Download Signaled drug delivery and transport across the blood

Signaled drug delivery and transport across the blood-brain
barrier
Peter Hinow∗
John N. J. Reynolds§
Ami Radunskaya†
Sean M. Mackay‡
Morgan Schroeder¶
Eng Wui Tan‡
Ian G. Tuckerk
September 30, 2015
Abstract
We use a mathematical model to describe the delivery of a drug to a specific
region of the brain. The drug is carried by liposomes that can release their cargo
by application of focused ultrasound. Thereupon, the drug is absorbed through the
endothelial cells that line the brain capillaries and form the physiologically important blood-brain barrier. We present a compartmental model of a capillary that is
able to capture the complex binding and transport processes the drug undergoes
in the blood plasma and at the blood-brain barrier. We apply this model to the
delivery of levodopa (L-dopa, used to treat Parkinson’s disease) and doxorubicin
(an anticancer agent). The goal is to optimize the delivery of drug while at the
same time minimizing possible side effects of the ultrasound.
Keywords: liposomal delivery, focused ultrasound, mathematical modeling.
1
Introduction
The challenge for drug delivery science is summarized by the following: to deliver the
drug specifically (or at least selectively) to the target site, in the required amount, at
the required rate, at the appropriate time and for the required time period. So called
responsive nanocarriers loaded with drug are of interest in meeting the challenge [1].
Ideally, responsive nanocarriers are designed to release some or all their drug load
when stimulated by an external trigger (e.g. ultrasound, near infrared) or by an internal
∗
Department of Mathematical Sciences, University of Wisconsin - Milwaukee, PO Box 413, Milwaukee, WI 53201, USA; Corresponding author; email: [email protected], phone: (++1) 414 229 6886
†
Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA
‡
Department of Chemistry, University of Otago, Dunedin, New Zealand
§
Department of Anatomy and The Brain Health Research Centre, University of Otago, Dunedin,
New Zealand
¶
Department of Biology, University of Oregon, Eugene, OR 97403, USA
k
New Zealand’s National School of Pharmacy, University of Otago, Dunedin, New Zealand
1
trigger (e.g. local pH, a specific enzyme) [2]. If the external stimulus is focused at the
target site for drug release, it will trigger release of drug from the responsive drugloaded nanoparticles as they are carried by the blood through that region. Sophisticated
targeted delivery using magnetic resonance guided high intensity focused ultrasound
(MR-HIFU) has been used for targeted release of cytotoxic drugs from heat-sensitive
liposomes [3, 4].
Although it may be useful to use focused ultrasound (FUS) to trigger release of
drug from circulating liposomes in a target region when the drug is required, research is
required to ensure therapeutic concentrations are achieved in the tissue after application
of FUS. The relationship between intravenous drug and the drug concentration in the
tissue after FUS treatment may lead to adjustments to clinically prescribed dosages [5].
The dose that is delivered to the target site will depend on the liposomes (concentration,
drug loading, responsiveness), the FUS characteristics, the properties of the drug, the
transfer processes to the target, and processes interfering with transfer to the target.
Safety considerations limit the intensity and choice of frequency of the FUS [6].
Delivery of drugs to the brain has special challenges because of the selective barrier
function of the blood brain barrier (BBB). The neurovascular unit of the BBB consists
of the capillary endothelial cells, astrocytes, pericytes and neurons. Because of the presence of tight junctions and efflux transporters, this structure is far less permeable than
non-brain capillaries [7] although it has carriers for the uptake of selected molecules
(e.g. glucose, large aromatic amino acids). This selective permeability poses a major
challenge to the delivery of drugs, such as those used to treat dementia, movement disorders and brain cancers [8]. Drug released from responsive liposomes into the blood in the
target region must cross the BBB. If the drug is transferred by an active process (e.g. as
for L-dopa), the local high concentration may saturate the transporter compromising
transfer. Conversely, a localized high concentration may be beneficial for transfer of a
drug which is subject to P-glycoprotein (P-gp) efflux at the BBB, but plasma protein
binding of released drug may reduce brain uptake. Further, HIFU, particularly in the
presence of microbubbles, has been demonstrated to increase the permeability of the
BBB [9, 10].
In this paper we describe a model of a single capillary to predict the uptake of drug
into the tissue supplied by the capillary, where the drug is released from ultrasound
(US)-sensitive liposomes. We anticipate that this model will form the basis of a larger
model for the cerebral vasculature [11, 12]. Previous mathematical models have focused
on delivery of oxygen to the brain [13, 14] and drugs in single capillaries [15]. Using
a spatial advection-diffusion type model (partial differential equation), Richardson et
al. [15] investigated the delivery of a drug with the help of a magnetic carrier particle. Compartmental (ordinary differential equation) models for drug transfer across the
blood-brain barrier were proposed recently by Liu et al. [16] and Ball et al. [17]. In
the latter paper, a full body pharmacokinetic model was constructed in order to scale
results obtain in rats to humans. It is also worthwhile to mention the work of Gugat
[18] who used a system of hyperbolic partial differential equations to model contaminant
transport in a water distribution network.
2
Mathematical modeling has long proved its usefulness in pharmaceutical science, in
particular when concerned with the understanding of drug delivery, release and transport processes; see [19, 20, 21] for some recent reviews. Our earlier work addressed
predicting the release kinetics of matrix tablets [22, 23] and the modulation of the
permeability of liposome membranes by binding of bile salts [24]. An important contribution of mathematical modeling is to help understand the release from the delivery
device [1] and drug transport in the tissue [25, 26] where experiments are often difficult,
expensive and possibly fraught with ethical problems. The mathematical tools include
ordinary differential equations, partial differential equations, and cellular automata, depending on the problem at hand [21]. While it is desirable to have experimental data
available to test the predictions of a mathematical model, the process of modeling and
numerical simulations themselves are valuable as they often raise questions and drive
future research.
