Time Domain Fluorescent Diffuse Optical Tomography: analytical expressions

S. Lam; F. Lesage; X. Intes

doi:10.1364/OPEX.13.002263

Optics Express
Vol. 13,
Issue 7,
pp. 2263-2275
(2005)
•https://doi.org/10.1364/OPEX.13.002263

Time Domain Fluorescent Diffuse Optical Tomography: analytical expressions

S. Lam, F. Lesage, and X. Intes

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S. Lam, F. Lesage, and X. Intes, "Time Domain Fluorescent Diffuse Optical Tomography: analytical expressions," Opt. Express 13, 2263-2275 (2005)

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Abstract

Light propagation in tissue is known to be favored in the Near Infrared spectral range. Capitalizing on this fact, new classes of molecular contrast agents are engineered to fluoresce in the Near Infrared. The potential of these new agents is vast as it allows tracking non-invasively and quantitatively specific molecular events in-vivo. However, to monitor the bio-distribution of such compounds in thick tissue proper physical models of light propagation are necessary. To recover 3D concentrations of the compound distribution, it is necessary to perform a model based inverse problem: Diffuse Optical Tomography. In this work, we focus on Fluorescent Diffuse Optical Tomography expressed within the normalized Born approach. More precisely, we investigate the performance of Fluorescent Diffuse Optical Tomography in the case of time resolved measurements. The different moments of the time point spread function were analytically derived to construct the forward model. The derivation was performed from the zero order moment to the second order moment. This new forward model approach was validated with simulations based on relevant configurations. Enhanced performance of Fluorescent Diffuse Optical Tomography was achieved using these new analytical solutions when compared to the current formulations.

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Fig. 1. Typical TPSF and respective moments. The IFS curve corresponds to a typical instrument response function.

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Fig. 2. Configuration used for the simulations herein. The source (detectors) locations are depicted by red (blue) dots

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Fig. 3. Results of repartition of energy, meantimes and variance of 1,000 randomly generated noised TPSF.

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Fig. 4. Example of sensitivity matrices. a) and b) correspond respectively to

m_{0}^{λ 2}

(r _s,r _d) and

m_{0}^{λ 2}

(r _s,r _d)·

m_{2}^{λ 2}

(r _s,r _d) for a 6cm thick slab with source-detector facing each other and a 0.1 µM background of Cy 7. c) and d) correspond to the same parameters for a 0.1 µM background of Cy 5.5. Last, e) and f) correspond to the same parameters for a 0.1 µM background of Cy 3B.

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Fig. 5. Reconstruction from synthetic data for Cy 7: a) 0th moment only, b) 0^th, 1^st and 2^nd moments; Cy 5.5 : c) 0^th moment only, d) 0^th, 1^st and 2^nd moments; and Cy 3B: e) 0^th moment only, f) 0^th, 1^st and 2^nd moments. The quantitative values are expressed in µM.

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Fig. 6. Reconstruction from synthetic data for Cy 5.5 using all three moments noisy data.

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Tables (3)

Table 1. Parameters used in the simulations

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Table 2. Fluorochrome investigated herein.

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Table 3. Noise model used. The standard deviations are expressed in percent of the mean.

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Equations (14)

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\frac{1}{v} \frac{\partial}{\partial t} Φ (r, t) - D Δ^{2} Φ (r, t) + μ_{a} Φ (r, t) = S (r, t)

\frac{\partial}{\partial t} N_{ex} (r, t) = - \frac{1}{τ} N_{ex} (r, t) + σ \cdot Φ^{λ 1} (r, t) [N_{tot} (r, t) - 2 N_{ex} (r, t)]

\frac{N_{ex} (r, ω)}{τ} = \frac{σ \cdot N_{tot} (r)}{1 - i ω τ} \cdot Φ^{λ 1} (r, ω)

Φ^{λ 2} (r_{s}, r_{d}, ω) = η ∭_{volume} N_{ex} (r, ω) \cdot Φ^{λ 2} (r, r_{d}, ω) \cdot d^{3} r

Φ^{λ 2} (r_{s}, r_{d}, ω) = ∭_{volume} Φ^{λ 1} (r_{s}, r, ω) \cdot \frac{Q_{eff} \cdot N_{tot} (r)}{1 - i ω τ} \cdot Φ^{λ 2} (r, r_{d}, ω) \cdot d^{3} r

