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The Need for Mathematical Modelling of Spatial Drug Distribution Within the Brain Esmée Vendel1, Vivi Rottschäfer1 and Elizabeth C

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The Need for Mathematical Modelling of Spatial Drug Distribution Within the Brain Esmée Vendel1, Vivi Rottschäfer1 and Elizabeth C Vendel et al. Fluids Barriers CNS (2019) 16:12 https://doi.org/10.1186/s12987-019-0133-x Fluids and Barriers of the CNS REVIEW Open Access The need for mathematical modelling of spatial drug distribution within the brain Esmée Vendel1, Vivi Rottschäfer1 and Elizabeth C. M. de Lange2* Abstract The blood brain barrier (BBB) is the main barrier that separates the blood from the brain. Because of the BBB, the drug concentration-time profle in the brain may be substantially diferent from that in the blood. Within the brain, the drug is subject to distributional and elimination processes: difusion, bulk fow of the brain extracellular fuid (ECF), extra-intracellular exchange, bulk fow of the cerebrospinal fuid (CSF), binding and metabolism. Drug efects are driven by the concentration of a drug at the site of its target and by drug-target interactions. Therefore, a quantita- tive understanding is needed of the distribution of a drug within the brain in order to predict its efect. Mathemati- cal 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-specifc and drug-specifc properties that afect 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 flls the current gaps in predicting the local drug distribution within the brain. Keywords: Mathematical modeling, Drug transport, Brain extracellular fuid, Blood–brain barrier, Pharmacokinetics Introduction of the cerebrospinal fuid (CSF) and metabolism. Fur- Te blood–brain barrier (BBB) separates the blood thermore, the drug may bind to specifc binding sites from the brain. Te BBB is formed by the brain capil- (targets) and non-specifc binding sites (brain tissue lary endothelial cells that constitute the walls of the components). Consequently, the drug concentration- brain capillaries. Multiprotein complexes called tight time profle in the brain may be substantially diferent junctions are located between adjacent brain capillary from that in the blood [1], while also local diferences in endothelial cells and seal the intercellular space, thereby drug concentration-time profles within the brain may limiting intercellular difusion. In addition, transport arise. Te local concentration-time profles within the across the BBB is afected by transporters and helper brain are highly important, since drug efects within the molecules located at the brain capillary endothelial central nervous system (CNS) are driven by the concen- cells that move compounds from the blood to the brain tration-time profle of a drug at the site of its target: the or from the brain to the blood. Consequently, the drug drug needs to be distributed to its target in sufcient concentration-time profle in the brain may be substan- concentrations and duration in order to optimally inter- tially diferent from that in the blood [1]. Once in the act with its target and elicit the desired efect. To predict brain, the drug is subject to distribution and elimination a drug’s efect, therefore, a quantitative understanding processes: difusion, bulk fow of the brain extracellu- is needed on brain target site distribution. However, as lar fuid (ECF), extra-intracellular exchange, bulk fow the human brain is inaccessible for sampling, measur- ing drug concentration-time profles is highly restricted. Mathematical models are a helpful tool to describe and *Correspondence: [email protected] understand the impact of processes that govern drug 2 Leiden Academic Centre for Drug Research, Einsteinweg 55, distribution within the brain. Moreover, while direct 2333CC Leiden, The Netherlands Full list of author information is available at the end of the article measurement of spatial drug distribution within the © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creat iveco mmons .org/ publi cdoma in/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Vendel et al. Fluids Barriers CNS (2019) 16:12 Page 2 of 33 brain is restricted, mathematical models allow the pre- Te blood in the brain capillaries is separated from the diction of the spatial distribution of a drug within the brain tissue by the BBB (Fig. 1a). Te brain capillaries brain. To adequately predict the drug distribution of a reunite to form veins, from which the blood is carried drug into and within the brain, a mathematical model away from the brain back into the heart. Te brain tis- should include all of the above mentioned factors that sue (brain parenchyma) consists of the brain ECF and govern the concentration-time profles of a drug within the brain cells. Te brain ECF surrounds the cells and the brain. However, currently existing models focus circulates within the brain tissue. Te CSF circulates on just one or a few of these processes. Te aim of this between the sub-arachnoid space (located between the review is to provide an overview of the current state of dura mater, a layer of connective tissue surrounding the the art in modelling drug distribution into and within brain tissue, and the brain tissue, see Fig. 1), the brain the brain and highlight the need for novel methods ventricles, and the spine (Fig. 1). Te blood is separated that provide a more complete description of the local from the CSF by the blood–CSF barrier (BCSFB) and the drug distribution within the brain. We frst summarise blood–arachnoid barrier. Te BCSFB is located between the factors afecting the drug distribution within the the blood in the brain capillaries and the CSF in ventri- brain (in “Factors afecting drug distribution within the cles of the brain (Fig. 1b). Te blood–arachnoid barrier brain”). Ten, we give an overview of currently avail- is positioned between the blood in the dura mater and able models on the distribution of compounds into and the CSF in the sub-arachnoid space (Fig. 1c). Te entire within the brain and of models that integrate two or brain contains many potential binding sites for endog- more of these aspects (in “Existing models on the local enous compounds (that originate within the body) and distribution of drugs in the brain”). Finally, in “Te need exogenous compounds (that originate outside the body). for a refned mathematical model on spatial drug distri- In addition, metabolic enzymes residing in the brain may bution within the brain”, we discuss how we can improve chemically convert substances into new molecules. In or combine current models to develop a comprehensive the subsequent sections, we highlight the properties of model for improved prediction of drug distribution into the brain vascular network, the brain barriers, the brain and within the brain. tissue (including the brain ECF and the brain cells), the CSF, the fuid movement within the brain, binding and Factors afecting drug distribution within the brain metabolism. Te distribution of a drug within the brain determines the local concentration of drug that is available to bind The brain vascular network to its target and thereby induce an efect. Both the struc- An extensive network of vasculature supplies the brain tural properties of the brain and those of the drug afect with oxygen and nutrients (Fig. 2, left). Te brain surface the distribution of the drug within the brain. In this sec- is perfused with large arteries and veins that carry oxy- tion, we frst discuss the brain-specifc and drug-specifc gen and nutrients to the brain (Fig. 2, middle). Te larger properties. Ten, we describe the processes that afect brain arteries branch out into smaller arterioles that pen- local drug distribution within the brain. Tese processes etrate the brain cortex and merge into the brain micro- depend on both the brain-specifc and drug-specifc circulation, consisting of the brain capillary beds (Fig. 2, properties. Finally, we discuss how spatial variations in right). Te brain capillaries that make up the capillary drug distribution processes may lead to spatial difer- beds surround the brain tissue. Waste products are car- ences in drug concentration-time profles within the ried away from the capillary beds by the venules. Te ven- brain. ules merge into the veins, which lead the blood and the waste products it contains back to the heart. Te brain Brain‑specifc properties capillaries have a large surface area: they are the main site Te brain-specifc properties are the structural proper- for the exchange of oxygen and nutrients with the brain ties of the brain. Te structural properties most impor- tissue [2]. Te brain capillary network is very dense and tant for drug distribution within the brain are highlighted it is estimated that each neuron is perfused with its own in Fig. 1. Blood is supplied to the brain by arteries feed- capillary [3]. Te average distance between the capillaries ing the anterior (front) or posterior (back) part of the in the rat brain is only about 50 µm [4–7]. Te brain cap- brain. Te arteries branch out into smaller brain capil- illaries are separated from the brain by the brain barriers, laries. At the level of the brain capillaries, compounds which will be discussed in the next section.
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