1. Simulation and Modeling of the Impacts of Nanoparticle Size on Nanoparticle
Mediated Drug Delivery to Brain Tumor
Daniel Kurniawan
Department of Chemical and Biomolecular Engineering, National University of Singapore, 4
Engineering Drive 4, Singapore 117576, Singapore
ABSTRACT
Nanoparticle can be used as a polymeric carrier to effectively deliver a chemotherapeutic drug
for brain tumor treatments by bypassing the blood brain barrier. In this study, we employ a com-
putational fluid dynamics simulation to study the influence of nanoparticle size to the temporal
and spatial nanoparticle concentration profile in brain. The study used the convection-diffusion
model to describe the nanoparticle transport. Idealized geometry brain and tumor model is used as
the domain of the simulation. Literature review is conducted to establish the correlation between
nanoparticle size with both nanoparticle diffusion coefficient and drug release rate in order to show
its effect on the concentration profile. The temporal variation analysis shows that smaller nanopar-
ticle can reach a steady-state concentration condition at a faster rate with a lower volume-averaged
concentration. The spatial variation analysis shows that smaller nanoparticle size encompasses a
larger, albeit insignificant, volume of distribution at steady state with a smaller maximum nanopar-
ticle concentration in both brain and tumor region.
INTRODUCTION
In 2016, nearly 78,000 new cases of brain tumors are expected to be diagnosed in which nearly
25,000 of them are malignant (American Brain Tumor Association , 2015). The most common
strategy to treat brain tumors usually involves surgical removal, chemotherapy, and radiotherapy.
For chemotherapy, though the drug is biologically effective to treat brain tumor, drug transport
properties are often overlooked. Effective delivery of therapeutic agents to tumor cells is essen-
tial to the success of chemotherapy, especially the brain tumor treatment because of the various
obstacles.
One challenging problem that drug delivery by systemic administration is facing is the blood
brain barrier (BBB). The current practice to bypasses the BBB is to use a polymeric system infused
directly into the tumor site, achieving a localized drug delivery. This study particularly focuses on
the usage of nanoparticles as a drug-loaded polymeric carrier which will be infused directly into
the tumor site by a needle to enhance the treatment outcome. With the advance of nanotechnology,
polymeric nanoparticles can be employed as carriers to transport the entrapped or adsorbed drugs
across the BBB. Also, nanoparticles systems may also be useful in protecting the drug from a
biological degradation prior to its release. As a result, more sustained release of therapeutic drug
to treat brain tumor can be achieved. The primary goal of this study is to analyze the temporal
1

2. and spatial nanoparticle concentration profile infused in brain by the means of computational fluid
dynamic (CFD) simulations.
The emphasis of this study is to conduct a parametric study to examine the impact of nanopar-
ticle size to nanoparticle concentration distribution in brain. Two important transport properties
which are directly influenced by nanoparticle sizes are the nanoparticle diffusivity coefficient and
the drug release rate. Therefore, it is necessary to study the impact of nanoparticle on both proper-
ties first. With an established correlation between nanoparticle sizes and both nanoparticle diffu-
sivity and drug release rate, we will then examine how changing both parameters simultaneously,
based on the correlation, will impact the temporal and spatial nanoparticle concentration profile.
MATHEMATICAL MODEL
Equation
The brain interstitial fluid flow is described by coupling the modified continuity and momentum
equation for fluid flow in a porous medium. The continuity equation for incompressible interstitial
fluid in the brain tissue is:
· v = FV (1)
where v is superficial interstitial fluid velocity vector and FV is the rate of fluid gain from the
capillary bed per unit volume of tissue. Fluid removal by the lymphatic system is not included
because brain tissue lacks a well-defined lymphatic system. The fluid gain is assumed to be a
non-uniformly distributed source, depending on the pressure difference between blood vessels
and interstitial fluid. The constitutive equation for FV follows Starling’s law (Baxter , 1989):
FV = Lp(S/V )[pv − pi − σ(πv − πi)] (2)
where Lp is hydraulic conductivity of the microvascular wall, pV is vascular pressure, pi is inter-
stitial fluid pressure, S/V is available exchange area of the blood vessels per unit volume of tissue,
σ is the osmotic reflection coefficient for plasma proteins, and πv and πi are osmotic pressures of
blood plasma and interstitial fluid, respectively.
