Breast reduction is one of the most common procedures in breast surgery.  The aim of this work is to develop a computational model allowing one to forecast the final breast geometry according to the incision marking parameters. This model can be used in surgery simulators that provide preoperative planning and training, allowing the study of the errors origin in breast reduction

Clain, Stéphane

Machado, Gaspar J.

Pereira, Rui M. S.

Smirnov, Georgi

Universidade do Minho: RepositoriUM

11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)6th European Conference on Computational Fluid Dynamics (ECFD VI)E. On˜ate, J. Oliver and A. Huerta (Eds)SOFT TISSUE MODELLING FOR ANALYSIS OF ERRORSIN BREAST REDUCTION SURGERYSTE´PHANE CLAIN1, GASPAR J. MACHADO2, RUI M. S. PEREIRA3AND GEORGI SMIRNOV41 Centre of Mathematics, Department of Mathematics and Applications, University of Minho,Campus de Azure´m, Guimara˜es, Portugal, clain@math.uminho.pt2 Centre of Mathematics, Department of Mathematics and Applications, University of Minho,Campus de Azure´m, Guimara˜es, Portugal, gjm@math.uminho.pt3 Centre of Mathematics and collaborator with the Centre of Physics, Department ofMathematics and Applications, University of Minho, Campus de Azure´m, Guimara˜es,Portugal, rmp@math.uminho.pt4 Centre of Physics, Department of Mathematics and Applications, University of Minho,Campus de Gualtar, Braga, Portugal, smirnov@math.uminho.ptKey words: Soft Tissue Modelling, Calculus of Variations, FEM, Plastic Surgery.Abstract. Breast reduction is one of the most common procedures in breast surgery.The aim of this work is to develop a computational model allowing one to forecast thefinal breast geometry according to the incision marking parameters. This model can beused in surgery simulators that provide preoperative planning and training, allowing thestudy of the errors origin in breast reduction.1 INTRODUCTIONAlthough the number of works on the breast soft tissue modelling has increased sig-nificantly during the last few years [1, 2, 4, 5, 6], the development of an adequate breastmodel still continues to be an unsolved problem. The main areas of the breast modellingapplication are aesthetic surgery [4] and medical imaging analysis [2]. The breast reduc-tion surgery simulator presented here is based on the numerical solution of the so calledgeneral problem of plastic surgery. From the mathematical point of view this is a problemof calculus of variations with unusual boundary conditions, known as knitting conditions.The breast tissue is considered as a hyperelastic material. Although most of soft tissuesare incompressible, we consider the breast as a compressible Neo-Hookean material. Thecomplex structure of the breast involves several tissues that form it. Its elastic propertiescannot be readily deduced from the elastic properties of the tissues forming it. The com-pressibility of the breast is an experimental fact. (The breast reduction surgery whereonly a part of the skin is removed from the breast, diminishes the breast volume without1Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi Smirnovremoving any part of its internal tissue.) The skin has elastic properties very differentfrom those of the breast gland and fat tissue, and therefore, must be modelled differently.The breast tissue is modelled using three-dimensional elements and the skin is modeledusing two-dimensional elements. The skin function is to contain the jelly-like breast tis-sue. Its modelling is a very important point in the understanding and forecasting of theresults of breast reduction.A realistic breast model is impossible without so called Chassaignac’s space. This tissueplays a special role in the breast mobility and is responsible for the connection betweenthe breast and the chest. We model it as a mass-spring system [3]. The determinationof the elastic parameters is carried out from the breast geometry observation [4]. Thismethodology is based on the fact that when the patient changes her position (upright,prone, on the back, etc.), the breast geometry also changes and these transformationsdepend on the elastic properties of the breast. All necessary measurements do not needspecial equipment and can be fulfilled in a usual consulting room.The paper is organized as follows. We present in Section 2 the mathematical formu-lation of the two problems we will address : the breast under gravitational force and thesuturing problem. In Section 3 we detail the numerical algorithms to solve them. Numer-ical