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Residual stresses are generated in living tissues by processes of growth and adaptation and they significantly influence the mechanical behaviour of the tissues. Thus, to effectively model the elastic response of the tissues relative to a residually stressed configuration the residual stresses need to be incorporated into the constitutive equations. The purposes of this paper are (a) to summarise a general elastic constitutive formulation that includes residual stress, (b) to specify the tensors needed for the three-dimensional implementation of the theory in a nonlinear finite element code, and (c) to use the theory and its implementation to evaluate the wall stress distribution in an abdominal aortic aneurysm (AAA) using patient specific geometry and material model parameters. The considered material is anisotropic with two preferred directions indicating the orientation of the collagen fibres in the aortic tissue. The method described in this paper is general and can be used, by specifying appropriate energy functions, to investigate other residually stressed biological systems.

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The Holzapfel-Gasser-Ogden (HGO) model for anisotropic hyperelastic behaviour of collagen fibre reinforced materials was initially developed to describe the elastic properties of arterial tissue, but is now used extensively for modelling a variety of soft biological tissues. Such materials can be regarded as incompressible, and when the incompressibility condition is adopted the strain energy Ψ of the HGO model is a function of one isotropic and two anisotropic deformation invariants. A compressible form (HGO-C model) is widely used in finite element simulations whereby the isotropic part of Ψ is decoupled into volumetric and isochoric parts and the anisotropic part of Ψ is expressed in terms of isochoric invariants. Here, by using three simple deformations (pure dilatation, pure shear and uniaxial stretch), we demonstrate that the compressible HGO-C formulation does not correctly model compressible anisotropic material behaviour, because the anisotropic component of the model is insensitive to volumetric deformation due to the use of isochoric anisotropic invariants. In order to correctly model compressible anisotropic behaviour we present a modified anisotropic (MA) model, whereby the full anisotropic invariants are used, so that a volumetric anisotropic contribution is represented. The MA model correctly predicts an anisotropic response to hydrostatic tensile loading, whereby a sphere deforms into an ellipsoid. It also computes the correct anisotropic stress state for pure shear and uniaxial deformations. To look at more practical applications, we developed a finite element user-defined material subroutine for the simulation of stent deployment in a slightly compressible artery. Significantly higher stress triaxiality and arterial compliance are computed when the full anisotropic invariants are used (MA model) instead of the isochoric form (HGO-C model).

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One of the least studied universal deformations of incompressible nonlinear elasticity, namely the straightening of a sector of a circular cylinder into a rectangular block, is revisited here and, in particular, issues of existence and stability are addressed. Particular attention is paid to the system of forces required to sustain the large static deformation, including by the application of end couples. The influence of geometric parameters and constitutive models on the appearance of wrinkles on the compressed face of the block is also studied. Different numerical methods for solving the incremental stability problem are compared and it is found that the impedance matrix method, based on the resolution of a matrix Riccati differential equation, is the more precise.

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In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging-derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.

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Inverse estimation of biomechanical parameters of soft tissues from non-invasive measurements has clinical significance in patient-specific modelling and disease diagnosis. In this paper, we propose a fully nonlinear approach to estimate the mechanical properties of the human gallbladder wall muscles from in vivo ultrasound images. The iteration method consists of a forward approach, in which the constitutive equation is based on a modified Hozapfel-Gasser-Ogden law initially developed for arteries. Five constitutive parameters describing the two orthogonal families of fibres and the matrix material are determined by comparing the computed displacements with medical images. The optimisation process is carried out using the MATLAB toolbox, a Python code, and the ABAQUS solver. The proposed method is validated with published artery data and subsequently applied to ten human gallbladder samples. Results show that the human gallbladder wall is anisotropic during the passive refilling phase, and that the peak stress is 1.6 times greater than that calculated using linear mechanics. This discrepancy arises because the wall thickness reduces by 1.6 times during the deformation, which is not predicted by conventional linear elasticity. If the change of wall thickness is accounted for, then the linear model can used to predict the gallbladder stress and its correlation with pain. This work provides further understanding of the nonlinear characteristics of human gallbladder.

