A spatially distributed computational model of brain cellular metabolism

A comprehensive review on modeling aspects of infusion-based drug delivery in the brain

Brain disorders represent an ever-increasing health challenge worldwide. While conventional drug therapies are less effective due to the presence of the blood-brain barrier, infusion-based methods of drug delivery to the brain represent a promising option. Since these methods are mechanically controlled and involve multiple physical phases ranging from the neural and molecular scales to the brain scale, highly efficient and precise delivery procedures can significantly benefit from a comprehensive understanding of drug-brain and device-brain interactions. Behind these interactions are principles of biophysics and biomechanics that can be described and captured using mathematical models. Although biomechanics and biophysics have received considerable attention, a comprehensive mechanistic model for modeling infusion-based drug delivery in the brain has yet to be developed. Therefore, this article reviews the state-of-the-art mechanistic studies that can support the development of next-generation models for infusion-based brain drug delivery from the perspective of fluid mechanics, solid mechanics, and mathematical modeling. The supporting techniques and database are also summarized to provide further insights. Finally, the challenges are highlighted and perspectives on future research directions are provided. STATEMENT OF SIGNIFICANCE: Despite the immense potential of infusion-based drug delivery methods for bypassing the blood-brain barrier and efficiently delivering drugs to the brain, achieving optimal drug distribution remains a significant challenge. This is primarily due to our limited understanding of the complex interactions between drugs and the brain that are governed by principles of biophysics and biomechanics, and can be described using mathematical models. This article provides a comprehensive review of state-of-the-art mechanistic studies that can help to unravel the mechanism of drug transport in the brain across the scales, which underpins the development of next-generation models for infusion-based brain drug delivery. More broadly, this review will serve as a starting point for developing more effective treatments for brain diseases and mechanistic models that can be used to study other soft tissue and biomaterials.

A spatially distributed model of brain metabolism highlights the role of diffusion in brain energy metabolism

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes and the cerebral blood flow. A comprehensive mathematical model of brain metabolism should account not only for the biochemical processes and the interaction of neurons and astrocytes, but also the diffusion of metabolites. In the present article, we present a computational methodology based on a multidomain model of the brain tissue and a homogenization argument for the diffusion processes. In our spatially distributed compartment model, communication between compartments occur both through local transport fluxes, as is the case within local astrocyte-neuron complexes, and through diffusion of some substances in some of the compartments. The model assumes that diffusion takes place in the extracellular space (ECS) and in the astrocyte compartment. In the astrocyte compartment, the diffusion across the syncytium network is implemented as a function of gap junction strength. The diffusion process is implemented numerically by means of a finite element method (FEM) based spatial discretization, and robust stiff solvers are used to time integrate the resulting large system. Computed experiments show the effects of ECS tortuosity, gap junction strength and spatial anisotropy in the astrocyte network on the brain energy metabolism.

Measurement and Modeling of Transport Across the Blood-Brain Barrier

The blood-brain barrier (BBB) is a dynamic regulatory barrier at the interface of blood circulation and the brain parenchyma, which plays a critical role in protecting homeostasis in the central nervous system. However, it also significantly impedes drug delivery to the brain. Understanding the transport across BBB and brain distribution will facilitate the prediction of drug delivery efficiency and the development of new therapies. To date, various methods and models have been developed to study drug transport at the BBB interface, including in vivo brain uptake measurement methods, in vitro BBB models, and mathematic brain vascular models. Since the in vitro BBB models have been extensively reviewed elsewhere, we provide a comprehensive summary of the brain transport mechanisms and the currently available in vivo methods and mathematic models in studying the molecule delivery process at the BBB interface. In particular, we reviewed the emerging in vivo imaging techniques in observing drug transport across the BBB. We discussed the advantages and disadvantages associated with each model to serve as a guide for model selection in studying drug transport across the BBB. In summary, we envision future directions to improve the accuracy of mathematical models, establish noninvasive in vivo measurement techniques, and bridge the preclinical studies with clinical translation by taking the altered BBB physiological conditions into consideration. We believe these are critical in guiding new drug development and precise drug administration in brain disease treatment.

