Computational optical biopsy

Open Access

Research

Computational optical biopsy

Yi Li*

1

_{, Ming Jiang}

2

_{and Ge Wang}

1,2

Address: 1_{Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA and} 2_{Bioluminescence Tomography Laboratory, Departments}

of Radiology and Biomedical Engineering, University of Iowa, Iowa City, IA 52242, USA Email: Yi Li* - [email protected]; Ming Jiang - [email protected]; Ge Wang - [email protected] * Corresponding author

Molecular imagingoptical imagingfluorescent imagingbioluminescent imaginginverse problemdiffusion approximation

Abstract

Optical molecular imaging is based on fluorescence or bioluminescence, and hindered by photon scattering in the tissue, especially in patient studies. Here we propose a computational optical biopsy (COB) approach to localize and quantify a light source deep inside a subject. In contrast to existing optical biopsy techniques, our scheme is to collect optical signals directly from a region of interest along one or multiple biopsy paths in a subject, and then compute features of an underlying light source distribution. In this paper, we formulate this inverse problem in the framework of diffusion approximation, demonstrate the solution uniqueness properties in two representative configurations, and obtain analytic solutions for reconstruction of both optical properties and source parameters.

Introduction

Gene therapy is a breakthrough in the modern medicine, which promises to cure diseases by modifying gene expression. A key for development of gene therapy is to

monitor the in vivo gene transfer and its efficacy in the

mouse model. Traditional biopsy methods are invasive, insensitive, inaccurate, inefficient, and limited in the extent. To map the distribution of the administered gene, reporter genes such as those producing luciferase are being used to generate light signals within a living mouse, which can be externally measured [1]. A cooled highly sensitive CCD camera has been built to take a 2D view of expression of the bioluminescent reporter luciferases. Such a 2D image of photon emission is then superim-posed onto a 2D visible light picture of the mouse for localization of the reporter gene activity. In addition to gene therapy, this new imaging tool has great potentials in

various other biomedical applications as well. An in vivo

bioluminescence tomography system integrated with an X-ray CT/micro-CT scanner is recently reported in [2,3]. The novel concept is to collect emitted photons from mul-tiple 3D directions with respect to a living mouse marked by bioluminescent reporter luciferases, and reconstruct an

internal bioluminescent source distribution based on both

the outgoing bioluminescent signals and the CT/micro-CT volume of the mouse. Then, the 3D bioluminescent source distribution and the corresponding CT/micro-CT volume are registered of anatomical and pathological structures, such as the lung and various tumors.

Optical imaging of small animals based on fluorescent/ bioluminescent probes promises great opportunities for translational research and eventually clinical applications, because fluorescent/ bioluminescent signals directly reveal molecular and cellular activities, and are sensitive, specific, non-ionizing, non-invasive and cost-effective.

Published: 14 June 2005

BioMedical Engineering OnLine 2005, 4:36 doi:10.1186/1475-925X-4-36

Received: 01 April 2005 Accepted: 14 June 2005

This article is available from: http://www.biomedical-engineering-online.com/content/4/1/36

© 2005 Li et al; licensee BioMed Central Ltd.

Pure optical imaging cannot detect the molecular activi-ties triggered by biomarkers because the light generated are generally out of the visible spectrum. Despite the progress in optical molecular imaging of small animals, little research has been done for optical molecular imag-ing of patients. A light source induced by either fluores-cence or bioluminesfluores-cence probes is usually weak, and would be often deep inside a body should it be used in patients. Optical methods for in vivo imaging are all faced with the problem of limited transmission of light through tissues [[1], p. 237]. Because the human body absorbs and scatters photons in the visible and near infrared ranges with the mean-free-path in the sub-millimeter domain, such a source cannot be effectively detected on the body surface [4].

In this paper, we propose a computational optical biopsy (COB) method [5] to supplement and enhance the capa-bilities of fluorescent molecular tomography and biolu-minescence tomographic, especially for their potential uses in patients. In order to detect the light source in a region of interest deep inside a subject, we can use a fiber-optical probe to detect the light source directly in the sub-ject along one or multiple biopsy paths and next to com-pute the parameters or features of the embedded light source.

