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

Abstract

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.

1 Introduction

The brain capillary bed is the major site of drug exchange between the blood and the brain. Blood flows from the general blood circulation into the brain capillary bed by a feeding arteriole and back by a draining venule. The rate at which drug molecules within the blood are exposed to the brain is determined by the brain capillary blood flow rate. Drug exchange between the blood plasma in the brain capillaries and the brain extracellular fluid (ECF) is controlled by the blood-brain barrier (BBB).

Drug distribution into and within the brain has been extensively summarized in a recent review [1]. In short, the BBB has great impact on the relationship between the concentration-time profiles of unbound drug in the blood plasma (blood plasma pharmacokinetics (PK)) and in the brain ECF (brain ECF PK). The BBB consists of brain endothelial cells that are held closely together by tight junctions. Unbound drug may cross the BBB by passive and/or active transport [210]. Passive transport is bidirectional and occurs by diffusion through the BBB endothelial cells (transcellular transport) and through the BBB tight junctions between the endothelial cells (paracellular transport). Passive transport is quantified by the BBB permeability, which is the speed by which a compound passively crosses the BBB, and depends on the properties of both the drug and the brain. Active transporters located at the BBB move compounds either inward (in the direction of the brain ECF, active efflux) or outward (in the direction of the blood plasma, active efflux). Once having crossed the BBB, drug distributes within the brain ECF by diffusion. Diffusion within the brain ECF is hindered by the brain cells [11, 12]. This hindrance is described by the so-called tortuosity and leads to an effective diffusion that is smaller than normal (in a medium without obstacles). Moreover, a fluid flow, the brain ECF bulk flow, is present. The brain ECF bulk flow results from the generation of brain ECF by the BBB and drainage into the cerebrospinal fluid (CSF). Both diffusion and brain ECF bulk flow are important for the distribution of a drug to its target site, which is the site where a drug exerts its effect. In order to do induce an effect, a drug needs to bind to specific binding sites (targets). Only unbound drug, i.e. drug that is not bound to any components of the brain, can interact with its target [13, 14]. This is a dynamic process of association and dissociation, the so-called drug binding kinetics. These association and dissociation rates may affect the concentration of unbound drug at the target site [15, 16]. While the drug dissociation rate has been thought of as the most important determinant of the duration of interactions between a drug and its binding site [17], a more recent study shows that the drug association rate is equally important [16].

A number of models integrating several of the discussed processes of drug distribution into and within the brain is available, see for example [11, 12, 1825] and [26]. The most recent and comprehensive brain drug distribution model is the physiologically-based pharmacokinetic model for the rat and for human [27, 28]. This model takes multiple compartments of the central nervous system (CNS) into account, including plasma PK, passive paracellular and transcellular BBB transport, active BBB transport, and distribution between the brain ECF, intracellular spaces, and multiple CSF sites, on the basis of CNS-specific and drug-specific parameters. However, it does not take into account distribution within brain tissue (brain ECF).

Much is still unknown on the spatial distribution of a drug within the brain and quantitative data on the processes governing brain spatial-temporal drug transport are lacking. The purpose of the present study is therefore to gain insight into the processes governing spatial drug distribution within the brain. Here, we developed a 3D brain unit model, in which local brain drug distribution is explicitly taken into account. The 3D brain unit model encompasses blood plasma PK, the BBB, brain ECF, brain ECF bulk flow, diffusion, and binding to specific and non-specific binding sites [29, 30] within the brain. This 3D piece of brain tissue can be considered the smallest physiological unit of the brain in terms of drug transport. Within the 3D brain unit, drug is carried along with the blood plasma by the brain capillary blood flow and as such presented to the brain ECF. Drug distributes between the blood plasma and the brain ECF by transport across the BBB. Thereafter, drug distribution within the brain ECF is affected by diffusion, bulk flow and binding. We describe the distribution of drug within the brain ECF by a partial differential equation (PDE) and couple this to two ordinary differential equations (ODEs) to account for specific and non-specific drug binding.

The model builds on a proof-of-concept 2D brain unit model [31]. The 2D model is a basic model covering many essential aspects of drug distribution within the brain, including passive BBB transport, diffusion, brain ECF bulk flow, specific binding of a drug at its target site and non-specific binding of a drug to components of the brain. Here, brain cells are implicitly implemented by describing the hindrance the cells impose on the transport of a drug within the brain ECF in a tortuosity term, λ. There, λ is defined as , with D being the normal diffusion coefficient and D* the effective diffusion coefficient [12]. The 2D brain unit model has enabled the study of the effect of drug properties and brain tissue characteristics on the distribution of a drug within the brain ECF and on its specific and non-specific binding behaviour of the drug.

The current 3D brain unit model further improves the prediction of drug distribution within the brain. The third dimension improves the realistic features of the model as the brain is also 3D. Moreover, the third dimension allows the brain ECF to be not entirely surrounded by capillaries, such that the brain ECF is a continuous medium, like in reality. Then, the brain capillary blood flow and active transport across the BBB, which are both important mechanisms of drug transport into the brain, are included. Here, we focus on one single brain unit. This allows for a thorough characterisation of drug distribution within one 3D brain unit before expanding to a larger scale.

