Modeling of the contrast-enhanced perfusion test in liver based on the multi-compartment flow in porous media

One-Dimensional Blood Flow Modeling in the Cardiovascular System. From the Conventional Physiological Setting to Real-Life Hemodynamics

Research in the dynamics of blood flow is essential to the understanding of one of the major driving forces of human physiology. The hemodynamic conditions experienced within the cardiovascular system generate a highly variable mechanical environment that propels its function. Modeling this system is a challenging problem that must be addressed at the systemic scale to gain insight into the interplay between the different time and spatial scales of cardiovascular physiology processes. The vast majority of scientific contributions on systemic-scale distributed parameter-based blood flow modeling have approached the topic under relatively simple scenarios, defined by the resting state, the supine position, and, in some cases, by disease. However, the physiological states experienced by the cardiovascular system considerably deviate from such conditions throughout a significant part of our life. Moreover, these deviations are, in many cases, extremely beneficial for sustaining a healthy life. On top of this, inter-individual variability carries intrinsic complexities, requiring the modeling of patient-specific physiology. The impact of modeling hypotheses such as the effect of respiration, control mechanisms, and gravity, the consideration of other-than-resting physiological conditions, such as those encountered in exercise and sleeping, and the incorporation of organ-specific physiology and disease have been cursorily addressed in the specialized literature. In turn, patient-specific characterization of cardiovascular system models is in its early stages. As for models and methods, these conditions pose challenges regarding modeling the underlying phenomena and developing methodological tools to solve the associated equations. In fact, under certain conditions, the mathematical formulation becomes more intricate, model parameters suffer greater variability, and the overall uncertainty about the system's working point increases. This paper reviews current advances and opportunities to model and simulate blood flow in the cardiovascular system at the systemic scale in both the conventional resting setting and in situations experienced in everyday life.

Computational exploration of injection strategies for improving medicine distribution in the liver

BACKGROUND AND OBJECTIVES

The liver, a vital metabolic organ, is always susceptible to various diseases that ultimately lead to fibrosis, cirrhosis, acute liver failure, chronic liver failure, and even cancer. Optimal and specific medicine delivery in various diseases, hepatectomy, shunt placement, and other surgical interventions to reduce liver damage, transplantation, optimal preservation, and revival of the donated organ all rely on a complete understanding of perfusion and mass transfer in the liver. This study aims to simulate the computational fluid dynamics of perfusion and the temporal-spatial distribution of a medicine in a healthy liver to evaluate the hemodynamic characteristics of flow and medicine transport with the purpose of more effective liver treatment.

METHODS

Patient-specific geometries of parenchyma and hepatic artery, portal vein, and hepatic vein vessels of a healthy liver were segmented and reconstructed from the abdominal computed tomography scan images. Mesh was generated for the comprehensive combined model using unstructured tetrahedral elements. Transient pressure values were applied as boundary conditions at the portal vein and hepatic artery inlets, and pressure outlet boundary condition was assumed at the hepatic vein outlet. Medicine injection was done through the portal vein. The liver parenchyma was assumed to be a porous medium. Finally, computational fluid dynamics (CFD) simulation was performed to investigate blood perfusion, medicine distribution, and saturation time.

RESULTS

The velocity parameter values calculated for the hepatic artery, portal vein, and hepatic vein vessels were consistent with the physiological ranges. The mass flow rate was higher in the portal vein than in the hepatic artery, which was consistent with high perfusion through the portal vein. The portal pressure gradient was 8.53 mmHg. From a pharmacokinetic viewpoint, medicine distribution in porous tissue was a heterogeneous process. Medicine distribution was higher at end-diastolic pressure than at peak-systolic pressure which showed the influence of hepatic artery flow. The tissue was saturated faster at first 40 s and with decreasing porosity, saturation time decreased, and distribution improved.

CONCLUSION

The right lobe included a higher number of vascular terminals due to its larger volume, and the flow rate was higher in this lobe compared to the left lobe. This showed the significant effect of the right lobe on the overall function of the body. Recirculation flow zones along hepatic artery and portal vein branches emphasized the sensitivity of downstream vessels. Rotational flow and potential vortex formation at the hepatic vein outlet may indicate a risk of plaque and clot formation in this region. The heterogeneous distribution of medicine indicated the importance of injection time in treating liver diseases. The percentage of tissue porosity affected the saturation time, so adjusting the medicine dose and injection time could be challenging in treatments.

