Numerical simulation and identification of macroscopic vascularised liver behaviour: Case of indentation tests

Abstract

Mini-invasive surgery restricts the surgeon information to two-dimensional digital representation without the corresponding physical information obtained in previous open surgery. To overcome these drawbacks, real time augmented reality interfaces including the true mechanical behaviour of organs depending on their internal microstructure need to be developed. For the case of tumour resection, we present here a finite element numerical study of the liver mechanical behaviour including the effects of its own vascularisation through numerical indentation tests in order extract the corresponding macroscopic behaviour. The obtained numerical results show excellent correlation of the corresponding force-displacement curves when compared with macroscopic experimental data available in the literature.

Keywords

Liver vascularisation finite element model numerical simulation indentation tests real time

1. Introduction

Through mini-invasive techniques, surgery experienced huge progress in terms of patient comfort as well as complication decrease. However, these improvements came with a drastic complexification of the surgeon work. In particular, for liver tumour resection, per-surgery organ visualisation went from a three-dimensional global direct view of the liver to a display of the organ on two-dimensional screens. Up to the surgeon to rebuild, in real time, the third dimension from the different screens. This induced an additional mental strain to the surgeon while simultaneously reducing considerably the quantity of visual and useful information available. The present project led by IRCAD (Institut de Recherche contre les Cancers de l’Appareil Digestif) aims to create optimal surgery conditions for mini-invasive liver surgery by providing an augmented reality deforming in real-time accordingly to the patient organ displacements [1,2]. This image will be enhanced with additional transparency superimposed information such as the vascularisation and tumour positions. The resulting augmented reality 3D image will not only be a huge help for per-surgery actions and decision, but will also allow to build pre-operative specific tools to improve the surgical planning steps [3] or help surgeon students for virtual training.

2. Model development

The constitution of a real-time 3D image of the liver in augmented reality requires high precision to help the surgeons. In order to be compliant with an expected medical precision error below 2 mm, we make use of a numerical model, based on finite element (FE) analyses [4] integrating the true mechanical behaviour of the biological tissues (liver and vascularisation) and optimisation procedure. The purpose is to integrate the specific biological mechanical behaviour of the liver as well as its microstructure and determine the way they affect the macroscopic behaviour. Although the integration of the microstructural behaviour into a macroscopic response is sometimes complex [5–12], the use of continuous model [7,8] enable precise determination of the overall mechanical behaviour.

To build a compliant mechanical model, the two following steps are applied: first, the geometry of the liver including its vascularisations is extracted from patient imaging data (MRI or CT scan) and integrated within the numerical FE model (see Fig. 1). At the same time, mechanical properties of the different tissue constituting the liver (liver, vessels wall), are extracted from experimental tests from non-invasive techniques [13,14]. From this, a complete geometry, including all mechanical properties for each internal structures, is built. Then, a homogenised FE numerical model [15] is developed for the surgeon, providing precise deformations and displacements. In this study, liver and vessels are modelled using linear elastic material, with a Young modulus of 3 kPa for the liver, 620 kPa for the vessels, and Poisson coefficient of 0.49 [16]. To validate the macroscopic behaviour of the numerical model depending on its integrated vascularisation, numerical indentation tests are simulated and compared to experimental macroscopic information available in the literature [17].

Fig. 1.

FE model of Liver including its vascularisation.

In a second stage, since a homogenized FE numerical model is extremely slow to compute, an optimisation procedure is required. The complete model is processed first through a computational learning method (Locally Liner Embedding – LLE [18–20]). Here, the patient liver and vascularisation specific geometries are treated as parameters and, for each new patient geometry, can be expressed as a combination of all the known geometries in the data-base. Also, since the deformations applied to the liver are bounded to the scope of surgical motions, they can similarly be expressed as parameters. Then, considering the set of parameters describing the geometries, the mechanical properties and the possible external constrains applied during a surgery, solutions are computed for all given values of the different parameters using a Proper Generalized Decomposition (PGD) Method [21,22]. This allows computing all sets of corresponding internal and external deformed positions of the liver. All these states are then stored in a data-base and will be used to reconstruct, through a real time linear combination, all the local real deformations through the whole organ during the surgery.