In the remainder of this article, the mathematical representation of the characteristics of the 3D brain unit is introduced (section 2). There, we formulate the model (section 2.1) and the mathematical descriptions of the drug distribution within the blood plasma of the brain capillaries (section 2.2) and within the brain (section 2.3). In section 2.4 we formulate the model boundary conditions that describe drug exchange between the blood plasma and the brain ECF by passive and active BBB transport, as well as drug transport at the boundaries of the unit. In section 3, we study the effect of several factors on drug distribution within the brain ECF. In section 3.1, we evaluate the effect of the brain capillary blood flow velocity on local brain ECF PK in the 3D brain unit. Next, we evaluate the effect of active influx and efflux on local brain ECF PK (section 3.2). Then, in section 3.3 we show how the interplay between the brain capillary blood flow velocity, passive BBB permeability and active transport affects drug concentrations within the 3D brain unit. Finally, in section 4 we conclude our work and discuss future perspectives.

2 The 3D brain unit

The 3D brain unit represents the smallest piece of brain tissue that contains all physiological elements of the brain. The 3D brain unit is part of a larger network of 3D brain units, but here we focus on just one 3D brain unit that is fed by an arteriole and drained by a venule (Fig 1, left). The 3D brain unit is a cube in which the brain capillaries (represented by red rectangular boxes on the ribs) surround the brain ECF (Fig 1, left). The segments of red rectangular boxes protruding from the vertices from the 3D brain unit are parts of brain capillaries from neighbouring units. As such, each vertex connects three incoming brain capillaries to three outgoing brain capillaries, with the exception of the vertex connected to the arteriole and the vertex connected to the venule. These connect the arteriole to three outgoing brain capillaries and three incoming brain capillaries to the venule, respectively.

A single 3D brain unit (Fig 1, middle) has a blood-plasma-domain (red) consisting of multiple sub-domains. These include the brain capillary domain where drug enters the unit (indicated by Uin in Fig 1), the domains representing the x-directed, y-directed and z-directed brain capillaries (indicated by Ux1−x4, Uy1−y4 and Uz1−z4 in Fig 1) and the brain capillary domain where drug leaves the unit (indicated by Uout in Fig 1). Drug within the blood plasma is transported by the brain capillary blood flow. The brain capillary blood flow splits at the vertices of the unit, where brain capillary branching occurs (Fig 1, right).

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Fig 1. Sketch of the 3D model brain unit.

Left: The structure represented by the 3D brain unit. An arteriole carries blood plasma (containing drug) into a brain capillary bed, that is connected to a venule that drains the blood plasma. The brain capillaries (red) surround the brain ECF (blue). Middle: the 3D brain unit and its sub-domains. The unit consists of a brain-ECF-domain (blue) and a blood-plasma-domain (red). The blood-plasma-domain is divided into several subdomains: Uin is the domain where the dose of absorbed drug enters the 3D brain unit, Ux1-x4, Uy1-y4 and Uz1-z4 are the domains representing the x-directed, y-directed and z-directed capillaries, respectively. Right: Directions of transport in the model. The drug enters the brain capillaries in Uin. From there, it is transported through the brain capillaries by the brain capillary blood flow in the direction indicated by the small arrows. Drug in the brain capillary blood plasma exchanges with the brain ECF by crossing the BBB. Drug within the brain ECF is, next to diffusion, transported along with brain ECF bulk flow (indicated by the bold arrow).

https://doi.org/10.1371/journal.pone.0238397.g001

In developing the model, we make the following assumptions about drug distribution within the brain capillaries:

Assumptions 1.

(i) The drug concentration within the blood plasma changes as a function of time depending on dose, bioavailability, the rate of absorption (in case of oral administration), distribution volume and elimination into and from the blood plasma.

(ii) The blood carrying the drug flows into 3D brain unit by a feeding arteriole and leaves via a draining venule ( Fig 1, left).

(iii) The drugs enters the brain unit in the domain Uin ( Fig 1, middle), and drug concentrations in Uin are unaffected by the brain capillary blood flow.

(iv) The brain capillary blood flow is directed away from Uin ( Fig 1, right).

(v) In the blood plasma, drug transport by diffusion is negligible compared to drug transport by the brain capillary blood flow.

(vi) The brain capillary blood flow velocity is by default equal in all brain capillaries.

(vii) Drug within the blood plasma does not bind to blood plasma proteins. All drug within the blood plasma is in an unbound state and is able to cross the BBB.

Drug within the blood plasma of the brain capillaries crosses the BBB to exchange with the brain ECF. The BBB is located at the border between the brain capillaries (red) and the brain ECF (blue), see Fig 1. Drug exchange between the blood plasma and the brain ECF is described by passive and active transport across the BBB in both directions. Here, we assume that active influx transporters move a compound from the blood plasma directly into the brain ECF and that active efflux transporters move a compound from the brain ECF directly into the blood plasma.

Within the brain ECF, we formulate:

Assumptions 2.

(i) Drug within the brain ECF is transported by diffusion and brain ECF bulk flow.

(ii) Cells are not explicitly considered, but only by taking the tortuosity (hindrance on diffusion imposed by the cells) into account.

(iii) The brain ECF bulk flow is unidirectional. It is pointed in the x-direction, see the bold arrow in Fig 1 (right).

(iv) All drug distributes within the brain ECF and we only have extracellular binding sites.

(v) The total concentration of specific and non-specific binding sites is constant.

(vi) The specific and non-specific binding sites are evenly distributed over the 3D brain unit and do not change position.

(vii) The specific and non-specific binding sites lie on the outside of cells and the drug does not have to cross cell membranes in order to bind to binding sites.

(viii) Drug binding is reversible and drugs associate and dissociate from their binding sites.