In this paper, we use ordinary differential equations to describe a compartmental
model of blood flow in a capillary. This modeling framework is commonly used as a first
step in the development of a mathematical description of a complex network describing
blood flow in the brain. The differential equations of the model are derived by balancing
gain and loss terms for each substance due to transport into or out of the compartment
and to chemical reactions among the constituents. The system of equations is then
solved numerically using open source software. In many cases, when parameters in
the model are unknown, experimental data can be used to fit the model to the data
[24]. Ordinary differential equations are also amenable to tools from control theory and
optimization [27], which has found many applications for example in improvement of
cancer chemotherapy protocols.
Here we propose a compartmental ordinary differential equation model that describes the delivery of a drug to the brain from responsive liposomes. The model considers the interactions of the free drug with proteins in the plasma and the brain tissue
[28] and nonlinear active transport mechanisms [17]. We focus on the transport of two
drugs, L-dopa, a drug that readily crosses the BBB via a carrier [29] and doxorubicin
that transfers poorly. L-dopa, a precursor of dopamine, is used to treat Parkinson’s
disease. However, long term exposure to the drug can produce fluctuations that reduce
the beneficial effects of the treatment [30]. It is being investigated for possible targeted
delivery [31]. Doxorubicin, used to treat various cancers, is delivered in liposomes to
diminish cardiac side effects. Recent work by Aryal et al. [32] and Graham et al. [33] has
used FUS in the presence of microbubbles to increase the permeability of the BBB to
doxorubicin thereby improving the survival of rats with implanted gliomas. Previously,
Enden and Schroeder [1] studied drug release from liposomes subjected to low frequency
ultrasound. Herein we take the further step to consider the subsequent transport and
clearance of the drug in the vascular network serving the brain.
This paper is organized as follows. In Section 2 we present a general ordinary differential equation model for the delivery of a drug to the brain by way of carrier particles
and their subsequent triggered release. This model is intended as a “blueprint” for future work not only by our own group but hopefully also by other researchers. We then
3
give further details and specify this model for the two drugs L-dopa and doxorubicin in
Section 3. In Section 4 we describe in vitro experiments to determine the permeability
of liposomes exposed to ultrasound and we discuss the parametrization of our mathematical model. The results of this research are presented in Section 5. First we report
the release of carboxyfluorescein from sonosensitive dioleoylphosphatidylethanolamine
(DOPE) liposomes [34] under different ultrasound parameters. The remainder of that
section contains numerical simulations of liposomal drug delivery for both L-dopa and
doxorubicin. The discussion in Section 6 ends the paper.
2
The general model
For the purpose of mathematical modeling, we assume that
1. the transport of the drug across the blood-brain barrier happens in a capillary
network that is replaced by a single “surrogate” vessel,
2. this vessel serves a volume of brain tissue with a well-defined boundary,
3. the injection of the liposomes happened sufficiently long ago such that they are
uniformly distributed in the blood, and
4. the blood vessel is entirely contained in the volume to which the ultrasound signal
is applied.
A schematic depiction of the ultrasound mediated drug release is given in Figure 1.
Let u denote the concentration of liposomes and u0 its baseline blood concentration,
which we at least initially assume to be constant. The capillary volume is sufficiently
small so that the destruction of liposomes in this volume does not alter significantly
their concentration in the total blood volume. The liposomes and their contents are
identified as one, i.e. whether the ultrasound destroys the liposomes or merely causes
them to leak is insignificant for the purpose of modeling, as only the source of drug, u,
gets depleted.
We denote by f the inverse of the transition time of the blood through the capillary
network (in s−1 ). The differential equation for u is given by
du
= f (u0 − u(t)) − h(s(t))u(t),
dt
(1)
where s(t) ≥ 0 is the time-dependent ultrasound signal. The function h is a non-negative
monotone increasing function that models the effect of the ultrasound on the liposomes,
which is discussed further in Section 3. If s ≡ 0, then this equation has the steady state
solution u = u0 . We also take u0 as initial condition for u.
Let α ≥ 1 be the number of drug molecules per liposome, which we call the “drug
loading coefficient” of the liposomes. We assume that α is constant initially. Once
released from the liposome carriers, the drug may bind to plasma proteins in the blood
and is subsequently unable to cross the BBB [28]. Note that this loss is independent
4
cell uptake, clearance
brain parenchyma
passive transport
active transport
blood-brain barrier
capillary
blood flow
ultrasound
Figure 1: The general mechanism of ultrasound-mediated drug release and transport
across the blood-brain barrier. Yellow circles represent drug-loaded liposomes, green
ellipses represent plasma protein and purple circles represent the drug molecules.
of the flow rate. We let k1 denote the binding rate constant of drug to plasma protein
and assume that there is always enough plasma protein, so that no competition for
binding occurs. Transport across the BBB can occur either by passive mechanisms due
to concentration differences only, or by active transport. We denote the rate constant for
the passive transport in Fick’s law by k2 > 0. The active transport occurs at a maximal
rate k3 and K is the drug concentration at which the active transport occurs at half
the maximal rate. Then we have the following equation for the drug concentration v in
the capillary
dv
k3
= αh(s(t))u(t) − (f + k1 )v(t) − k2 (v(t) − w(t)) −
v(t),
dt
K + v(t)
(2)
where w(t) is the concentration of free drug in the brain tissue. We assume that there
is no active transport in the direction from the brain to the blood. We note, however,
that active transport from brain tissue to the blood could be incorporated into future
models of drugs that are subject to efflux at the blood-brain-barrier.
Let V denote the ratio of the brain tissue volume to the volume of the capillary.
Then we have for w
dw
k3
V
= k2 (v(t) − w(t)) +
v(t).
(3)
dt
K + v(t)
The equation for w contains no terms for metabolism or elimination of free drug from
the brain. These can be added if desired and will be further discussed in Section 3. The
5
initial conditions for all variables are
u(0) = u0 ,
v(0) = 0,
w(0) = 0.
