Physically based deformable object modeling and soft tissue deformation

Bilgisayar destekli sanal ameliyat benzetimleri ve farklı alanlardaki uygulamaları, birçok parametrenin olası en doğru biçimde belirlendiği ve uygulandığı matematiksel modellerle mümkündür. Bu parametreler benzetimi yapılacak biyolojik yapıların geometrik sınırlarının, organ özelliklerinin, farklı kuvvetler altındaki davranışlarının ve en önemlisi dokuya ait malzeme parametrelerinin olası en doğru biçimde belirlenmesi ile elde edilir. Bu çalışma yumuşak dokuların doğrusal olmayan, viskoelastik davranışlarının matematiksel olarak modellenmesi ve önerilen bu modelin sonlu elemanlar yöntemi kullanılarak sayısal olarak çözülmesine odaklanmıştır. Örnek organ olarak seçilen insan karaciğerine ait malzeme katsayıları, hastane ortamında canlı dokular üzerinde yapılan deney sonuçları kullanılarak doğrusal olmayan eniyileme yöntemi ile elde edilmiştir. Uygulanan genel yaklaşımlardan farklı olarak benzetim sonuçlarının çözümden önce yani modelleme aşamasında yapılan ön basitleştirmelerden etkilenmemesi ilkesi benimsenmiştir. Bu nedenle, modelleme aşamasında uygulanan ve gerçeklikten uzaklaşılması ile sonuçlanan geleneksel basitleştirme adımları uygulanmamış, bu yaklaşım yerine tam çözümü veren benzetim sonuçları üzerinde model indirgeme teknikleri uygulanmıştır. İlk aşama olarak yumuşak dokuların önemli özelliklerini içeren bir model önerilmiş, sonrasında oluşturulan bu model sonlu elemanlar yöntemi kullanılarak analiz edilmiştir. Analiz sonrası elde edilen sonuçlar bir sonraki aşamada kullanılmak üzere saklanmıştır. İkinci adımda saklanan sonuçlar üzerinde Karhunen Loeve model indirgeme yöntemi kullanılarak, ilk adımda elde edilen sonuçlara yakın değerleri gerçek zamanlı sunabilen bir çözüm elde edilmiştir. Son adımda ise basitleştirilmemiş model ile indirgenmiş model sundukları hızlar ve sonuçların doğrulukları açısından karşılaştırılmıştır. Anahtar Kelimeler: Yumuşak dokular, sonlu elemanlar yöntemi, model indirgeme, viskoelastisite.As indicated by The Institute of Medicine Report, 44,000 to 98,000 people die annually as a result of procedure mistakes in surgery. The main reason cited was the young surgeons' insufficient experience of new techniques before facing real-world surgical operations. With the use of virtual surgery, surgeons have the opportunity to test different critical surgical procedures in a low- cost, ethically sound environment. The virtual surgery simulators equipped with physically based modeling engines are outstanding candidates for the simulation of deformable human organs. The real time soft tissue simulation has gained a great interest recently as a result of advancements in areas such as surgery planning and the surgical simulations. Linear deformation models may not provide the required accuracy in such areas whilst nonlinear models do not serve the real time needs. Therefore, there is a common need for a computationally simplified yet accurate, nonlinear, large deformable viscoelastic model of soft tissues to be used in these real time applications. Computer aided surgery and surgery simulation applications require accurate form and feature description as well as proper material and behavior descriptions of the biological tissues in a mathematically formulated model. This study focuses on mathematical formulation and numerical implementation of a nonlinear viscoelastic model of soft tissues using Finite Element Method (FEM). As an object, a human liver is selected in our study. The necessary material parameters are extracted from results of in vivo material tests on human liver using a nonlinear optimization method. Due to the technology limitations, today's physics based surgery simulators are forced to use different simplification methods to obtain satisfactory visual effects and response time in delivering an acceptable visual and haptic user experience. Thus, previous approaches to this problem involve techniques in the simplification of the mathematical models prior to obtain the necessary numerical solutions. The model simplification is necessary to achieve the required computational speeds however; these simplifications will inevitably result in reduced accuracy. Unlike the above mentioned approaches, our research is based on the opposite view that accuracy should not unduly influenced due to premature model simplifications prior to the analysis. Therefore, simplification step that is applied in the modeling phase, resulting in inaccurate outcome, is displaced to be applied after the solution phase. This aims to limit the degradation of already calculated values. As the first step, a complete soft tissue model is created. Subsequently this model is analyzed using Finite Element Method (FEM) and the results are stored and are used as the input to the next stage. Then, the Karhunen-Loeve decomposition technique is used to simplify the previously obtained data resulting in the final simulated model. Simplification technique is necessary to achieve acceptable real-time update rates due to the computing power available. Finally, a complete (unsimplified) model is compared with the simplified model in terms of accuracy and speed. Since soft tissues undergo large deformations under applied load and required volume preservation behavior, the use of linear strain tensor is not suitable for the accurate modeling of the resulting large deformations involved. Limitation of using a linear strain tensor is overcome by the use of nonlinear Green strain tensor in our customized FEM code. Standard material tests on human liver reveal the material nonlinearity relationship between applied forces and the resulting displacements. This nonlinear behavior is necessitated the use of hyperelastic strain energy density function in our FEM implementation. Viscoelastic behavior is also a predominant feature of soft tissues that must be included in any soft tissue deformation simulation. Therefore, the quasi-linear viscoelastic behavior of a human liver is added to our implementation to provide for this need. To implement time dependent viscoelastic behavior in improved the reduced order model, the surface nodes and its neighboring nodes are determined inside the Radius of Influence (ROI). Then, a constant unit force is applied to each of surface nodes for a period of 20 seconds. It is assumed that the creep response of the soft tissue lasts for 20 seconds due to the limitations of data storage and higher computational costs. Our proposed model, when employed, results in a final constitutive equation that successfully produces accurate results while catering for the whole previously mentioned phenomenon namely; geometric and material nonlinearity, volume preservation, viscoelasticity. Keywords: Real-time simulation; deformable models; model order reduction; nonlinear simulations; soft tissue; viscoelasticity