Time Domain Fluorescent Diffuse Optical ... - OSA Publishing

Time Domain Fluorescent Diffuse Optical Tomography: analytical expressions S. Lam, F. Lesage and X. Intes ART, Advanced Research Technologies Inc. 2300 Alfred Nobel, Saint-Laurent (QU) – H4S 2A4 – Canada [email protected]

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. ©2005 Optical Society of America OCIS codes: (170.5280) Photon migration; (170.3660) Light propagation in tissues; (170.5270) Photon density waves; (260.2510) Fluorescence; (170.6920) Time-resolved Imaging.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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Received 24 January 2005; revised 25 February 2005; accepted 15 March 2005

4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2263

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1. Introduction Optical techniques based on the Near-Infrared (NIR) spectral window have made significant progress in biomedical research in recent years. The relative low absorption and low scattering in the 600-1000 nm spectral range allow detection of photons that have traveled through several centimeters of biological tissue [1]. Coupled with accurate models of light propagation, NIR techniques enable imaging of deep tissue with boundary measurements using non-ionizing, low dose radiation. The interest in NIR techniques is fueled by the ability of the techniques to monitor functional tissue parameters such as oxy- and deoxy-hemoglobin [2,3] and the development of appropriate low cost instrumentation [4]. Based on these qualities, NIR optical imaging is expected to play a key role in breast cancer detection, characterization [6-11] and monitoring through therapy [12]; brain functional imaging [13,] and stroke monitoring [15,16]; muscle physiological and peripheral vascular disease imaging [17]. For all these applications, NIR techniques rely on endogenous contrast such as tissue hemodynamics. Another potential application of NIR techniques is to monitor exogenous contrast [18]. Especially, we see the emergence of an optical molecular imaging field that bears great promises in clinical applications [19]. NIR fluorescence optical imaging is rapidly evolving as a new modality to monitor functional and/or molecular events in either human or animal tissue. The developments of new contrast agents that target specific molecular events [20,21] are particularly promising. By specifically binding [23,24] or being activated in tumors [25], molecular probe detection can be achieved in the early stages of molecular changes prior to structural modification [26]. Moreover, the endogenous fluorescence in the NIR spectral window is weak leading to good fluorescence sensitivity [27]. As of today, NIR molecular imaging is confined to small animal models [28] and the translation to human imaging is foreseen as imminent. However, the technical problems encountered in imaging larger tissue volumes are challenging. Besides sensitive instrumentation, robust and accurate models for fluorescent light propagation are needed. Tomographic algorithms in the continuous mode [29] and in the frequency domain [30,31] have been proposed. Both numerical and analytical models exist and have been applied successfully to experimental data. In this work, we propose an analytical fluorescent diffuse optical algorithm extended to the time domain. For this purpose, we derive analytical solutions of the heterogeneous fluorescent diffusion equation for the 0th, the 1st and the 2nd moments of the fluorescent time point spread function (FTPSF). The formalism of the fluorescent normalized Born approximation [29] is used to obtain the mathematical expressions of the forward model employed for the diffuse optical tomography (DOT) procedure. The algorithm is tested with synthetic data mimicking relevant breast geometry. These simulations highlight the advantage of incorporating higher moments in the fluorescent DOT problem. 2. Theory In this section we describe the theoretical framework of light propagation in biological tissue and the specific theoretical frame we used to obtain the analytical solutions of the fluorescent time domain diffusion equation. 2.1. Light propagation in tissue Light propagation in tissue is well modeled by the diffusion equation. In the time domain the mathematical expression modeling light propagation in a homogenous medium is: 1 ∂ Φ (r, t ) − D∇ 2 Φ (r, t ) + µ a Φ(r, t ) = S(r, t ) v ∂t

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(1)



where Φ (r, t ) is the photon fluence rate, D is the diffusion coefficient expressed as D = 1 3µ 's with µ s' being the reduced scattering coefficient, µ a is the linear absorption

coefficient, v is the speed of light in the medium and S(r, t ) is the source term (assumed to be a δ function in our case). From Eq. (1), we can estimate the value of the field in each position in the investigated medium. In turn, the knowledge of the value of the field locally allows modeling accurately the reemission of a fluorescent field by endogenous or exogenous markers. Indeed, the fluorescent field is due to excited molecules that reemit photons at a constant wavelength. This phenomenon of reemission can be modeled as a source term embedded in the medium. The propagation from these sources to the detector is then modeled in the same frame as in Eq. (1). The temporal behavior of the excited fluorophore population at a given point is expressed by [32]: ∂ 1 N ex (r, t ) = − N ex (r, t ) + σ ⋅ Φ λ1 (r, t )[N tot (r, t ) − 2 N ex (r, t )] ∂t τ