2.1 Formulation of the 3D brain unit

The 3D brain unit is a cubic domain, U, that represents a piece of brain tissue. We define U = {(x,y,z) ∈ ∣ 0≤x≤ xr ∧ 0≤y≤yr ∧ 0≤z≤zr}. There, xr, yr and zr are constants that represent the length of one unit and are defined as dcap+2r, with dcap the distance between the brain capillaries and r the brain capillary radius. In one brain unit, the brain capillaries, the BBB and the brain ECF are represented by the subsets UplU, UBBBU and UECFU, respectively, such that U = UplUBBBUECF.

Within Upl, we define Uin as the domain where the blood plasma, containing drug, enters the 3D brain unit from a feeding arteriole. We define Uout as the domain where the blood plasma, containing drug, leaves the 3D brain unit to a draining venule. Additionally, we define the x-directed, y-directed and z-capillaries as the sets {Uxi,i = 1,‥,4}, {Uyi,i = 1,‥,4} and {Uzi,i = 1,‥,4}. The brain capillaries are divided by the lines x = y (or y = z or x = z) and x+y = yr (or y+z = zr or x+z = zr), for which an example is shown in Fig 2. The only exceptions for this are the brain capillaries adjacent to Uin and Uout, see below.

The definitions of the regions are as follows:

Ux1 = {(x,y,z) ∈ U ∣ r≤x<xr-y, r≤x<xr-z ∧ 0≤y<r ∧ 0≤z<r}

Ux2 = {(x,y,z) ∈ U ∣ yr-y<x≤y ∧ z≤x<xr-z ∧ yr≥y>yr-r ∧ 0≤z<r}

Ux3 = {(x,y,z) ∈ U ∣ y≤x<xr-y ∧ zr-z<x≤z ∧ 0≤y<r ∧ zr≥z>zr-r}

Ux4 = {(x,y,z) ∈ U ∣ yr-y<x≤y ∧ zr-z<x≤z ∧ yr≥y>yr-r ∧ zr≥z>zr-r}

Uy1 = {(x,y,z) ∈ U ∣ r≤y<yr-z ∧ r≤y≤yrx ∧ 0≤x<r ∧ 0≤z<r}

Uy2 = {(x,y,z) ∈ U ∣ z≤y<yr-z ∧ xr-x≤y<x ∧ xr≥x>xr-r ∧ 0≤z<r}

Uy3 = {(x,y,z) ∈ U ∣ zr-z<y≤z ∧ x<y≤yr-x ∧ 0≤x<r ∧ zr≥z>zr-r}

Uy4 = {(x,y,z) ∈ U ∣ zr-z≤y<z ∧ xr-x<y≤x ∧ xr≥x>xr-r ∧ zr≥z>zr-r}

Uz1 = {(x,y,z) ∈ U ∣ r≤z≤zr-x ∧ r≤z≤zr-y ∧ 0≤x<r ∧ 0≤y<r}

Uz2 = {(x,y,z) ∈ U ∣ x<z≤zr-x ∧ yr-y≤z<y ∧ 0≤x<r ∧ yr≥y>yr-r}

Uz3 = {(x,y,z) ∈ U ∣ xr-x≤z<x ∧ y<z≤zr-y ∧ xr≥x>xr-r ∧ 0≤y<r}

Uz4 = {(x,y,z) ∈ U ∣ xr-x≤z<x ∧ yr-y≤z<y ∧ xr≥x>xr-r ∧ yr≥y>yr-r}

Uin = {(x,y,z) ∈ U ∣ 0≤x<r ∧ 0≤y<r ∧ 0≤z<r}

Uout = {(x,y,z) ∈ U ∣ xr-r≤x<xr ∧ yr-r≤y<yr ∧ zr-r≤z<zr}.

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Fig 2. Front view of the 3D brain unit.

Definitions of Upl are given. The x-directed, y-directed and z-capillaries are divided by the lines x = y (or y = z or x = z) and x+y = yr (or y+z = zr or x+z = zr). The only exceptions for this are the brain capillaries adjacent to Uin and the brain capillaries adjacent to Uout.

https://doi.org/10.1371/journal.pone.0238397.g002

The BBB is represented by a subset UBBBU, such that UBBB = ∂ Upl\∂ U. This denotes the border between the blood plasma and the brain ECF, located at distance r from the edges of the 3D brain unit.

The brain ECF is represented by a subset UECFU, such that UECF = U\(UplUBBB).

Within U we define the following quantities describing drug concentration:

Cpl (x,y,z,t): Upl x

CECF (x,y,z,t): UECFx

B1 (x,y,z,t): UECFx

B2 (x,y,z,t): UECF x

Here, Cpl is the concentration of unbound drug in the blood plasma, CECF is the concentration of unbound drug in the brain ECF, B1 is the concentration of drug in the brain ECF bound to specific binding sites and B2 is the concentration of drug in the brain ECF bound to non-specific binding sites.

2.2 Description of drug distribution in Upl

Based on assumptions 1(i) and 1(iii), we describe the concentration of unbound drug within Uin by including parameters related to single oral administration [32] using the Bateman function [33]: (1) where F is the bioavailability of the drug, ka the absorption rate constant of the drug, ke the elimination rate constant of the drug, Dose the molar amount of orally administered drug, and Vd the distribution volume, which relates the total amount of drug in the body to the drug concentration in the blood plasma. We focus on single oral administration but can also study other choices.

