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

TLDR: A 3D brain unit model is developed, which builds on the previous proof-of-concept 2Dbrain unit model, to understand the factors that govern local unbound and bound drug PK within the brain, and shows that all the above mentioned factors affect brain ECF PK in an interdependent manner.

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 trancellular 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. Author summary Insights on how a drug distributes within the brain over both time and space are still limited. Here, we develop a ‘3D brain unit model’ in order to understand the factors that control drug concentrations within a small piece of brain tissue, the 3D brain unit. In one 3D brain unit, the brain capillaries, which are the smallest blood vessels of the brain, surround the brain extracellular fluid (ECF). The blood-brain barrier (BBB) is located between the brain capillaries and the brain ECF. The model describes the impact of brain capillary blood flow, transport across the BBB, diffusion, flow and drug binding on the distribution of a drug within the brain ECF. We distinguish between free (unbound) drug and drug that is bound to binding sites within the brain. We show that all of the above mentioned factors affect drug concentrations within brain ECF in an interdependent manner. The 3D brain unit model that we have developed is an excellent tool to increase our understanding of how local drug concentrations within the brain ECF are affected by brain transport and binding processes.

Figures

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.

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.

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.

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.

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.

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, yr 2 , zr 2 ), Middle: (x,y,z) = ( xr 2 , yr 2 , zr 2 ). 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.

Full text

RESEARCH ARTICLE
A 3D brain unit model to further improve
prediction of local drug distribution within the
brain
Esme
´
e Vendel
ID
1
, Vivi Rottscha
¨
fer
1
*, Elizabeth C. M. de Lange
2
*
1 Mathematical Institute, Leiden University, Leiden, The Netherlands, 2 Leiden Academic Centre for Drug
Research, Leiden University, Leiden, The Netherlands
* vivi@math.leidenuniv.nl (VR); ecmdelange@lacdr.leidenuniv.nl (EL)
Abstract
The development of drugs targeting the brain still faces a high failure rate. One of the rea-
sons is a lack of quantitative understanding of the complex processes that govern the phar-
macokinetics (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 concen-
tration-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 pro-
cesses 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 arteri-
ole 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 con-
trolled by the blood-brain barrier (BBB).
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OPEN ACCESS
Citation: Vendel E, Rottscha¨fer V, de Lange ECM
(2020) A 3D brain unit model to further improve
prediction of local drug distribution within the
brain. PLoS ONE 15(9): e0238397. https://doi.org/
10.1371/journal.pone.0238397
Editor: Stefan Liebner, Institute of Neurology
(Edinger-Institute), GERMANY
Received: March 20, 2020
Accepted: August 15, 2020
Published: September 23, 2020
Copyright: © 2020 Vendel et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All relevant data are
within the manuscript.
Funding: The authors received no specific funding
for this work.
Competing interests: The authors have declared
that no competing interests exist.

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 [2–10]. 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 permeabil-
ity, 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 com-
pounds either inward (in the direction of the brain ECF, active efflux) or outward (in the direc-
tion 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, 18–25] 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 quantita-
tive 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.
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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
ffiffiffiffi
D
D

