Modelling biomechanical problems in the framework of porous media mechanics: diabetic foot and biologically inspired scaffolds

The Theory of Porous Media is applied in this thesis to two different biomechanical problems. First, a computational model of the diabetic foot is developed by taking advantage of patient specific geometries and time-dependent loads experimentally measured. Differently from the state-of-the-art hyperelastic behavior, the plantar tissue is modelled as a fully saturated porohyperelastic porous medium. In this way, time becomes a variable with a real physical meaning, allowing to consider the peculiarities of the subject specific gait cycles. In addition, thanks to the two different phases constituting the medium (i.e. solid and fluid phases), total stress can be split into interstitial fluid pressure and effective stress contributions. Thus, the position and the value of peaks is different, whether total or effective stress are considered, and the peaks in effective counterparts are increasing with the number of gait cycles, as a result of the fluid-structure interaction. By taking advantage of these results we hypothesized that the development of the ulceration phenomenon in the diabetic foot is guided by the excess of an average effective pressure limit. Due to the lack in literature of specific ulceration triggering stress value, maximum average pressure values found for a healthy subject are used as triggering values for the diabetic foot model. As a result, the development of the ulcers within the plantar tissue is mimicked, showing ulcer positions that are in good agreement with the one observed by clinicians. The second application is the scaffold for the spinal fusion: starting from experimental results, a computational model of tissue differentiation is developed, guided by mechanical stimuli applied. The model obtained is capable to predict the tissue formation in post-surgery period, reducing the velocity of tissue transformation depending on stress values found in literature. It can be used by clinicians as a tool for defining subject specific scaffold dimensions and after surgery loading guidelines. The last biomechanical application is the three layer composite scaffold for the articular defect repair. Due to the complexity of the problem, a scaffold with final tissue developed is taken into account and its behavior as a function of loads applied is studied. By taking advantage of the experimental loads from the diabetic foot, we hypothesized to place the scaffold in a knee joint. Thus, it is loaded with vertical and shear loads, studying the development of effective stresses and interstitial fluid pressure within the domain. The numerical results allowed us to suggest an enhancement of the composite scaffold configuration, to avoid stress peaks patterns in the bone layer of the scaffold. The Theory of Porous Media allowed us to make a step forward with respect to the state of the art modelling in different biomechanical fields, by developing computational models that can be used by physicians as effective tools in conjunction with experimental results

Modelling biomechanical problems in the framework of porous media mechanics: diabetic foot and biologically inspired scaffolds