Several optical biopsy needle systems are already in oper-ation [6-8], which have indicated the physical feasibility of this COB project. While similar to existing optical biopsy procedures in using fiber-optic probes [6-8], the proposed COB system and methods depend on not only optical devices but also advanced modeling and computa-tion techniques to reconstruct an underlying source inten-sity distribution and extract its features of interest such as source center and effective intensity. There are several dis-tinctions that substantiate our innovations. Our COB approach relies on sophisticated signal modeling and esti-mation from data collected along a number of biopsy paths, while other biopsy/endoscopic techniques perform direct anatomical imaging on speci c spots only. Our COB targets source intensity distributions triggered by probes instead of tissue/vascular properties that are concerned by other biopsy/endoscopic methods. Our COB intends to sense both fluorescent and bioluminescent sources, not just fluorescent sources as some optical endoscopic/spec-troscopic techniques are designed for.

Mathematically we will Consider

where u₀ is the average photon flux in all directions,

, µ_αare positive constants with µ_a

and µs being the absorption and scattering constants

respectively in and f is either a measure or a L∞∩L1

function, which represents the light source.

Single point source

We will first consider the case where the source term

, where δis the Dirac operator.

In this case we have

where C is a constant such that

or

i.e.,

Assume that the needle insertion follows a straight line l :

, where is the direction of the insertion

and one point along the insertion (Figure 1).

Let υbe a direction that needle tip detects light from. Then

the measurement along the line l is given by

and = (t₁, t₂, t₃). υ= (υ1, υ2, υ3) which has a fixed angle θwith the needle direction , i.e., cos (θ) = t1υ1 + t2υ2 +

t3υ3

−

(

∇

)

+ =

( )

∞

( )

   

div D in

= 0,

3

u u f

u

0 0

0

1

µα \

D x

( )

+

(

)

= 1

3 µ_α µ’_s

\3

f =

∑

_iN₌₁λ δ_{i x}i x

( )

u x

x x x x

D

i

eff i

eff

0 i 2

i=1 N

= C

exp

( )

⋅

−

(

−

)

( )

=

∑

λ

µ

µ µα

,

lim

exp ,

]

ε→ + =ε − ∂_∂

(

µ

)



  



   

  



 

 =

( )

∫

0

1 3

C

eff

[ _v

D

x x dS

x x

C D

= 1

( )

4π , 4

u x

D _{x x x x}

i

i i

i N

0

1

1

4 5

( )

=

−

(

−

)

( )

=

∑

π

λ µ

exp _eff

.

G G G

x s

( )

= +0 st Gt

G 0

m s D u

v

D u

s

x st ( )

( ) = − ⋅ ∂

∂

= − ⋅ ∇ ⋅ _{= +}

0

0 0

6 ν G JG G

G

such that

Due to the notation and translation invariant properties of (1) and the design of the insertion needle, we may assume, without loss of generality, that

so that υ1 = 0, (υ2υ3) = (cos α, sin α) and .

Then (7) becomes

Now let .

Assuming that each insertion line can detect sources from

two different angles α1, α2 such that α1 - α2 ≠kπ, k ∈ ± ,

then

is non-singular. Therefore appropriate combination α1

and α2 depending on γof the measurements m(s) would

single out , so that without loss of generality, we

may assume that

Prinicpal of computational optical biopsy

Figure 1

A needle insertion path, where t is the direction of the insertion, 0 one point along the insertion, v a direction in which the

nee-dle tip detects light, s₁; s₂ are points along the path

m s

x x

x x x x

x x v

i i

i i

i N

i

( )=

(

+ −

)

−

(

−

)

< − > (

=

∑

1 4

1

7

3 1 π

µ λ

µ eff

eff

exp

, ))

G

x s

( )

=s

(

1 0 0

)

=

( )

2

, , and θ π. 8

xi =

(

x x x₁i, i₂, ₃i

)

m s

s x x x x x

i i i i i i

( )=

+

(

−

)

+ +

 

 _ − −

1 4

1 1

2

2 3 2 3

2 2

π

λ µeff cos α siin

exp eff

α

µ

(

)

−

(

)

+ + 







(

−

)

+ +

s x1i xi xi s xi xi x

2 2 3

3 2

1 2

2 3

2 2 / 2 _ii

i N

2

1

9

 

 _

⋅ ( )

=

∑

wi = x₂i2 +x₃i2

`

cos cos

sin sin

α α

α α

1

1 2

2

 

 

 

which are the measurements such that <x - xi_, υ_{> = , k}

= 2, 3 after combinations of α1 and α2 mentioned above

and define

Theorem 2.1

(Single point source) If N = 1, i.e.: if

, then one

inser-tion would uniquely determine λ, x1_{, and} µ

eff, provided that the

insertion line does not go through the point of the source. (See remark.)