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Finite strain analyses of the left ventricle provide important information on heart function and have the potential to provide insights into the biomechanics of myocardial contractility in health and disease. Systolic dysfunction is the most common cause of heart failure; however, abnormalities of diastolic function also contribute to heart failure, and are associated with conditions including left ventricular hypertrophy and diabetes. The clinical significance of diastolic abnormalities is less well understood than systolic dysfunction, and specific treatments are presently lacking. To obtain qualitative and quantitative information on heart function in diastole, we develop a three-dimensional computational model of the human left ventricle that is derived from noninvasive imaging data. This anatomically realistic model has a rule-based fibre structure and a structure-based constitutive model. We investigate the sensitivity of this comprehensive model to small changes in the constitutive parameters and to changes in the fibre distribution. We make extensive comparisons between this model and similar models that employ different constitutive models, and we demonstrate qualitative and quantitative differences in stress and strain distributions for the different constitutive models. We also provide an initial validation of our model through comparisons to experimental data on stress and strain distributions in the left ventricle.

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Estimation of biomechanical parameters of soft tissues from noninvasive measurements has clinical significance in patient-specific modeling and disease diagnosis. In this work, we present a quasi-nonlinear method that is used to estimate the elastic moduli of the human gallbladder wall. A forward approach based on a transversely isotropic membrane material model is used, and an inverse iteration is carried out to determine the elastic moduli in the circumferential and longitudinal directions between two successive ultrasound images of gallbladder. The results demonstrate that the human gallbladder behaves in an anisotropic manner, and constitutive models need to incorporate this. The estimated moduli are also nonlinear and patient dependent. Importantly, the peak stress predicted here differs from the earlier estimate from linear membrane theory. As the peak stress inside the gallbladder wall has been found to strongly correlate with acalculous gallbladder pain, reliable mechanical modeling for gallbladder tissue is crucial if this information is to be used in clinical diagnosis.

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This paper studies human gallbladder (GB) smooth muscle contractions. A two-state cross-bridge model was used to estimate the apparent attachment and detachment rate constants, as well as increased Ca2+ concentration from the peak active stress during the isometric contraction. The active stress was computed from a mechanical model based entirely on non-invasive routine ultrasound scans. In the two-state cross-bridge model, the two apparent rate constants, representing the total attached/detached cross-bridges, respectively, were estimated using active stress prediction for 51 subjects undergoing cholecystokinin-provocation test, together with estimates from the four-state cross-bridge model for a swine carotid, bovine tracheal and guinea pig GB smooth muscles. The study suggests that the apparent rate constants should be patient-specific, i.e. patients with a lower stress level are characterized by smaller apparent rate constants. In other words, the diseased GB may need to develop fast cycling cross-bridges to compensate in the emptying process. This is a first step towards more quantitative and non-invasive measures of GB pain, and may provide useful insight in understanding GB motility and developing effective drug treatments.

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This study investigates the potential correlation between acalculous biliary pain and mechanical stress during the bile-emptying phase. This study is built on the previously developed mathematical model used to estimate stress in the gallbladder wall during emptying [Li, W. G., X. Y. Luo, et al. Comput. Math. Methods Med. 9(1):27-45, 2008]. Although the total stress was correctly predicted using the previous model, the contribution from patient-specific active stress induced by the cholecystokinin (CCK) test was overlooked. In this article, we evaluate both the active and passive components of pressure in a gallbladder, which undergoes isotonic refilling, isometric contraction and emptying during the infusion of CCK. The pressure is estimated from in vivo ultrasonographical scan measurements of gallbladder emptying during CCK tests, assuming that the gallbladder is a thin ellipsoidal membrane. The passive stress is caused by the volume and shape changes during refilling at the gallbladder basal pressure, whereas the active stress arises from the pressure rise during the isometric gallbladder contraction after the CCK infusion. The effect on the stress estimates of the gallbladder to the liver is evaluated to be small by comparing numerical simulations of a gallbladder model with and without a rigid 'flat top' boundary. The model was applied to 51 subjects, and the peak total stress was found to have a strong correlation with the pain stimulated by CCK, as measured by the patient pain score questionnaires. Consistent with our previous study for a smaller sample, it is found that the success rate in predicting of CCK-induced pain is over 75%.