The Theory of Porous Media is applied in this thesis to two different biomechanical problems. First, a computational model of the diabetic foot is developed by taking advantage of patient specific geometries and time-dependent loads experimentally measured. Differently from the state-of-the-art hyperelastic behavior, the plantar tissue is modelled as a fully saturated porohyperelastic porous medium. In this way, time becomes a variable with a real physical meaning, allowing to consider the peculiarities of the subject specific gait cycles. In addition, thanks to the two different phases constituting the medium (i.e. solid and fluid phases), total stress can be split into interstitial fluid pressure and effective stress contributions. Thus, the position and the value of peaks is different, whether total or effective stress are considered, and the peaks in effective counterparts are increasing with the number of gait cycles, as a result of the fluid-structure interaction. By taking advantage of these results we hypothesized that the development of the ulceration phenomenon in the diabetic foot is guided by the excess of an average effective pressure limit. Due to the lack in literature of specific ulceration triggering stress value, maximum average pressure values found for a healthy subject are used as triggering values for the diabetic foot model. As a result, the development of the ulcers within the plantar tissue is mimicked, showing ulcer positions that are in good agreement with the one observed by clinicians. The second application is the scaffold for the spinal fusion: starting from experimental results, a computational model of tissue differentiation is developed, guided by mechanical stimuli applied. The model obtained is capable to predict the tissue formation in post-surgery period, reducing the velocity of tissue transformation depending on stress values found in literature. It can be used by clinicians as a tool for defining subject specific scaffold dimensions and after surgery loading guidelines. The last biomechanical application is the three layer composite scaffold for the articular defect repair. Due to the complexity of the problem, a scaffold with final tissue developed is taken into account and its behavior as a function of loads applied is studied. By taking advantage of the experimental loads from the diabetic foot, we hypothesized to place the scaffold in a knee joint. Thus, it is loaded with vertical and shear loads, studying the development of effective stresses and interstitial fluid pressure within the domain. The numerical results allowed us to suggest an enhancement of the composite scaffold configuration, to avoid stress peaks patterns in the bone layer of the scaffold. The Theory of Porous Media allowed us to make a step forward with respect to the state of the art modelling in different biomechanical fields, by developing computational models that can be used by physicians as effective tools in conjunction with experimental results.La Teoria dei Mezzi Porosi è stata applicata in questa tesi a due differenti applicazioni nell’ambito della biomeccanica. In primis è stato sviluppato un modello computazionale per il piede diabetico, sfruttando geometrie patient specific e carichi dipendenti dal tempo misurati sperimentalmente. Contrariamente al comportamento iperelastico usualmente utilizzato in letteratura, il tessuto plantare è stato modellato come mezzo poroso poroiperelastico completamente saturo. In tale maniera è stato possibile considerare la variabile tempo come una reale variabile fisica, permettendoci di tenere in considerazione le peculiarità del ciclo del passo del soggetto. Inoltre, grazie alla presenza delle due fasi che costituiscono il mezzo (fase solida e fase fluida), è stato possibile suddividere il contributo di stress totali in tensioni efficaci e pressione del fluido interstiziale. Di conseguenza la posizione dei picchi e la loro intensità risulta differente, a seconda che si consideri la componente totale o la controparte effettiva dello stress, e l’effetto dell’interazione fluido struttura determina un andamento crescente dei picchi con il numero dei cicli del passo compiuti. Tenendo conto dei risultati così ottenuti, si è ipotizzato che lo sviluppo del processo di ulcerazione nel piede diabetico sia guidato dal superamento di un valore limite di pressione effettiva media. A causa dell’assenza in letteratura di valori tensionali di soglia per l’innesco dell’ulcera, i valori massimi di pressione media efficace riscontrati nel modello del piede sano sono stati impiegati come valori limite per il piede diabetico. I risultati numerici così ottenuti hanno mostrato posizioni di innesco e sviluppo dell’ulcera nel tessuto plantare che sono in buon accordo con quelle riscontrate nella realtà dai clinici. La seconda applicazione considerata è stato lo scaffold per spinal fusion: partendo dai risultati sperimentali, si è sviluppato un modello computazionale di differenziazione tissutale guidato da stimoli meccanici applicati. Il modello così ottenuto ha permesso di predire la formazione tissutale nella fase post operatoria, riducendo la velocità di trasformazione dei tessuti in funzione di specifici valori tensionali trovati in letteratura. In tale maniera è stato possibile offrire ai medici uno strumento con cui si possa definire un dimensionamento patient specific dello scaffold e si possano sviluppare delle linee guida di carico per la fase post operatoria. L’ultima applicazione in ambito biomeccanico analizzata è lo scaffold composito tristrato per la riparazione di difetti articolari. A causa della complessità del fenomeno si è considerato uno scaffold con tessuti target sviluppati e si è studiato il suo comportamento soggetto a differenti carichi applicati. Ipotizzando di porre lo scaffold in una articolazione del ginocchio, abbiamo utilizzato i carichi sperimentali ottenuti per il piede diabetico. Abbiamo così caricato lo scaffold con i carichi verticali e taglianti, studiando l’andamento della pressione del fluido e le tensioni efficaci nel tempo all’interno dei tre strati che lo compongono. I risultati numerici ottenuti ci hanno permesso di ipotizzare un miglioramento del layout dello scaffold composito, finalizzato a ridurre picchi tensionali che hanno origine nello strato osseo di quest’ultimo. Grazie all’applicazione della Teoria dei Mezzi Porosi è stato dunque possibile compiere alcuni passi in avanti rispetto allo stato dell’arte in differenti campi della biomeccanica computazionale, permettendoci di fornire ai clinici utili strumenti che possono essere utilizzati a supporto dei risultati sperimentali

Rheological models for elasto-plastic behavior of superconducting strands

The main scope of this paper is to present rheological models that can simulate the non-linear behavior of superconducting strands at different temperatures. In detail, these models estimate the values of the elastic modulus (E) and the elastic-plastic tangent modulus (H) in a very simple and efficient way. Superconducting strands are composite materials; they are usually made of a normal metal matrix where superconducting filaments are embedded. The wire mechanics is studied by taking into consideration both the longitudinal behavior (along the axis of the wire) and the transversal one (on the plane section of the wire itself). Concerning the longitudinal axis, the constitutive materials are represented by a system of mechanisms arranged in parallel: normal metals are represented by a series of springs and frictional devices, while superconducting filaments are considered as single springs, because of their elastic behavior. With respect to the transversal plane, two models are developed. In the first one all materials are considered as a composition of systems of mechanisms (springs and frictional devices) arranged in series. In the second one, which is more accurate, the cross section of the wire is subdivided into different stripes. Each stripe is represented again by a composition of mechanisms and is considered in parallel with the other stripes. Different wires are taken into consideration and their elastic modulus and elastic-plastic tangent modulus are obtained. The three models developed for each strand are compared with numerical results obtained in previous works with the finite element method and virtual testing technique