Proof

In this case since

Differentiate (13) with respect to s, we have

such that the critical point of (s) uniquely identifies x₁.

Next with x₁ identified, we obtain

or

And by taking the derivatives of and evaluating them at

x₁, we obtain that

where z = µeffw.

From (18) we obtain

and plugging it into (19) we have

If f (z) is monotone in z > 0, then (21) would give us a

unique solution z > 0.

We compute and obtain

i.e., we have uniquely solved by the information from

the values of µeff, w and therefore λby (17).

Next if we let x₂ = w cos β, x₃ = w sin β, then we get

which could uniquely determine βby the sign conditions

of , k = 2.3.

That is, by now, we are able to determine, (x₁, x₂, x₃), µeff

and λ; i.e. complete information about the source.

Remark

In case that the insertion line goes through the point of the source, which is verified by having each

m s

x s x w

s x w

k

i ki i i

i i ( )= +

(

−

)

+       −

(

)

+     1 4 1 1 2 1 2 3 2 2 π

λ µeff // , , , , 2 1 2 2 10 2 3 ⋅ 

(

−

)

+   _ ( ) = ∈ −∞ ∞( )    = ∑

exp µeff s x w

k s i i i N      

x_ki

m_kA≡ m_k

( )

s s dsA A "

( )

−∞ ∞

∫

, for integer = 0,1,2, 11

u x

D x x x x

0 _{1 1}

1 4

( )

= −

(

−

)

π λ µ

exp _eff

m s

x s x w

s x w

k k ( )= + ( − ) +     − ( ) +    1 4 1 1 2 2

12 2 3 2 π λ µ µ eff e exp /

fff s x w

s − ( ) +    ∈ −∞ ∞( ) ( )

12 2

12 , , ,

m s m s m s

w s x w

s x ( )≡ ( )+ ( )= + ( − ) +    − ( ) 2 2 3 2 2 2 2

12 2 2 12 1 16 1 π

λ µeff

++ 

   ( − ) + 

( )

w23 s x12 w2

13 exp 2µeff

.

m s’

( )

( )14

= − + ( − ) + 

 

(

( − ) +

)

+ + ( −

λ2 2 µ µ µ

12 2 2 12 2 1

1 3 3

w eff s x w eff x x w eff s x)) +

  − ( ) +     ( − ) +  2 2 2

12 2 4

12 2

8 2

w

s x w s x w

s

π exp µ

.

eff

−−

( x1) ( )15

m

m x w w

w w

1

2 2 2

2 6 1 16 16

( )

=

(

+

)

⋅

(

)

( )

λ µ π µ eff eff exp 2

λ π µ

µ

2 2 4

2 1 16 1 17 =

(

)

+

(

)

( )

( )

w w w m x

exp 2 _eff

eff

m

m x

m x b

z z z w " , 1 1 2 2

2 3 3

1 18

( )

( )

≡ = + +

(

)

+

(

)

( )

m x

m x a

z z z z

z w 4 1 1 3 2 2 4

6 2 2 9 18 12

1 19 ( )

_{( )}

( )

≡ = +

(

)

(

+ + +

)

+

(

)

( )

w z z z b =

(

+ +

)

+

(

)

( )

2 3 3

1 20

2

2 3

2 2 9 18 12

3 3 21 2 3 2 2 2 a b

z z z z

z z f z =

(

+

)

(

+ + −

)

+ +

(

)

=

( )

.

( )

′

( )

= −

(

+ + +

)

+ +

(

)

<

f z

z z z z

z z

3 2

2

9 27 24

2 3 3

0,

m

tanβ =

( )

( )

m s m s 3 0 2 0

{some single point}

then only the two components of the point source, orthogonal to the insertion can be identified, which is the best possible. On the other hand, in practice, such events would have probability zero!

Single ball source

Next we consider ball sources, i.e., we assume that

where χΩ is the characteristic function of Ω and

.