Application of a multicomponent model of convectional reaction-diffusion to description of glucose gradients in a neurovascular unit

A supply of glucose to a nervous tissue is fulfilled by a cerebrovascular network, and further diffusion is known to occur at both an arteriolar and a microvascular level. Despite a direct relation, a blood flow dynamic and reaction-diffusion of metabolites are usually considered separately in the mathematical models. In the present study they are coupled in a multiphysical approach which allows to evaluate the effects of capillary blood flow changes on near-vessels nutrient concentration gradients evidently. Cerebral blood flow (CBF) was described by the non-steady-state Navier-Stokes equations for a non-Newtonian fluid whose constitutive law is given by the Carreau model. A three-level organization of blood–brain barrier (BBB) is modelled by the flux dysconnectivity functions including densities and kinetic properties of glucose transporters. The velocity of a fluid flow in brain extracellular space (ECS) was estimated using Darcy’s law. The equations of reaction-diffusion with convection based on a generated flow field for continues and porous media were used to describe spatial-time gradients of glucose in the capillary lumen and brain parenchyma of a neurovascular unit (NVU), respectively. Changes in CBF were directly simulated using smoothing step-like functions altering the difference of intracapillary pressure in time. The changes of CBF cover both the decrease (on 70%) and the increase (on 50%) in a capillary flow velocity. Analyzing the dynamics of glucose gradients, it was shown that a rapid decrease of a capillary blood flow yields an enhanced level of glucose in a near-capillary nervous tissue if the contacts between astrocytes end-feet are not tight. Under the increased CBF velocities the amplitude of glucose concentration gradients is always enhanced. The introduced approach can be used for estimation of blood flow changes influence not only on glucose but also on other nutrients concentration gradients and for the modelling of distributions of their concentrations near blood vessels in other tissues as well.

Current Approaches of Building Mechanistic Pharmacodynamic Drug-Target Binding Models

Mechanistic pharmacodynamic models that incorporate the binding kinetics of drug-target interactions have several advantages in understanding target engagement and the efficacy of a drug dose. However, guidelines on how to build and interpret mechanistic pharmacodynamic drug-target binding models considering both biological and computational factors are still missing in the literature. In this chapter, current approaches of building mechanistic PD models and their advantages are discussed. We also present a methodology on how to select a suitable model considering both biological and computational perspectives, as well as summarize the challenges of current mechanistic PD models.

MITODYN: An Open Source Software for Quantitative Modeling of Mitochondrial and Cellular Energy Metabolic Flux Dynamics in Health and Disease

Mitochondrial respiratory chain (RC) transforms the reductive power of NADH or FADH2 oxidation into a proton gradient between the matrix and cytosolic sides of the inner mitochondrial membrane, that ATP synthase uses to generate ATP. This process constitutes a bridge between carbohydrates' central metabolism and ATP-consuming cellular functions. Moreover, the RC is responsible for a large part of reactive oxygen species (ROS) generation that play signaling and oxidizing roles in cells. Mathematical methods and computational analysis are required to understand and predict the possible behavior of this metabolic system. Here we propose a software tool that helps to analyze individual steps of respiratory electron transport in their dynamics, thus deepening understanding of the mechanism of energy transformation and ROS generation in the RC. This software's core is a kinetic model of the RC represented by a system of ordinary differential equations (ODEs). This model enables the analysis of complex dynamic behavior of the RC, including multistationarity and oscillations. The proposed RC modeling method can be applied to study respiration and ROS generation in various organisms and naturally extended to explore carbohydrates' metabolism and linked metabolic processes.

Brain energetics, mitochondria, and traumatic brain injury

We review current thinking about, and draw connections between, brain energetics and metabolism, and between mitochondria and traumatic brain injury. Energy is fundamental to proper brain function. Its creation in a useful form for neurons and glia, and consistently in response to the brain's high energy needs, is critical for physiological pathways. Dysfunction in the mechanisms of energy production is at the center of neurological and neuropsychiatric pathologies. We examine the connections between energetics and mitochondria - the organelle responsible for almost all the energy production in the cell - and how secondary pathologies in traumatic brain injury result from energetic dysfunction. This paper interweaves these topics, a necessity since they are closely coupled, and identifies where there exist a lack of understanding and of data. In addition to summarizing current thinking in these disciplines, our goal is to suggest a framework for the mathematical modeling of mechanisms and pathways based on optimal energetic decisions.

A 3D brain unit model to further improve prediction of local drug distribution within the brain

The development of drugs targeting the brain still faces a high failure rate. One of the reasons is a lack of quantitative understanding of the complex processes that govern the pharmacokinetics (PK) of a drug within the brain. While a number of models on drug distribution into and within the brain is available, none of these addresses the combination of factors that affect local drug concentrations in brain extracellular fluid (brain ECF). Here, we develop a 3D brain unit model, which builds on our previous proof-of-concept 2D brain unit model, to understand the factors that govern local unbound and bound drug PK within the brain. The 3D brain unit is a cube, in which the brain capillaries surround the brain ECF. Drug concentration-time profiles are described in both a blood-plasma-domain and a brain-ECF-domain by a set of differential equations. The model includes descriptions of blood plasma PK, transport through the blood-brain barrier (BBB), by passive transport via paracellular and transcellular routes, and by active transport, and drug binding kinetics. The impact of all these factors on ultimate local brain ECF unbound and bound drug concentrations is assessed. In this article we show that all the above mentioned factors affect brain ECF PK in an interdependent manner. This indicates that for a quantitative understanding of local drug concentrations within the brain ECF, interdependencies of all transport and binding processes should be understood. To that end, the 3D brain unit model is an excellent tool, and can be used to build a larger network of 3D brain units, in which the properties for each unit can be defined independently to reflect local differences in characteristics of the brain.

