A physiologically-based flow network model for hepatic drug elimination I: regular lattice lobule model

Enlarging the thermal coagulation volume during thermochemical ablation with alternating acid-base injection by shortening the injection interval: A computational study

BACKGROUND AND OBJECTIVES

Thermochemical ablation (TCA) is a cancer treatment that utilises the heat released from the neutralisation of acid and base to raise tissue temperature to levels sufficient to induce thermal coagulation. Computational studies have demonstrated that the coagulation volume produced by sequential injection is smaller than that with simultaneous injection. By injecting the reagents in an ensuing manner, the region of contact between acid and base is limited to a thin contact layer sandwiched between the distribution of acid and base. It is hypothesised that increasing the frequency of acid-base injections into the tissue by shortening the injection interval for each reagent can increase the effective area of contact between acid and base, thereby intensifying neutralisation and the exothermic heat released into the tissue.

METHODS

To verify this hypothesis, a computational model was developed to simulate the thermochemical processes involved during TCA with sequential injection. Four major processes that take place during TCA were considered, i.e., the flow of acid and base, their neutralisation, the release of exothermic heat and the formation of thermal damage inside the tissue. Equimolar acid and base at 7.5 M was injected into the tissue intermittently. Six injection intervals, namely 3, 6, 15, 20, 30 and 60 s were investigated.

RESULTS

Shortening of the injection interval led to the enlargement of coagulation volume. If one considers only the coagulation volume as the determining factor, then a 15 s injection interval was found to be optimum. Conversely, if one places priority on safety, then a 3 s injection interval would result in the lowest amount of reagent residue inside the tissue after treatment. With a 3 s injection interval, the coagulation volume was found to be larger than that of simultaneous injection with the same treatment parameters. Not only that, the volume also surpassed that of radiofrequency ablation (RFA); a conventional thermal ablation technique commonly used for liver cancer treatment.

CONCLUSION

The numerical results verified the hypothesis that shortening the injection interval will lead to the formation of larger thermal coagulation zone during TCA with sequential injection. More importantly, a 3 s injection interval was found to be optimum for both efficacy (large coagulation volume) and safety (least amount of reagent residue).

An in silico assessment on the potential of using saline infusion to overcome non-confluent coagulation zone during two-probe, no-touch bipolar radiofrequency ablation of liver cancer

No-touch bipolar radiofrequency ablation (bRFA) is known to produce incomplete tumour ablation with a 'butterfly-shaped' coagulation zone when the interelectrode distance exceeds a certain threshold. Although non-confluent coagulation zone can be avoided by not implementing the no-touch mode, doing so exposes the patient to the risk of tumour track seeding. The present study investigates if prior infusion of saline into the tissue can overcome the issues of non-confluent or butterfly-shaped coagulation. A computational modelling approach based on the finite element method was carried out. A two-compartment model comprising the tumour that is surrounded by healthy liver tissue was developed. Three cases were considered; i) saline infusion into the tumour centre; ii) one-sided saline infusion outside the tumour; and iii) two-sided saline infusion outside the tumour. For each case, three different saline volumes were considered, i.e. 6, 14 and 22 ml. Saline concentration was set to 15% w/v. Numerical results showed that saline infusion into the tumour centre can overcome the butterfly-shaped coagulation only if the infusion volume is sufficient. On the other hand, one-sided infusion outside the tumour did not overcome this. Two-sided infusion outside the tumour produced confluent coagulation zone with the largest volume. Results obtained from the present study suggest that saline infusion, when carried out correctly, can be used to effectively eradicate liver cancer. This presents a practical solution to address non-confluent coagulation zone typical of that during two-probe bRFA treatment.

An in silico derived dosage and administration guide for effective thermochemical ablation of biological tissues with simultaneous injection of acid and base

BACKGROUND AND OBJECTIVES

Thermochemical ablation (TCA) is a thermal ablation technique involving the injection of acid and base, either sequentially or simultaneously, into the target tissue. TCA remains at the conceptual stage with existing studies unable to provide recommendations on the optimum injection rate, and reagent concentration and volume. Limitations in current experimental methodology have prevented proper elucidation of the thermochemical processes inside the tissue during TCA. Nevertheless, the computational TCA framework developed recently by Mak et al. [Mak et al., Computers in Biology and Medicine, 2022, 145:105494] has opened new avenues in the development of TCA. Specifically, a recommended safe dosage is imperative in driving TCA research beyond the conceptual stage.