For such source we have where

where

– the total intensity of the

source in the form of λχ_{B(o, r)}. (24)

Again we will discuss the single ball source first, i.e. N = 1,

so that we have

Theorem 3.1

(Single ball source) If N = 1 ; i.e. if u₀ is given by (25), then one insertion would uniquely determine M (λ3, r1_{), x}1 _and µ

eff;

in case the line stays outside of B (x1_{, r}1_{). In case the insertion}

line enters the interior of B (x1_{, r}1_{), then x}1_{, r}1_, λ1 _and µ

eff are

uniquely determined by one insertion.

Proof Case 1

If the insertion line (refer to Theorem 1.1 in Section 2)

stays outside B (x1_{, r}1_{), then the detected u}

0 is given by

i.e., u₀ (x) behaves exactly as a single point source with an

intensity of M (λ1, r1_{) and hence Theorem 2.1 guarantees}

the result.

Case 2

If the insertion line enters the support of the ball source B

(x1_{, r}1_{) but misses the center point x}1_{, then again due to}

the rotation and translation invariant properties of (1), we may assume, without loss of generality that the insertion

is given by (8) and x1 _{= (x}

1, x2, x3) and hence the detection

given by (6) is now

m (s) (27)

where .

Again, assuming that each insertion line can detect

sources from two different angles α1, α2 such that α1 - α2 ≠

kπ, k ∈ ± , then

is non-singular. Therefore appropriate combination of the

two measurements depending only on α1 and α2 would

single out and , so that without loss of generality we

may assume that

m_k (s) = φ(s), where

If we let x₂ = w cos β, w₃ = w sin β, then

which would uniquely determine βby the above question

and the sign conditions if , k = 2.3.

Define

m s_k( )=0, _s \\

f _{i B x r} x

i N

i i

=

_{( )}

( )

( )

=

∑

λ χ _, , 22

1

B y r( , ) {= x \3:|x y− <| r}

u x r

r x x x x i a eff i eff i eff i eff i 0 1 1 ( )= − +

(

)

⋅ −

(

)

−     λ µ µ µ µ µ exp sinh            ( ) +

( )

⋅ − − ( ) = ∑ χ λ π _µ

B x r i

N

i i _i

eff

i i x

M r

D _{x x x x}

, , exp 1 1 4 1

ii B x ri i x

(

)

⋅ − ( )           ( ) ( ) 1 23

χ _, ,

M

( )

λ,r ≡

∫

rλτ ϕ τ τ2

( )

d

0

u x r

r x x x x a 0 1 1 1 1 1 1 1 ( )= − +

(

)

−

(

)

−      λ µ µ µ µ µ eff eff eff eff exp sinh      ( ) +

( )

⋅ −

(

−

)

− ( ) χ λ

π _µ χ

B x r

B x r

x

M r

D _{x x x x}

1 1

1

1 1 _{1 1}

1 4 1 1 , , , exp eff 1 1 25 ( )( )       ( ) x .

u x M r

D x x _eff x x

0 1 1 _{1 1}

1 4 1 26

( )

=

( )

⋅ −

(

−

)

( )

λ π _µ , exp , = − ∂ ∂ = − ∇ ⋅ = ( + ) ( − ) + = + D D

r s x w

S x st

υ

ν υ ν

λ µ µ µ

0 0 1

12 2

0

1

G GG

eff eff cosh efff eff effee

s−x w s x w

( ) +   −  ( − ) +     

12 2 12 2 3

sinhµ µ µfff

eff

r

B

s x w

x x M r

s

1

12 2 3 2 2 1 3 1 1 1 1 − ( ) +

(

)

× − − + ( ) + − /

( cosα sin )α χ λ, µ xx w x x x

s x w s

12 2 21 31 12 2

3 2 4 ( ) +   − − − ( ) +

(

)

( cos sin ) exp

/

α π µeff ( −− ) +

  −

( )

x w

B

12 2

1

28

χ

( )

w= x₂2+x₃2

` cos cos sin sin α α α α 1 1 2 2      

x1₂ x1₃

x1_k

= +

(

)

( − ) +  ( − ) +

 −

λ1 µeffr1 µeff s x12 w2cosh µeff s x12 w2 sinh µefff effeeff

s x w

s x w

M r B − ( ) +        − ( ) +

(

)