The triggered delivery of drug with liposomes and ultrasound has to be compared
with the direct delivery, where the drug is either injected into the bloodstream or taken
orally and then absorbed. The equation for this direct delivery method
dv
k3
= f v0 − (f + k1 )v(t) − k2 (v(t) − w(t)) −
v(t),
dt
K + v(t)
(4)
where v0 is the concentration of free drug in the blood when it arrives in the brain. The
equation for the drug concentration in the brain is the same as (3).
3
Details and Special Cases: L-dopa and Doxorubicin
Ultrasound can trigger the release of a drug from liposome carriers via thermal and
non-thermal effects. A careful study of the effect of ultrasound on release rates from
dierucoylphosphatidylcholine (DEPC) liposomes is reported in [35], where liposomes
loaded with fluorescent molecules were subjected to focused ultrasound at different frequencies and intensities, for varying amounts of time. Release rates were estimated as
the ultrasound parameters were varied. In Section 4 we describe a calibration experiment with a type of liposome that is sensitive to sonication below the inertial cavitation
threshold. Here we report the strength of the ultrasound signal as the peak negative
pressure, s. In our experiments we use a fixed ultrasound frequency of 1 M Hz, so that
s is equivalent to the mechanical index. Due to potential tissue damage, it is generally
agreed that a mechanical index of 1.9 should not be exceeded [6]. As a consequence of
this, all in vitro ultrasound release experiments were conducted at acoustic pressures less
than 1 M P a, see Figure 2 below. We assume the following about the general response
of these liposomes to ultrasound.
1. There is a threshold smin , below which the lipid molecules are in a gel phase. In
this phase the liposomes exhibit a minimum baseline rate of leakage.
2. For a given population of liposomes, the release rate increases linearly with ultrasound strength above this threshold.
Based on these assumptions, we use the following piecewise linear function
0
if 0 ≤ s < smin
h(s) = a0 +
a1 (s − smin ) if smin ≤ s.
(5)
We therefore have to determine three parameters that will be specific to the particular liposomes used, namely the ultrasound threshold smin , the release rate below this
threshold a0 , and the release response slope a1 . In order to model the background
release of drug, we let the loading coefficient, α, in equation (2) satisfy the equation
dα
= −a0 α
dt
6
(6)
so that α(t) = α0 e−a0 t , where α0 is the initial drug loading coefficient. This plays a role
since the duration of the treatment is comparable to the time scale of drug loss from
the liposomes (see Section 5.3 for further discussion). The activation threshold smin
may be lowered and the slope a1 may be increased in the presence of microbubbles. Ultrasonication of microbubbles in close proximity to cell membranes has been proposed
to cause transient openings in the lipid bilayer (sonoporation) [36], possibly through
inertial cavitation in some cases, although the exact mechanism is unclear. The clearance of liposomes from the blood also necessitates the addition of a differential equation
describing the evolution of u0
d
u0 = −k6 u0 (t)
(7)
dt
If the permeability of the blood-brain barrier changes under the influence of ultrasound, then we introduce a differential equation
d
0
if 0 ≤ s < σmin
k2 (t) = −c1 k2 (t) +
(8)
c2 (s − σmin ) if σmin ≤ s.
dt
Here σmin is the ultrasound threshold that causes the opening of the blood-brain barrier.
The constants c1 and c2 denote the closing and opening rates respectively, of the bloodbrain barrier. The value for k2 from this equation is then used in equations (2) and (3).
Note that in absence of ultrasound, equation (8) has only the equilibrium solution 0;
this is also the initial condition for k2 .
In some cases it may be of interest to track the metabolism or clearance of the
drug that has reached the brain. In this work, we will follow the metabolism of L-dopa
which is converted to dopamine. To this end, we introduce a new variable z which
is the concentration of metabolite in the brain. There are many ways in which the
metabolism can be modeled. We use the simplest of these, namely first-order kinetics.
Thus equation (3) is augmented and an equation for z is introduced as follows
V
dw
k3
= k2 (v(t) − w(t)) +
v(t) − V k4 w(t),
dt
K + v(t)
dz
= k4 w(t) − k5 z(t),
dt
(9)
where k4 is the rate constant of conversion, and k5 is the clearance rate constant for the
metabolite. The initial condition for z is z(0) = 0.
To summarize, in the following simulations we will use equations
• (1), (2), (6), (7) and (9) for L-dopa and equations
• (1) - (3) and (8) for doxorubicin, with the obvious modifications.
The transfer function of the ultrasound is given by (5) throughout the paper.
7
4
4.1
Materials and Methods
Liposome Experiments
All chemical reagents for liposome preparation were of analytical grade, obtained
from Sigma-Aldrich, and used without further purification. 1,2-Distearoyl-sn-glycero3-phosphocholine (DSPC), 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE),
and 1,2-distearoyl-sn-glycero-3-phosphoethanolamine-N-[methoxy(polyethylene glycol)2000] (ammonium salt) (DSPE-PEG2000) were obtained from Avanti Polar Lipids Inc.
Milli-QTM water with a resistivity of 18 M Ω cm−1 was used in the preparation of all solutions. Sodium phosphate buffer solution (20 mmol L−1 ) was prepared with Na2 HPO4
adjusted to pH 7.4 with HCl and degassed under vacuum. Polycarbonate membranes
and mini-extruder were obtained from Avanti Polar Lipids Inc. and used for the preparation of all liposomes. Fluorescence measurements were performed using a PerkinElmer LS50B luminescence spectrometer. Dynamic light scattering was conducted on
a Malvern Zetasizer Nano.
Sonosensitive DOPE liposomes were prepared using the lipid formulation
DOPE:DSPC:DSPE-PEG2000:Cholesterol in a 25 : 27 : 8 : 40 mol% ratio as previously
described by Evjen et al. [34]. Liposomes were prepared by adding chloroform solutions of DOPE (233 µL, 8 mg mL−1 ), DSPC (152 µL, 14 mg mL−1 ), DSPE-PEG2000
(321 µL, 7 mg mL−1 ), and cholesterol (774 µL, 2 mg mL−1 ) to a round-bottom flask.
