Four-Dimensional Vascular Tree Reconstruction Using Moving Grid Deformation

Rationale and Objectives

Thinned perforator flaps have been widely used in plastic surgery for greater survivability and decreased morbidity. However, quantitative analysis of three-dimensional (3D) blood flow direction and location has not been examined yet. Such information will benefit and guide the surgical thinning and dissection process. Toward this goal, this study was performed for 3D vascular tree reconstruction with the incorporation of temporal contrast-agent propagation information (three spatial dimensions plus one temporal dimension; ie, 4D).

Materials and Methods

A novel computational framework by adopting a moving grid deformation method is presented. To take advantage of temporal information of the bolus propagating, a sequential segmentation procedure is proposed. Moreover, the temporal evolution of the vascular tree (4D vascular tree) is reconstructed during the procedure.

Results

Eight anterolateral thigh perforator flaps from eight cadavers were used for this study. The age range is 60–80 years old and the gender includes four males and four females. The 3D nature of the vascular structure and 4D vascular tree evolving process are showed in comparison with maximum intensity projection images.

Conclusion

The proposed computational framework demonstrates effectiveness in the modeling of 4D vascular tree. Furthermore, it reveals the ability to detect small vessel tree structures that are beyond the limit of image resolution.

Perforator flaps have been increasingly used over the past decade in plastic surgery for greater survivability and decreased morbidity ( ). The majority of anatomic vascular studies on perforator flaps have used lead oxide injections on cadaver flaps followed by two-dimensional (2D) projection radiography to determine vascular territories. Although lead oxide treated specimens provide excellent images, the three-dimensional (3D) nature of the vascular structure is not revealed. To overcome this deficit, static computed tomography (CT) scanning can be used to provide 3D mappings of contrast enhanced vascular anatomy in the sagittal, coronal, and transverse views. However, none of these methods can provide insight information on the direction and location of blood flow within each layer of a perforator flap. In this study, we present a four-dimensional (4D) (ie, three spatial dimensions plus one temporal dimension) time sequence of perforator flaps showing the process of contrast agent propagation. To our best knowledge, no study has been carried out reporting the examination of 4D anatomy of perforator flaps.

The computational techniques used in traditional 3D vasculature tree reconstruction ( ) can be roughly grouped into six categories ( ): 1) pattern recognition techniques; 2) model-based approaches; 3) tracking-based approaches; 4) artificial intelligence–based approaches; 5) neural network–based approaches; and 6) miscellaneous tubelike object detection approaches. Unfortunately, no single method works for any applications. The most widely used model-based methods in vessel reconstruction are the geometric deformable models. One problem with geometric deformable models is the high computational cost. To improve the efficiency and accuracy, adaptive grid techniques have been adopted ( ) to help solve the partial differential equations involved in geometric deformable models.

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Materials and methods

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CT Radiography

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Moving Grid Generation by Deformation Method

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Step 1

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divv→(xt,yt,t)=−∂∂t(1m(,t)), d

→

(

)

−

∂

(

)

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curlv→(xt,yt,t)=0, c

→

(

)

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v→(xt,yt,t)⋅n→=0,for(xt,yt)∈∂Ω(t), v

→

(

)

⋅

→

(

)

∈

∂

(

)

where ∂Ω(t) ∂

(

) denotes the boundary of domain Ω(t) Ω

(

) . The Neumann boundary condition specifies the values the derivatives of a solution is to take on the boundary of the domain. In an image domain, the boundary constraint is zero which means the boundary points cannot move out or move inside to the image domain, but only move along the boundary. This is reflected as the zero vector field in the normal direction as shown in Eq 3 .

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divv→(x,y,t)=∇⋅v→(x,y,t), d

→

(

)

∇

⋅

→

(

)

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curlv→(x,y,t)=∇×v→(x,y,t). c

→

(

)

∇

→

(

)

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Step 2

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∂∂t(x,y,t)=m((x,y,t),t)v→((x,y,t),t), ∂

∂

(

)

(

)

→

(

)

where t∈[0,T] t

∈

[

] .