Additionally, based on assumptions 1(iv) and 1(v), we define: (2) (3) (4) with vblood the blood flow velocity within the brain capillaries and where the initial condition is given by (5)

2.3 Description of drug distribution in UECF

Based on assumptions 2, we describe the distribution of unbound and bound drug within UECF with the following system of equations: (6) with initial conditions (7) (8) where D is the diffusion coefficient in a free medium, λ the tortuosity, vECF the (x-directed) brain ECF bulk flow, B1max, the total concentration of specific binding sites within the brain ECF, k1on the association rate constant for specific binding, k1off the dissociation rate constant for specific binding, B2max the total concentration of non-specific binding sites within the brain ECF, k2on the association rate constant for non-specific binding and k2off the dissociation rate constant for non-specific binding.

2.4 Boundary conditions

We formulate boundary conditions that describe the change in concentration of drug at the boundary between the blood-plasma-domain (Uok) and the brain-ECF-domain (UECF), hence at UBB as well as at the boundaries of the 3D brain unit (Upl∩∂ U, UECF∩∂U).

2.4.1 Drug exchange between Upl and UECF.

We describe diffusive transport by the difference in drug concentrations in CECF and Cpl, multiplied by the BBB permeability, P. In addition, we model active transport into and out of the brain ECF with Michaelis-Menten kinetics, as they are well established and match with most available data on parameters related to BBB active transport, similar to the approach of [6]. In total, this leads to: (9) with u = Cpl, v = CECF, Ptrans being the permeability through the brain endothelial cells, ftrans the fraction of the area occupied by the brain endothelial cells, Dpara the diffusivity of a drug across the paracellular space, WPCS the width of the paracellular space, fpara the fraction of area occupied by the paracellular space, Tm-in the maximum rate of drug active influx, Tm-out the maximum rate of drug active efflux, Km-in the concentration of drug at which half of Tm-in is reached, Km-out the concentration of drug at which half of Tm-out is reached and SABBB the surface area of the BBB.

Based hereon, we describe the loss or gain of unbound drug in the brain ECF due to BBB transport with the following boundary conditions (only those for the x direction are given, the ones for the y and z directions are similar): (10) For the blood-plasma-domain, Upl, we use the reverse of (10) to describe drug transport across the BBB in the brain capillaries with the following boundary conditions: (11)

2.4.2 Drug exchange at the faces of the 3D brain unit.

We use additional boundary conditions to describe the drug concentrations at the sides of the domain. Since we assume that there is no diffusion in the blood plasma (see assumption 1(v)), we use the following boundary conditions: (12) for (x,y,z) ∈ UplUout ∩ ∂U at x = 0 and x = xr, (13) for (x,y,z) ∈ UplUout ∩ ∂U at y = 0 and y = yr, (14) for (x,y,z) ∈ UplUout ∩ ∂U at z = 0 and z = zr.

In addition, we define: (15) for (x,y,z) ∈ Uout ∩ ∂U.

We formulate the condition at the boundaries of the 3D brain unit as follows: (16) and where n is the normal vector on UECF∩∂U.

2.5 Model parameter values and units

The dimensions of the 3D brain unit are based on the properties of the rat brain. The model is suitable for data from human or other species as well, but we have chosen for the rat as for this species most data is available. The distance between the brain capillaries in the rat brain is on average 50 μm, while the brain capillaries have a radius of about 2.5 μm [3437]. Therefore, we set the radius of the brain capillaries, r, to 2.5 μm and the dimensions of the 3D brain unit in the x, y and z directions, xr, yr and zr respectively, to 55 μm.

In our model, we use Eqs (1)–(5) to describe drug concentration within the blood plasma, with boundary conditions described in Eqs (11)–(15). We describe the concentration of drug within the brain ECF with Eqs (6)–(8) with boundary conditions described in (9), (10) and (16). The range of values we use for the parameters in the model as well as their units are given in Table 1 below. This range is based on values found in the literature (from experimental studies), which we also give in the table. The literature does not provide values on the kinetic parameters related to non-specific binding kinetics (B2max, k2on and k2off). Therefore, we base the choices of these values on earlier articles that assume that drug binding to specific binding sites is stronger than to non-specific binding sites, while non-specific binding sites are more abundant [31, 38, 39].

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Table 1. 3D brain unit model parameters and their units, for rat brain.

The physiological range of values of the parameters is given. These are based on references from the literature. All parameters depend on both drug-specific and system-specific properties, except for dcap,r, vblood, vECF, Tm-in, Tm-out, SABBB,B1max and B2max, which depend on system-specific properties only.

https://doi.org/10.1371/journal.pone.0238397.t001

3 Model results

We study the distribution of a drug within the 3D brain unit by plotting its concentration-time profiles within the brain ECF (brain ECF PK). In addition, we study the distribution of the drug within the 3D brain unit. We first nondimensionalise the system of equations and boundary conditions by scaling all variables by a characteristic scale, see S1 Appendix for details. Next, in order to perform simulations, we discretise the nondimensionalised system spatially, using a well-established numerical procedure based on finite element approximations [66]. After weighing accuracy and computational cost, as well as taking into account that small changes in resolution of the computational mesh should not substantially affect the simulation results, we chose a resolution of 18 lines per dimension to proceed with in the simulations. We present the results using the parameters with dimensions. The output of the simulations are the concentrations of free, specifically bound and non-specifically bound drug, given in μmol L-1 over time (s).

The model can easily be used to study a specific drug by choosing the parameter values that are specific for this drug, provided that parameter values for this drug are known. In the present study, however, we choose to study generic parameter values that are in the middle of the physiological ranges given in Table 1. This allows us to perform a sensitivity analysis and study the effect of parameter values at both extremes of the physiological range on the behaviour of the model. We use, unless otherwise indicated, the parameter values that are given in Table 2.