The solvent was removed in vacuo over one hour to produce a dry thin lipid film on
the surface, which was rehydrated under sonication with 1 mL of a carboxyfluorescein
(CF) solution (100 mmol L−1 CF, Na2 HPO4 buffer 20 mmol L−1 , pH 7.4) to give a final
lipid concentration of 5 mmol L−1 . The lipid suspension was heated to 50◦ C, then first
R
extruded 11 times through 1000 nm polycarbonate membranes using an Avanti
MiniExtruder, and subsequently extruded a further 11 times through a 200 nm membrane
to give a unilamellar liposome suspension with an average diameter of approximately
200 nm.
The unentrapped solute was removed by size exclusion chromatography using a
R
Sephadex
G-100 gel column. The liposome suspension was collected and the phospholipid concentration was determined using the Stewart Assay [37], whereupon the
concentration was adjusted to 1 mmol L−1 . To confirm that the liposomes were formed
with minimal polydispersity and at the correct diameter, the suspension was measured
using dynamic light scattering (DLS).
Fluorescence measurements were performed using an excitation wavelength of
465 nm and an emission of 520 nm, slit widths of 2.5 nm, and an integration time
of 0.5 s. The fluorescence intensity measurements were recorded in a quartz fluorescence cuvette at a scattering angle of 90◦ . The quartz cuvette was filled with 2.94 mL
of pH 7.4 buffer and 60 µL of liposome suspension. The initial fluorescence intensity
of the resulting suspension was monitored for a duration of 30 s, after which the cuR
vette was removed, coupled to the ultrasound transducer using Aquasonic
ultrasound
transmission gel, and irradiated across the length of the cuvette. Ultrasound was applied at various intensities, producing acoustic pressures ranging from 0 to 0.64 M P a,
8
determined using a Precision Acoustics Ltd. 0.5 mm needle hydropohone with a 9 µm
PVdF membrane. A continuous wave was applied for a pulse duration of 3 s at 1 M Hz
(i.e. 3 · 106 cycles of ultrasound), after which the fluorescence intensity was measured
over 30 s. This was repeated for 5 pulses. To obtain the final fluorescence intensity,
100 µL of 10% Triton-X100 was added to the suspension. The fluorescence intensity
after each pulse was averaged and converted to carboxyfluorescein concentration using
a standard curve, which was created by measurement of the fluorescence intensity of
carboxyfluorescein at various concentrations ranging from 0 to 16 µmol L−1 .
4.2
Model Parametrization
The average diameter of a liposome is d = 2 · 10−7 m and so its volume is
Vlip =
πd3
= 4.2 · 10−18 L.
6
The solubility of L-dopa hydrochloride in water is cLD = 5 · 10−2 mol L−1 . The number
of L-dopa molecules encapsulated in one liposome by passive loading is therefore
αLD = NA · Vlip · cLD = 6 · 1023 mol−1 · 4.2 · 10−18 L · 5 · 10−2 mol L−1 = 1.26 · 105 .
Since commercially prescribed doxorubicin is loaded into liposomes by an “active” loading method, a 100-fold increase in the concentration of the drug in the liposome is
achievable [38], we therefore work with αDXR = 1.3 · 107 .
For the simulations presented in this paper, we use the baseline concentration of
liposomes in a human’s blood volume, which is
u0 =
1.25 · 1015
= 420 pmol L−1 ,
NA 5L
where 1.25 · 1015 is a typical dose. Results reported in [39] suggest that a half-life of
liposomes in the circulation of 24 hours is reasonable, yielding k6 = 8.3 · 10−6 s−1 .
According to Bär [40], the vascular volume fraction in the rat cerebral cortex is
0.015, which is largely independent of brain size between mammals [41, Figure 3C].
Thus we work with V = 65 (which is dimensionless). Kleinfeld et al. [42, Table 1]
report a mean red blood cell velocity in the capillaries of the rat cortex of 0.7 mm s−1 .
Even though capillary lengths in rat brains may vary greatly, we assume a “network
length” of 350 µm. Thus we obtain f = 2 s−1 .
The drug is subject to plasma binding from the moment of release from the liposomes. The equilibrium values of free drug have been reported by Khor and Hsu for
L-dopa [43] and by Greene et al. [44] for doxorubicin. The actual binding rate constants
are difficult to measure, and those for doxorubicin are not available in the literature.
Work of Huang et al. indicates that the half life time of radioactively marked L-6[18 F]Fluoro-DOPA is 30 s, [45, Figure 3], giving k1 = 0.02 s−1 . However, binding rates
that have been measured vary dramatically. For example, Sevilla et al. [46] estimate
values of k1 for emodin to be between 250 and 500 s−1 , while Paul et al. [47] find a much
9
K (µmol L−1 )
k3 (µmol L−1 s−1 )
28.2
0.3
34.2
0.35
101.6
1.1
Table 1: Parameter values governing the active transport of L-dopa.
slower binding rate of approximately 1.5 · 10−3 s−1 for the anti-HIV drug 1-phenylistan
(1-Pl). In the drug-specific simulations we use k1 = 0.6 s−1 for L-dopa and k1 = 1 s−1
for doxorubicin.