(2)

where N ex (r, t ) is the concentration of excited molecules at position r and time t , N tot (r, t ) is the concentration of total molecules of fluorophores (excited or not), τ is the radiative lifetime of the fluorescent compound (sec. or nanoseconds.), σ is the absorption cross section of the fluorophore (cm2) and Φ λ1 (r, t ) is the photon fluence rate (Nb photons.s-1.cm-2) at the excitation wavelength λ 1 . Considering that the number of excited molecule is low compared to the total molecules and taking the Fourier transform yield the expression for the concentration of excited molecules: N ex (r, ω) σ ⋅ N tot (r ) λ1 = ⋅ Φ (r, ω) τ 1 − iωτ

(3)

where ω1 is the angular frequency associated with t. Then, the total fluorescent field is the sum of the contributions of all the secondary fluorescent sources over the entire volume. In the case of a point source located at rs , the fluorescent field detected at a position r d is modeled by: Φ λ 2 (rs , rd , ω) = η ∫∫∫ N ex (r, ω) ⋅ Φ λ 2 (r, rd , ω) ⋅ d 3r

(4)

volume

where Φ λ 2 (r, rd , ω) represent a propagation term of the fluorescent field from the element of volume at r to the detector position rd et the reemission wavelength λ 2 . Then, by using Eq. (4) we obtain the complete expression: Φ λ 2 (rs , rd , ω) = ∫∫∫ Φ λ1 (rs , r, ω) ⋅ volume

Q eff ⋅ N tot (r ) λ 2 ⋅ Φ (r, rd , ω) ⋅ d 3r 1 − iωτ

(5)

where Q eff = q ⋅ η.σ is the quantum efficiency, product of q the quenching factor, η the

quantum yield and σ the absorption cross section of the fluorophore, Φ λ1 (r, t ) is the photon fluence rate at the excitation wavelength λ1 and at time t, and τ is the radiative lifetime of the fluorescent compound. Note that the product σ ⋅ N tot (r ) corresponds to the absorption coefficient of the fluorochrome and can be expressed also as ε ⋅ C tot (r ) where ε is the extinction coefficient of the fluorophore (cm-1. Mol-1) and C tot (r ) the concentration of the fluorochrome (Mol) at a position r .

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2.2. Fluorescent moment analytical expression. Following the derivation of Eq. (5) performed by O’Leary [33], Ntziachristos and Weissleder [29] proposed an approach to fluorescent diffuse optical tomography less susceptible to system source-detector couplings and tissue inhomogeneities. They cast the forward model in the frame of the normalized first order Born approximation that is mathematically expressed as: Φλ2 (rs , rd , ω) Q ⋅ C (r ) 1 = λ1 Φ0λ1 (rs , r, ω) ⋅ eff tot ⋅ Φ0λ 2 (r, rd , ω) ⋅ d3r (6) λ1 ∫∫∫ 1 − iωτ Φ0 (rs , rd , ω) Φ0 (rs , rd , ω) volume The difference between Eq. (5) and (6) resides in the normalization achieved with the homogeneous excitation field reaching the detector. The gain attained here is that the left hand side can be determined purely from measurements thereby canceling any source-detector couplings. Following the expression of M. O’Leary [33], this expression is used to construct the forward model for DOT and then the analytical expression of the weight function is: Φ λ 2 (rs , rd , ω) Q ⋅ C (r ) 1 D λ1 1 λ1 = G (rs , rv , ω) ⋅ eff tot v ⋅ λ 2 G λ 2 (rv , rd , ω)⋅ h 3 (7) λ1 λ1 λ1 1 − iωτ Φ 0 (rs , rd , ω) G (rs , rd , ω) voxels D D (ik r −r ) λj e j1 2 is the system’s Green’s function with k 2j = − vµ aλj + iω D , where G λj (r1 , r2 , ω) = r1 − r2