METHODS

The aforesaid computational TCA framework for sequential injection was applied and adapted to simulate TCA with simultaneous injection of acid and base at equimolar and equivolume. The developed framework, which describes the flow of acid and base, their neutralisation, the rise in tissue temperature and the formation of thermal damage, was solved numerically using the finite element method. The framework will be used to investigate the effects of injection rate, reagent concentration, volume and type (weak/strong acid-base combination) on temperature rise and thermal coagulation formation.

RESULTS

A higher injection rate resulted in higher temperature rise and larger thermal coagulation. Reagent concentration of 7500 mol/m³ was found to be optimum in producing considerable thermal coagulation without the risk of tissue overheating. Thermal coagulation volume was found to be consistently larger than the total volume of acid and base injected into the tissue, which is beneficial as it reduces the risk of chemical burn injury. Three multivariate second-order polynomials that express the targeted coagulation volume as functions of injection rate and reagent volume, for the weak-weak, weak-strong and strong-strong acid-base combinations were also derived based on the simulated data.

CONCLUSIONS

A guideline for a safe and effective implementation of TCA with simultaneous injection of acid and base was recommended based on the numerical results of the computational model developed. The guideline correlates the coagulation volume with the reagent volume and injection rate, and may be used by clinicians in determining the safe dosage of reagents and optimum injection rate to achieve a desired thermal coagulation volume during TCA.

An ultrasonically actuated needle promotes the transport of nanoparticles and fluids

Non-invasive therapeutic ultrasound (US) methods, such as high-intensity focused ultrasound (HIFU), have limited access to tissue targets shadowed by bones or presence of gas. This study demonstrates that an ultrasonically actuated medical needle can be used to translate nanoparticles and fluids under the action of nonlinear phenomena, potentially overcoming some limitations of HIFU. A simulation study was first conducted to study the delivery of a tracer with an ultrasonically actuated needle (33 kHz) inside a porous medium acting as a model for soft tissue. The model was then validated experimentally in different concentrations of agarose gel showing a close match with the experimental results, when diluted soot nanoparticles (diameter < 150 nm) were employed as delivered entity. An additional simulation study demonstrated a threefold increase in the volume covered by the delivered agent in liver under a constant injection rate, when compared to without US. This method, if developed to its full potential, could serve as a cost effective way to improve safety and efficacy of drug therapies by maximizing the concentration of delivered entities within, e.g., a small lesion, while minimizing exposure outside the lesion.

A computational framework to simulate the thermochemical process during thermochemical ablation of biological tissues

Thermochemical ablation (TCA) is a thermal ablation therapy that utilises heat released from acid-base neutralisation reaction to destroy tumours. This procedure is a promising low-cost solution to existing thermal ablation treatments such as radiofrequency ablation (RFA) and microwave ablation (MWA). Studies have demonstrated that TCA can produce thermal damage that is on par with RFA and MWA when employed properly. Nevertheless, TCA remains a concept that is tested only in a few animal trials due to the risks involved as the result of uncontrolled infusion and incomplete acid-base reaction. In this study, a computational framework that simulates the thermochemical process of TCA is developed. The proposed framework consists of three physics, namely chemical flow, neutralisation reaction and heat transfer. An important parameter in the TCA framework is the neutralisation reaction rate constant, which has values in the order of 10⁸ m³/(mol⋅s). The present study will demonstrate that since the rate constant impacts only the rate and direction of the reaction but has little influence on the extent of reaction, it is possible to replicate the thermochemical process of TCA by employing significantly smaller values of rate constant that are numerically tractable. Comparisons of the numerical results against experimental studies from the literature supports this. The aim of this framework is for researchers to advance and develop TCA to gain an in-depth understanding of the fundamental mechanisms of TCA and to develop a safe treatment protocol of TCA in the hope of advancing TCA into clinical trials.