The brain tissue is assumed to be a rigid porous medium. The momentum equation for fluid
flow through tissue is assumed to be:
ρ
∂v
∂t
+ v · v = − pi + µ 2
v −
µ
K
v (3)
where t is time, ρ and µ are density and viscosity of the interstitial fluid and K is the Darcy
permeability of the tissue.
Nanoparticle availability is governed by a generic convection-diffusion equation with a source
/ elimination term. Thus, the nanoparticle conservation equation, to be coupled with the interstitial
fluid flow equations, is as follows:
∂C
∂t
= Di
2
C − · (vC) + S (4)
where C is the nanoparticle concentration , Di is nanoparticle diffusivity in interstitial space and
S is the source / elimination term.
2

3. (a) (b) (c)
Figure 1. The idealized model geometry used in the simulation, showing the presence of ventricle
(shown in blue), tumor (green), and the remaining brain tissue (red). (a) The complete 3-D sim-
ulation domain of brain. (b) Zoomed-in isometric view of tumor region and needle. (c) The mid
cross sectional of simulation domain, containing all brain, tumor, and ventricle region, used for
representative 2-D visualization throughout this study.
The nanoparticle source / elimination term is contributed by drug release rate and nanoparticle
degradation rate. It is assumed that nanoparticles will be eliminated when drug is released from
nanoparticle or the nanoparticle itself is degraded. For simplicity, both drug release rate and
nanoparticle degradation rate follows the first-order kinetic model. Therefore, nanoparticle source
/ elimination term can be described by equation as follows:
S = −(krel + ke)C (5)
where krel and ke are the first-order elimination constant for nanoparticle concentration due to
drug release rate and degradation respectively.
Geometry
This study uses an idealized three-dimensional model to describe the brain geometry domain
used in the simulation. Fig. 1a and 1b depicts the 3-D geometry while Fig. 1c depicts its 2-D
representations. There are three main regions in the simulation domain: brain, tumor and ven-
tricle. Each region is modeled as a sphere with radius 66 mm, 10 mm and 18.5 mm respectively.
Spherical geometry is used as an idealized geometry model while keeping the volume region to
be as realistic as possible. In other words, the volume of the brain region in this simulation is
approximately equal to the average real brain volume of human.
The simulation also assume an idealized needle insertion position. Needle is inserted in the
center of tumor region (Fig. 1b). Needle wall penetrates both the brain and tumor region. The
inlet diameter of the needle is 1 mm long.
Model Parameter
The discussion about baseline values of parameters related to the interstitial fluid flow (Eq. (1),
(2), and (3)) is available on several previous studies on drug transport simulation (Arifin , 2009),
(Arifin , 2009). To study the effects of nanoparticle sizes on nanoparticle transport (Eq. (4) and
3

4. Eq. (5)), we will vary nanoparticle diffusivity (Di) and drug release rate (krel) since these two
parameters are directly affected by nanoparticle sizes. In this particular case study, the value of
the drug degradation constant (ke) is assumed to be 5 × 10−6
s−1
.
Nanoparticle Diffusivity Stokes-Einstein equation can model nanoparticle diffusivity in intersti-
tial fluid accurately. The study conducted by Cu (2009) shows that the Stokes-Einstein equation
can predict the diffusion coefficient of PLGA nanoparticles with diameter 170 ± 57 nm loaded
with green-fluorescent Coumarin-6 molecules in water. The special form of Stokes-Einstein equa-
tion for a spherical particles is as follows:
DAB =
kT
6πµBrA
(6)
where DAB is the diffusion coefficient of nanoparticle (A) in interstitial fluid (B), k is the Boltz-
mann’s constant, T is the absolute temperature, µB is the dynamic viscosity of interstitial fluid and
rA is the radius of the spherical nanoparticle. In our case study, it is assumed that T is the body
temperature (310.15 K) and the viscosity of interstitial fluid is 0.000 78 kg m−1
s−1
(Perry , 1996).