3. Results

Fig. 2.

Comparison of force–displacement indentation curves. (a) Effects of microstructure and vascularization on the force/displacement curve. (b) Comparison of numerical and experimental results. Symbols represent the mean values and vertical bars correspond to the variations of the different positions of loading. Blue and square data are extracted from Samur [17]. Red and circle data are the numerical results of the current work.

The numerical model validation is completely dependent on each of the patient’s mechanical response for the real time surgery follow up. Hence, to validate the macroscopic behavior as a function of the vascularisation distribution, we present here the comparing results from indentation tests obtained from the numerical model with literature data [17]. The results are presented on Fig. 2.

Experimental data [17] were obtained on three pigs’ livers (without knowledge of vascularisation distribution) by force-displacement measurements with an indenter tip of 8 mm radius at different indentation depths and with constant indentation speed of 1 mm/s. In comparison, the numerical results were obtained on a single liver model with identical indenter tip geometry and speed but for different depths and positions of indentation. This has the effect of changing the macroscopic liver rigidity as a function of the vascularisation distribution and can be compared (in range) to Samur’s experimental data.

The numerical results are in very good agreement with the experimental data confirming that the mechanical effects due to the differences in mechanical properties and distribution of the vascularisation is well integrated within the numerical model. Experimental results show an increase of the standard deviation measurements with increase of indentation depth. This is to be expected without knowledge of the internal liver microstructures. On the contrary, the standard deviations of the numerical model are shifted towards the higher values (Fig. 2(b)). This shows the influence of the indentation position where harder tissues (blood vessels) are localised (Fig. 2(a)).

Since the maximum indentation depth remains reasonably small (10 mm) compared to the liver thickness (between 100 mm and 200 mm), it seems appropriate to think that the measured variabilities are only due to the vascularisation distribution effects rather than the liver geometry.

Further work is required to evaluate the impact of indentation speed (viscosity effects), depth and indenter size. More accurate non-linear material behaviour would also be needed for deeper indentation validations (large deformations).

4. Conclusion

The construction of a precise augmented reality image of the liver in real time is strongly bounded to the associated numerical model providing the organ internal and external positions. Here we developed a macroscopic liver model integrating vascularisation. Numerical mechanical indentation tests were performed on a vascularised liver and validate the overall macroscopic behaviour compared with literature data. These results will enable the construction of a complete homogenised numerical model to be used as a real help for the surgeons.

Conflict of interest

The authors have no conflict of interest to report.

References

Hostettler

Nicolau

S.A.

Forest

Soler

and Remond

, Real Time Simulation of Organ Motions Induced by Breathing: First Evaluation on Patient Data, Biomedical Simulation Harders

and Székely

, eds, Springer, Berlin/Heidelberg, 2006, pp. 9–18.

Hostettler

Nicolau

S.A.

Soler

and Rémond

, Toward and accurate real-time simulation of internal organ motions during free breathing from skin motion tracking and an a priori knowledge of the diaphragm motion, Computer Assisted Radiology Surgery2 (2007), S100–S102.

Selle

Preim

Schenk

and Peitgen

H.O.

, Analysis of vasculature for liver surgical planning, IEEE Transaction Medical Imaging21 (2002), 1344–1357. doi:10.1109/TMI.2002.801166.

Zienkiewicz

and Taylor

, The Finite Element Method, McGraw-Hill, London, 1977.

Auffray

Dell’Isola

Eremeyev

V.A.

Madeo

and Rosi

, Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids, Mathematics and Mechanics of Solids20 (2015), 375–417. doi:10.1177/1081286513497616.

Auffray

Dell’Isola

Eremeyev

Madeo

Placidi

and Rosi

, Least action principle for second gradient continua and capillary fluids: A Lagrangian approach following Piola’s point of view, Advanced Structured Materials38 (2014), 89.