Quantifying fat zonation in liver lobules: an integrated multiscale in silico model combining disturbed microperfusion and fat metabolism via a continuum biomechanical bi-scale, tri-phasic approach

Metabolic zonation refers to the spatial separation of metabolic functions along the sinusoidal axes of the liver. This phenomenon forms the foundation for adjusting hepatic metabolism to physiological requirements in health and disease (e.g., metabolic dysfunction-associated steatotic liver disease/MASLD). Zonated metabolic functions are influenced by zonal morphological abnormalities in the liver, such as periportal fibrosis and pericentral steatosis. We aim to analyze the interplay between microperfusion, oxygen gradient, fat metabolism and resulting zonated fat accumulation in a liver lobule. Therefore we developed a continuum biomechanical, tri-phasic, bi-scale, and multicomponent in silico model, which allows to numerically simulate coupled perfusion-function-growth interactions two-dimensionally in liver lobules. The developed homogenized model has the following specifications: (i) thermodynamically consistent, (ii) tri-phase model (tissue, fat, blood), (iii) penta-substances (glycogen, glucose, lactate, FFA, and oxygen), and (iv) bi-scale approach (lobule, cell). Our presented in silico model accounts for the mutual coupling between spatial and time-dependent liver perfusion, metabolic pathways and fat accumulation. The model thus allows the prediction of fat development in the liver lobule, depending on perfusion, oxygen and plasma concentration of free fatty acids (FFA), oxidative processes, the synthesis and the secretion of triglycerides (TGs). The use of a bi-scale approach allows in addition to focus on scale bridging processes. Thus, we will investigate how changes at the cellular scale affect perfusion at the lobular scale and vice versa. This allows to predict the zonation of fat distribution (periportal or pericentral) depending on initial conditions, as well as external and internal boundary value conditions.

Periphery of porcine hepatic lobes has the smallest length density of hepatic sinusoids and bile canaliculi: A stereological histological study with implications for liver biopsies

BACKGROUND

Porcine liver is widely used in hepatologic research as a large animal model with many anatomical and physiological similarities with humans. However, only limited information on porcine liver spatial microstructure has been published, especially regarding the hepatic sinusoids and bile canaliculi. The aim of our study was to quantify the sinusoidal and bile canalicular network in healthy male and female porcine livers and to map the variability of these structures with heterogenous distribution to improve the evaluability of liver biopsy samples.

METHODS

Livers from 12 healthy piglets (6 females and 6 neutered males) were sampled into 36 tissue samples per organ, representing six hepatic lobes and three different regions related to the hepatic vasculature (peripheral, paracaval and paraportal region). Histological sections were processed with a random orientation of the cutting plane. The endothelium and the bile canaliculi were stained using Ricinus communis agglutinin I lectin histochemistry. The length densities of hepatic sinusoids LV(sinusoids,liver), of bile canaliculi LV(bile canaliculi,liver) and volume fraction VV(sinusoids,liver) and surface density SV(sinusoids,liver) of sinusoids were estimated using stereological methods. The newly acquired morphometric data were compared with previously published data on density of porcine hepatocytes and fractions of connective tissue.

RESULTS

The peripheral region had smallest LV(sinusoids,liver), smallest LV(bile canaliculi,liver) and greatest VV(sinusoids,liver). The six hepatic lobes had statistically comparable length densities of both sinusoids and bile canaliculi, but the left lateral lobe had smallest VV(sinusoids,liver). Regions with greater LV(sinusoids,liver) had also greater LV(bile canaliculi,liver) and SV(sinusoids,liver) and were accompanied by greater density of smaller hepatocytes. Regions with smaller LV(sinusoids,liver) and LV(bile canaliculi,liver) contained a greater fraction of interlobular connective tissue.

CONCLUSIONS

The length density of hepatic sinusoids is smaller in the peripheral regions of the porcine liver than in other regions related to the hepatic vasculature - paracaval and paraportal regions, and smaller in castrated males than in females. Greater length density of liver sinusoids was linked with greater local density of bile canaliculi, with local increase in the density of smaller hepatocytes and, simultaneously, with smaller fractions of hepatic connective tissue. The intrahepatic and inter-sexual variability of the porcine liver morphology needs to be taken into account when designing and interpreting experiments involving the histological quantification of the microvascular network. The complete primary morphometric data describing the distribution of morphometric parameters within porcine liver were made available in a form facilitating the power analysis to justify the minimal number of tissue samples or animals required when designing further histological evaluation studies. The macroscopic map of microvessels and bile canaliculi variability facilitates their assessment in liver biopsies in the pig.