In the following sections, we show the impact of the brain capillary blood flow velocity (vblood) in the absence of active transport (section 3.1), the impact of active transport (section 3.2) and the impact of vblood and active transport combined (section 3.3) on blood plasma and brain ECF PK and brain ECF drug distribution. We give the concentration-time profiles of unbound drug, specifically bound drug and non-specifically bound drug in the middle of UECF, where as well as those of unbound drug in the blood plasma in the middle of Ux1, where , on a log-scale versus time. Drug distribution profiles are given for cross-sections of the entire (x,y,z)-domain of the 3D brain unit for various times.

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Table 2. 3D brain unit model default parameter values and their units.

The values are for a hypothetical drug and are all within the physiological ranges given in Table 1.

https://doi.org/10.1371/journal.pone.0238397.t002

3.1 The effect of the brain capillary blood flow velocity on brain ECF PK within the 3D brain unit

The impact of the brain capillary blood flow velocity, vblood, on brain ECF PK within the 3D brain unit is evaluated. Parameters are as in Table 2 and we thus assume that there is no active transport, i.e. Tm-in = 0 and Tm-out = 0. Here, we focus on the effect of vblood on brain ECF PK in the middle of the 3D brain unit. We show the concentration-time profiles of unbound, specifically bound and non-specifically bound drug (CECF, B1 and B2, respectively) within the 3D brain unit on a larger time-scale, for several values of vblood. We do so for the default value of the passive permeability P (P = 0.1⋅10-7 m s-1), in Fig 3 (left), as well as for a high value of P (P = 100⋅10-7 m s-1), in Fig 3 (right). The lowest value of vblood is outside the known physiological ranges (see Table 1), but we choose it as vblood is predicted to mostly impact drug concentrations in the brain when P is much higher than vblood [67, 68]. The total passive permeability, P, includes both transcellular and paracellular permeability. The paracellular space may increase due to disruption of the tight junctions in certain disease conditions, thereby allowing larger molecules to pass through and increasing paracellular transport [69, 70]. We can tune our model and separate between transcellular and paracellular transport, as we do in S2 Appendix. In the current section we proceed with the total passive BBB permeability.

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Fig 3. The effect of the brain capillary blood flow velocity, vblood (m s-1), on the log PK of Cpl (red) and CECF (top), B1 (middle) and B2 (bottom) for a default (P = 0.1⋅10-7m s-1) (left) and a high (P = 100⋅10-7m s-1) (right) value of P.

Values of vblood are set at 0.05⋅10-4 m s-1, 0.5⋅10-4 m s-1, 5⋅10-4 m s-1, 50⋅10-4 m s-1 and 500⋅10-4 m s-1, as is depicted by different colours, where drug concentrations for the default value of vblood (vblood = 5⋅10-4 m s-1) are shown in blue. All other parameters are as in Table 2. The insets in each sub-figure show the PK for a shorter time.

https://doi.org/10.1371/journal.pone.0238397.g003

Fig 3 shows that vblood does not impact long-time behaviour of CECF, B1 and B2. The insets in Fig 3 demonstrate that vblood impacts short-time (t = 0-100 s) behaviour only when it has extremely low values (vblood≤0.5⋅10-4 m s-1), as depicted in the insets of Fig 3 by the yellow and purple lines, respectively. The impact of vblood on CECF, B1 and B2 is independent of the values of P (compare the left and right insets of Fig 3). The effects of P on drug concentrations within the brain ECF are similar to those found with our proof-of-concept 2D model [31]: for a high value of P, the attained values of CECF and B2 are higher and follow Cpl, while their decay is faster than for a low value of P. In addition, the ≥90% maximum value of B1, i.e. values of B1 that are more than 90% of the maximum value attained during the simulation (B1 ≥90% max(B1)), is attained shorter for a high value of P than for a low value of P.

From the results shown in Fig 3 we conclude that the effects of vblood on brain ECF PK are minimal. According to the Renkin-Crone equation [67, 68], the brain capillary blood flow affects drug influx, depending on the permeability of the BBB. This is also demonstrated by our model, and we show that vblood affects drug influx across the BBB in S3 Appendix.

The plots in Fig 4a show the changes in concentration of drug within the blood plasma over a short time-range (t = 5 to t = 25). There, Cpl is plotted along the capillaries starting at Uin (where drug enters the unit) to Uout (where drug exits the unit). We measure the distance from Uin, where the total distance between these points is 150 μm. Drug can be transported along several pathways, but in Fig 4a the values of Cpl are given along the pathway indicated in Fig 4b. When vblood = 0.5 (left), there are clear differences between Cpl in Uin (Distance = 0) and Cpl in the opposite corner (Distance = 150) at the time-points shown. However, as Cpl increases over time, the differences in Cpl become small relative to the value of Cpl. Fig 4c shows the distribution profiles of unbound drug within the 3D brain unit at t = 5 for different values of vblood. There, darker shades of red and blue correspond to higher concentrations of unbound drug in the blood plasma and the brain ECF, respectively. When vblood = 0.5⋅10-4 m s-1, the transport time of drug between Uin and the opposite corner is higher than when vblood = 5⋅10-4 m s-1. This is depicted in Fig 4c, where at t = 5, drug concentrations within Upl are equal for a high brain capillary blood flow velocity (vblood = 50⋅10-4 m s-1), while local differences in Cpl still exist for a low value of vblood (vblood = 0.5⋅10-4 m s-1). The value of vblood also affects local concentrations of CECF. For a low value of vblood (vblood = 0.5⋅10-4 m s-1), values of CECF at t = 5 are overall low, but highest in the corners closest to Uin. For higher values of vblood (vblood = 5⋅10-4 m s-1 and vblood = 50⋅10-4 m s-1), CECF at t = 5 is overall higher, but again highest in the corner close to Uin.