Hierarchical Modeling of the Liver Vascular System

The liver plays a key role in the metabolic homeostasis of the whole organism. To carry out its functions, it is endowed with a peculiar circulatory system, made of three main dendritic flow structures and lobules. Understanding the vascular anatomy of the liver is clinically relevant since various liver pathologies are related to vascular disorders. Here, we develop a novel liver circulation model with a deterministic architecture based on the constructal law of design over the entire scale range (from macrocirculation to microcirculation). In this framework, the liver vascular structure is a combination of superimposed tree-shaped networks and porous system, where the main geometrical features of the dendritic fluid networks and the permeability of the porous medium, are defined from the constructal viewpoint. With this model, we are able to emulate physiological scenarios and to predict changes in blood pressure and flow rates throughout the hepatic vasculature due to resection or thrombosis in certain portions of the organ, simulated as deliberate blockages in the blood supply to these sections. This work sheds light on the critical impact of the vascular network on mechanics-related processes occurring in hepatic diseases, healing and regeneration that involve blood flow redistribution and are at the core of liver resilience.

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.

How does saline backflow affect the treatment of saline-infused radiofrequency ablation?

BACKGROUND AND OBJECTIVE

Saline infusion is applied together with radiofrequency ablation (RFA) to enlarge the ablation zone. However, one of the issues with saline-infused RFA is backflow, which spreads saline along the insertion track. This raises the concern of not only thermally ablating the tissue within the backflow region, but also the loss of saline from the targeted tissue, which may affect the treatment efficacy.

METHODS

In the present study, 2D axisymmetric models were developed to investigate how saline backflow influence saline-infused RFA and whether the aforementioned concerns are warranted. Saline-infused RFA was described using the dual porosity-Joule heating model. The hydrodynamics of backflow was described using Poiseuille law by assuming the flow to be similar to that in a thin annulus. Backflow lengths of 3, 4.5, 6 and 9 cm were considered.

RESULTS

Results showed that there is no concern of thermally ablating the tissue in the backflow region. This is due to the Joule heating being inversely proportional to distance from the electrode to the fourth power. Results also indicated that larger backflow lengths led to larger growth of thermal damage along the backflow region and greater decrease in coagulation volume. Hence, backflow needs to be controlled to ensure an effective treatment of saline-infused RFA.

CONCLUSIONS

There is no risk of ablating tissues around the needle insertion track due to backflow. Instead, the risk of underablation as a result of the loss of saline due to backflow was found to be of greater concern.

The capillary fascicle in skeletal muscle: Structural and functional physiology of RBC distribution in capillary networks

KEY POINTS

The capillary module, consisting of parallel capillaries from arteriole to venule, is classically considered as the building block of complex capillary networks. In skeletal muscle, this structure fails to address how blood flow is regulated along the entire length of the synchronously contracting muscle fibres. Using intravital video microscopy of resting extensor digitorum longus muscle in rats, we demonstrated the capillary fascicle as a series of interconnected modules forming continuous columns that align naturally with the dimensions of the muscle fascicle. We observed structural heterogeneity for module topology, and functional heterogeneity in space and time for capillary-red blood cell (RBC) haemodynamics within a module and between modules. We found that module RBC haemodynamics were independent of module resistance, providing direct evidence for microvascular flow regulation at the level of the capillary module. The capillary fascicle is an updated paradigm for characterizing blood flow and RBC distribution in skeletal muscle capillary networks.

ABSTRACT

Capillary networks are the fundamental site of oxygen exchange in the microcirculation. The capillary module (CM), consisting of parallel capillaries from terminal arteriole (TA) to post-capillary venule (PCV), is classically considered as the building block of complex capillary networks. In skeletal muscle, this structure fails to address how blood flow is regulated along the entire length of the synchronously contracting muscle fibres, requiring co-ordination from numerous modules. It has previously been recognized that TAs and PCVs interact with multiple CMs, creating interconnected networks. Using label-free intravital video microscopy of resting extensor digitorum longus muscle in rats, we found that these networks form continuous columns of linked CMs spanning thousands of microns, herein denoted as the capillary fascicle (CF); this structure aligns naturally with the dimensions of the muscle fascicle. We measured capillary-red blood cell (RBC) haemodynamics and module topology (n = 9 networks, 327 modules, 1491 capillary segments). The average module had length 481 μm, width 157 μm and 9.51 parallel capillaries. We observed structural heterogeneity for CM topology, and functional heterogeneity in space and time for capillary-RBC haemodynamics within a module and between modules. There was no correlation between capillary RBC velocity and lineal density. A passive inverse relationship between module length and haemodynamics was remarkably absent, providing direct evidence for microvascular flow regulation at the level of the CM. In summary, the CF is an updated paradigm for characterizing RBC distribution in skeletal muscle, and strengthens the theory of capillary networks as major contributors to the signal that regulates capillary perfusion.