In terms of first-order kinetics, the rate of transport across the endothelial wall separating the brain from the blood compartment is given by Pθ SCblood , where Pθ is the
permeability coefficient, S the surface area and Cblood the concentration of drug in the
blood. It must be emphasized that this is a combination of active and passive transport which are presently indistinguishable. However, we assume that if a transporter
has been identified for a drug (such as L-dopa), then active transport dominates over
passive transport. Pardridge et al. [48, Table 5] report Pθ S = PS = 0.61 mL min−1 g −1
(per gram of rat brain). Pardridge et al. also report the radioactivity of [3 H]-labeled
L-dopa used in their study to be 5 µCi ml−1 and the specific activity of L-[3 H]dopa
as 38.9 µCi nmol−1 [48, Table 1]. The concentration in the blood can therefore be
determined as
µCi 1 nmol
Cblood = 5
≈ 0.13 µmol L−1 .
mL 38.9 µCi
Given that the experiment reported in [48] represented a very short elapsed time of 18 s,
we assume that the concentration of drug in the blood during the experiment is constant,
and equal to the initial concentration. We can then equate the active transport term
from Equation (2) to the general uptake term
dv
k3 v
=−
≈ −PS Cblood
dt
K +v
and solve this for k3 . Del Amo et al. [49, Table 5], report values of K ranging from 28.2
to 101.6 µmol L−1 compiled from various experimental sources. An intermediate value
of K from [29] is 34.2 µmol L−1 . Using the value of the initial concentration estimated
above for Cblood = v, we can estimate k3 , using a density of brain tissue as 1.05 g cm−3
[50]. The ranges for K and k3 for L-dopa are summarized in Table 1.
Parameters for the uptake of L-dopa in the brain and the subsequent metabolism
and clearance of dopamine are taken from [51], Model 3.
To parametrize equation (8) for doxorubicin, it should be noted that doxorubicin
has a relatively small molecular weight of 543 g mol−1 . Chen and Konofagou [9] report
that blood-brain barrier disruption can occur at acoustic pressures below 0.3 M P a for
molecules below 3 kDa. We select σmin = 0.1 M P a. Liu et al. [10] found that focused
ultrasound at 1 % duty cycle delivered for a mere 120 s resulted in accumulation of
epirubicine over a period of 3 − 6 h after the event. Additionally, Hynynen et al. [52]
found that the blood-brain barrier permeability, which was initially increased due to
ultrasound, decreased to about 15% of its original maximal value after 3 h. We use
these results to estimate values for c1 and c2 .
10
parameter
u0
f
α0
α
V
k1
k1
k3
K
k4
k5
k6
value
420 pmol L−1
2 s−1
1.3 · 105
1.3 · 107
65
0.6 s−1
1 s−1
see Table 1
see Table 1
5 · 10−4 s−1
4.8 · 10−4 s−1
8.3 · 10−6 s−1
interpretation
circulating liposome concentration
loss due to blood flow
initial drug loading coefficient, L-dopa
drug loading coefficient, DXR
brain volume served by capillary
binding rate of L-dopa in plasma
binding rate of doxorubicin in plasma
active transport rate, L-dopa
active transport saturation, L-dopa
L-dopa metabolism rate
dopamine clearance rate
liposome clearance rate
reference
this paper
[42]
this paper
this paper
[40]
see text
see text
[49, 48, 29]
[29]
[51]
[51]
[39]
Table 2: Parameter values used in this work. Note that u0 is an initial condition, not a
constant in the L-dopa simulations.
Numerical simulations were done with the open source packages scilab [53] using
the Euler backward scheme and in R [54] using an LSODA solver. All codes are available
from the authors upon request.
5
5.1
Results
Determination of the Response Function
In vitro experiments were performed in which carboxyfluorescein was released from
DOPE liposomes when exposed to varying intensities of ultrasound, the results of which
are described in Figure 2 (left). For the concentration of carboxyfluorescein released,
vCF (t), we have as in equation (2),
dvCF
= αh(s)u.
dt
In order to determine the total amount of carboxyfluorescein encapsulated within the
liposomes, and therefore the theoretical maximal concentration able to be released,
the liposomes were lysed with Triton-X100. Only a small amount of the encapsulated
content was released during the short accumulated time of ultrasound exposure (15 s),
thereby justifying the assumption that u is roughly constant in vitro for short times
(not shown). Moreover we find that the terminal concentration of carboxyfluorescein
is αu = 6.7 mmol L−1 . The transfer function parameters displayed in Table 3 were
obtained through fitting the function in (5). Note that the unit of a1 is M P a−1 s−1 .
5.2
General Numerical Simulations
To demonstrate the capabilities of our model, we begin with simulation of triggered
11
Figure 2: (Left) Release of carboxyfluorescein from DOPE liposomes at seven different ultrasound strengths with the corresponding linear fits. Release curves with an
ultrasound strength below 0.45 M P a are difficult to distinguish. (Right) The slopes as
functions of maximal negative ultrasound pressure with the optimal fit of the piecewise
linear response function from equation (5).
parameter
a0
a1
smin
σmin
c1
c2
value
2 · 10−4 s−1
6 · 10−3 M P a−1 s−1
0.45 M P a
0.15 M P a
2 · 10−4 s−1
1 M P a−1 s−2
interpretation
background release rate
release rate slope
lower ultrasound threshold
BBB opening threshold, DXR
BBB closing rate, DXR
BBB opening rate, DXR
reference
this paper
this paper
this paper
[9]
[10, 52]
[10]
Table 3: Ultrasound parameter values used in this work.
delivery of an “abstract” drug. The idea is that even without precise knowledge of
parameter values, it is possible to obtain qualitative results that will guide further investigations. Here we consider an example where the drug is subject to active transport
only and vary parameters that are associated with either the liposome formulation,
namely α, or the dosage of the liposomes delivered, namely u0 . In Figure 3 the area
under the curve of w is shown, when applying an ultrasound signal of 1 min duration,
pressure 0.6 M P a and a total time of 2 min. We use the parameters from Tables 1 and
2, except we have divided K by 10. This simulates the case where the transporters
saturate at low concentrations of drug. We observe a saturation at high drug loading
coefficients and liposome loads. Moreover, the level sets of the total delivered drug
coincide roughly with the hyperbolas αu0 = const. This indicates that, to a first approximation, increasing the loading capacity of the liposomes is equivalent to increasing
the dose of liposomes delivered.