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Fig 4. Changes in Cpl and CECF due to the effect of vblood.

While vblood is varied from 0.05⋅10-4 m s-1 to 50⋅10-4 m s-1, all other parameter values are as in Table 2. a) The pathway from Uin to Uout along which Cpl is plotted. b) Cpl is plotted against time (timepoints from 5 to 25) along the distance shown in (a). c) Distribution profiles of Cpl (red) and CECF (blue) of the 3D brain unit at t = 5. Darker shades of red and blue correspond to higher values of Cpl and CECF, respectively.

https://doi.org/10.1371/journal.pone.0238397.g004

3.2 The effect of active transport on the drug concentrations within the brain ECF

Active transport kinetics are regulated by the maximal transport rate (Tm) and the concentration of drug needed to reach half of the maximal transport rate (Km), see section 2.4.1. We first focus on active influx, such that Tm-out = 0. We vary Tm-in, which denotes the maximal rate of active transporters moving drug from the blood plasma into the brain ECF. Fig 5 shows the effects of increasing values of Tm-in (starting at Tm-in = 0, i.e. no active influx) on CECF (top), B1 (middle) and B2 (bottom). Fig 5 (top) reveals that an increased value of Tm-in correlates with increased concentrations of CECF. The time to the peak of CECF is not affected by the value of Tm-in. Fig 5 (middle) shows that Tm-in does affect the time during which the specific binding sites are saturated. We find that 90% max(B1) is attained longer for a higher Tm-in. Fig 5 (bottom) shows that higher values of Tm-in correlate with higher values of B2 and thus a greater occupancy of non-specific binding sites. The non-specific binding sites within the brain ECF become saturated with drug when Tm-in is sufficiently high (Tm-in = 100⋅10-7 μmol s-1). To evaluate the effect of active efflux on drug concentrations within the brain ECF, we repeat our simulations with Tm directed outward, i.e. with Tm-out = 0-100⋅10-7 μmol s-1 and Tm-in = 0. Fig 6 (top) shows that CECF decreases faster for higher values of Tm-out, corresponding to more active efflux. Fig 6 (middle) reveals that Tm-out affects the time during which specific binding sites are saturated: the time at which B1 attains 90% max(B1) is smaller for a high value of Tm-out. For sufficiently high values of Tm-out, the binding sites do not become saturated. Fig 6 (bottom) shows that B2 is similarly affected by active efflux as CECF.

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Fig 5. The effect of active influx on the log concentration-time profiles of drug in the brain ECF, relative to those in the blood plasma.

Top: unbound drug in the brain ECF (CECF) compared to unbound drug in the blood plasma (Cpl, red curve). Middle: drug bound to its target sites (B1). Bottom: drug bound to non-specific binding sites (B2). The value of Tm-in is changed from 0 to 100⋅10-7 μmol s-1. The rest of the parameters are as in Table 2.

https://doi.org/10.1371/journal.pone.0238397.g005

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Fig 6. The effect of active efflux on the log concentration-time profiles of drug in the brain ECF, relative to those in the blood plasma.

Top: unbound drug in the brain ECF (CECF) and unbound drug in the blood plasma (Cpl, red curve). Middle: drug bound to its target sites (B1). Bottom: drug bound to non-specific binding sites (B2). The value of Tm-out is changed from 0 to 100⋅10-7 μmol s-1. The rest of the parameters are as in Table 2.

https://doi.org/10.1371/journal.pone.0238397.g006

3.3 The effect of the brain capillary blood flow velocity in the presence of active transport

In section 3.1 we have shown that both the passive BBB permeability, P, and the brain capillary blood flow velocity, vblood, affect dug brain ECF PK in the absence of active transport. Here, we study how P and vblood combined with active transport affect drug PK within the brain ECF. Fig 7 shows the log plot of CECF for vblood = 5⋅10-4 m s-1 (top) and vblood = 0.5⋅10-4 m s-1 (bottom) and for P = 0.1⋅10-7 m s-1 (left) and P = 100⋅10-7 m s-1 (right) in the presence of active influx, i.e. for various values of Tm-in (Tm-out = 0). Note that the vertical scale is the same in all plots. Fig 7 shows how P and vblood affect the impact of Tm-in on brain ECF PK. A smaller value of vblood only slightly reduces CECF when Tm-in is sufficiently high (Tm-in≥10⋅10-7 μmol s-1), see Fig 7, left. An increase in P does reduce the impact of Tm-in on CECF substantially (Fig 7, right). When the BBB is very permeable, like for drugs that easily cross the BBB, such as phenytoin [27, 71], active influx needs to be fast to have any effect, as drug can easily pass the BBB to flow back into the blood plasma. As shown in Fig 7, right, in the presence of a high value of P, Tm-in only (slightly) affects CECF when it is 10⋅10-7 μmol s-1 or higher.

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Fig 7. The log concentration-time profiles of unbound drug in brain ECF (CECF) with 1000x increased permeability P (left to right, 0.1⋅10-7 m s-1 to 100⋅10-7 m s-1) or 10x decreased flow vECF (top to bottom, 5⋅10-4m s-1 to 0.5⋅10-4 m s-1) in the presence of active influx compared to the concentration of unbound drug in the blood plasma (Cpl, red curve).

