{c} + μ_{a_{xi}} + μ_{a_{xf}}^{el}) L_{j}] Φ_{x}^{el} dΩ + \frac{1}{γ} \int_{Γ^{el}} Φ_{x}^{el} L_{j} dΓ] =

\sum_{el = 1}^{M} [\frac{1}{γ} \int_{Γ^{el}} S L_{j} dΓ]

\sum_{el = 1}^{M} [K_{1}^{el} + K_{2}^{el} + K_{3}^{el}] Φ_{x}^{el} = \sum_{el}^{M} r^{el}

S = S δ (\vec{r} - {\vec{r}}_{S})

\int_{Γ^{el}} S δ (r - r_{s}) d Γ = {\begin{matrix} S r_{s} \in Γ^{el} \\ 0 otherwise \end{matrix}

r^{el} = \frac{1}{γ} \int_{Γ^{el}} L_{j} Sδ (\vec{r} - {\vec{r}}_{s}) dΓ = \frac{1}{γ} L_{j} (r_{s}) S

{(r^{el})}_{j} = \frac{1}{γ} S

\sum_{el = 1}^{M} [\int_{Ω^{el}} [D_{m}^{el} (\frac{\partial Φ_{m}^{el}}{\partial x} \frac{\partial L_{j}}{\partial x} + \frac{\partial Φ_{m}^{el}}{\partial y} \frac{\partial L_{j}}{\partial y}) + (\frac{i ω}{c} + μ_{a_{m}}) Φ_{m}^{el} L_{j}] d Ω + \frac{1}{γ} \int_{Γ^{el}} Φ_{m}^{el} L_{j} dΓ] =

\sum_{el = 1}^{el} [\int_{Ω^{el}} Φ_{x}^{el} μ_{a_{x \to m}}^{el} ϕ (1 + ω τ^{el}) L_{j} d Ω]

K {\bar{Φ}}_{x, m} = b

(N_{B} - 1) N_{S} \geq N

\frac{(N_{B} - 1) N_{S}}{2} \geq N

E_{x, m} (μ_{a_{xf}}) = \frac{1}{2} \sum_{l = 1}^{N_{s}} \underset{j \neq l}{\sum_{j = 1}^{N_{B}}} (\frac{{({({(Φ_{x, m})}_{l})}_{j})}_{c} - {({({(Φ_{x, m})}_{l})}_{j})}_{me}}{{({({(Φ_{x, m})}_{l})}_{j})}_{me}}) (\frac{{({({(Φ_{x, m}^{*})}_{l})}_{j})}_{c} - {({({(Φ_{x, m}^{*})}_{l})}_{j})}_{me}}{{({({(Φ_{x, m}^{*})}_{l})}_{j})}_{me}})

\nabla E_{x} = \frac{\partial E_{x}}{(\partial μ_{a_{xf}})} = Re \sum_{j \in d Ω} (〈 (\frac{{(Φ_{x}^{*})}_{c} - {(Φ_{x}^{*})}_{me}}{{(Φ_{x})}_{me} {(Φ_{x}^{*})}_{me}}), \frac{\partial {(Φ_{x})}_{c}}{(\partial μ_{a_{xf}})} 〉)

\nabla E_{m} = \frac{\partial E_{m}}{(\partial τ)} = Re \sum_{j \in d Ω} (〈 (\frac{{(Φ_{m}^{*})}_{c} - {(Φ_{m}^{*})}_{me}}{{(Φ_{m})}_{me} {(Φ_{m}^{*})}_{me}}), \frac{\partial {(Φ_{m})}_{c}}{(\partial τ)} 〉)

\nabla {E'}_{m} = \frac{\partial E_{m}}{(\partial μ_{a_{x \to m}})} = Re \sum_{j \in d Ω} (〈 (\frac{{(Φ_{m}^{*})}_{c} - {(Φ_{m}^{*})}_{me}}{{(Φ_{m})}_{me} {(Φ_{m}^{*})}_{me}}), \frac{\partial {(Φ_{m})}_{c}}{(\partial μ_{a_{x \to m}})} 〉)

E_{x, m} ({\bar{μ}}_{a_{k}} + d) = E_{x, m} ({\bar{μ}}_{a_{k}}) + Q (d)

Q (d) = g_{k}^{T} d + \frac{1}{2} d^{T} G_{k} d

G_{k} d = - g_{k}

\frac{∥ r_{i} ∥}{∥ g_{k} ∥} \leq min (\frac{1}{k}, ∥ g_{k} ∥)

G (x) d = \frac{1}{σ} [g (x + σ d) - g (x)]

R_{1} = 0.01

R_{k + 1} = 2 R_{k} if λ_{k} \geq 1.0

R_{k + 1} = \frac{1}{3} R_{k} if λ_{k} < 1.0

\begin{matrix} Given & x_{i}, i = 1, \dots, n \\ For & i = n + 1, \dots, P \\ then if F_{i} is binary & x_{i} = F_{i} (x_{j}, x_{k}), j, k < i \\ and if F_{i} is unary & x_{i} = F (x_{j}), . j < i \\ f (x) = x_{n + P} \end{matrix}