× +

12 2

3

12 2 3 2 1 1 µ χ λ µ / ,, exp / r

s x w

s x w s x

1 1

2 2

12 2 3 2 1 1 4

( )

+ ( − ) +   − ( ) +

(

)

( − ) µ π µ eff

eff 22 2

1 +    − ( )⋅ w B χ

tanβ =

( )

,

( )

( )

m s m s 3 0 2 0 29

(s) = m₂ (s)2 _{+ m}

3 (s)2 = w2φ2 (s). (30)

Note that since in this case we assume that the insertion

line enters the interior of B (x1_{, r}1_{) but misses the center}

x1_{, we have 0 <w <r}1 _{and the line would enter and leave B}

(x1_{, r}1_{) when}

which is exactly where our measurement M_k (s), or

equiv-alently φ(s) fails to be differentiable. Therefore let sin and

s_out be two points where we observe the jump

discontinu-ity of 1 _(s).

Then we have

(32) together with (29) uniquely identify x₁ and establish

a relation between r1 and w. Then we are able to evaluate

(x1) to get

i.e.,

Next by taking the derivatives of and evaluating them

at x₁, we obtain that

>From (35) we obtain

and plugging it into (36) we get

which can be veri ed to be an increasing function.

There-fore (38) has a unique solution z and (37) then uniquely

de nes the w which in turn defines λ1 uniquely by (34) and

r1 _{uniquely by (32).}

Discussions and conclusion

We have demonstrated the modeling of computational optical biopsy with the diffusion optics, the solution uniqueness properties in two typical configurations and provide explicit formulas for the reconstruction of both optical properties and source parameters. Mathematically, one single insertion will be enough to estimate the above parameters. However, physically, more measurements will guarantees the robustness of the estimates. The needle trajectory can also be dynamically restarted and opti-mized towards the center of the source based on the suc-cessive estimate. Further investigation on multiple point sources and multiple ball sources is undergoing. For example the double sources probelm can be handled in such a way that by using the moments defined (11) one could reduce the problem to a equivelent one with a few freedom and thus a numerical optimization technique will be used to solve the inverse problems.

Contributions

The three authors made about equal contributions in this work.

6 Acknowledgements

This work is partially supported by the National Institutes of Health (EB001685 and EB002667).

References

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29(17):2034-2036. m

s x w r

s x r w

−

(

)

+ =

= ± −

( )

  



1

2 2 1

1 1 2

2 31

or

m

x s s

r w s s

in out

out in

1

2

2

2

32

1 2 2

= +

= + −



  

 

  

( )

.

m

m x w x

w r w w w

1 2 2 1

2

12 1

2

1 ( )= ⋅

( )

=

(

+

)

( )−

ϕ

λ µeff µeff cosh µeff sinh((µeff )

(

)

( )

2

6 1 6

2

33 µeffexp µeff

,

r w

λ µ ω µ

µ µ ω µ ω µ

1 2

1 4 6

1 2

2 1

=

(

)

⋅ ⋅

+

(

)

(

)

−

exp

cosh sinh

eff eff

eff eff eff

r

r eeff

m x

ω

(

)



 

( )⋅ _{( )}

2 1 34

m

m x

m x b

z z z z z

z z z z

" _{sinh cosh sinh}

cosh cosh

1 1

2

3 3

( )

( )

≡ = ω

( )

(

−

( )

+

( )

+

(( )

)

⋅

( )

( )

35

m x

m x a

z z z z z z z

4 1 1

2 3

3 15 15 6

( )_{( )}

( ) ≡ =

− sinh( )+ cosh( )− sinh( )+ cosh( )) 

 

− ( )+ ( )

( )⋅ ( )

w sinh z zcosh z 36

w z z z z z

b z z z

=

( )

−

( )

+

( )

−

( )

+

( )

(

)

( )

3 3

37 2

sinh cosh sinh

sinh cosh

a b

z z z z z z z z

3

15 15 6

2

2

=−sinh( )+ cosh( ) − sinh( )+ cosh( )− sinh( )++ ( )

( )− ( )+ ( )



 

( )

= ( )

z z

z z z z z

f z

3

2 2

3 3

38 cosh

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7. Li XD, Chudoba C, et al.: Imaging needle for optical coherence tomography. Optics Letters 2000, 25(20):1520-1522.