∑

(

)

at the considered wavelength λ j ∈ [λ1 ,λ 2 ] . 10

0

Meas Fit IFS

Normalized counts

1st moment 2nd momenttN 10

10

t1

-1

0th moment

-2

0

1

2

3

4

5

6

7

8

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

The Eq. (7) is defined in the frequency domain. We propose to find similar analytical expressions in the time domain. Such analytical solutions for the absorption case have been proposed in the past for the 0th, 1st and 2nd moment of the TPSF [34]. The correspondence of these moments to the TPSF is illustrated in Fig. 1. The 0th moment corresponds to the integration of the counts (equivalent to the continuous mode), the 1st moment corresponds to the mean time of arrival of the photon and the 2nd moment to the variance of arrival of the photon. The normalized moments of order k >1 of a distribution function p(t ) are defined by [35]:

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+∞

+∞

−∞

−∞

m k = t k = ∫ t k ⋅ p(t )dt

()

∫ p t dt

(8)

In our notations, for k=0, we use the simple integration without division. We employed this formalism in the case of the normalized first order born approximation. Hence the normalized 0th moment is expressed as : m 0λ 2 (rs , rd ) = Φ λN2 (rs , rd , ω = 0 ) =

∑G

λ1

voxels

(rs , rv , ω = 0) ⋅ G λ 2 (rv , rd , ω = 0) × Q eff h 3 × C (r ) tot v G λ1 (rs , rd , ω = 0 ) Dλ2

(9)

This expression corresponds to Eq. (7) for the continuous mode. From now on, we assume that the excitation and emission wavelengths are similar ( λ1 = λ 2 = λ ), i.e. typically the shift incurred under the fluorescent process will not be significant and the same propagator describes the propagation of light for both excitation and emission. Then normalizing the 1st and the 2nd moment to this first moment yields the analytical solutions: Normalized 1st moment

⎫ ⎧⎛ r − r + rs − rd r −r ⎞ ⎪ ⎪⎜ τ + s v − v d ⎟× ⎜ λ λ λ ⎟ ⎪⎪ ⎪ ⎪ 2 . v D 2 . v D µ µ a a ⎠ m 0λ 2 (rs , rd ) ⋅ m1λ 2 (rs , rd ) = ⎬ ⎨⎝ 3 voxels ⎪ λ ⎪ G (rs , rv , ω = 0) ⋅ G λ (rv , rd , ω = 0) Q eff h × λ × C tot (rv )⎪ ⎪ λ ⎪⎭ G (rs , rd , ω = 0) D ⎪ ⎩

∑

(10)

Normalized 2nd moment ⎧⎛ rs − rv + rs − rd rv − rd ⎪⎜ τ 2 + + ⎪⎜ 4.v 2 µ λa µ λa D λ 4.v 2 µ λa µ λa D λ ⎪⎜ 2 ⎪⎪⎜ ⎧ ⎧ rs − rv rv − rd ⎫⎪ rs − rv + rv − rd rv − rd λ2 λ2 ⎪ λ2 m 0 (rs , rd )⋅ m 2 (rs , rd ) = ⎨⎜⎜ + ⎪⎨τ + + − ⎬ − t (rs , rd ) ⋅ ⎨τ + λ λ λ λ λ λ voxels⎪⎜ 2.v µ a D ⎪⎭ 2.v µ a D 2.v µ aλ D λ ⎪ ⎪ 2.v µ a D ⎩ ⎪⎝ ⎩ ⎪ G λ (r , r , ω = 0 ) ⋅ G λ (r , r , ω = 0 ) Q h 3 s v v d ⎪× × effλ × C tot (rv ) G λ (rs , rd , ω = 0 ) D ⎩⎪

∑

⎞⎫ ⎟⎪ ⎟⎪ ⎟⎪ ⎫ ⎟⎪ ⎪ ⎟⎪ ⎬ ⎟⎬ ⎟⎪ ⎪ ⎭ ⎠⎪ ⎪ ⎪ ⎪ ⎭

(11)

Where t λ 2 (rs , rd ) corresponds to the fluorescent mean time for the particular source-detector pair considered. For simplicity, we do not present in this paper the full derivation of these analytical solutions. The fluorescent DOT problem in the time domain is based on the analytical expression derived above and summarized in the set of linear equations: m λ0 2 (rs1 , rd1 )

λ2

W11m0

m λ0 2 (rs1 , rd1 ) ⋅ m λ2 2 (rs1 , rd1 )

m λ0 2 (rsm , rdm ) ⋅ m λ2 2 (rsm , rdm )

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C tot (rv1 )

