Correction to: A computational fluid dynamics approach to determine white matter permeability The article “A computational fluid dynamics approach to determine white matter permeability” written by Marco Vidotto, Daniela Botnariuc, Elena De Momi and Daniele Dini was originally published electronically on the publisher’s Internet portal (currently SpringerLink) on 20 February 2019 without open access.

Reduced-order modeling of blood flow for noninvasive functional evaluation of coronary artery disease Abstract We present a novel computational approach, based on a parametrized reduced-order model, for accelerating the calculation of pressure drop along blood vessels. Vessel lumina are defined by a geometric parametrization using the discrete empirical interpolation method on control points located on the surface of the vessel. Hemodynamics are then computed using a reduced-order representation of the parametrized three-dimensional unsteady Navier–Stokes and continuity equations. The reduced-order model is based on an offline–online splitting of the solution process, and on the projection of a finite volume full-order model on a low-dimensionality subspace generated by proper orthogonal decomposition of pressure and velocity fields. The algebraic operators of the hemodynamic equations are assembled efficiently during the online phase using the discrete empirical interpolation method. Our results show that with this approach calculations can be sped up by a factor of about 25 compared to the conventional full-order model, while maintaining prediction errors within the uncertainty limits of invasive clinical measurement of pressure drop. This is of importance for a clinically viable implementation of noninvasive, medical imaging-based computation of fractional flow reserve.

Modeling left ventricular dynamics with characteristic deformation modes Abstract A computationally efficient method is described for simulating the dynamics of the left ventricle (LV) in three dimensions. LV motion is represented as a combination of a limited number of deformation modes, chosen to represent observed cardiac motions while conserving volume in the LV wall. The contribution of each mode to wall motion is determined by a corresponding time-dependent deformation variable. The principle of virtual work is applied to these deformation variables, yielding a system of ordinary differential equations for LV dynamics, including effects of muscle fiber orientations, active and passive stresses, and surface tractions. Passive stress is governed by a transversely isotropic elastic model. Active stress acts in the fiber direction and incorporates length–tension and force–velocity properties of cardiac muscle. Preload and afterload are represented by lumped vascular models. The variational equations and their numerical solutions are verified by comparison to analytic solutions of the strong form equations. Deformation modes are constructed using Fourier series with an arbitrary number of terms. Greater numbers of deformation modes increase deformable model resolution but at increased computational cost. Simulations of normal LV motion throughout the cardiac cycle are presented using models with 8, 23, or 46 deformation modes. Aggregate quantities that describe LV function vary little as the number of deformation modes is increased. Spatial distributions of stress and strain change as more deformation modes are included, but overall patterns are conserved. This approach yields three-dimensional simulations of the cardiac cycle on a clinically relevant time-scale.

Linking microvascular collapse to tissue hypoxia in a multiscale model of pressure ulcer initiation Abstract Pressure ulcers are devastating injuries that disproportionately affect the older adult population. The initiating factor of pressure ulcers is local ischemia, or lack of perfusion at the microvascular level, following tissue compression against bony prominences. In turn, lack of blood flow leads to a drop in oxygen concentration, i.e, hypoxia, that ultimately leads to cell death, tissue necrosis, and disruption of tissue continuity. Despite our qualitative understanding of the initiating mechanisms of pressure ulcers, we are lacking quantitative knowledge of the relationship between applied pressure, skin mechanical properties as well as structure, and tissue hypoxia. This gap in our understanding is, at least in part, due to the limitations of current imaging technologies that cannot simultaneously image the microvascular architecture, while quantifying tissue deformation. We overcome this limitation in our work by combining realistic microvascular geometries with appropriate mechanical constitutive models into a microscale finite element model of the skin. By solving boundary value problems on a representative volume element via the finite element method, we can predict blood volume fractions in response to physiological skin loading conditions (i.e., shear and compression). We then use blood volume fraction as a homogenized variable to couple tissue-level skin mechanics to an oxygen diffusion model. With our model, we find that moderate levels of pressure applied to the outer skin surface lead to oxygen concentration contours indicative of tissue hypoxia. For instance, we show that applying a pressure of 60 kPa at the skin surface leads to a decrease in oxygen partial pressure from a physiological value of 65 mmHg to a hypoxic level of 31 mmHg. Additionally, we explore the sensitivity of local oxygen concentration to skin thickness and tissue stiffness, two age-related skin parameters. We find that, for a given pressure, oxygen concentration decreases with decreasing skin thickness and skin stiffness. Future work will include rigorous calibration and validation of this model, which may render our work an important tool toward developing better prevention and treatment tools for pressure ulcers specifically targeted toward the older adult patient population.

