This study suggests a method for computing the constitutive model for veins in vivo from clinically registered ultrasound images. The vein is modelled as a hyperelastic, incompressible, thin-walled cylinder and the membrane stresses are computed using strain energy. The material parameters are determined by tuning the membrane stress to the stress obtained by enforcing global equilibrium.  In addition to the mechanical model, the study also suggests a preconditioning of the pressure-radius signal. The preconditioning computes an average pressure-radius cycle from all consecutive cycles in the registration and removes, or reduces undesirable disturbances. In order to overcome this problem, an approach is proposed that allows constitutive equations to be determined from clinical data by means of reasonable assumptions regarding in situ configurations and stress states of vein walls. The approach is based on a two-dimensional Fung-type stored-energy function that captures the characteristic nonlinear and anisotropic responses of vein

Fortuny, G.

Herrera, B.

Marcé Nogué, Jordi

Marimón, F.

English

UPCommons. Portal del coneixement obert de la UPC

741   DETERMINING THE CONSTITUTIVE PARAMETERS OF THE HUMAN FEMORAL VEIN IN SPECIFIC PATIENTS G. FORTUNY*, B. HERRERA*, F. MARIMÓN+ AND J. MARCÉ† * Department of Computer Engineering and Mathematics Universitat Rovira i Virgili Av/ Universitat 1, 43204 Reus, Spain e-mail: gerard.fortuny@urv.cat, http://deim.urv.cat/~gerard.fortuny/  +Department of Medicine and Surgery Universitat Rovira i Virgili C/ Sant Llorenç 21, 43201 Reus, Spain  †Department of Strength of Materials and Structural Engineering,  Universitat Politècnica de Catalunya,  C/ Colom 11, 08222 Terrassa, Spain e-mail: jordi.marce@upc.edu Key words: Femoral vein; Constitutive equations; Venous live tissue; Human; Parameter identification. Abstract. This study suggests a method for computing the constitutive model for veins in vivo from clinically registered ultrasound images. The vein is modelled as a hyperelastic, incompressible, thin-walled cylinder and the membrane stresses are computed using strain energy. The material parameters are determined by tuning the membrane stress to the stress obtained by enforcing global equilibrium.  In addition to the mechanical model, the study also suggests a preconditioning of the pressure-radius signal. The preconditioning computes an average pressure-radius cycle from all consecutive cycles in the registration and removes, or reduces undesirable disturbances. In order to overcome this problem, an approach is proposed that allows constitutive equations to be determined from clinical data by means of reasonable assumptions regarding in situ configurations and stress states of vein walls. The approach is based on a two-dimensional Fung-type stored-energy function that captures the characteristic nonlinear and anisotropic responses of veins.    1 INTRODUCTION 4.5% of the population is at risk of suffering a venous thromboembolism disease, with an approximate mortality rate of 11% ([2, 3]). Our general objective consists of studying a XI International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XI E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) 742G. Fortuny, B. Herrera, F. Marimón and J. Marcé. 2serious pathology that has important consequences: deep vein thrombosis (DVT). The problems involved in modeling venous tissue have been largely ignored by biomechanics researchers, most of whose efforts have instead focused on determining constitutive models of the arterial tissue ([7, 11]). Venous and arterial walls have a similar structure and composition, the main difference between their respective walls being the thickness and fiber orientation of the medial zone. In this study, we determine a constitutive model of the venous wall tissue in its real life location inside the human body. In the future, we intend to study diseased venous walls and their relation to the origins of DVP. Some studies have modulated the mechanical properties of venous walls ([1, 9]), although these studies have only looked at these properties in laboratory conditions and in non-live tissue. Constitutive equations can be determined from experimental data regarding the diameter of a vessel segment that is subject to internal pressure and external axial force, and the load-free reference geometry of the vessel segment, including the wall thickness.In the present study, if the membrane stresses are to be computed, two assumptions need to be made to overcome the limitations of the clinical data ([10]), (i) The in vivo conditions, the axial stretch of the vessel and the axial external force are constant and independent from internal pressure  (ii) The ratio between the axial and circumferential stress is known at one internal pressure P.2. METHODS 2.1. Original Data To carry out the present study, we have used images captured by projecting ultrasound in real time, which is a typical method for clinically registering the pressure and radius. The ultrasound probe is lineal to 4.5 MHz and uses the MyLab Xview 70 high resolution image projection system (Esaote, Genoa, Italy). This new-generation ultrasound tool eliminates particles whilst preserving the information needed for diagnosis. We have used a time sequence of 10 seconds to register in a file of a healthy 40 year-old person. With this observation, we also obtain the internal pressure P.2.2. Theoretical framework In general, veins and arteries have similar walls and a structure in three distinct layers: the intima, the media  and the adventitia. The media is the middle layer of the vein and consists of a complex three-dimensional network of smooth muscle cells, and elastin and collagen fibrils ([4]). The mechanical properties of venous and arterial walls are different; for example, the pulsating behaviour of the arterial walls is absent from the venous walls. In venous walls the media layer is thinner than the arterial wall. Also, the fibre orientation of venous walls is not clear. Consequently, we use a non-fibre oriented model to approximate the constitutive equation of the venous wall. 743G. Fortuny, B. Herrera, F. Marimón and J. Marcé. 