12
Figure 3: Area under the curve of w when α and u0 vary. Parameter values are taken
from Tables 1 and 2, except that K from Table 1 was divided by 10. The level curves
of equal area under the curve are roughly hyperbolas.
5.3
Delivery of L-dopa
L-dopa is a drug used to treat Parkinson’s disease (PD), a condition characterized by
marked deficits of dopamine in the striatum of the brain due to progressive death of
dopamine-producing cells in the midbrain [55]. Dopamine itself does not cross the bloodbrain barrier, however L-dopa is a substrate of the large neutral amino acid transporters
LAT1 and LAT2 and therefore readily enters the brain, particularly through the action
of LAT1 [56, 49]. In dopamine neurons, L-dopa is converted to dopamine through the
action of dopa decarboxylase in the surviving dopamine neurons. It is also converted
in the periphery, hence is administered along with a dopa decarboxylase inhibitor to
maximize the availability for brain transport and minimize peripheral side effects [57].
The plasma half-life of L-dopa is 50-90 min [58, 59, 60], and storage of dopamine
is altered in the Parkinsonian brain [61], hence L-dopa is usually administered to PD
patients in multiple doses over the course of a day. As the disease progresses, fluctuations
in the levels of brain dopamine occur, which are thought to lead over time to the
side effects of motor fluctuation and abnormal movements (dyskinesias). This has led
to recent attempts to administer L-dopa and other dopamine agonists in a manner
that leads to more continuous stimulation of dopamine receptors [62]. These include
continuous enteral infusion of L-dopa or parenteral infusion of dopamine agonists [57].
One recent approach which has shown promise has been administration of L-dopa and
analogues using liposomes. Liposomes containing L-dopa when injected systemically in
a rat model of PD elevated dopamine in the striatum to much higher levels than can
be achieved using equimolar systemic L-dopa [63]. In addition, targeting the liposome
to the brain vasculature was utilized in order to release L-dopa specifically into areas of
interest, resulting in increased dopamine levels in a mouse model of PD. This increased
13
dopamine level resulted in enhanced behavioral performance on the rotarod apparatus,
compared with free L-dopa alone [31]. Thus there is evidence that maintaining the
levels of dopamine in the striatum for as high and for as long as possible will improve
the treatment of PD using L-dopa.
In the present study, we determine if frequent release of L-dopa from liposomes using
ultrasound, with an intermittent pattern of stimulation to reduce the unwanted heating
effects of ultrasound, results in good achievable L-dopa levels in the brain. Figure
4 shows a simulation with a 10% duty cycle, with ultrasound pulses of width 10 ms
(i.e. 10,000 cycles at 1 M Hz) and a total ultrasound delivery time of 1 s (100 pulses)
over a time period of 10 seconds.
The ultrasound is initiated one minute after the liposomes are injected and is stopped
10 s later, 70 s after the liposomes are injected. Figure 4 shows the results of a simulation
when there is no leakage of drug from the liposomes (a0 = 0 in equation (5)). This
simulation suggests that, with “ideal” liposomes that release no drug in the absence of
ultrasound, small amounts of drug can be delivered to a targeted area. In the upper
right panel of the figure, the concentration of free drug in the capillary, v(t), is shown.
This concentration quickly drops to zero when the ultrasound is turned off. The lower
panel shows the kinetics of the drug concentrations in the tissue. The concentration of
free drug in tissue, w(t), has a sharp peak two seconds after the ultrasound is turned
off, while the concentration of metabolized or bound drug, z(t), does not reach its peak
for more than 30 minutes. If the liposomes are assumed to have a baseline leakage, the
effect of very short periods of ultrasound is relatively small compared to the effect of
the baseline release. In the subsequent simulations, we use longer sonication periods
and include the baseline release as measured in the calibration experiment described in
Section 5.1.
In order to determine the effect of fractionated dosing of ultrasound on liposomal
release, treatment simulations with duty cycles decreasing from 50% to 1%, and pulse
widths decreasing from 1 min to 100 ms were performed, the results of which are shown
in Figure 5. We note that changing the duty cycle decreases the maximum drug in the
brain, but also evens out the delivery. Fractionating the doses without changing the
duty cycle has little effect. In other words, it is the total amount of sonication delivered
in a fixed amount of time that affects the delivery of drug to the brain. However,
fractionating the ultrasound may decrease harmful thermal effects.
Once calibrated, the mathematical model can be used to predict concentrations of
bound or free drug when repeated doses of liposomes are administered. In Figure 6 we
show two representative simulations. In these simulations we model repeated injections
of liposomes that are cleared from the system. The concentration of metabolized or
bound drug in tissue can be maintained within a prescribed range by specifying an
ultrasound application frequency and by repeated doses of liposomes. In Figure 6 we
compare two different delivery protocols. In both cases, the total sonication time is two
minutes per dose. In one case, the liposome boosts are given every two hours, with 5
ms pulses of ultrasound, and in the other case the liposome boosts are given every two
and a half hours, with 10 ms pulses. We see that the former delivery protocol decreases
14
v(t): free drug in plasma
5
10
µmolL−1
1.0
0
0.0
0.5
MPa
1.5
15
2.0
s(t): ultrasound intensity
60
62
64
66
68
70
60
62
64
time (s)
66
68
70
time (s)
w(t)
z(t)
0.000 0.005 0.010 0.015 0.020
µmolL−1
Free drug, w(t), and bound drug, z(t), in brain
0
100
200
300
400
time (min)
Figure 4: (Top left) Ultrasound signal pressure during the application of focused ultrasound: 10% duty cycle, with 100 pulses of width 10 ms, separated by 90 ms with the
ultrasound off. The ultrasound frequency is 1M Hz, and assuming a peak rarefactional
pressure of 1.5 M P a, the mechanical index is 1.5. (Top right) Concentration of free
L-dopa in plasma during the same time period. For this simulation, we assume that
there is no baseline release from the liposomes: a0 = 0 in equation (5). (Bottom) The
concentration of free drug in the brain, w(t), peaks relatively quickly at 72 seconds.