Breaking the state of the heart: meshless model for cardiac mechanics Abstract Cardiac modeling has recently emerged as a promising tool to study pathophysiology mechanisms and to predict treatment outcomes for personalized clinical decision support. Nevertheless, achieving convergence under large deformation and defining a robust meshing for realistic heart geometries remain challenging, especially when maintaining the computational cost reasonable. Smoothed particle hydrodynamics (SPH) appears to be a promising alternative to the finite element method (FEM) since it removes the burden of mesh generation. A point cloud is used where each point (particle) contains all the physical properties that are updated throughout the simulation. SPH was evaluated for solid mechanics applications in the last decade but its capacity to address the challenge of simulating the mechanics of the heart has never been evaluated. In this paper, a total Lagrangian formulation of a corrected SPH was used to solve three solid mechanics problems designed to test important features that a cardiac mechanics solver should have. SPH results, in terms of ventricle displacements and strains, were compared to results obtained with 11 different FEM-based solvers, by using synthetic cardiac data from a benchmark study. In particular, passive dilation and active contraction were simulated in an ellipsoidal left ventricle with the exponential anisotropic constitutive law of Guccione following the direction of fibers. The proposed meshless method is able to reproduce the results of three benchmark problems for cardiac mechanics. Hyperelastic material with fiber orientation and high Poisson ratio allows wall thickening/thinning when large deformation is present.

On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon Abstract In the past years, many attempts have been made in order to model the process of bone remodeling. This process is complex, as it is governed by not yet completely understood biomechanical coupled phenomena. It is well known that bone tissue is able to self-adapt to different environmental demands of both mechanical and biological origin. The mechanical aspects are related to the functional purpose of the bone tissue, i.e., to provide support to the body and protection for the vitally important organs in response to the external loads. The many biological aspects include the process of oxygen and nutrients supply. To describe the biomechanical process of functional adaptation of bone tissue, the approach commonly adopted is to consider it as a ‘feedback’ control regulated by the bone cells, namely osteoblasts and osteoclasts. They are responsible for bone synthesis and resorption, respectively, while osteocytes are in charge of ‘sensing’ the mechanical status of the tissue. Within this framework, in  Lekszycki and dell’Isola (ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 92(6):426–444, 2012), a model based on a system of integro-differential equations was introduced aiming to predict the evolution of the process of remodeling in surgically reconstructed bones. The main idea in the aforementioned model was to introduce a scalar field, describing the biological stimulus regulating the interaction among all kinds of bone cells at a macroscale. This biological field was assumed to depend locally on certain deformation measures of the (reconstructed) bone tissue. However, biological knowledge suggests that this stimulus, after having been produced, ‘diffuses’ in bone tissue, so controlling in a complex way its remodeling. This means that the cells which are target of the stimulus may not be located in the same place occupied by the cells producing it. In this paper, we propose a model which intends to explain the diffusive nature of the biological stimulus to encompass the time-dependent and space–time displaced effects involved in bone reconstruction process. Preliminary numerical simulations performed in typical cases are presented. These numerical case studies suggest that the ‘diffusive’ model of stimulus is promising: we plan to continue these kinds of studies in further investigations.

A multiscale synthesis: characterizing acute cartilage failure under an aggregate tibiofemoral joint loading Abstract Knee articular cartilage is characterized by a complex mechanical behavior, posing a challenge to develop an efficient and precise model. We argue that the cartilage damage, in general, can be traced to the fibril level as a plastic deformation, defined as micro-defects. To investigate these micro-defects, we have developed a detailed finite element model of the entire healthy tibiofemoral joint (TF) including a multiscale constitutive model which considers the structural hierarchies of the articular cartilage. The net model was simulated under physiological loading conditions to predict joint response under 2000 N axial compression and damage initiation under high axial loading (max 7 KN) when the TF joint flexed to 30°. Computed results sufficiently agreed with earlier experimental and numerical studies. Further, initiation and propagation of damage in fibrils were computed at the tibial cartilage located mainly in the superficial and middle layers. Our simulation results also indicated that the stiffer the fibril is (higher cross-link densities), the higher the contact stress required to elicit a fibril yield and the higher the rate of yielding as a function of increased contact stress. To the best of our knowledge, this is the first model that combines macro-continuum joint mechanics and micromechanics at the tissue level. The computational construct presented here serves as a simulation platform to explore the interplay between acute cartilage damage and micromechanics characteristics at the tropocollagen level.