3In the femoral vein, the variation range of the pressure–inner diameter and the wall thickness at mean pressure is taken from clinical data. A two-dimensional (membrane) model describing the biaxial (i.e. circumferential and axial) response is to be determined. Employing a least-squares approach this can be achieved by minimizing the sum of the squared errors W: ( ) ( )[ ]∑ −+−=iizzzziW2mod2mod σσσσ θθθθ(1)In Eq.(1), the index i denotes the ith of the n data points, modθθσ  and modzzσ are the circumferential and axial Cauchy stresses predicted by the model, and θθσ  and zzσ  are the mean circumferential and axial Cauchy stresses of the wall computed directly from experimental data by enforcing equilibrium. Following the theory of hyperelasticity ([6]) the principal model stresses: θθθθ λψλσ∂∂=mod ;            zzzz λψλσ∂∂=mod(2)may be derived from the two-dimensional SEF ),( zλλψψ θ= and can be expressed in terms of the principal stretches θλ  and zλ  associated with the circumferential and axial directions, respectively. The circumferential and axial stretches are defined as mm Dd /=θλ  and Zzz /=λ ; whereby dm and Dm denote the actual and the referential (unloaded) mid-wall diameter of the vessel, and z and Z denote the actual and the referential length of a vessel segment. Note that Eq. (2) is only valid if the stress tensor and the strain tensor are coaxial. This is the case in the present study, which is restricted to axisymmetric geometry and boundary conditions. For ψ  a two-dimensional Fung-type SEF proposed by Von Maltzahn et al.(1984) is used as follows: )1(2−= QeCψ , (3)where22 2 zzzzzzz EcEEcEcQ ++= θθθθθθθ (4)744G. Fortuny, B. Herrera, F. Marimón and J. Marcé. 4The SEF ψ  incorporates four constitutive parameters, C, θθc , zcθ  and zzc . The circumferential and axial Green–Lagrange strains θθE  and zzE  can be expressed in terms of the circumferential and axial stretches, that is, )1( 221 −= θθθ λE  and )1(221 −= zzzE λ , respectively. Fung-type SEFs have been used successfully to model the mechanical responses of numerous veins from various species and anatomical sites. Substituting Eq.(4) for Eq.(2) leads to explicit expressions for modθθσ  and modzzσ  as functions of the principal stretches θλ , zλ and the constitutive parameters C, θθc , zcθ  and zzc . Note that ψ  is convex if and only if zzz ccc θθθ <2 ,0>θθc  and 0>zzc  (for a derivation see [6]). The circumferential and axial mean wall stress θθσ  and zzσ  can be determined by enforcing global equilibrium. Thus, hrP=θθσ , ππσ)2(2hrhFPrzz ++= (5)where h is the actual wall thickness, r is the actual inner radius, P is the transmural pressure and F is the external axial force. Substituting these expressions in θθσσ /zzk =  the external axial force F can be determined explicitly as: APkkPrF +−= )12(2 π (6)where, according to assumption (ii), the stress ratio k is known for a particular pressure Passociated with the actual radio r and πrhA = .2.3. Calculus The constitutive parametersC , θθc , zcθ  and zzc  have to be determined as variables from a nonlinear zero function in order to determine the minimum error function. We consider the function W as the function of error, we want to obtain the minimum of this function or the zero of this function’s derivative. To do so we will use the Levenberg-Marquardt method for the non-linear least squares problems [8]. Thus, we will considererC , θθc , zcθ  and zzc  as variables in the function W, that is ),,,( zzz cccCW θθθ , so that its minimum will be found in values that verify that 0),,,( =∇ zzz cccCW θθθthat is: 745G. Fortuny, B. Herrera, F. Marimón and J. Marcé. 50),,,(0),,,(0),,,(0),,,(=∂∂=∂∂=∂∂=∂∂zzzzzzzzzzzzzzzcccCcWcccCcWcccCcWcccCCWθθθθθθθθθθθθθθθ(7)where W∂ / x∂  are the partial derivative of function W by the variable x. Now we need to apply the Newton-Raphson method to the function W∇  in dimension 4: 10)())((' )()()1()( ≥=∇+−∇ + kXWXXXW kkkk (8)where 'W∇  is the Jacobian 44×  matrix. We take ),,,( 0000)0( zzz cccCX θθθ=  where ),,,( 0000 zzz cccC θθθ  are the values obtained in [10] for arteries. This is a linear system of equations for )1( +kX , and given that 'W∇  is a non-singular matrix, it can be solved using a normal lineal system method. A simple verification shows the local minimum property of the value obtained (we apply a small perturbation to our final value). 