\frac{\partial f}{\partial x_{i}} = \sum_{j} \frac{\partial f}{\partial x_{j}} \frac{\partial F_{j}}{\partial x_{i}} j > i

\begin{matrix} Given & x_{i,} i = 1, \dots, P \\ set & {\hat{x}}_{i} = 0, i = 1, ., n + P - 1 and {\hat{x}}_{n + P} = 1 \\ for & i = n + P, \dots, n + 1 \end{matrix}

\begin{matrix} then, if F_{i} is binary, & {\hat{x}}_{j} = {\hat{x}}_{j} + {\hat{x}}_{i} \frac{\partial F_{i}}{\partial x_{j}} i > j \\ and & {\hat{x}}_{k} = {\hat{x}}_{k} + {\hat{x}}_{i} \frac{\partial F_{i}}{\partial x_{k}} i > j \\ else if F_{i} is unary, & {\hat{x}}_{j} = {\hat{x}}_{j} + {\hat{x}}_{i} \frac{\partial F_{i}}{\partial x_{j}} i > j \\ derivatives & g_{i} = {\hat{x}}_{i} i = 1, \dots, n \end{matrix}

{({\hat{μ}}_{a_{xf}})}_{p} = \frac{\partial E_{x, m}}{\partial {(μ_{a_{xf}})}_{p}} = \frac{\partial E_{x, m}}{\partial K} \frac{\partial K}{\partial {(μ_{a_{xf}})}_{p}} + \frac{\partial E_{x, m}}{\partial b} \frac{\partial b}{\partial {(μ_{a_{xf}})}_{p}}

= \sum_{el} \sum_{i, j} {({\hat{K}}^{el})}_{i, j} (\frac{\partial K_{i, j}^{el}}{\partial {({μ_{a}}_{xf})}_{p}}) + \sum_{el} \sum_{j} {\hat{b}}_{j} \frac{\partial b_{j}}{\partial {({μ_{a}}_{xf})}_{p}}

\hat{K} = \frac{\partial E_{x, m}}{\partial K} = \frac{\partial E_{x, m}}{\partial {\bar{Φ}}_{x, m}} \frac{\partial {\bar{Φ}}_{x, m}}{\partial K} = {\hat{\bar{Φ}}}_{x, m} \frac{\partial {\bar{Φ}}_{x, m}}{\partial K} = {\hat{\bar{Φ}}}_{x, m} \frac{\partial Φ_{x, m}}{\partial {\bar{μ}}_{a_{xf}}} \frac{\partial {\bar{μ}}_{a_{xf}}}{\partial K}

K {\bar{Φ}}_{x, m} = b

\frac{\partial K}{\partial {\bar{μ}}_{a_{xf}}} {\bar{Φ}}_{x, m} + K \frac{\partial {\bar{Φ}}_{xf}}{\partial {\bar{μ}}_{a_{xf}}} = 0

\frac{\partial K}{\partial {\bar{μ}}_{a_{xf}}} = - \frac{K \frac{\partial {\bar{Φ}}_{x, m}}{\partial {\bar{μ}}_{xf}}}{{\bar{Φ}}_{x, m}}

\frac{\partial {\bar{μ}}_{xf}}{\partial K} = - K^{- 1} \frac{{\bar{Φ}}_{x, m}}{\frac{\partial {\bar{Φ}}_{x, m}}{\partial {\bar{μ}}_{a_{x, m}}}}

\hat{K} = - {\hat{\bar{Φ}}}_{x, m} K^{- 1} Φ_{x, m} = - {\bar{v}}^{T} {\bar{Φ}}_{x, m}

\bar{v} = \hat{\bar{Φ}} K^{- 1}

K \bar{v} = {\hat{\bar{Φ}}}_{x, m}

\hat{b} = \frac{\partial E_{x, m}}{\partial b} = \frac{\partial E_{x, m}}{\partial {\bar{Φ}}_{x, m}} \frac{\partial {\bar{Φ}}_{x, m}}{\partial b} = {\hat{\bar{Φ}}}_{x, m} \frac{\partial {\bar{Φ}}_{x, m}}{\partial b} = {\hat{\bar{Φ}}}_{x, m} \frac{\partial {\bar{Φ}}_{x, m}}{\partial \bar{τ}} \frac{\partial \bar{τ}}{\partial b}

K {\bar{Φ}}_{x, m} = b

K \frac{\partial {\bar{Φ}}_{x, m}}{\partial \bar{τ}} = \frac{\partial b}{\partial \bar{τ}}

\frac{\partial \bar{τ}}{\partial b} = K^{- 1} \frac{1}{\frac{\partial {\bar{Φ}}_{x, m}}{\partial \bar{τ}}}

K \hat{b} = {\hat{\bar{Φ}}}_{x, m}

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Optics Express