Role of saline concentration during saline-infused radiofrequency ablation: Observation of secondary Joule heating along the saline-tissue interface

Infusion of saline prior to radiofrequency ablation (RFA) is known to enlarge the thermal coagulation zone. The abundance of ions in saline elevate the electrical conductivity of the saline-saturated region. This promotes greater electric current flow inside the tissue, which increases the amount of RF energy deposition and subsequently enlarges the coagulation zone. In theory, infusion of higher concentration of saline should lead to larger coagulation zone due to the greater number of ions. Nevertheless, existing studies on the effects of concentration on saline-infused RFA have been conflicting, with the exact role of saline concentration yet to be fully elucidated. In this paper, computational models of saline-infused RFA were developed to investigate the role of saline concentration on the outcome of saline-infused RFA. The elevation in tissue electrical conductivity was modelled using the microscopic mixture model, while RFA was modelled using the coupled dual porosity-Joule heating model. Results obtained indicated that the presence of a concentration threshold to which no further elevation in tissue electrical conductivity and enlargement in thermal coagulation can occur. This threshold was determined to be at 15% NaCl. Analysis of the Joule heating distribution revealed the presence of a secondary Joule heating site located along the interface between wet and dry tissue. This secondary Joule heating was responsible for the enlargement in coagulation volume and its rapid growth phase during ablation.

The liver, a functionalized vascular structure

The liver is not only the largest organ in the body but also the one playing one of the most important role in the human metabolism as it is in charge of transforming toxic substances in the body. Understanding the way its blood vasculature works is key. In this work we show that the challenge of predicting the hepatic multi-scale vascular network can be met thanks to the constructal law of design evolution. The work unveils the structure of the liver blood flow architecture as a combination of superimposed tree-shaped networks and porous system. We demonstrate that the dendritic nature of the hepatic artery, portal vein and hepatic vein can be predicted, together with their geometrical features (diameter ratio, duct length ratio) as the entire blood flow architectures follow the principle of equipartition of imperfections. At the smallest scale, the shape of the liver elemental systems-the lobules-is discovered, while their permeability is also predicted. The theory is compared with good agreement to anatomical data from the literature.

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.

Liver Bioreactor Design Issues of Fluid Flow and Zonation, Fibrosis, and Mechanics: A Computational Perspective

Tissue engineering, with the goal of repairing or replacing damaged tissue and organs, has continued to make dramatic science-based advances since its origins in the late 1980’s and early 1990’s. Such advances are always multi-disciplinary in nature, from basic biology and chemistry through physics and mathematics to various engineering and computer fields. This review will focus its attention on two topics critical for tissue engineering liver development: (a) fluid flow, zonation, and drug screening, and (b) biomechanics, tissue stiffness, and fibrosis, all within the context of 3D structures. First, a general overview of various bioreactor designs developed to investigate fluid transport and tissue biomechanics is given. This includes a mention of computational fluid dynamic methods used to optimize and validate these designs. Thereafter, the perspective provided by computer simulations of flow, reactive transport, and biomechanics responses at the scale of the liver lobule and liver tissue is outlined, in addition to how bioreactor-measured properties can be utilized in these models. Here, the fundamental issues of tortuosity and upscaling are highlighted, as well as the role of disease and fibrosis in these issues. Some idealized simulations of the effects of fibrosis on lobule drug transport and mechanics responses are provided to further illustrate these concepts. This review concludes with an outline of some practical applications of tissue engineering advances and how efficient computational upscaling techniques, such as dual continuum modeling, might be used to quantify the transition of bioreactor results to the full liver scale.

Shape-shifting thermal coagulation zone during saline-infused radiofrequency ablation: A computational study on the effects of different infusion location

BACKGROUND AND OBJECTIVE

The majority of the studies on radiofrequency ablation (RFA) have focused on enlarging the size of the coagulation zone. An aspect that is crucial but often overlooked is the shape of the coagulation zone. The shape is crucial because the majority of tumours are irregularly-shaped. In this paper, the ability to manipulate the shape of the coagulation zone following saline-infused RFA by altering the location of saline infusion is explored.