3. RESULTS We obtain values for the constitutive parametersC , θθc , zcθ  and zzc ; and (using the constitutive model) we can obtain information on the unloaded referential geometry (inner diameter D) and the in situ boundary force F for each observation (see Table 1). Table 1: Computed inner diameter D; external axial force F; constitutive parameters C, θθc , zcθ  and zzc .With the specified constitutive parameters summarized in Table 1, the SEFs turn out to be convex, which is a crucial property that ensures mechanically and mathematically reliable behaviour. Specific constitutive equations are obtained by substituting the constitutive parameters in Eq.(2). The relations between the pressure and the inner diameter computed from the constitutive models are approximately the same as the observed values (Fig.1). The biaxial response (Fig.2) shows the anisotropy and nonlinearity between the axial and circumferential stretches.  D(mm) F(N) C(kPa) θθc zcθ zzcFemoral Vein 8.8 1.14 14.37 2.08 1.39 1.01 746G. Fortuny, B. Herrera, F. Marimón and J. Marcé. 6Figure 1:  Pressure–inner diameter cycles (marked by squares) from the values for the femoral vein. The solid line indicates the pressure–inner diameter relation predicted by the constitutive model. Figure 2:  Circumferential (thick lines) and axial (thin lines) Cauchy stress contours in the (mid-wall) stretch plane for the femoral vein from a non pathologic subject.4. CONCLUSIONS This is the first attempt to provide constitutive equations for human veins and the femoral vein in particular. The mechanical behaviour of human arteries has already been described, but this is not the case for human veins. Consequently, we have used the results obtained in [9] to develop a new constitutive model for human veins.  In this study we have tried to show a simple method for determining constitutive parameters for the biaxial stretch states of human vein walls (Fig.2) in a specific subject (Fig.1). The proposed approach is based on providing information about the axial values (axial stretch, external axial force and axial stress) that is not contained in clinical data. Evaluating the predictive capability of constitutive equations requires changes in the boundary conditions, that is, in the pressure and stretches. However, it is possible to alter the boundary conditions in a tolerable way. Obviously, future research in this area could compare several constitutive model approximations in various subjects, each one from a different risk population. Even so, the particular conditions for a specific patient can change; blood pressure can be elevated through exercise or diminished by pharmacological methods. The specificity of this method allows a specific model for each patient. These models try to characterize the nonlinear anisotropic material responses and the in-situ boundary conditions, because the original data are measurements of stretches in-situ. This fact will promote future research into potential thrombosis risk factors and parameters for monitoring pharmacological therapies, etc. Despite inherent limitations the present approach demonstrates a reasonable way to determine constitutive equations for human veins that would otherwise not be available.747G. Fortuny, B. Herrera, F. Marimón and J. Marcé. 75. ACKNOWLEDGMENTS Support was provided by Universitat Rovira i Virgili  grant: AIRE2010-01 REFERENCES [1] Ackroyd JS, Pattison M, Browse NL. 1985. A study of the mechanical properties of fresh and preserved human femoral vein wall and valve cusps. Br J Surg 72, 117-19. [2] Cohen AT, Agnelli G, Anderson FA, Arcelus JI, Bergqvist D, Brecht JG, Greer IA, Heit JA, Hutchinson JL, Kakkar AK, Mottier D, Oger E, Samama MM, Spannagl M 2007. Venous thromboembolism (VTE) in Europe. The number of VTE events and associated morbidity and mortality. Thromb Haemost 98(4), 756-64. [3] Cushman M, Tsai AW, White RH, Heckbert SR, Rosamond WD, Enright P, Folsom AR. 2004. Deep vein thrombosis and pulmonary embolism in two cohorts: the longitudinal investigation of thromboembolism etiology. Am J Med 117(1),19-25.[4] Fawcett D.W. (Bloom Fawcett), 1995, Tratado de Histologia. McGraw-Hill Interamericana, Madrid 438-443 [5]   Fung, Y.C., Fronek, K., Patitucci, P., 1979. Pseudoelasticity of arteries and the choice of its mathematical expression.Am J Physiol 237 (5), H620–631. [6] Holzapfel, G.A., 2000. Nonlinear Solid Mechanics, A Continuum Approach for Engineering.Wiley, Chichester. [7] Holzapfel, G.A., Gasser, T.C., Ogden, R.W., 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models.J Elasticity 61, 1–48.[8] Marquardt D. 1963. An algorithm for least squares estimation on nonlinear parameters. SIAM. J.Appl. Math. 11, 431-441 [9] Opgaard OS, Edvinsson L. 1997. Mechanical properties and effects of sympathetic co-transmitters on human coronary arteries and veins. Basic Res Cardiol 92, 168-80.[10] Schulze-Bauer C.A.J., Holzapfel G.A. 2003 Determination of constitutive equations for human arteries from clinical data. Journal of Biomechanics 36, 165–169 [11] Stålhand J. 2009. Determination of human arterial wall parameters from clinical data. Biomech Model Mechanobiol 8, 141–148 [12] von Maltzahn, W.W., Warriyar, R.G., Keitzer, W.F., 1984. Experimental measurements of elastic properties of media and adventitia of bovine carotid arteries. J Biomech 17 (11),839–847.

Determining the constitutive parameters of the human femoral vein in specific patients

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Determining the constitutive parameters of the human femoral vein in specific patients

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