We will use this inverse relationship to establish the correlation between diffusion coefficient and
the radius of any type of nanoparticle.
Drug Release Rate Drug release rate coefficient (krel) in Eq. (5) can be determined by looking at
the commonly documented drug release profile of a nanoparticle. The drug release rate, following
the first-order kinetic law, can be described as follows:
r =
∂C
∂t
= −
1
DL
∂Cd
∂t
= −krelC (7)
where r is the release rate, Cd is the concentration of drug, and DL is the drug loading, which is
a measure of the amount of drug loaded per mass of nanoparticles.
Hence, we can get the corresponding model of the drug release profile by solving Eq. (7)
for Cd. With this assumption of the first order kinetic law, drug release profile should be an
exponential equation as follows:
Cd = A(1 − e−krelt
) (8)
where A is a lumped solution constant and krel is the coefficient that we are interested in. We
will fit the drug release profile available from literature to obtain krel coefficient dependency to
nanoparticle sizes. Note that krel will not only be influenced by nanoparticle size exclusively,
it also depends on many other important factor such as nanoparticle formulation and drug type.
Therefore, available data from literature must be chosen carefully to isolate the relationship be-
tween nanoparticle sizes and drug release rate.
Pandey (2016) conducted a study to evaluate the performance of Tamoxifen embedded PLGA
nanoparticles (PLGA-Tmx) as an anticancer drug vehicle. In the study, similar PLGA-Tmx
nanoparticles are prepared with varying dimension of 17 nm to 30 nm by changing the con-
centration of polymer, emulsifier, and drug. In this case, PLGA concentration of 10 mg mL−1
,
20 mg mL−1
and 30 mg mL−1
used for the nanoparticle fabrication resulted in particles with diam-
eter of 19.8 nm, 21.4 nm and 24.2 nm respectively. Drug release profile of different nanoparticle
size is recorded (Fig. 2a).
4

5. (a) (b)
Figure 2. Invitro sustained release of drug fabricated using different PLGA concentration, which
resulted in different PLGA-Tmx size; PLGA concentration of 10 mg mL−1
, 20 mg mL−1
and
30 mg mL−1
used for the nanoparticle fabrication resulted in particles with diameter of 19.8 nm,
21.4 nm and 24.2 nm respectively. (a) The original figure of drug release profile in the literature.
(b) Best exponential model fit of the drug release profile data.
The cumulative drug release profile were best fitted with the exponential equation in Eq. (8)
leading to a time constant value (krel) of 1.72 × 10−5
s−1
(r2
= 0.86) for PLGA-Tmx nanoparticle
with a diameter of 19.8 nm prepared from 10 mg mL−1
of PLGA. On the other hand, a lower best-
fit values of time constant (krel) are observed from nanoparticle with bigger sizes: 1.14 × 10−5
s−1
and 8.43 × 10−6
s−1
(r2
= 0.94; 0.96) for PLGA-Tmx nanoparticle with a diameter of 21.4 nm
and 24.2 nm prepared from 20 mg mL−1
and 30 mg mL−1
of PLGA respectively. The higher
values of time constant (krel) implies that drug can be released from the nanoparticle carriers at a
faster rate. It is qualitatively consistent with the fact that smaller nanoparticle will have a larger
surface area per unit volume to release the loaded drugs; hence, it is capable of releasing drugs at
a faster rate. Fig. 2b shows the comparison between experimental data and the fitted data. This
dependency result between krel and nanoparticle sizes will be used as a model parameter in the
simulation. Linear interpolation is used to obtain krel of nanoparticles with different sizes.
Boundary Conditions
Interstitial fluid flow is solved using a pressure based boundary condition in the ventricle sur-
face and brain periphery. The ventricle surface will be the high pressure domain while brain
periphery will be the low pressure domain. In particular, interstitial fluid is constantly infused at
a constant pressure (pventricle = 1447.4 Pa ). At the same time, the outermost boundary condition
is the pressure at the arachnoid villi (pouter = 667 Pa) where the fluid is removed (Kimelberg ,
2004). Along the internal boundaries between brain and tumor region, conditions of continuity
are imposed. At the needle wall in both brain and tumor region, no-slip conditions are applied.
