Mass transport is a fundamental process in living organisms, responsible for the delivery of oxygen, nutrients, and drugs from blood vessels to tissues, as well as the removal of waste products back to the vascular and lymphatic systems. This exchange occurs across various biological barriers, including blood vessel walls and cellular membranes, and involves complex mechanical and biochemical interactions. Traditional experimental and clinical methods have provided valuable insights into mass transport, but due to the complexity and heterogeneity of biological systems, computational models are essential for deeper understanding. Here, we give a brief review of a smeared modeling concept, termed the Kojic Transport Model (KTM), for gradient-driven physical fields within composite media such as biological tissue. The basic idea of the KTM is the formulation of the composite smeared finite element (CSFE). This element is composed of volumetric domains with different physical fields that are coupled by connectivity elements at each FE node representing biological barriers. The domains include 1D fields represented by 3D continuum form by formulation of the consistent transport tensors The KTM has been implemented in our finite element package PAK and demonstrated to be accurate and robust in modeling. We have selected models of mass transport (diffusion and perfusion) in the liver, pancreas, and tumor which grows over time.
Understanding the biomechanical response of a femoral bone under physiological loads is critical for assessing bone health and fracture risk, particularly in aging populations affected by osteopenia and osteoporosis. This study investigates the mechanical behavior of the femoral bone under standing loading conditions across three bone states: healthy bone, bone with osteopenia, and bone with osteoporosis. In-house finite element solver, PAK, was used to calculate von Mises stress and displacement values for each condition. The influence of reduced bone density on structural integrity was examined by comparing results among the three models. Displacement values increased from 0.724 mm in the healthy model to 1.02 mm in bone with osteopenia and 1.6 mm in bone with osteoporosis. For von Mises stress, the values indicate a slight decrease in stress values with bone density loss. Results obtained by the inhouse solver were validated against those obtained from the commercial solver Nastran, showing strong agreement in both stress and displacement values. The findings highlight the importance of early diagnosis of bone diseases and suggest that material properties based on patient-specific data could further enhance the predictive power of such simulations.
Aortic dissection is a severe clinical condition that demands a thorough understanding of the interactions between the true and false lumens. In this study, numerical simulations were performed for two patients, generating two preoperative and two postoperative models. The analyses were carried out using the PAKSF solver, a specialized tool for fluid-structure interaction problems. The objective was to evaluate the influence of virtual surgical interventions by examining essential hemodynamic parameters, including shear stress, pressure, and flow velocity in both lumens. The findings indicate that virtual surgical modeling can serve as a valuable tool for surgical planning, providing improved insight into post-intervention hemodynamic alterations. Moreover, this methodology supports patient-specific treatment strategies, contributing to optimized outcomes. Overall, the results highlight the potential of numerical simulations in clinical practice, particularly for the management of complex vascular disorders..
The extremely complex biological and biochemical processes are mainly treated by laboratory and clinical investigations. In recent decades, the application of computational modeling has become more important in research and applications in the medical sciences. Despite enormous efforts and achievements in this modeling, there is still a need for new efficient and reliable methods, particularly within today's hot field - Artificial Intelligence (AI). In this report, we present a brief description of a methodology that we believe offers a basis for modeling gradient-driven physical fields in composite media, such as tissue. This methodology is based on the concept of multiscale smeared physical fields, termed the Kojic Transport Model (KTM), that is published in several journal papers and summarized in a recent book (Kojic et al. 2022). Our KTM includes modeling of partitioning, blood flow, molecular transport within the tissue, a multiscale-multiphysics model of coupling electrical field and ion concentration, and a model of convective-diffusive transport within the lung parenchyma. We present here two typical examples for the illustration of our KTM application.