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∫Ω(t)(−∂∂t(1m(,t)))dxdy=0 ∫

(

)

(

−

∂

(

)

for all t . Furthermore, the monitor function m(,t) m

(

) should be normalized under the following condition,

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∫Ω(t)1m(,t)dxdy=|Ω(0)|,forallt, ∫

(

)

(

)

(

)

for

all

where | Ω(0) | means the area over the domain Ω at time t = 0. The mathematical proof of the deformation method can be found elsewhere ( ).

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Vascular Tree Reconstruction using Moving Grid with Deformation

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Initial segmentation

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Vascular grid plot generation

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m0()={0.3,whenpoint(x,y)belongstoinitialsegmentedvesselregion;1.0,else m

(

)

{

0.3

when

point

(

)

belongs

initial

segmented

vessel

region

;

1.0

else

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m(,t)=1−tT+tT⋅m0(), m

(

)

−

⋅

(

)

where t∈[0,T] t

∈

[

] is the dummy time variable used in the grid deformation process same as the one occurred in Eq 1–7 . By this definition, we have m(,0)=1 m

(

)

1 and m(,T)=m0() m

(

)

(

) . This implies the grid is deformed from uniform to the desired one in which the size of a grid cell is governed by the objective monitor function m0() m

(

) . The normalization of m(,t) m

(

) is carried out using Eq 7 for all t∈[0,T] t

∈

[

] .

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Vascular vessel extraction

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v→T=<xT−x0,yT−y0>T, v

→

−

where (x0,y0) (

) is the initial uniform grid, and (xT,yT) (

) is the grid plot using the deformation method.

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F(xT,yT)=∇⋅v→T. F

(

)

∇

⋅

→

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Results

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Demonstration of the Proposed Method

Initial vascular tree segmentation

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Grid plot generation

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Vascular vessel extraction

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4D vascular tree evolution

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Results for Datasets ALT2-ALT8

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Discussion

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References

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4. Condurache A.P., Aach T., Grzybowski S., et. al.: Imaging and analysis of angiogennesis for skin transplantation by microangiography. Proc IEEE Conf Image Proc 2005; 2: pp. 1250-1253.
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11. Xue J., Liao G.: Least-squares finite element method on adaptive grid for PDEs with shocks. Numer Meth Part Diff Eq 2006; 22: pp. 114-127.
12. Zegeling P.A.: Moving grid techniques.Thompson J.F.Soni B.K.Weatherill N.P.1999.CRC Press LLCBoca Raton, FL:
13. Terzopoulos D., Vasilescu M.: Sampling and reconstruction with adaptive meshes. Proc Comp Vision Patt Recognition 1991; 1: pp. 70-75.
14. Vasilescu M., Terzopoulos D.: Adaptive meshes and shells. Proc Comp Vision Patt Recognition 1992; 1: pp. 829-832.
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16. Moser J.: Volume elements of a Riemann manifold. Trans AMS 1965; 120: pp. 286-294.
17. Semper B., Liao G.: A moving grid finite element method using grid deformation. Numer Meth Part Diff Eq 1995; 11: pp. 603.

Four-Dimensional Vascular Tree Reconstruction Using Moving Grid Deformation

Rationale and Objectives

Materials and Methods

Results

Conclusion

Materials and methods

CT Radiography

Moving Grid Generation by Deformation Method

Step 1

Step 2

Vascular Tree Reconstruction using Moving Grid with Deformation

Initial segmentation

Vascular grid plot generation

Vascular vessel extraction

Results

Demonstration of the Proposed Method

Initial vascular tree segmentation

Grid plot generation

Vascular vessel extraction

4D vascular tree evolution

Results for Datasets ALT2-ALT8

Discussion

References

Further Reading

Abstracts of Funded National Institutes of Health Grants (December)

Computer-Aided Diagnosis Scheme for Detection of Lacunar Infarcts on MR Images

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