tests are given in the fourth section and the study ends with a conclusion and someperspectives.2 PROBLEM STATEMENT2.1 MotivationThe breast reduction surgery consists of the following steps. First, the nipple is placedin a new position and the breast is incised by two planes orthogonal to the chest andby an oblique plane (Fig. 1(a)). Next, the tissues incised by the orthogonal planes aresutured to each other (Fig. 1(b)). After that, the tissues incised by the oblique planeare sutured to the chest (Fig. 1(c)). In this work we want to develop a computationaltool to simulate this suturing process in order to predict the format of the breast after areduction surgery.2.2 Breast under gravitational forceDue to the gravitational force action, the breast is deformed with respect to an initialnull-gravity configuration Bgrav and f : Bgrav → R3 characterizes the deformation of thebody, i.e. for a given point p of the initial configuration, f(p) is the new position afterapplying the gravity force.The boundary of Bgrav is Γ = Γ1 ∪ Γ2 ∪ ΓIm. The set Γ1 represents the part of Bgrav(subset of S = {p : h(p) = 0}) containing the Chassaignac space, while Γ2 stands forthe skin. At last, the breast is fixed on its lower part — inframammary fold. We shallrepresent it by ΓIm (see Fig. 2).2Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi SmirnovFigure 1: Scheme of breast reduction surgery.Figure 2: Free surface of the breast (left — front view) and Chassaignac space and inframammary fold(right — back view).The problem consists of minimizing the functionalJgrav(f) =∫BgravWbr(∇f(p))dp−∫Bgravρ ag · f(p)dp+∫Γ2Wsk(∇f||(p))dSp +∫Γ1c‖f(p)− p‖dSp, (1)where Wbr : R3×3 → R is the volume strain-energy density and Wsk : R2×2 → R standsfor the skin strain-energy density. Moreover, the breast displacement is subject to theconstraintsf(p) = p, in ΓIm, (2)h(f(p)) = 0, in Γ1. (3)3Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi SmirnovWe consider the breast as a hyperelastic neo-Hookean compressible material with thestrain-energy density function given byWbr(F ) =µbr2(tr(FF t)− 3− 2 ln (det(F )))+ λbr2(det(F )− 1)2 ,where F = ∇f is the Jacobi matrix of f and (λbr, µbr) are the Lame´ parameters for thebreast.For the skin strain-energy density Wsk, we consider the gradient along the tangentplane of the surface, denoted by ∇f||(p). So, we haveWsk(F||) =µsk2(tr(F||F t||)− 2− 2 ln(det(F||)))+λsk2(det(F||)− 1)2,where F|| = ∇f|| is the Jacobi matrix of the superficial (skin) displacement f|| detailed inthe next section, and (λsk, µsk) are the Lame´ parameters for the skin.At last, vector ag stands for the gravity vector. The density ρ is assumed to be constant.The mass-spring model of the Chassaignac space is characterized by the constant c.2.3 Suturing problemIn order to mathematically model the suturing problem, we denote by Bcut the do-main which corresponds to the incised breast and prescribe the conditions to performthe suturing. As a first step, we neglect the gravity force assuming that the suturingprocess is mainly independent of the external field. We also neglect the effect due tothe Chassaignac space. We introduce the boundaries Γ =⋃4i=1 Γi to model the suturing.Boundary Γ1 is fixed and remains on the chest, while Γ2 is the surface associated to theskin. Boundary Γ3 = Γ+ ∪ Γ− corresponds to the first incision and is composed by twosurfaces Γ± (incised tissues) to be sutured to each other. The surface Γ4 corresponds tothe second incision where the tissues have to be sutured to a fixed surface characterizedby S = {p : h(p) = 0} (see Fig. 3). In our model the interface between the chest and thebreast corresponds to the plane x = 0 while the symmetry plane of the first incision isy = 0, as shown in Fig. 3, left panel.Assuming that the gravity and the Chassaignac space contributions are negligible, theproblem writes as the minimization of the functionalJcut(f) =∫BcutWbr(∇f(p))dp+∫Γ2Wsk(∇f||(p))dSp (4)subject to the constraintsf(p) = p, in Γ1, (5)f(p)− f(g(p)) = 0, in Γ+, (6)h(f(p)) = 0, in Γ4. (7)4Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi SmirnovFigure 3: Geometrical representation of the breast: front view (left) and back view (right).The expressions for the strain-energy densities Wbr and Wsk are the ones presented in theprevious section and f(Bcut) is the new position of the body after the suturing one has todetermine. The map g : Γ+ → Γ− corresponds to the suturing operation since it providesthe point-to-point correspondance between the left and right