Effect of collagenase–gelatinase ratio on the mechanical properties of a collagen fibril: a combined Monte Carlo–molecular dynamics study Abstract Loading in cartilage is supported primarily by fibrillar collagen, and damage will impair the function of the tissue, leading to pathologies such as osteoarthritis. Damage is initiated by two types of matrix metalloproteinases, collagenase and gelatinase, that cleave and denature the collagen fibrils in the tissue. Experimental and modeling studies have revealed insights into the individual contributions of these two types of MMPs, as well as the mechanical response of intact fibrils and fibrils that have experienced random surface degradation. However, no research has comprehensively examined the combined influences of collagenases and gelatinases on collagen degradation nor studied the mechanical consequences of biological degradation of collagen fibrils. Such preclinical examinations are required to gain insights into understanding, treating, and preventing degradation-related cartilage pathology. To develop these insights, we use sequential Monte Carlo and molecular dynamics simulations to probe the effect of enzymatic degradation on the structure and mechanics of a single collagen fibril. We find that the mechanical response depends on the ratio of collagenase to gelatinase—not just the amount of lost fibril mass—and we provide a possible mechanism underlying this phenomenon. Overall, by characterizing the combined influences of collagenases and gelatinases on fibril degradation and mechanics at the preclinical research stage, we gain insights that may facilitate the development of targeted interventions to prevent the damage and loss of mechanical integrity that can lead to cartilage pathology.

Assessment of the energy-related cost function over a range of walking speeds Abstract Cost funtions are needed for calculation of muscle forces in musculoskeletal models. The behavior of the energy-related cost function, proposed by Praagman et al. (J Biomech 39(4):758–765, 2006. https://doi.org/10.1016/j.jbiomech.2004.11.034) (CFP), can be used as an optimization criteria in musculoskeletal models for studying gait. In particular, in this work, its performance is compared against two empirical phenomenological models at different walking speed conditions. Also, the sensitivity of the CFP function to model parameters, such as muscle mass, maximal isometric muscle force, optimal muscle fiber length and maximum muscle velocity of the contractile element, was analyzed. The obtained results showed that CFP presents different behavior (in terms of the normalized root-mean-squared deviation (NRMSD) and the coefficient of multiple correlation (CMC)) for different muscles. Also, it provided estimates with median of NRMSD between 0.176 and 0.299 and median of CMC between 0.703 and 0.865 both metrics for slow, free and fast walking speed, which could be considered as acceptable results. Furthermore, the results indicated that CFP is insensitive to changes in muscle mass and relatively sensitive to maximal isometric muscle force. However, CFP presented a noisy behavior on estimations of muscle energy rate for some muscle as compared to phenomenological models. Finally, estimations by CFP during gait are within the values obtained by the empirical phenomenological models.

Refeeding reverses fasting-induced remodeling of afferent nerve activity in rat small intestine Abstract Intestinal afferents play an important role in coordinating intestinal motor control. Fasting induces morpho-mechanical intestinal remodeling. This study aimed to characterize the effect of fasting and refeeding on mechanosensitivity in mesenteric afferent nerves in isolated Sprague–Dawley rat jejunum. A control group fed ad libitum, a group fasted for 7 days and a group refed 7 days after 7 days fasting were studied. Jejunal segments were used for electrophysiological, histomorphological and mechanical studies. Mesenteric afferent nerve firing was recorded during a ramp distension up to 40 mmHg luminal pressure. Multiunit afferent recordings were separated into low threshold and wide-dynamic-range single-unit activity. Intestinal deformation (strain), bowel distension load (stress) and firing frequency of mesenteric afferent nerve bundles [spike rate increase ratio (SRIR)] were compared among groups. Fasting induced intestinal histomorphometric remodeling, which was reversed by refeeding. The firing frequency increased with distension in all groups. SRIR was largest in the fasting group (P < 0.05). Compared to the control group, fasting increased afferent activity in whole nerve bundles and wide-dynamic-range units at high degrees of distensions (P < 0.05 at pressure 40 mmHg; P < 0.05 at strain 1.2; P < 0.01 at stress 8 kPa). Refeeding reversed the fasting-induced afferent hypersensitivity and the shift between receptor subtypes. In conclusion, refeeding reversed fasting-induced remodeling.