The value of of Tm-in is changed from 0 to 100⋅10-7 μmol s-1, as depicted by various colours. The rest of the parameters are as in Table 2.

https://doi.org/10.1371/journal.pone.0238397.g007

Fig 8 shows the log profiles of CECF for vblood = 5⋅10-4 m s-1 (top) and vblood = 0.5⋅10-4 m s-1 (bottom) and for P = 0.1⋅10-7 m s-1 (left) and P = 100⋅10-7 m s-1 (right) in the presence of active efflux, i.e. for various values of Tm-out (Tm-in = 0). Fig 8 reveals that vblood does not affect the impact of Tm-out on CECF. This is expected, as vblood mainly affects Cpl, while active efflux depends on CECF. The passive permeability P does affect the impact of Tm-out on CECF. If P is high, drug can easily flow across the BBB back into the brain ECF, following the concentration gradient between the blood plasma and the brain ECF, thereby countering the effect of Tm-out. Fig 8 (top right) shows that for a high P, CECF is only affected by Tm-out when its value is higher than 10⋅10-7 μmol s-1. The values of CECF in the presence of active efflux and a high passive BBB permeability, P, are unaffected by vblood (Fig 8, right).

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Fig 8. The PK on log-scale of unbound drug in brain ECF (CECF) with 1000x increased permeability P (left to right, 0.1⋅10-7 m s-1 to 100⋅10-7 m s-1) and 10x decreased blood flow velocity vblood (top to bottom, 5⋅10-4 m s-1 to 0.5⋅10-4 m s-1) in the presence of active efflux compared to the concentration of unbound drug in the blood plasma (Cpl, red curve).

The value of Tm-out is changed from 0 to 100⋅10-7 μmol s-1, as indicated by the different colours. The rest of the parameters are as in Table 2.

https://doi.org/10.1371/journal.pone.0238397.g008

Next, we study how the drug distribution within the 3D brain unit is affected by vblood, P, Tm-in and Tm-out. Fig 9 shows cross-sections (for and z = 0) of the 3D brain unit at t = 5, in which the distribution of Cpl and CECF is plotted. The values of Cpl and CECF are represented by shades of red and blue, respectively, where darker shades indicate higher concentrations. In Fig 9a (left) we give a plot for a default P and vblood (Fig 9a, left). Then, we decrease vblood (Fig 9a, middle) or increase P (Fig 9a, right). For a lower vblood, relative differences of Cpl over space increase (Fig 9a, middle).

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Fig 9. The distribution profiles at cross-sections (at ) of the 3D brain unit at t = 5 of unbound drug in brain ECF with lower brain capillary blood flow velocity (vblood = 0.5⋅10-4 m s-1, middle column), higher passive BBB permeability (P = 100⋅10-7 m s-1, right column), presence of active influx (middle row, Tm-in = 1⋅10-7 μmol s-1) and presence of active efflux (bottom row, Tm-out = 1⋅10-7 μmol s-1) at t = 5.

Parameters are as in Table 2.

https://doi.org/10.1371/journal.pone.0238397.g009

Additionally, due to the decrease in Cpl, local differences in CECF become more apparent. A larger value of P results in an increased exchange of drug between the blood plasma and the brain ECF, such that CECF becomes higher (Fig 9a, right).

Fig 9b shows that the presence of active influx (Tm-in = 1⋅10-7 μmol s-1) increases CECF. As a consequence, local differences within UECF become relatively small. With a low value of vblood, local differences in Upl become apparent (Fig 9b, middle). Finally, Fig 9c shows that with active efflux, CECF becomes smaller than when no active efflux is present, except for when P is high and more pronounced.

Values of CECF are given in the table in Fig 10c in order to show the differences within the 3D brain unit more clearly. There, values of CECF are given for four different locations within the 3D brain unit for several values of vblood and P and t = 500. The table again (as in Figs 7, 8 and 9) shows that vblood and P affect the impact of Tm-in and Tm-out on CECF. It provides additional information on the distribution of CECF within the 3D brain unit. In general, CECF is higher in the corners relative to the edge and middle within the 3D brain unit. The extent of these local concentration differences depends on the values of Tm-in and Tm-out. The differences are largest when Tm-out = 1⋅10-7 μmol s-1, depicted in the lowest line of each sub-table. There, CECF in corner 2 is higher than in corner 1. In addition, in the presence of active influx, the values of CECF are lower in corner 2 than in corner 1. Again, the extent of this difference depends on the value of Tm-in.

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Fig 10. Values of CECF (10-3 μ mol L-1) at several locations within the brain unit for different values of P and vblood at t = 500.

a) Locations within the 3D brain unit. Corner 1: (x,y,z) = (r,r,r), Corner 2: (x,y,z) = (xr-r,yr-r,zr-r), Edge: (x,y,z) = (0, ,), Middle: (x,y,z) = (,,). b) Values of CECF are shown for a low ((P = 0.01⋅10-8 m s-1), default (P = 0.1⋅10-8 m s-1) and high (P = 1⋅10-8 m s-1) value of P in the top, middle and bottom table, respectively. Within each table, concentrations are given for several values of vblood (vblood = 0.5⋅10-4 m s-1, vblood = 5⋅10-4 m s-1 and vblood = 50⋅10-4 m s-1, left to right), Tm-in (Tm-in = 0, Tm-in = 1⋅10-7 μmol s-1, Tm-in = 10⋅10-7 μmol s-1 and Tm-in = 100⋅10-7 μmol s-1) and Tm-out (Tm-out = 0, Tm-out = 1⋅10-7 μmol s-1, Tm-out = 10⋅10-7 μmol s-1 and Tm-out = 100⋅10-7 μmol s-1) at different locations. When Tm-in is changed, Tm-out = 0 and vice versa. c) Colour legend. In each table, colours are relative to the value of CECF in the middle of the unit in the absence of active transport for vblood = 5⋅10-4 m s-1, of which the colour is denoted by “Default”. The intensity of green corresponds to the extent of increase, and the intensity of red corresponds to the extent of decrease of CECF compared to the default. Other parameters are as in Table 2.