Sensitivity analysis on elastic-plastic properties of superconducting strands

Scientific and medical equipment may require the generation of high magnetic fields. To this purpose superconducting (SC) magnets wound with SC wires are mainly used. Superconducting wires are made of SC filaments embedded in a bronze/copper matrix. Bronze and copper typically show elastic-plastic behavior with hardening, while filaments remains in elastic conditions. Because of their manufacturing process, SC wires experience a wide thermal variation: from reaction temperature (900-950K) to working conditions (4-7K). Due to its heterogeneity, a straightforward numerical simulation of a coil, taking into account all the details of the microstructure, would result in an enormous number of unknowns. In some previous works homogenization methods were proposed to address this problem. In particular in [D.P. Boso, Superconductor Science & Technology, 2013] it is presented that our model can reproduce well the stress and strain curves of wires by means of virtual testing and finite element method. However, it is very difficult to find a consistent set of thermal and elastic-plastic material data over the whole temperature range needed. In this work we address this problem. We investigate how the variability and uncertainty on the measured samples reflect on the obtained homogenized behavior. A parametric study is performed, with variations up to 30% on the elastic properties of each component to see the overall effects on the elastic behavior. Concerning the plastic field, yielding stress is increased of 30% and tangent moduli are increased up to 300%. Variations of the properties of a single material as well as simultaneous variations of the properties of different components are considered. As far as the elastic field is concerned, some regularity is found for a single increment: the leading factor for the global behavior is the volume fraction. Out of the regularities found in the results, it is possible to obtain a relationship to correlate the variation of the Young modulus of one material to the simultaneous variation of the Young modulus of different components. With respect to the plastic field, very different performances between the wire longitudinal behavior and the cross-section behavior are observed. Along the longitudinal axis SC filaments show their influence on the global plastic behavior, while in the wire cross-section bronze and copper materials lead the nonlinear mechanics

A Poroelastic Model for Plantar Tissue During Gait: Main Features and Perspectives

Recently a new computational model, based on the thermodynamically constrained averaging theory [1], has been proposed to predict tumor initiation and proliferation [2-4]. A similar mathematical approach is proposed here to study plantar tissue mechanics. The foot tissue is modeled as an elastic porous medium, in large strain regime and completely filled by a fluid phase. In detail, the tissue cells and their extracellular matrix form the solid skeleton with pores saturated by the interstitial fluid. Transport of nutrients and possible delivery of drugs within the microvasculature are also considered by introducing an effective diffusion coefficient, which is estimated from the real degree of vascularization of the tissue. Furthermore, the tissue may become necrotic depending on the stress level and/or oxygenation degree; thus if an ulcer is formed, the tissue comprises a healthy and a necrotic fraction. The primary variables of the model are: the interstitial fluid pressure pf, the displacement vector of the solid phase us, and the mass fraction of oxygen dissolved in the interstitial fluid, \u3c9Of . With respect to these primary variables, the governing equations are discretized in space by the finite element method [5], in time by using the \u3b8-Wilson method and then solved numerically. Considering the interstitial fluid allows mimicking the viscoelastic behavior of the plantar tissue observed experimentally by Gefen [6]. This is shown in the simulated cases, where a foot during stance and several gait cycles are modeled. The presented examples integrate experimental data at different scales (patient specific foot geometry, tissue elasticity and permeability, possible tissue vasculopathy, global forces measured during gait, etc.) and allow validating the developed modeling procedure by comparisons between numerical and measured plantar pressures. Being the global response of the bi-phase system viscoelastic, it is shown that the duration of stance as well as of each of gait cycle has an influence on tissue strain and stress fields. In Figure 1 discretization of the foot, load history and total stress field at two instants are shown. Since the time scale of ulceration is much larger that the typical time increment used to model a gait cycle, a multiscale approach in time is under development. It will allow accomplishing this modeling framework, which final aim is the prediction of risk of foot ulceration in diabetic patient