Modeling a 3-D multiscale blood-flow and heat-transfer framework for realistic vascular systems

Modeling of biological domains and simulation of biophysical processes occurring in them can help inform medical procedures. However, when considering complex domains such as large regions of the human body, the complexities of blood vessel branching and variation of blood vessel dimensions present a major modeling challenge. Here, we present a Voxelized Multi-Physics Simulation (VoM-PhyS) framework to simulate coupled heat transfer and fluid flow using a multi-scale voxel mesh on a biological domain obtained. In this framework, flow in larger blood vessels is modeled using the Hagen–Poiseuille equation for a one-dimensional flow coupled with a three-dimensional two-compartment porous media model for capillary circulation in tissue. The Dirac distribution function is used as Sphere of Influence (SoI) parameter to couple the one-dimensional and three-dimensional flow. This blood flow system is coupled with a heat transfer solver to provide a complete thermo-physiological simulation. The framework is demonstrated on a frog tongue and further analysis is conducted to study the effect of convective heat exchange between blood vessels and tissue, and the effect of SoI on simulation results.

A 1D-0D-3D coupled model for simulating blood flow and transport processes in breast tissue

In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system necessitating the use of different flow models for different geometries and flow regimes. In total, we consider four different model types. First, a system of 1D nonlinear hyperbolic partial differential equations (PDEs) is considered to simulate blood flow in larger arteries with highly elastic vessel walls. Second, we assign 1D linearized hyperbolic PDEs to model the smaller arteries with stiffer vessel walls. The third model type consists of ODE systems (0D models). It is used to model the arterioles and peripheral circulation. Finally, homogenized 3D porous media models are considered to simulate flow and transport in capillaries and tissue within the breast volume. Sink terms are used to account for the influence of the venous and lymphatic systems. Combining the four model types, we obtain two different 1D-0D-3D coupled models for simulating blood flow and transport processes: The first model results in a fully coupled 1D-0D-3D model covering the complete path from the aorta to the breast combining a generic arterial network with a patient specific breast network and geometry. The second model is a reduced one based on the separation of the generic and patient specific parts. The information from a calibrated fully coupled model is used as inflow condition for the patient specific sub-model allowing a significant computational cost reduction. Several numerical experiments are conducted to calibrate the generic model parameters and to demonstrate realistic flow simulations compared to existing data on blood flow in the human breast and vascular system. Moreover, we use two different breast vasculature and tissue data sets to illustrate the robustness of our reduced sub-model approach.

Rigorous mathematical optimization of synthetic hepatic vascular trees

In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure with the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.

Geometrical model of lobular structure and its importance for the liver perfusion analysis

A convenient geometrical description of the microvascular network is necessary for computationally efficient mathematical modelling of liver perfusion, metabolic and other physiological processes. The tissue models currently used are based on the generally accepted schematic structure of the parenchyma at the lobular level, assuming its perfect regular structure and geometrical symmetries. Hepatic lobule, portal lobule, or liver acinus are considered usually as autonomous functional units on which particular physiological problems are studied. We propose a new periodic unit-the liver representative periodic cell (LRPC) and establish its geometrical parametrization. The LRPC is constituted by two portal lobulae, such that it contains the liver acinus as a substructure. As a remarkable advantage over the classical phenomenological modelling approaches, the LRPC enables for multiscale modelling based on the periodic homogenization method. Derived macroscopic equations involve so called effective medium parameters, such as the tissue permeability, which reflect the LRPC geometry. In this way, mutual influences between the macroscopic phenomena, such as inhomogeneous perfusion, and the local processes relevant to the lobular (mesoscopic) level are respected. The LRPC based model is intended for its use within a complete hierarchical model of the whole liver. Using the Double-permeability Darcy model obtained by the homogenization, we illustrate the usefulness of the LRPC based modelling to describe the blood perfusion in the parenchyma.

A nonlinear multi-scale model for blood circulation in a realistic vascular system

In the last decade, numerical models have become an increasingly important tool in biological and medical science. Numerical simulations contribute to a deeper understanding of physiology and are a powerful tool for better diagnostics and treatment. In this paper, a nonlinear multi-scale model framework is developed for blood flow distribution in the full vascular system of an organ. We couple a quasi one-dimensional vascular graph model to represent blood flow in larger vessels and a porous media model to describe flow in smaller vessels and capillary bed. The vascular model is based on Poiseuille's Law, with pressure correction by elasticity and pressure drop estimation at vessels' junctions. The porous capillary bed is modelled as a two-compartment domain (artery and venous) using Darcy's Law. The fluid exchange between the artery and venous capillary bed compartments is defined as blood perfusion. The numerical experiments show that the proposed model for blood circulation: (i) is closely dependent on the structure and parameters of both the larger vessels and of the capillary bed, and (ii) provides a realistic blood circulation in the organ. The advantage of the proposed model is that it is complex enough to reliably capture the main underlying physiological function, yet highly flexible as it offers the possibility of incorporating various local effects. Furthermore, the numerical implementation of the model is straightforward and allows for simulations on a regular desktop computer.