METHODS

A 3D model of the liver tissue was developed. Saline infusion was described using the dual porosity model, while RFA was described using the electrostatic and bioheat transfer equations. Three infusion locations were investigated, namely at the proximal end, the middle and the distal end of the electrode. Investigations were carried out numerically using the finite element method.

RESULTS

Results indicated that greater thermal coagulation was found in the region of tissue occupied by the saline bolus. Infusion at the middle of the electrode led to the largest coagulation volume followed by infusion at the proximal and distal ends. It was also found that the ability to delay roll-off, as commonly associated with saline-infused RFA, was true only for the case when infusion is carried out at the middle. When infused at the proximal and distal ends, the occurrence of roll-off was advanced. This may be due to the rapid and more intense heating experienced by the tissue when infusion is carried out at the electrode ends where Joule heating is dominant.

CONCLUSION

Altering the location of saline infusion can influence the shape of the coagulation zone following saline-infused RFA. The ability to 'shift' the coagulation zone to a desired location opens up great opportunities for the development of more precise saline-infused RFA treatment that targets specific regions within the tissue.

A fully coupled porous media and channels flow approach for simulation of blood and bile flow through the liver lobules

Two dimensional, steady state, and incompressible blood and bile flows through the liver lobules are numerically simulated. Two different geometric models A and B are proposed to study the effects of lobule structure on the fluid flow behaviour. In Model A, the lobule tissue is represented as a hexagonal shape porous medium with a set of flow channels at its vertices accounting for the hepatic artery, portal and central veins along with bile ductules. Model B is a channelized porous medium constructed by adding a set of flow channels, representing the bile canaliculies and lobule sinusoids, to Model A. The bile and blood flow through the lobule is simulated by the finite element approach, based on the Darcy/Brinkman equations in the lobule tissue and the Navier-Stokes (or Stokes) equations in the flow channels. In Model B, a transmission factor on the boundaries of the bile canaliculies is introduced to connect the bile and blood flows. First, a single regular lobule is utilized to exhibit the fluid flow pattern through the liver lobule represented by proposed geometric models. Then, the model is extended to a group of liver lobules to demonstrate the flow through a liver slice represented by irregular lobules. Numerical results indicate that the Darcy and Brinkman equations provide nearly the same solutions for Model A and similar solutions with a little difference for Model B. It is shown that the existence of sinusoids and bile canaliculies inside the liver lobules has noticeable effects on its fluid flow pattern, in terms of pressure and velocity fields.

Better prediction of the local concentration-effect relationship: the role of physiologically based pharmacokinetics and quantitative systems pharmacology and toxicology in the evolution of model-informed drug discovery and development

Model-informed drug discovery and development (MID3) is an umbrella term under which sit several computational approaches: quantitative systems pharmacology (QSP), quantitative systems toxicology (QST) and physiologically based pharmacokinetics (PBPK). QSP models are built using mechanistic knowledge of the pharmacological pathway focusing on the putative mechanism of drug efficacy; whereas QST models focus on safety and toxicity issues and the molecular pathways and networks that drive these adverse effects. These can be mediated through exaggerated on-target or off-target pharmacology, immunogenicity or the physiochemical nature of the compound. PBPK models provide a mechanistic description of individual organs and tissues to allow the prediction of the intra- and extra-cellular concentration of the parent drug and metabolites under different conditions. Information on biophase concentration enables the prediction of a drug effect in different organs and assessment of the potential for drug-drug interactions. Together, these modelling approaches can inform the exposure-response relationship and hence support hypothesis generation and testing, compound selection, hazard identification and risk assessment through to clinical proof of concept (POC) and beyond to the market.

A quantitative systems pharmacology approach, incorporating a novel liver model, for predicting pharmacokinetic drug-drug interactions