Metastasis involves the dissemination of circulating tumor cells (CTCs) from the primary tumor, their survival in the bloodstream, and eventual colonization of distant organs. Adhesive interactions with platelets, leukocytes, and endothelial cells are critical to CTC survival and extravasation. Platelet adhesion enhances CTC viability and promotes metastasis. To investigate the biomechanical conditions governing CTC arrest, we developed a computational platform within the PAK software, integrating 2D axisymmetric solid-fluid models. This framework evaluates CTC transit through capillaries under physiological conditions and examines how capillary pressure gradients, CTC size and stiffness, and platelet size influence arrest dynamics. A parametric analysis was conducted to assess the effects of platelet number, CTC mechanical properties, and ligand-receptor bond stiffness on CTC trajectory, axial positioning, and endothelial adhesion. Both resting and thrombin-activated platelets were modeled, incorporating experimental adhesion force data via 1D finite element truss elements to simulate ligand-receptor interactions. The study establishes quantitative relationships between cellular and molecular parameters and the mechanical thresholds for CTC arrest, providing insights into early metastatic events and improving predictive models of metastasis progression. This paper is a summary of our previous publications as a document of our research related to 50 years of the development of our PAK finite element software
This study presents the use of computational modeling to improve understanding of hypertrophic cardiomyopathy (HCM) and its progression to heart failure. A review of related work highlights advances and challenges in computational modeling and integration of clinical data for model development and validation. Using parametric model of the left heart ventricle, the multiscale Fluid-Solid Interaction (FSI) is performed within the PAK Finite Element (FE) software, simulating cardiac cycles in two patients with clinically diagnosed HCM, and analyzing biomechanical parameters such as pressure, velocity, displacement, and pressure-volume (PV) loops. The results show strong agreement with clinical data, particularly for left ventricular ejection fraction (LVEF). Our findings demonstrate the value of computational tools for patient-specific, non-invasive assessment of cardiac function and disease progression. Continuous development of PAK software and its integration with clinical data will further enhance its role in cardiovascular research, supporting diagnosis, treatment planning, and personalized care in cardiomyopathies and heart failure.
Two-equation turbulent models are widely used in finite element analysis of fluid flow. The PAK program package was upgraded from the classical laminar flow to the turbulent k-omega model. The turbulence model was validated using the case of fluid flow in a backward-facing step channel. The analysis results show good agreement with experimental data reported in the literature. The code was subsequently tested on a case of fluid flow in a coronary artery bifurcation, with particular emphasis on the stenosis resulting from the patient's disease.
The interaction between the Finite Element Method (FEM) and Smoothed Particle Hydrodynamics (SPH) is crucial in multiphysics simulations of granular-structure systems, impacts, and material behavior. This paper presents a penalty-based contact algorithm with Coulomb friction for FEM-SPH coupling. The method accounts for static and dynamic friction and ensures stable force transfer between SPH particles and FEM nodes. A numerical example of granular pile sliding on a steel plate demonstrates the approach. The results confirm that the proposed algorithm provides an efficient and robust framework for FEM-SPH interaction, improving the applicability of SPH in structural, geomechanical, and fluid-structure simulations.
This paper presents the integration of muscle fatigue modeling into the finite element solver PAK, based on an extended version of Hill's phenomenological model. While traditional muscle models focus primarily on force generation, this study incorporates fatigue dynamics to provide a more realistic representation of muscle performance over time. By extending Hill's three-component model to take into account different types of muscle fibers and their distinct fatigue characteristics, we improve the accuracy of computational muscle simulations. The proposed approach employs functionally graded materials (FGM) to model heterogeneous muscle structures and utilizes an incremental-iterative finite element scheme to calculate equilibrium configurations. The developed model is validated through comparisons with experimental data, demonstrating its ability to capture key aspects of muscle contraction, force production, fatigue progression, and recovery. The implementation of this model in PAK provides a powerful computational tool for biomechanical research, with potential applications in rehabilitation engineering, sports science, and musculoskeletal system simulations. Future work will focus on refining fatigue mechanisms and extending the model to simulate full musculoskeletal interactions. This study contributes to the advancement of computational biomechanics by enabling more accurate and physiologically relevant simulations of muscle function.