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The implantation and development of intravenously injected Trichinella spiralis newborn larvae were examined in different strains of inbred mice by determining muscle larvae burden. This was compared to the numbers of muscle larvae that established after a natural infection during which a quantitative assessment of intestinal newborn larvae production was made. In most inbred strains of mice, newborn larvae do not all successfully implant in muscle. Mice of the DBA/1 strain are the most resistant to successful implantation, and C3H mice are the most permissive. This pattern is evident in the strains studied whether newborn larvae are injected intravenously or are produced by intestinal adults. Thus, after a natural infection, 100% of intestinally produced newborn larvae implanted in C3H mice, whereas in NFR 68% and DBA/1 mice 62% successfully matured in muscle. Immunity to newborn larvae could be demonstrated as early as 10 days after exposure to this stage of the life cycle. This immunity was protective against a complete challenge infection given 9 days after newborn larvae had been injected intravenously. Protection against newborn larvae was identical in male and female mice or in mice from 1 to 9 months of age. We conclude that there are two mechanisms by which mice impair newborn larvae establishment or development in muscle. The first appears to be nonimmunological (non-specific resistance), and the second is immunological. Genetically determined variation in strain-specific expression is apparent with both mechanisms. In strains displaying high intrinsic "resistance" (DBA/1), this process is likely to account for most of the 38% reduction in newborn larvae establishment in a primary infection. However, immunity against newborn larvae develops quickly enough to have a significant effect on migratory larvae in primary infections where adults persist in the intestine (e.g., the B10 congenic mice), or when high adult worm burdens delay adult worm rejection. Muscle larvae burden, therefore, reflects systemic nonspecific resistance to newborn larvae as well as immunological processes that occur in the intestine and systemically.

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The nematode Trichinella spiralis is rejected from the intestine at a time that is characteristic for each inbred strain of mouse. Previous work (R. G. Bell et al. 1982a) had empirically identified strong, intermediate, and weak phenotypes (NFR, C3H/He, and C57B1/10 mice, respectively) in mice infected with 400 muscle larvae. It is shown that this classification applies to another eight inbred strains: SWR, DBA/2, DBA/1, LP, Bub/Bn--all intermediate, and NZB/BIN, C57L, A, and Mus molossinus--all weak. This phenotypic classification consistently applies with infections of 400-800 muscle larvae. Below doses of 300 muscle larvae, the strain designation of phenotype does not consistently apply. By this it is meant that the relative rejection rate changes for certain strains so that eventually some strains that were strong (NFR) or intermediate (AKR) responders to 400 muscle larvae become weak responders to 50 muscle larvae. Other strains increase their relative rejection time (B10 . BR, B10 . Q) while many do not change (NFS, C3Heb/Fe, DBA/2, DBA/1). The phenomenon is most apparent in inbred parental strains rather than in F1 crosses, and it represents a phenotypic variation in rejection time that is dependent on dose. It is also demonstrated that time of rejection is directly proportional to dose in all inbred and F1 mouse strains that we have examined. Analysis of F1 crosses shows that most have the rejection time of the strongest responding parental line, suggesting simple genetic control of strong, intermediate, and weak responses. Two F1 crosses invalidated this theory. The DBA/1 X C3H/He (intermediate X intermediate) showed a strong response. The additive effects of parental rejection phenotype indicated that these lines could not be genetically identical for intermediate responsiveness. Similarly, the NFR (strong) X B10 . BR (weak) F1 showed intermediate rejection, indicating partial dominance of C57B1/10 genes over the strong responder NFR strain. Neither the primary expulsion time phenotype, phenotypic variation to low doses, or the rejection characteristics of F1 crosses could be ascribed to genes linked to the major histocompatibility complex.