side of the cut.3 NUMERICAL SCHEME FOR THE BREAST3.1 Mesh and discretisationThe breast tissue is modelled using a three-dimensional integral energy while the skineffect is represented by a two-dimensional integral. The domain B ⊂ R3 denotes thebody Bgrav or the body Bcut corresponding to the initial configuration before applying thegravitational field or the suturing operation respectively.To design the numerical scheme, we denote by T a mesh of B constituted of I non-overlapping tetrahedrons cells τi, i = 1, . . . , I, where we represent by |τi| its volume, andN vertices Pn = (Pnx, Pny, Pnz), n = 1, . . . , N . We denote by Pij = (Pijx, Pijy, Pijz) ∈ R3,j = 1, 2, 3, 4, the vertices of tetrahedron τi and by Tk, k = 1, . . . , K, the faces of thetetrahedrons of the mesh that belong to Γ2 and by Pkj = (Pkjx, Pkjy, Pkjz) ∈ R3, j = 1, 2, 3,the vertices of Tk. To discretize the new position function f , for each node Pn, we associatean approximation fn and denote by fh the usual continuous linear piecewise function.Notice that no displacement at node Pn means fn = fh(Pn) = Pn.3.2 Triple integrals of the functionalsFor a new configuration characterized by the approximation p ∈ B → fh(p) ∈ R3, theinternal energy on tetrahedron τi is given byWτi = |τi|(µ2[tr(FiFti )− 2− 2 ln(det(Fi))]+λ2[det(Fi)− 1]2),5Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi Smirnovwhere Fi is the 3× 3 matrix solution of the linear systemfi2 − fi1 = Fi(Pi2 − Pi1), fi3 − fi1 = Fi(Pi3 − Pi1), fi4 − fi1 = Fi(Pi4 − Pi1).The total internal energy is approximated by Jh1 =∑τiWτi .The second volume integral of (1) is approximated byJh2 =∑τi|τi|4ρag ·(fi1 + fi2 + fi3 + fi4).3.3 Surface integrals of the functionalsWe now detail the tangential gradient ∇f|| we use to compute the skin energy. Thediscrete piecewise linear function fh transforms a triangle Tk with vertices OAB into atriangle T ′k with vertices O′A′B′. Since the translation and the rotation does not changethe stress due to the deformation, we assume that O′B′ is colinear to OB and A′ belongsto the same plane than triangle OAB. Function f|| is a two-dimensional function locallygiven by f||(O) = O, f||(A) = A′, f||(B) = B′. The Jacobian matrix of f|| is the constantmatrixJf|| = A =[a bc d].Using the assumption, one has[a bc d] [ ‖OB‖0]=[ ‖O′B′‖0],[a bc d] [ ‖OA‖ cos(α)‖OA‖ sin(α)]=[ ‖O′A′‖ cos(α′)‖O′A′‖ sin(α′)],where α = ∠(OA,OB) and α′ = ∠(O′A′, O′B′). The first linear system gives c = 0 anda = ‖O′B′‖‖OB‖ . Substituting these expressions in the second linear system we obtaind =‖O′A′‖ sin(α′)‖OA‖ sin(α) , c =‖O′A′‖ cos(α′)− a‖OA‖ cos(α)‖OA‖ sin(α) .The superficial energy on triangle T for the skin is then given byWT = |T |(µsk2[tr(Jf||Jf t||)− 2− 2 ln(det(Jf||))]+λsk2[det(Jf||)− 1]2)and the whole superficial energy is approximated byJh3 =∑T∈Γ2WT .To compute an approximation of the fourth integral of (1) we useJh4 =∑Ti∈Γ1c|Ti|3(‖fi1 − Pi1‖+ ‖fi2 − Pi2‖+ ‖fi3 − Pi3‖),where c is assumed to be constant.6Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi Smirnov3.4 Optimization problem for the breast under gravitational forceThe unknowns of the discretized version of the minimization of (1) subject to theconstraints (2) and (3) form a subset of fn = (fnx, fny, fnz), n = 1, . . . , N . Indeed,constraint (2) implies that the vertices in ΓIm are fixed and considering S as the verticalplane x = 0, constraint (3) imposes a null displacement in the x coordinates of the verticesin Γ1. We gather all the discrete unknowns in vector Xh and denote byJhgrav(Xh) = Jh1 + Jh2 + Jh3 + Jh4the numerical energy functional we have to minimize with respect to vector Xh.Remark. From vector Xh and the boundary conditions, we deduce a unique continuouslinear piecewise function fh we use in the integral computation. Vector Xh only containsthe unknown displacement components. For example, if point P1 is fixed, the componentsf1x, f1y, f1y for the displacement are not integrated in vector Xh since f1 = P1. In thesame way, the displacement of point P2 on plane x = 0 is fulfilled setting f2x = 0. Inconsequence, only components f2y and f2y belong to vector Xh.In order to determine the minimizer X¯h of the discrete functional Jhgrav(Xh), we employthe conjugate gradients method. To this end, one has to compute an approximation of thederivative of the discrete functional in order to X`, ` = 1, . . . ,#Xh. Since the function isnonlinear and has complex structure, numerical derivatives are computed in a very simpleway. For example, the derivatives with