\frac{Φ^{λ 2} (r_{s}, r_{d}, ω)}{Φ_{0}^{λ 1} (r_{s}, r_{d}, ω)} = \frac{1}{Φ_{0}^{λ 1} (r_{s}, r_{d}, ω)} ∭_{volume} Φ_{0}^{λ 1} (r_{s}, r, ω) \cdot \frac{Q_{eff} \cdot C_{tot} (r)}{1 - i ω τ} \cdot Φ_{0}^{λ 2} (r, r_{d}, ω) \cdot d^{3} r

\frac{Φ^{λ 2} (r_{s}, r_{d}, ω)}{Φ_{0}^{λ 1} (r_{s}, r_{d}, ω)} = \frac{D^{λ 1}}{G^{λ 1} (r_{s}, r_{d}, ω)} \sum_{voxels} \frac{1}{D^{λ 1}} G^{λ 1} (r_{s}, r_{v}, ω) \cdot \frac{Q_{eff} \cdot C_{tot} (r_{v})}{1 - i ω τ} \cdot \frac{1}{D^{λ 2}} G^{λ 2} (r_{v}, r_{d}, ω) \cdot h^{3}

m_{k} = 〈 t^{k} 〉 = \int_{- \infty}^{+ \infty} t^{k} \cdot p (t) dt ⁄ \int_{- \infty}^{+ \infty} p (t) dt

m_{0}^{λ 2} (r_{s}, r_{d}) = Φ_{N}^{λ 2} (r_{s}, r_{d}, ω = 0) = \sum_{voxels} \frac{G^{λ 1} (r_{s}, r_{v}, ω = 0) \cdot G^{λ 2} (r_{v}, r_{d}, ω = 0)}{G^{λ 1} (r_{s}, r_{d}, ω = 0)} \times \frac{Q_{eff} h^{3}}{D^{λ 2}} \times C_{tot} (r_{v})

m_{0}^{λ 2} (r_{s}, r_{d}) \times m_{1}^{λ 2} (r_{s}, r_{d}) = \sum_{voxels} {\begin{matrix} (τ + \frac{∣ r_{s} - r_{v} ∣ + ∣ r_{s} - r_{d} ∣}{2 . v \sqrt{μ_{a}^{λ} D^{λ}}} - \frac{∣ r_{v} - r_{d} ∣}{2 . v \sqrt{μ_{a} D^{λ}}}) \times \\ \frac{G^{λ} (r_{s}, r_{v}, ω = 0) \cdot G^{λ} (r_{v}, r_{d}, ω = 0)}{G^{λ} (r_{s}, r_{d}, ω = 0)} \times \frac{Q_{eff} h^{3}}{D^{λ}} \times C_{tot} (r_{v}) \end{matrix}}

m_{0}^{λ 2} (r_{s}, r_{d}) \cdot m_{2}^{λ 2} (r_{2}, r_{d}) = \sum_{voxels} {\begin{matrix} (\begin{matrix} τ^{2} + \frac{∣ r_{s} - r_{v} ∣ + ∣ r_{s} - r_{d} ∣}{4 . v^{2} μ_{a}^{λ} \sqrt{μ_{a}^{λ} D^{λ}}} + \frac{∣ r_{v} - r_{d} ∣}{4 . v^{2} μ_{a}^{λ} \sqrt{μ_{a}^{λ} D^{λ}}} \\ + {τ + \frac{∣ r_{s} - r_{v} ∣}{2 . v \sqrt{μ_{a}^{λ} D^{λ}}} + \frac{∣ r_{v} - r_{d} ∣}{2 . v \sqrt{μ_{a}^{λ} D^{λ}}}}^{2} - t^{- λ 2} (r_{s}, r_{d}) \cdot {τ + \frac{∣ r_{s} - r_{v} ∣ + ∣ r_{v} - r_{d} ∣}{2 . v \sqrt{μ_{a}^{λ} D^{λ}}} - \frac{∣ r_{v} - r_{d} ∣}{2 . v \sqrt{μ_{a}^{λ} D^{λ}}}} \end{matrix}) \\ \times \frac{G^{λ} (r_{s}, r_{v}, ω = 0) \cdot G^{λ} (r_{v}, r_{d}, ω = 0)}{G^{λ} (r_{s}, r_{d}, ω = 0)} \times \frac{Q_{eff} h^{2}}{D^{λ}} \times C_{tot} (r_{v}) \end{matrix}}