The need for mathematical modelling of spatial drug distribution within the brain

The blood brain barrier (BBB) is the main barrier that separates the blood from the brain. Because of the BBB, the drug concentration-time profile in the brain may be substantially different from that in the blood. Within the brain, the drug is subject to distributional and elimination processes: diffusion, bulk flow of the brain extracellular fluid (ECF), extra-intracellular exchange, bulk flow of the cerebrospinal fluid (CSF), binding and metabolism. Drug effects are driven by the concentration of a drug at the site of its target and by drug-target interactions. Therefore, a quantitative understanding is needed of the distribution of a drug within the brain in order to predict its effect. Mathematical models can help in the understanding of drug distribution within the brain. The aim of this review is to provide a comprehensive overview of system-specific and drug-specific properties that affect the local distribution of drugs in the brain and of currently existing mathematical models that describe local drug distribution within the brain. Furthermore, we provide an overview on which processes have been addressed in these models and which have not. Altogether, we conclude that there is a need for a more comprehensive and integrated model that fills the current gaps in predicting the local drug distribution within the brain.

Mathematical Biology Models of Parkinson's Disease

Parkinsons disease (PD) is a progressive neurodegenerative disease with substantial and growing socio-economic burden. In this multifactorial disease, aging, environmental, and genetic factors contribute to neurodegeneration and dopamine (DA) deficiency in the brain. Treatments aimed at DA restoration provide symptomatic relief, however, no disease modifying treatments are available, and PD remains incurable to date. Mathematical modeling can help understand such complex multifactorial neurological diseases. We review mathematical modeling efforts in PD with a focus on mechanistic models of pathogenic processes. We consider models of α-synuclein (Asyn) aggregation, feedbacks among Asyn, DA, and mitochondria and proteolytic systems, as well as pathology propagation through the brain. We hope that critical understanding of existing literature will pave the way to the development of quantitative systems pharmacology models to aid PD drug discovery and development.

Improving the Prediction of Local Drug Distribution Profiles in the Brain with a New 2D Mathematical Model

The development of drugs that target the brain is very challenging. A quantitative understanding is needed of the complex processes that govern the concentration–time profile of a drug (pharmacokinetics) within the brain. So far, there are no studies on predicting the drug concentration within the brain that focus not only on the transport of drugs to the brain through the blood–brain barrier (BBB), but also on drug transport and binding within the brain. Here, we develop a new model for a 2D square brain tissue unit, consisting of brain extracellular fluid (ECF) that is surrounded by the brain capillaries. We describe the change in free drug concentration within the brain ECF, by a partial differential equation (PDE). To include drug binding, we couple this PDE to two ordinary differential equations that describe the concentration–time profile of drug bound to specific as well as non-specific binding sites that we assume to be evenly distributed over the brain ECF. The model boundary conditions reflect how free drug enters and leaves the brain ECF by passing the BBB, located at the level of the brain capillaries. We study the influence of parameter values for BBB permeability, brain ECF bulk flow, drug diffusion through the brain ECF and drug binding kinetics, on the concentration–time profiles of free and bound drug.

A computational model integrating brain electrophysiology and metabolism highlights the key role of extracellular potassium and oxygen

The human brain is a small organ which uses a disproportionate amount of the total metabolic energy production in the body. While it is well understood that the most significant energy sink is the maintenance of the neuronal membrane potential during the brain signaling activity, the role of astrocytes in the energy balance continues to be the topic of a lot of research. A key function of astrocytes, besides clearing glutamate from the synaptic clefts, is the potassium clearing after neuronal activation. Extracellular potassium plays a significant role in triggering neuronal firing, and elevated concentration of potassium may lead to abnormal firing patterns, e.g., seizures, thus emphasizing the importance of the glial K⁺ buffering role. The predictive mathematical model proposed in this paper elucidates the role of glial potassium clearing in brain energy metabolism, integrating a detailed model of the ion dynamics which regulates neuronal firing with a four compartment metabolic model. Because of the very different characteristic time scales of electrophysiology and metabolism, care must be taken when coupling the two models to ensure that the predictions, e.g., neuronal firing frequencies and the oxygen-glucose index (OGI) of the brain during activation and rest, are in agreement with empirical observations. The temporal multi-scale nature of the problem requires the design of new computational tools to ensure a stable and accurate numerical treatment. The model predictions for different protocols, including combinations of elevated activation and ischemic episodes, are in good agreement with experimental observations reported in the literature.