The concentration of drug taken up and metabolized, z(t), peaks much later at approximately 35 min. All parameters used in this simulation, with the exception of a0 , are
shown in Tables 1 and 2.
15
1.5
2
3
50% duty cycle
10% duty cycle
1% duty cycle
0
1
z(t) µmolL−1
1.0
0.5
0.0
s(t) (MPa )
50% duty cycle
10% duty cycle
1% duty cycle
0
50
100
150
200
0
200
time (min)
400
600
800
2.5
time (min)
1.5
1.0
0.5
0.0
pulse width 1 min
pulse width 10 s
pulse width 1 s
pulse width 0.1 s
z(t) µmolL−1
2.0
pulse width 1 min
pulse width 10 s
pulse width 1 s
pulse width 0.1 s
1
2
3
4
0
200
400
600
800
time (min)
time (min)
Figure 5: (Top) A comparison of three duty cycles, 50%, 10% and 1%. Note that the
graph does not show the full duration of delivery at 1% duty cycle. In all three cases, the
pulse width is 1 min and total sonication time is 8 min. The pulse repetition frequency
is 0.5, 0.1 and 0.01 min−1 , respectively. Lowering the duty cycle decreases the total
amount of drug delivered to the brain, and also flattens out the concentration profile of
drug in the brain. (Bottom) A comparison of four pulse-widths. In all cases, the duty
cycle is 10% and the total sonication time is 8 min (the full signal is not shown). The
fractionation of the ultrasound pulses has very little effect on z(t), the concentration of
bound drug in the brain.
16
1.5
1.0
0.5
z(t) (µmolL−1)
0.0
10 ms pulses, injections every 2.5 hours
5 ms pulses, injections every 2 hours
0
5
10
15
20
time (h)
Figure 6: A comparison of two different protocols for the administration of repeated
doses of liposomes and ultrasound. In both cases the total sonication time is 2 minutes
per liposome dose, and the duty cycle is 2%. If liposomes are re-injected every two
hours, the concentration of bound drug in the brain tissue, z(t), can be maintained
between 0.91 and 1.25 µmolL−1 . As the time between injections is increased, the pulsewidth of the ultrasound must be increased in order to maintain similar concentrations
of bound drug.
the amplitude of the fluctuations in concentration.
The simulation results in Figure 3 suggest that the mass of drug delivered to the
brain depends largely only on the total dose (drug in liposomes × number of liposomes)
of drug, and not on the concentration of drug in the liposomes. This can be tested
by making batches of liposomes with two different drug loadings (low and high) and
administering doses of liposomes such that the total mass of drug is the same, then
measuring brain concentrations over time, in the presence and absence of ultrasound.
This can be done at different ultrasound intensities and duty cycles (Figure 5) to test
this aspect of the model. It could be done with an actively absorbed drug first and if
an effect is seen then look at a passively absorbed drug to test if the effect is due to
localized “saturation” of the carrier.
The model considers the issue of the kinetics of drug binding to plasma proteins.
This is an issue which is usually considered to be of no importance since free drug in
plasma is in equilibrium with protein bound drug, within the time frames of standard
pharmacokinetic processes. However, in the case of triggered drug release, the drug
17
Figure 7: Uptake of L-dopa after direct delivery to the blood, together with data points
from [45].
molecules move from a protein-free environment within liposomes into plasma during
flow through a capillary. Thus the relative kinetics of protein binding, permeation
(active and passive) across the endothelium, and blood flow in the capillary become
important. More research is required to measure the kinetics of protein binding before
this aspect of the model can be tested experimentally.
When delivered directly into the bloodstream, the drug will immediately bind to the
plasma protein and therefore will only achieve a much lower concentration in the brain
vasculature. In Figure 7 we present simulations of the system of equations (4) and (3).
Huang et al. [45] show the uptake of radioactively labeled L-6-[18 F]Fluoro-DOPA in the
striatum and cerebellum after an intravenous bolus injection. They show that after
20 − 30 min, the maximal activity, corresponding to maximal concentration, has been
reached. We obtain a good qualitative agreement with this observation. However, we
have not attempted to match this too closely since we do not have enough information
how these experiments were carried out, nor do we have a calibration curve between
concentration and radioactivity.
5.4
Delivery of Doxorubicin
Doxorubicin and its derivatives are anticancer drugs that interfere with DNA replication
and thereby lead to cell death. Their adverse side effects include suppression of white
blood cell production in the bone marrow and cardiomyopathy, possibly leading to
heart failure. Over the past years, the delivery of doxorubicin encapsulated in liposomes,
known as Doxil, has become increasingly common. In the case of brain cancer treatment,
there are no known active transporters for doxorubicin to cross the BBB, making the
treatment of brain tumors with this drug particularly difficult. Recent works have
18
shown that the application of focused ultrasound in combination with microbubbles can
locally and temporally increase the permeability of the BBB, increasing the potential
for treating malignant gliomas [9, 32, 64].
Aryal and collaborators [32] report experimental work in which rats implanted with
gliomas showed improved survival when treated with liposomal doxorubicin and focused
ultrasound. In fact, only rats in the group treated with both doxorubicin and ultrasound
survived, compared with those treated with the drug or ultrasound alone, or those who
received no treatment. In these experiments, ultrasound is delivered in the form of brief
pulses that are repeated over a few seconds, in order to avoid overheating the brain
tissue. Mathematically, we can model the effect of ultrasound on the temperature of
nearby tissue using an equation similar to that describing the change of permeability
of the blood-brain barrier, Equation (8). The change in the permeability constant, k2 ,
can therefore be used as a surrogate to monitor thermal effects in an attempt to avoid
damage to nearby tissue. In Figure 8, bursts of ultrasound (20 s on, 10 min apart)
are simulated. The rapid opening and slow restoration of the blood-brain barrier may
result in a reverse transport of drug from the brain to the blood, if the concentration
of drug becomes higher in the brain. Hence, at steady state, the concentrations in both
compartments are equal.