λ2

W1mn0

m λ0 2 (rsm , rdm ) m λ0 2 (rs1 , rd1 ) ⋅ m1λ 2 (rs1 , rd1 ) m λ0 2 (rsm , rdm ) ⋅ m1λ 2 (rsm , rdm )

m 0λ 2 m1 λ 2 λ2 m 0 ⋅m1 11

W W =

λ2

m0 Wmn

m 0λ 2 ⋅m1λ 2

W1n

⋅

m 0λ 2 ⋅m1λ 2 m1 m 0λ 2 ⋅m 2λ 2 11

(12)

m 0λ 2 ⋅m1λ 2 mn m 0λ 2 ⋅m λ2 2 1n

W

W

W

W

λ2

Wmm10

⋅m 2λ 2

λ2

m0 Wmn

⋅m λ2 2

C tot (rvn )



λ2

λ2

λ2

λ2

λ2

where Wijm0 , Wijm0 ⋅m1 and Wijm0 ⋅m 2 , the weight function for the ith source-detector pair and the jth voxel are directly derived respectively from Eqs. (9), (10) and (11). In this inverse problem, the object function is defined as the fluorophore concentration. For the cases presented herein, we implemented boundary conditions using the extrapolated boundary conditions [36] and image sources. 2.3. Inverse problem Many different approaches exist to tackle the inverse problem. In this work we choose to employ the algebraic reconstruction technique (ART) due to its modest memory requirements for large inversion problems and the calculation speed it attains. Algebraic techniques are well known and broadly used in the biomedical community [37]. These techniques operate on a system of linear equations such as the ones seen in Eq. (12). We can rewrite Eq. (12) as: b = A⋅x

(13)

where b is a vector holding the measurements for each source-detector pair, A is the matrix of the forward model (weight matrix), and x is the vector of unknowns (object function). ART solves this linear system by sequentially projecting a solution estimate onto the hyperplanes defined by each row of the linear system. The technique is used in an iterative scheme and the projection at the end of the kth iteration becomes the estimate for the (k+1)th iteration. This projection process can be expressed mathematically as [38]: x (jk +1) = x (jk ) + ξ

∑a x a ∑a a ∑ (k )

bi −

ij

j

i

ij ij

ij

(14)

i

i

where x (jk ) is the kth estimate of jth element of the object function, b i the ith measurement, a ij the i-jth element of the weight matrix A and ξ the relaxation parameter. The relaxation parameter adjusts the projection step for each iteration. A small ξ value makes the inversion more robust but also slows down convergence. The selection of ξ is most of the time, done empirically [39]. We have chosen ξ =0.1 based on previous studies [40]. Also, a positive constraint was imposed on the object function. This hard constraint is adequate with fluorescent measurements as long as negative concentrations are unphysical. The stopping criterion of the iterative inversion algorithm was set a posteriori using the reconstructions of Fig. 5 for the Cy 5.5 case. The criterion retained was the number of iterations, which was fixed to 500 and selected using the merit functions described in [40]. 2.3. Simulations We tested the formulation derived in Section 2.2 with simulations. First we constructed a synthetic phantom with parameters relevant to the softly compressed human breast in dimension (6cm thickness) and for the optical endogenous properties. Second we simulated a homogeneous fluorochrome distribution over the volume with 1 cm3 heterogeneities exhibiting a contrast of 10 in concentration. The different parameters of the simulations are provided in Table 1. The fluorescent signal is dependent on the intrinsic characteristics of the fluorochrome employed. Simulations were carried out with three representative compounds: Cy 7, Cy 5.5 and Cy 3B. These fluorochromes were selected due to the span of lifetimes they do exhibit, which is characteristic of cyanine dyes [22]. The different properties of these fluorochromes are provided in Table 2.

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Table 1. Parameters used in the simulations

µ λa 1 (cm-1)

0.06

Dimensions (cm)

9x6x9

µ λa 2 (cm-1)

0.06

C background (µM)

0.1

10.00

C inclusion (µM)

1.0

10.00

Voxel size (cm)

0.36x0.3x0.36

µ 'sλ1 (cm-1) µ 'sλ 2 (cm-1)

a)

b)

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

The synthetic phantom was probed with a 25x25 constellation of source detectors. This constellation was distributed evenly 1.5 cm apart in both dimensions. The phantom configuration is provided in Fig. 2. Table 2. Fluorochrome investigated herein.

Compound

τ (ns)

ε (cm-1.M-1)

η (%)

Cy 7