https://doi.org/10.1371/journal.pone.0238397.g010

4 Discussion

We have developed a mathematical model that describes the local distribution of a drug within a 3D brain unit as an extension of our earlier 2D proof-of-concept model [31]. The 3D brain unit is represented as a cube. This new model provides an important step towards more realistic features of the brain. The 3D representation allows for the brain ECF to be represented as a continuous medium. The brain capillary blood flow and active transport across the BBB have been explicitly incorporated. This enables us to more realistically predict the impact of the interplay of cerebral blood flow, BBB characteristics, brain ECF diffusion, brain ECF bulk flow and brain (target) binding on drug distribution within the brain. Altogether our model allows the study of the effect of a large amount of parameters values (summarized in Table 1) on drug distribution within the 3D brain unit.

The current modelling work is based on certain assumptions (Assumptions 1 and 2). We will shortly discuss their probability and impact (see [72]) below. Assumptions 1(i), 1(ii), 1(iv), 1(v), 2(i), 2(iii), 2(v), 2(vi) and 2(viii) are based on actual physiological processes, adapted to the simplified geometry of the 3D brain unit. Therefore, these assumptions are unlikely to be violated, but the impact of violation would be high on the results of the simulations. Assumptions 1(iii) and 1(vi) are known to be more complex in real, but are expected to have a small impact when violated. Assumption 1(vii) is not violated for drugs that do not bind plasma proteins. However, for drugs that do bind plasma proteins, the assumption is likely violated with an impact to be investigated in future work. In similar fashion, assumptions 2(ii), 2(iv) and 2(vii) are not violated for drugs that do not cross cells, but it is likely that for drugs that do, they are violated with an impact to be investigated in future work.

In the present work, we have investigated the properties of the 3D brain unit with a sensitivity analysis and thus looking at hypothetical compounds. The advantage of studying the model in this way is that it allows us to investigate a wider range of parameter values than an existing compound would have allowed. Moreover, the hypothetical compound has parameter values that are within and on the extremes of the reported physiological ranges and we therefore believe that it is an accurate representation of reality. The study has focused on the effect of the newly implemented brain properties on brain ECF concentrations a drug within the brain. It is shown that the brain capillary blood flow velocity and the passive BBB permeability affect the concentration of a drug within the brain, and, as anticipated [73, 74] that a low brain capillary blood flow velocity affects the short-term, but not the long-term concentration-time profiles of Cpl and CECF (Figs 3 and 4). In addition to the confirmation of these earlier reported results, our model enables the study of the effects of BBB permeability in conjunction with other factors, like active BBB transport, on brain ECF PK. It was found that passive BBB permeability has a high impact on brain ECF PK, even when drug is actively transported across the BBB. Moreover, the BBB permeability and, in smaller extent, the brain capillary blood flow velocity affect the impact of active influx on drug PK within the brain ECF (Figs 7 and 8). Interestingly, the brain capillary blood flow velocity, passive BBB permeability and active transport do not only affect the concentration of drug within the brain ECF, but also its distribution within the brain ECF (Figs 9 and 10). The local differences observed within the 3D brain unit exist on a relatively small time-scale. It is anticipated that in certain cases, like those of high drug-target binding or active transport, these differences may also exist on a larger time-scale, but this requires further investigation.

To ensure the quality of a mathematical model, the model predictions are ideally compared to experimental data. Validation of the presented model however, describing spatial drug distribution within the brain ECF, is not straightforward. As experimental data on spatial drug distribution within brain ECF are not yet available on the level of detail as predicted by our model, we show results that are new. The results of our simulation are therefore a hypothesis and serve as a lead for experiments. For the present work, it is already possible to validate parts of the model. For example, in the current manuscript, we have compared our results on the effect of brain capillary blood flow on BBB influx with the well-established Renkin-Crone equation. The results were shown to agree, which supports our hypothesis that our basic description of blood plasma PK is realistic. Ideally, a thorough interplay between theoretical and experimental work is developed in future, leading to a gain in knowledge in spatial drug distribution on the most efficient way possible.

Taken together, the current 3D brain unit model shows the impact of drug-specific and brain-specific parameters on drug distribution within the brain ECF. The added value is that all these factors can now be studied in conjunction to understand the interdependencies of multiple brain parameter values and drug properties, as was shown in this work. This makes this single 3D brain unit model suitable for the next step, which is to mount up multiple units to represent a larger volume of brain tissue, in which the brain tissue properties for each unit can be defined independently. With the establishment of the current 3D brain unit model, we are now ready to incorporate intra-extracellular exchange and drug binding to intracellular binding sites in future modelling work. As the current model is in 3D, the units can be built up, and drug distribution within the brain ECF can be described, in all possible directions. The units may be given different systemic properties (such as the BBB permeability or drug target concentration), to represent the heterogeneity of the brain in a 3D manner.

Supporting information

S2 Appendix. The effect of paracellular permeability on PK within the brain ECF.

https://doi.org/10.1371/journal.pone.0238397.s002

(PDF)

S3 Appendix. The Renkin-Crone equation and the 3D brain unit model. [67, 68, 75].

https://doi.org/10.1371/journal.pone.0238397.s003

(PDF)

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