6
Discussion
The application of liposomes to the site-specific delivery of therapeutic agents has grown
over the past few decades. Recently, liposomes have also been applied to the treatment of neurological disorders, such as Parkinson’s disease and glioblastoma multiform,
where systemic administration of the required therapeutic agent is either ineffective,
or results in debilitating side effects. Even with the advances in targeted delivery, due
to the complexity of the in vivo experiments required to assess drug delivery to the
brain, precise measurements of the drug concentrations delivered are scarcely reported
in the literature. With the growing success of mathematical modeling in pharmaceutical applications, it is our goal to complement the complex nature of in vivo animal
drug delivery and disease models with predictive mathematical modeling tools for the
optimization of delivery technologies. In regards to ultrasound assisted drug delivery,
the relative simplicity of conducting in silico experiments compared with their in vivo
counterparts allows for easy, time efficient optimization of the ultrasound parameters
required to achieve successful and meaningful in vitro and in vivo results. Our present
physiologically-based compartmental model has the advantage of great conceptual simplicity, to which a user may add gain or loss terms, or even additional variables to fit a
given situation of interest. The numerical implementation of the model is equally simple, as many libraries of ordinary differential equations exist and can be implemented
in Open Source software.
As an example of a drug delivery system sensitive to ultrasound, the response function of liposomes containing 25% DOPE (DOPE25) to ultrasound was determined under
relatively low acoustic pressures. DOPE25 was used as DOPE has previously been re-
19
Figure 8: Concentrations of doxorubicin in the blood, in the brain and permeability of
the blood-brain barrier. The ultrasound signal s is plotted in red in the fourth panel,
together with the threshold σmin for opening of the BBB (dashed line).
ported to act as a sonosensitizing agent in liposome formulations, due to its cone-shaped
geometry and subsequent non-bilayer forming nature [65]. A clear threshold is evident
for the acoustic pressure required in order for the liposomes to release their encapsulated solute in response to ultrasound, see Figure 2. This threshold likely coincides with
the energy required to induce a lamellar to reverse hexagonal phase transition, which is
thought to be the mechanism behind the sonosensitivity of DOPE [65].
The release of L-dopa and doxorubicin from liposomes and subsequent delivery to
the brain has also been simulated. The model has been carefully parameterized from
available literature. Furthermore, L-dopa and doxorubicin highlight the differences in
transport mechanisms across the BBB; L-dopa relies on active transport and readily
crosses the blood-brain barrier whereas doxorubicin relies solely on passive diffusion
and therefore does not ordinarily cross.
Substrate-receptor binding rate constants are scarcely reported in the literature,
likely due to the experimental difficulty in obtaining them. Therefore in cases where
there were no direct reports of the constants required for the model, estimates were
made from studies reporting similar parameters.
20
We have demonstrated a simple compartmental model of drug delivery to the brain
utilizing sonosensitive liposomes as a delivery vehicle. The present work provides a
mathematical setting that is amenable to the optimization of ultrasound-mediated drug
delivery under a range of optimality criteria. In general, overheating of sensitive brain
tissue should be avoided to minimize tissue damage and achieve a successful therapeutic
outcome. Optimization to achieve this is possible by employing a total energy constraint
(the area under the curve of s(t)), or by a “reporter variable” such as temperature (for
which k2 was taken as a surrogate). The resulting simulations suggest that lowering
the duty cycle of the ultrasound applied decreases the total amount of drug delivered
to the brain, whilst fractionation of the ultrasound pulses has very little effect on the
concentrations of free (w(t)) and bound (z(t)) drug in the brain. In the case of L-dopa,
the goal is to achieve a relatively constant level of drug or its metabolites in the brain,
for which the model suggests that ultrasound should be applied at low duty cycles over
a long time period. However this may be difficult to achieve in a clinical environment
due to the mechanical constraints of delivering this level of ultrasound over prolonged
periods. The criteria for the design of an optimal liposome formulation is also evident
for the delivery of a therapeutic agent on demand in response to ultrasound. Ideally,
sonosensitive liposome should possess a high drug loading capacity, thermal stability,
a low ultrasound threshold to induce release, and minimal passive leakage below this
threshold.
In future work, experimental validation of the model will be undertaken using the
case of L-dopa delivery to the brain and its subsequent conversion to dopamine, by
measuring these enhanced dopamine levels in vivo. The physiologically-based compartmental model also does not yet take into account the spatial extent of the brain region
at which the drug is released. Introduction of a spatial component requires a formulation of the model using partial differential equations, thereby greatly increasing the
mathematical complexity; a similar example of which can be seen in work by Gugat [18].
Nevertheless, numeral simulations of flow through networks are available and may be
beneficial, making it possible to account for the dynamical regulation of blood flow to
different regions of the brain depending on the local demand for oxygen and nutrients.
Conflict of interests
The authors declare no conflict of interest.
Acknowledgments
We are grateful for financial support from the grant “Collaborative Research: Predicting the Release Kinetics of Matrix Tablets” (DMS 1016214 to Peter Hinow and DMS
1016136 to Ami Radunskaya) of the National Science Foundation of the United States
of America. Morgan Schroeder received an award from the Support for Undergraduate
Research Fellows (SURF) program at the University of Wisconsin - Milwaukee. John
Reynolds received a Rutherford Discovery Fellowship from the Royal Society of New
21
Zealand. Sean Mackay received a doctoral scholarship from the University of Otago.
We acknowledge the kind hospitality of the University of Otago during several collaborative visits and during a sabbatical stay of Ami Radunskaya. Peter Hinow was partly
supported from the Simons Foundation grant “Collaboration on Mathematical Biology”
during a visit to Pomona College. The funding bodies had no influence on study design,
collection, analysis and interpretation of data.
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