respect to direction X` is given by∂Jhgrav∂X`=Jhgrav(X` + `)− Jhgrav(X`),where ` is a vector of zeros except value  for the `−th entry.3.5 Optimization for the suturing problemFor the suturing problem, the reference mesh corresponds to the body Bcut at theinitial position (before the suturing). Constraint (6) implies that for any vk on Γ±, weset fky = 0 to enforce the suturing in the symmetric plane y = 0. Constraint (7) impliesthat for any vk on Γ4, we set fkx = 0. At last, constraint (5) is similar to (2). We thenintroduce the functionalJhcut(Xh) = Jh1 + Jh3which integrates the constraints while vector Xh only contains the unknown displacementcomponents.4 NUMERICAL TESTSIn this section we present some numerical results. First, we consider a breast beforethe surgery and apply the gravitational field. The breast is considered as a symmetric7Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi Smirnovbody and the plane of symmetry is orthogonal to the chest and passes through the nippledividing the breast in two equal parts. It is assumed that in the neutral state (i.e. ifall forces are equal to zero) the breast is a spherical cap. The nipple pedicle is notmodelled. The mesh is in Fig. 4 (left) and the result is in Fig. 5 (left). Then, we considerthe reduction surgery. To this end, we consider the incised breast where we apply thesuturing. Numerically speaking, we consider the mesh given in Fig. 4 (centre and right)where we apply the optimization procedure for the suturing problem and after that weapply the optimization problem for the breast under gravitational force. We present inFig. 5 (right) the final result. Notice that the nipple position is very high. This is theresult of an error committed during the surgery planning. Namely, the angle between thecutting planes of the first cut is too small. The result corresponding to the right choiceof parameters is shown in Fig. 6.Figure 4: Mesh for the gravitational field (left) and meshes for the suturing problem (centre and right).5 CONCLUSIONThe software developed for breast reduction modelling allows one to forecast the finalbreast geometry according to the incision marking parameters. The comparison of oursimulations with a real surgery gives satisfactory results. Although the model is of lowprecision, we were able to verify that it is sufficient for a satisfactory analysis of errorsfrequently done during breast reduction surgery and allows to understand how to avoidor correct them.ACKNOWLEDGEMENTSThe authors are greatful to Augusta Cardoso and Igor Vasilevskiy. This work wassupported with the Portuguese Funds from the “Fundac¸a˜o para a Cieˆncia e a Tecnolo-gia” through the Project EXPL/MAT-NAN/0606/2013. The authors were also sup-ported by the Research Centre of Mathematics of the University of Minho with the Por-8Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi SmirnovFigure 5: Breast before (left) and after (right) reduction surgery.Figure 6: Breast before (left) and after (right) reduction surgery.tuguese Funds from the “Fundac¸a˜o para a Cieˆncia e a Tecnologia”, through the ProjectPEstOE/MAT/UI0013/2014.REFERENCES[1] N. Ayache and H. Delingette (Eds.). Surgery Simulation and Soft Tissue Modeling.Lecture Notes in Computing Science. Springer, Berlin, 2003.[2] F.S. Azar, D.X. Metaxas and M.D. Schnall. Methods for modeling and predictingmechanical deformations of the breast under external perturbations. Medical ImageAnalysis, Vol. 6, 1-27, 2002.[3] A. Cardoso, H. Costa, V. Sa´ and G. Smirnov. On the importance of Chassaignac’sspace in breast modelling. In: IV European Conference on Computational Mechanics,9Ste´phane Clain, Gaspar J. Machado, Rui M. S. Pereira and Georgi SmirnovParis, May, 16-21, France, 2010.[4] A. Cardoso, G. Coelho, H. Zenha, V. Sa´, G. Smirnov and H. Costa. ComputerSimulation of Breast Reduction Surgery. Aesth. Plast. Surg. Vol. 37, 68-76, 2013.[5] H. Delingette, Towards realistic soft tissue modeling in medical simulation. In: Pro-ceedings of IEEE: Special Issues on Surgery Simulation, April, 512-523, 1998.[6] A. Perez del Palomar, B. Calvo, J. Herrero, J. Lo´pez and M. Doblare´. A finite elementmodel to accurately predict real deformations of the breast. Medical Engeneering &Phisics, Vol. 30, 1089-1097, 2008.10

Soft tissue modelling for analysis of errors in breast reduction surgery

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Soft tissue modelling for analysis of errors in breast reduction surgery

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