All pharmaceutical companies are required to assess pharmacokinetic drug-drug interactions (DDIs) of new chemical entities (NCEs) and mathematical prediction helps to select the best NCE candidate with regard to adverse effects resulting from a DDI before any costly clinical studies. Most current models assume that the liver is a homogeneous organ where the majority of the metabolism occurs. However, the circulatory system of the liver has a complex hierarchical geometry which distributes xenobiotics throughout the organ. Nevertheless, the lobule (liver unit), located at the end of each branch, is composed of many sinusoids where the blood flow can vary and therefore creates heterogeneity (e.g. drug concentration, enzyme level). A liver model was constructed by describing the geometry of a lobule, where the blood velocity increases toward the central vein, and by modeling the exchange mechanisms between the blood and hepatocytes. Moreover, the three major DDI mechanisms of metabolic enzymes; competitive inhibition, mechanism based inhibition and induction, were accounted for with an undefined number of drugs and/or enzymes. The liver model was incorporated into a physiological-based pharmacokinetic (PBPK) model and simulations produced, that in turn were compared to ten clinical results. The liver model generated a hierarchy of 5 sinusoidal levels and estimated a blood volume of 283 mL and a cell density of 193 × 106 cells/g in the liver. The overall PBPK model predicted the pharmacokinetics of midazolam and the magnitude of the clinical DDI with perpetrator drug(s) including spatial and temporal enzyme levels changes. The model presented herein may reduce costs and the use of laboratory animals and give the opportunity to explore different clinical scenarios, which reduce the risk of adverse events, prior to costly human clinical studies.

Building a 3D Virtual Liver: Methods for Simulating Blood Flow and Hepatic Clearance on 3D Structures

In this paper, we develop a spatio-temporal modeling approach to describe blood and drug flow, as well as drug uptake and elimination, on an approximation of the liver. Extending on previously developed computational approaches, we generate an approximation of a liver, which consists of a portal and hepatic vein vasculature structure, embedded in the surrounding liver tissue. The vasculature is generated via constrained constructive optimization, and then converted to a spatial grid of a selected grid size. Estimates for surrounding upscaled lobule tissue properties are then presented appropriate to the same grid size. Simulation of fluid flow and drug metabolism (hepatic clearance) are completed using discretized forms of the relevant convective-diffusive-reactive partial differential equations for these processes. This results in a single stage, uniformly consistent method to simulate equations for blood and drug flow, as well as drug metabolism, on a 3D structure representative of a liver.

A physiologically-based flow network model for hepatic drug elimination III: 2D/3D DLA lobule models

Background

One of the major issues in current pharmaceutical development is potential hepatotoxicity and drug-induced liver damage. This is due to the unique metabolic processes performed in the liver to prevent accumulation of a wide range of chemicals in the blood. Recently, we developed a physiologically-based lattice model to address the transport and metabolism of drugs in the liver lobule (liver functional unit).

Method

In this paper, we extend our idealized model to consider structural and spatial variability in two and three dimensions. We introduce a hexagonal-based model with one input (portal vein) and six outputs (hepatic veins) to represent a typical liver lobule. To capture even more realistic structures, we implement a novel sequential diffusion-limited aggregation (DLA) method to construct a morphological sinusoid network in the lobule. A 3D model constructed with stacks of multiple 2D sinusoid realizations is explored to study the effects of 3D structural variations. The role of liver zonation on drug metabolism in the lobule is also addressed, based on flow-based predicted steady-state O₂ profiles used as a zonation indicator.

Results

With this model, we analyze predicted drug concentration levels observed exiting the lobule with their detailed distribution inside the lobule, and compare with our earlier idealized models. In 2D, due to randomness of the sinusoidal structure, individual hepatic veins respond differently (i.e. at different times) to injected drug. In 3D, however, the variation of response to the injected drug is observed to be less extreme. Also, the production curves show more diffusive behavior in 3D than in 2D.

Conclusion

Although, the individual producing ports respond differently, the average lobule production summed over all hepatic veins is more diffuse. Thus the net effect of all these variations makes the overall response smoother. We also show that, in 3D, the effect of zonation on drug production characteristics appears quite small. Our new biophysical structural analysis of a physiologically-based 3D lobule can therefore form the basis for a quantitative assessment of liver function and performance both in health and disease

Electronic supplementary material

The online version of this article (doi:10.1186/s12976-016-0034-5) contains supplementary material, which is available to authorized users.

A physiologically-based flow network model for hepatic drug elimination II: variable lattice lobule models

We extend a physiologically-based lattice model for the transport and metabolism of drugs in the liver lobule (liver functional unit) to consider structural and spatial variability. We compare predicted drug concentration levels observed exiting the lobule with their detailed distribution inside the lobule, and indicate the role that structural variation has on these results. Liver zonation and its role on drug metabolism represent another aspect of structural inhomogeneity that we consider here. Since various liver diseases can be thought to produce such structural variations, our analysis gives insight into the role of disease on liver function and performance. These conclusions are based on the dominant role of convection in well-vascularized tissue with a given structure.