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Trypanosoma musculi infections were given to mice of different strains before, at the same time, and after an infection with 400 Trichinella spiralis. Examined parameters of the host response to T. spiralis were worm rejection, antifecundity responses, development of immunological memory, and muscle larvae burden. After dual infection, each mouse strain showed characteristic effects on resistance to T. spiralis. This was due to a dynamic interaction between the genes controlling rejection of T. spiralis and those influencing T. musculi growth. C3H mice develop high trypanosome parasitemias. This impairs worm expulsion and the development of memory to T. spiralis when Trypanosoma infections take place on the same day or 7 days before. The C57B1/6 mouse develops low parasitemias and T. musculi infections on the same day, or 7 days before T. spiralis, delaying worm rejection only slightly despite the overall weak capacity of B6 mice to expel worms. NFR-strain mice are strong responders to T. spiralis and also develop low parasitemias. Trypanosome infections on the same day, or after T. spiralis, produce a delay in worm rejection; the former is comparable to C3H mice. However, NFR mice alone showed enhanced rejection of worm when T. musculi infections preceded T. spiralis by 7 days. An unusual feature of C3H mice was that T. musculi infections 7 days before T. spiralis increased antifecundity responses at the same time that worm expulsion was inhibited. Trypanosome infections can therefore modulate distinct antihelminth immune responses in different directions simultaneously. The different outcomes of dual infections compared with single infections provides another selective mechanism by which genetic polymorphisms can be established and maintained in the vertebrate host.

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Inbred mice infected with Trypanosoma musculi displayed wide variations in peak blood parasitemia. The most susceptible mice were C3H and A strain, while Balb/c, C57B1/6, and the related congenic B10 strains were the most resistant. The effect of an intestinal infection with either Trichinella spiralis or Heligmosomoides polygyrus on proliferation of T. musculi was investigated. T. spiralis infections given at the same time or up to 45 days before a T. musculi infection always caused an increase in blood parasitemia in C3H mice. Maximum increases were observed when T. spiralis infections preceded T. musculi by 5-10 days. In all mouse strains examined, dual infections increased maximum parasitemia by two- to four-fold, regardless of the degree of resistance of that mouse strain to either T. musculi or T. spiralis. This suggested that the immunological "cost" of a T. spiralis infection was the same for strains that were strong or weak responders to a primary infection with T. spiralis. In contrast, infection with H. polygyrus did not promote T. musculi parasitemia over the level of a single infection. The increase in blood parasitemia in T. spiralis-infected mice was largely due to the intestinal adult worm, but migratory larvae and mature muscle larvae also stimulated increased parasitemias. The increase in parasitemia was proportionate to the dose of T. spiralis, and the sex of the host did not affect the blood trypanosome level.

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Rats were immunized with a Trichinella spiralis infection restricted by chemotherapy to the intestine (the T/M regime) or with a complete infection that resulted in the deposition of muscle larvae. After an oral challenge infection, rapid expulsion could be demonstrated in both groups within 20 min and with 100% recovery of the infectious dose from the stomachs and intestines of infected animals. Immune and nonimmune groups were distinguished by the large numbers of worms in the intestinal lumens of immune rats and large numbers of worms in the intestinal walls in nonimmune rats. Infectious larvae persisted in the stomach lumens for longer in the immune rats. There was no quantitative difference in worm distribution in the intestine during rapid expulsion in rats immunized with the T/M regime or those given a complete infection. However, in the complete infection group 69% of the luminal worms were trapped in mucus; this did not occur during rapid expulsion in rats immunized with the T/M regime. Mucus trapping was observed only when muscle larvae had matured to the infectious stage in muscle (28 days after the primary infection). Complete infection rats challenged at 14 or 21 days did not display significant mucus trapping of larvae in the intestinal lumen. We conclude that (i) mucus trapping is not essential for rapid expulsion and (ii) mucus trapping is produced by systemic exposure to target antigens of the infectious larvae.

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In rats and some inbred mouse strains, one immune response, rapid expulsion, confers up to 95% protection against a challenge infection with Trichinella spiralis. Strain analysis in mice has shown that only three inbred strains, all originating from Swiss-line mice at the National Institutes of Health, Bethesda, Md., express rapid expulsion. Crosses between responder strain mice (NFR/N) and nonresponders (C3H/HeJ or B10 X BR) have indicated that rapid expulsion is dominant and autosomal (Bell et al., Exp. Parasitol. 53:301-314, 1982). In this study a segregation analysis of rapid expulsion in the F2 and backcross conformed to the Mendelian ratios expected of a single gene. This gene was not linked to the major histocompatibility complex (MHC) (chromosome 17) or the gene for albinism (c/c locus on chromosome 7). This locus has not previously been identified as conferring resistance to any infectious agent, and we have therefore designated the gene Ihe-1 (intestinal helminth expulsion 1).

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