Madeo

Dell’Isola

and Darve

, A continuum model for deformable, second gradient porous media partially saturated with compressible fluids, Journal of the Mechanics and Physics of Solids61 (2013), 2196–2211. doi:10.1016/j.jmps.2013.06.009.

Dell’Isola

Sciarra

and Vidoli

, Generalized Hooke’s law for isotropic second gradient materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences465 (2009), 2177–2196. doi:10.1098/rspa.2008.0530.

Sciarra

Dell’Isola

Ianiro

and Madeo

, A variational deduction of second gradient poroelasticity, Part I: General theory, Journal of Mechanics of Materials and Structures3 (2008), 507–526. doi:10.2140/jomms.2008.3.507.

10.

Madeo

Dell’Isola

Ianiro

and Sciarra

, A variational deduction of second gradient poroelasticity II: An application to the consolidation problem, Journal of Mechanics of Materials and Structures3 (2008), 607–625. doi:10.2140/jomms.2008.3.607.

11.

Sciarra

Dell’Isola

and Coussy

, Second gradient poromechanics, International Journal of Solids and Structures44 (2007), 6607–6629. doi:10.1016/j.ijsolstr.2007.03.003.

12.

Dell’Isola

Sciarra

and Batra

R.C.

, A second gradient model for deformable porous matrices filled with an inviscid fluid, Solid Mechanics and Its Applications125 (2005), 221–229. doi:10.1007/1-4020-3865-8_25.

13.

Hostettler

George

Rémond

Nicolau

S.A.

Soler

and Marescaux

, Bulk modulus and volume variation measurement of the liver and the kidneys in vivo using abdominal kinetics during free breathing, Computer Methods and Programs Biomedicine100 (2010), 149–157. doi:10.1016/j.cmpb.2010.03.003.

14.

Nierenberger

George

Baumgartner

Rémond

Ahzi

Wolfram

Kahn

J.L.

and Rahman

R.A.

, Towards building a multiscale mechanical model for the prediction of acute subdural hematomas, in: ASME 11th Biennial Conference on Engineering Systems Design and Analysis, 2012, pp. 261–266.

15.

Rahman

R.A.

George

Baumgartner

Nierenberger

Rémond

and Ahzi

, An asymptotic method for the prediction of the anisotropic effective elastic properties of the cortical vein: Superior sagittal sinus junction embedded within a homogenized cell element, Journal Mechanics Materials Structures7 (2012), 593–611. doi:10.2140/jomms.2012.7.593.

16.

Nava

Mazza

Furrer

Villiger

and Reinhart

W.H.

, In vivo mechanical characterization of human liver, Medical Image Analysis12 (2008), 203–216. doi:10.1016/j.media.2007.10.001.

17.

Samur

Sedef

Basdogan

Avtan

and Duzgun

, A robotic indenter for minimally invasive characterization of soft tissues, International Congress Series1281 (2005), 713–718. doi:10.1016/j.ics.2005.03.117.

18.

Roweis

S.T.

and Saul

L.K.

, Nonlinear dimensionality reduction by locally linear embedding, Science290 (2000), 2323–2326. doi:10.1126/science.290.5500.2323.

19.

Gonzalez

Aguado

J.V.

Cueto

Abisset-Chavanne

and Chinesta

, kPCA-based Parametric Solutions within the PGD Framework, Archives of Computational Methods in Engineering (2016), 1–18. In press. doi:10.1007/s11831-016-9173-4.

20.

Gonzalez

Cueto

and Chinesta

, Computational patient avatars for surgery planning, Annals Biomedical Engineering44 (2016), 35–45.

21.

Chinesta

Keunings

and Leygue

, The Proper Generalized Decomposition for Advanced Numerical Simulations, Springer International Publishing, 2014.

22.

Niroomandi

Gonzalez

Alfaro

Cueto

and Chinesta

, Model order reduction in hyperelasticity: A proper generalized decomposition approach, International Journal for Numerical Methods in Engineering96 (2013), 129–149.