p
, with D being the normal dif-
fusion 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 distri-
bution 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 mecha-
nisms 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 neigh-
bouring 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 con-
nected 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 multi-
ple sub-domains. These include the brain capillary domain where drug enters the unit (indi-
cated by U
in
in Fig 1), the domains representing the x-directed, y-directed and z-directed
brain capillaries (indicated by U
x1−x4
, U
y1−y4
and U
z1−z4
in Fig 1) and the brain capillary
domain where drug leaves the unit (indicated by U
out
in Fig 1). Drug within the blood plasma
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is transported by the brain capillary blood flow. The brain capillary blood flow splits at the ver-
tices of the unit, where brain capillary branching occurs (Fig 1, right).
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 vol-
ume 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 U
in
( Fig 1, middle), and drug concentra-
tions in U
in
are unaffected by the brain capillary blood flow.
(iv) The brain capillary blood flow is directed away from U
in
( 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 diffu-
sion imposed by the cells) into account.
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: U
in
is the domain where the dose of absorbed drug enters the 3D brain unit, U
x1-x4
, U
y1-y4
and
U
z1-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 U
in
. 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
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(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) 2 R
3
j 0x x
r
^ 0yy
r
^ 0zz
r
}. There, x
r
, y
r
and z
r
are constants that represent
the length of one unit and are defined as d
cap
+2r, with d
cap
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 U
pl
U, U
BBB
U and U
ECF
U, respectively, such
that U = U
pl
[ U
BBB
[ U
ECF
.
Within U
pl
, we define U
in
as the domain where the blood plasma, containing drug, enters
the 3D brain unit from a feeding arteriole. We define U
out
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 {U
xi
,i = 1,‥,4}, {U
yi
,i = 1,‥,4} and {U
zi
,
i = 1,‥,4}. The brain capillaries are divided by the lines x = y (or y = z or x = z) and x+y = y
r
(or
y+z = z
r
or x+z = z
r
), for which an example is shown in Fig 2. The only exceptions for this are
the brain capillaries adjacent to U
in
and U
out
, see below.
The definitions of the regions are as follows:
U
x1
= {(x,y,z) 2 U j rx<x
r
-y, rx<x
r
-z ^0y<r ^ 0z<r}
U
x2
= {(x,y,z) 2 U j y
r
-y<xy ^ zx<x
r
-z ^y
r
y>y
r
-r ^ 0z<r}
U
x3
= {(x,y,z) 2 U j yx<x
r
-y ^ z
r
-z<xz ^ 0y<r ^ z
r
z>z
r
-r}
U
x4
= {(x,y,z) 2 U j y
r
-y<xy ^ z
r
-z<xz ^ y
r
y>y
r
-r ^ z
r
z>z
r
-r}
U
y1
= {(x,y,z) 2 U j ry<y
r
-z ^ryy
r
x ^0x<r ^ 0z<r}
U
y2
= {(x,y,z) 2 U j zy<y
r
-z ^x
r
-xy<x ^ x
r
x>x
r
-r ^ 0z<r}
U
y3
= {(x,y,z) 2 U j z
r
-z<yz ^ x<yy
r
-x ^ 0x<r ^ z
r
z>z
r
-r}
U
y4
= {(x,y,z) 2 U j z
r
-zy<z ^ x
r
-x<yx ^ x
r
x>x
r
-r ^ z
r
z>z
r
-r}
U
z1
= {(x,y,z) 2 U j rzz
r
-x ^ rzz
r
-y ^ 0x<r ^ 0y<r}
U
z2
= {(x,y,z) 2 U j x<zz
r
-x ^y
r
-yz<y ^ 0x<r ^ y
r
y>y
r
-r}
U
z3
= {(x,y,z) 2 U j x
r
-xz<x ^ y<zz
r
-y ^x
r
x>x
r
-r ^ 0y<r}
U
z4
= {(x,y,z) 2 U j x
r
-xz<x ^ y
r
-yz<y ^ x
r
x>x
r
-r ^ y
r
y>y
r
-r}
U
in
= {(x,y,z) 2 U j 0x<r ^ 0y<r ^ 0z<r}
U
out
= {(x,y,z) 2 U j x
r
-rx<x
r
^ y
r
-ry<y
r
^ z
r
-rz<z
r
}.
The BBB is represented by a subset U
BBB
U, such that U
BBB
= @ U
pl
\@ 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 U
ECF
U, such that U
ECF
= U\(U
pl
[U
BBB
).
Within U we define the following quantities describing drug concentration:
C
pl
(x,y,z,t): U
pl
x R
þ
! R
þ
;
C
ECF
(x,y,z,t): U
ECF
x R
þ
! R
þ
;
B
1
(x,y,z,t): U
ECF
x R
þ
! R
þ
;
B
2
(x,y,z,t): U
ECF
x R
þ
! R
þ
:
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "A 3d brain unit model to further improve prediction of local drug distribution within the brain" ?

In this article the authors show that all the above mentioned factors affect brain ECF PK in an interdependent manner. 

Q2. What are the future works mentioned in the paper "A 3d brain unit model to further improve prediction of local drug distribution within the brain" ?

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. With the establishment of the current 3D brain unit model, the authors are now ready to incorporate intra-extracellular exchange and drug binding to intracellular binding sites in future modelling work. 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.