Degradation of structural integrity and fatigue life estimation remain critical challenges in engineering, including biomedical applications. Fracture mechanics and crack propagation prediction are highly sensitive to material parameters, with the Stress Intensity Factor being the most significant physical parameter for the estimation of crack stress fields. This paper applies a fatigue crack growth model and structural integrity assessment using advanced numerical methods. The model calculates Stress Intensity Factor via the J-Equivalent Domain method, implemented in the in-house PAK software. Crack growth is simulated using the Extended Finite Element Method, incorporating discontinuous functions and asymptotic crack-tip displacement fields through Partition of Unity and Fast Marching-Level Set methods, which eliminates explicit crack meshing. The approach is validated through case studies from both classic engineering and biomedical structures.
Solvent removal by diffusion from a polymer solution occurs in various technological processes, such dry spinning of synthetic silk-like fibers. There, it is important to determine the diffusion coefficient which is dependent on the solvent concentration, assuming the validity of Fick's law. In order to prove the internal diffusion within the dope while traveling through the spider canal, we performed the experimental investigation and developed specific finite element models. We first summarize our methodology of the numerical computation of such diffusion coefficient, published in our reference (Koji & cacute; et al., 2006). The diffusion coefficient is determined by matching the mass of the solution computed by the finite element (FE) model, and the mass measured using a pan-weighing experiment, in which a small amount of the polymer solution is placed in a pan and allowed to evaporate into the air. The second part of this report, according to our reference (Koji & cacute; et al., 2004), is devoted to exploring the process in which a spider generates a fiber with the extraordinary strength of several orders higher than any technologically produced fiber. It was hypothesized that the governing process within the spider canal where the elongation flow of the dope occurs is the radial diffusion of the water with zero concentration at the canal wall. The computational model for the water diffusion from the dope is developed with the corresponding boundary conditions to confirm our hypothesis.
Finite element (FE) modeling the motion of deformable bodies within a fluid, as well as the interaction between the bodies, has been a continuous challenge over the years. There are in principle two approaches-loose and strong coupling between the FE models of the two domains, with their advantages and drawbacks. We here present our concept of strong coupling which we thoroughly tested in the past, particularly with respect to the accuracy of the model, Theoretical background and computational procedure are briefly described and then illustrated on two typical applications, among numerous in our references. One is related to the microfluidic chip and another-to conditions of the passing of the circulating tumor cells (CTCs) through a capillary narrowing; both are significant in biomedical engineering. Regarding the first example, we have that in recent years, attention has been paid to the creation of the so-called microfluidic chips. These devices are used for testing very small blood samples. There, blood is passed through a micro-device in which cells are separated according to cell stiffness under precisely controlled laminar flow conditions. The development of the microfluidic chip can be significantly accelerated and simplified by building its computational replica-a computer program designed for modeling the chip's geometry and topology, as well as simulating the processes occurring within it. In the second example, we explored the parameter space, where a relationship between the capillary blood pressure gradient and the circulating tumor cells (CTCs) mechanical properties (size and stiffness) was determined for the CTC arrest in a capillary. The presented computational platform, along with the derived pressure-size-stiffness relationship, serves as a valuable tool for investigating the biomechanical conditions underlying capillary arrest of circulating tumor cells (CTCs) and CTC clusters, enabling predictive modeling of metastatic progression based on biophysical CTC parameters, and supporting the rational design of size-based CTC isolation technologies that account for large CTC deformations under high pressure gradients.
This paper contains two parts - the first is related to the mechanics of lung microstructure, and the second is a formulation of a multiscale-multiphysics model of the lung. The second part considers the entire lung that relies on the mechanics of the microstructure and recently developed a multiscale composite finite element of the lung tissue (MSCL). The MSCL is further extended to include airflow, blood flow, and mass transport by diffusion, as a General Lung Finite Element (GLFE). The microstructural mechanics relies on the generally accepted Wilson-Bachofen model of balance between internal and external forces of the lung supporting system. Besides the stresses in the tissue, the effects of the surfactant play an important role in the alveolated microstructural system, vital for lung functioning. Our microstructural model demonstrates a geometric hysteresis within a duct. The multiscale-multiphysics model for airflow and blood flow is based on our smeared concept (Kojic Transport Model, KTM) where the subdomains within our Composite Smeared Finite Element (CSFE) represent the airway generations; and capillary, extracellular space and cells for blood flow and mass transport by blood. This computational methodology is built into our finite element code PAK.