Modelling the impact of clot fragmentation on the microcirculation after thrombectomy

Many ischaemic stroke patients who have a mechanical removal of their clot (thrombectomy) do not get reperfusion of tissue despite the thrombus being removed. One hypothesis for this 'no-reperfusion' phenomenon is micro-emboli fragmenting off the large clot during thrombectomy and occluding smaller blood vessels downstream of the clot location. This is impossible to observe in-vivo and so we here develop an in-silico model based on in-vitro experiments to model the effect of micro-emboli on brain tissue. Through in-vitro experiments we obtain, under a variety of clot consistencies and thrombectomy techniques, micro-emboli distributions post-thrombectomy. Blood flow through the microcirculation is modelled for statistically accurate voxels of brain microvasculature including penetrating arterioles and capillary beds. A novel micro-emboli algorithm, informed by the experimental data, is used to simulate the impact of micro-emboli successively entering the penetrating arterioles and the capillary bed. Scaled-up blood flow parameters-permeability and coupling coefficients-are calculated under various conditions. We find that capillary beds are more susceptible to occlusions than the penetrating arterioles with a 4x greater drop in permeability per volume of vessel occluded. Individual microvascular geometries determine robustness to micro-emboli. Hard clot fragmentation leads to larger micro-emboli and larger drops in blood flow for a given number of micro-emboli. Thrombectomy technique has a large impact on clot fragmentation and hence occlusions in the microvasculature. As such, in-silico modelling of mechanical thrombectomy predicts that clot specific factors, interventional technique, and microvascular geometry strongly influence reperfusion of the brain. Micro-emboli are likely contributory to the phenomenon of no-reperfusion following successful removal of a major clot.

A 3D-1D coupled blood flow and oxygen transport model to generate microvascular networks

In this work, we introduce an algorithmic approach to generate microvascular networks starting from larger vessels that can be reconstructed without noticeable segmentation errors. Contrary to larger vessels, the reconstruction of fine-scale components of microvascular networks shows significant segmentation errors, and an accurate mapping is time and cost intense. Thus there is a need for fast and reliable reconstruction algorithms yielding surrogate networks having similar stochastic properties as the original ones. The microvascular networks are constructed in a marching way by adding vessels to the outlets of the vascular tree from the previous step. To optimise the structure of the vascular trees, we use Murray's law to determine the radii of the vessels and bifurcation angles. In each step, we compute the local gradient of the partial pressure of oxygen and adapt the orientation of the new vessels to this gradient. At the same time, we use the partial pressure of oxygen to check whether the considered tissue block is supplied sufficiently with oxygen. Computing the partial pressure of oxygen, we use a 3D-1D coupled model for blood flow and oxygen transport. To decrease the complexity of a fully coupled 3D model, we reduce the blood vessel network to a 1D graph structure and use a bi-directional coupling with the tissue which is described by a 3D homogeneous porous medium. The resulting surrogate networks are analysed with respect to morphological and physiological aspects.

An image-based geometric model for numerical simulation of blood perfusion within the liver lobules

An image-based numerical algorithm is presented for simulating blood flow through the liver tissue. First, a geometric model is constructed by applying image processing techniques on a real microscopic image of a liver tissue. Then, incompressible blood flow through liver lobules is simulated. Effects of tissue heterogeneity and deformity, presence/absence of the second central vein in a particular lobule, and apparent sinusoids density in the liver cross section on the blood flow are investigated. Numerical results indicate that the existence of thick low permeability vascular septum, high permeability sinusoids, and lobule tissue heterogeneity can considerably affect interlobular and intralobular blood flow.

A new framework for assessing subject-specific whole brain circulation and perfusion using MRI-based measurements and a multi-scale continuous flow model

A large variety of severe medical conditions involve alterations in microvascular circulation. Hence, measurements or simulation of circulation and perfusion has considerable clinical value and can be used for diagnostics, evaluation of treatment efficacy, and for surgical planning. However, the accuracy of traditional tracer kinetic one-compartment models is limited due to scale dependency. As a remedy, we propose a scale invariant mathematical framework for simulating whole brain perfusion. The suggested framework is based on a segmentation of anatomical geometry down to imaging voxel resolution. Large vessels in the arterial and venous network are identified from time-of-flight (ToF) and quantitative susceptibility mapping (QSM). Macro-scale flow in the large-vessel-network is accurately modelled using the Hagen-Poiseuille equation, whereas capillary flow is treated as two-compartment porous media flow. Macro-scale flow is coupled with micro-scale flow by a spatially distributing support function in the terminal endings. Perfusion is defined as the transition of fluid from the arterial to the venous compartment. We demonstrate a whole brain simulation of tracer propagation on a realistic geometric model of the human brain, where the model comprises distinct areas of grey and white matter, as well as large vessels in the arterial and venous vascular network. Our proposed framework is an accurate and viable alternative to traditional compartment models, with high relevance for simulation of brain perfusion and also for restoration of field parameters in clinical brain perfusion applications.