∣ \begin{matrix} m_{0}^{λ 2} (r_{s 1}, r_{d 1}) \\ ⋮ \\ m_{0}^{λ 2} (r_{sm}, r_{dm}) \\ m_{0}^{λ 2} (r_{s 1}, r_{d 1}) \cdot m_{1}^{λ 2} (r_{s 1}, r_{d 1}) \\ ⋮ \\ m_{0}^{λ 2} (r_{sm}, r_{dm}) \cdot m_{1}^{λ 2} (r_{sm}, r_{dm}) \\ m_{0}^{λ 2} (r_{s 1}, r_{d 1}) \cdot m_{2}^{λ 2} (r_{s 1}, r_{d 1}) \\ ⋮ \\ m_{0}^{λ 2} (r_{sm}, r_{dm}) \cdot m_{2}^{λ 2} (r_{sm}, r_{dm}) \end{matrix} ∣ = ⋮ ∣ \begin{matrix} W_{11}^{m_{0}^{λ 2}} & \dots & W_{\ln}^{m_{0}^{λ 2}} \\ ⋮ & ⋱ & ⋮ \\ W_{ml}^{m_{0}^{λ 2}} & \dots & W_{mn}^{m_{0}^{λ 2}} \\ W_{11}^{m_{0}^{λ 2} \cdot m_{1}^{λ 2}} & \dots & W_{\ln}^{m_{0}^{λ 2} \cdot m_{1}^{λ 2}} \\ ⋱ & ⋮ \\ W_{m 1}^{m_{0}^{λ 2} \cdot m_{1}^{λ 2}} & \dots & W_{mn}^{m_{0}^{λ 2} \cdot m_{1}^{λ 2}} \\ W_{11}^{m_{0}^{λ 2} \cdot m_{2}^{λ 2}} & \dots & W_{\ln}^{m_{0}^{λ 2} \cdot m_{2}^{λ 2}} \\ ⋮ & ⋱ & ⋮ \\ W_{m 1}^{m_{0}^{λ 2} \cdot m_{2}^{λ 2}} & \dots & W_{mn}^{m_{0}^{λ 2} \cdot m_{2}^{λ 2}} \end{matrix} ∣ \cdot ∣ \begin{matrix} C_{tot} (r_{vl}) \\ ⋮ \\ ⋮ \\ ⋮ \\ ⋮ \\ ⋮ \\ ⋮ \\ ⋮ \\ ⋮ \\ C_{tot} (r_{vn}) \end{matrix} ∣

b = A \cdot x

x_{j}^{(k + 1)} = x_{j}^{(k)} + ξ \frac{b_{i} - \sum_{i} a_{ij} x_{j}^{(k)}}{\sum_{i} a_{ij} a_{ij}} \sum_{i} a_{ij}

$µ_{a}^{λ 1}$ (cm^-1)	0.06	Dimensions (cm)	9×6×9
$µ_{a}^{λ 2}$ (cm^-1)	0.06	C_background (µM)	0.1
µ $'_{a}^{λ 2}$ (cm^-1)	10.00	C_inclusion (µM)	1.0
µ $'_{a}^{λ 2}$ (cm^-1)	10.00	Voxel size (cm)	0.36×0.3×0.36

Compound	τ(ns)	ε (cm^-1.M^-1)	η(%)
Cy 7	<0.3	200 000	0.28
Cy 5.5	1.0	190 000	0.23
Cy 3-B	2.8	130 000	0.67

Measure	Energy	Meantime	Variance
σ(%)	2	0.2	2

$µ_{a}^{λ 1}$ (cm^-1)	0.06	Dimensions (cm)	9×6×9
$µ_{a}^{λ 2}$ (cm^-1)	0.06	C_background (µM)	0.1
µ $'_{a}^{λ 2}$ (cm^-1)	10.00	C_inclusion (µM)	1.0
µ $'_{a}^{λ 2}$ (cm^-1)	10.00	Voxel size (cm)	0.36×0.3×0.36

Compound	τ(ns)	ε (cm^-1.M^-1)	η(%)
Cy 7	<0.3	200 000	0.28
Cy 5.5	1.0	190 000	0.23
Cy 3-B	2.8	130 000	0.67

Abstract

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Tables (3)

Equations (14)

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