At the inlet of the needle, nanoparticle is injected with a constant fluid flow velocity condition at
0.000 106 m s−1
.
The boundary conditions for nanoparticle concentration involves no-flux conditions on both
ventricle surface and brain periphery. Along the internal boundaries between brain and tumor
region, conditions of continuity are imposed. No-flux conditions are also applied at the needle
5

6. Table 1. Simulation Group
Group Size (nm) Di (m2
s−1
) krel (s−1
) ke (s−1
)
1 19.8 2.94 × 10−11
1.72 × 10−5
5 × 10−6
2 20.6 2.83 × 10−11
1.37 × 10−5
5 × 10−6
3 21.4 2.72 × 10−11
1.14 × 10−5
5 × 10−6
4 22.8 2.55 × 10−11
9.70 × 10−6
5 × 10−6
5 24.2 2.41 × 10−11
8.42 × 10−6
5 × 10−6
Figure 3. Temporal variation of volume-averaged nanoparticle concentration in tumor region
wall in both brain and tumor region. At the inlet of the needle, nanoparticle is injected at a
constant nanoparticle concentration at 4 kg m−3
SIMULATION METHOD
Based on the correlation study on nanoparticle transport parameters, we run 5 different groups
of simulation to observe the effect of nanoparticle size on nanoparticle concentration temporal
and spatial profile. Table 1 completely summarized the simulation parameters for nanoparticle
transport for each group.
Simulation were conducted using ANSYS Fluent 16.2. The simulation is carried out using Eq.
(1), (2), and (3) to solve for interstitial fluid transport and Eq. (4) and (5) to solve for nanoparticle
transport. In this study, the simulation is carried out in a complete 3-D domain. However, for
this specific ideal geometry cases (symmetric sphere), 2-D axisymmetric model, with the top-half
circle of Fig. 1c as the simulation domain, can actually be used to reduce the computational effort.
6

7. Table 2. Simulation Group Exponential Fit [Model: y = A(1 − e−kt
)]
Group Size (nm) A (kg m−3
) k (s−1
) R2
1 19.8 0.870 8.81 × 10−5
0.97
2 20.6 0.896 8.68 × 10−5
0.97
3 21.4 0.914 8.61 × 10−5
0.97
4 22.8 0.928 8.54 × 10−5
0.97
5 24.2 0.938 8.51 × 10−5
0.97
RESULTS AND DISCUSSION
Temporal Variation of Volume-averaged Nanoparticle Concentration Profile
To examine the transient profile, we will look at the temporal variation of volume-averaged
nanoparticle concentration in tumor region. Volume-averaged nanoparticle concentration in tu-
mor region for all 5 simulation groups is plotted in Fig. 3. Exponential model (similar to Eq.
(8)) is employed to gain a better understanding of the transient concentration profile. The result
of exponential best-fit for each group is summarized in Table 2 with A as the lumped solution
constant and k as the time constant value.
As nanoparticle size become larger, it can be clearly observed that the steady state averaged
concentration profile increases; the value of A, which corresponds to the concentration value at
time infinity, increases as the nanoparticle size increases. It can be explained by the fact that
smaller nanoparticle sizes corresponds to a higher release rate coefficient. It implies that smaller
nanoparticle does not need as high concentration as the larger one to achieve a similar elimination
rate required in the steady state.
In the exponential model, time constant k indicates how long does it take for the volume-
averaged concentration to reach steady state. The higher values of time constant (k) implies that
steady state condition of nanoparticle concentration can be achieved at a faster rate. The best
exponential fit shows that smaller nanoparticle has a higher time constant; i.e. it is faster for
a smaller nanoparticle to reach its steady state condition. This constant might be an important
parameter to control if one wants to design a drug that can be released in a sustained manner.
Spatial Variation of Steady-state Nanoparticle Concentration Profile
To examine the spatial profile, we will look at the spatial variation of steady-state nanoparticle
concentration. Comparison of steady state nanoparticle concentration profile between simulation
group 1, 3 and 5 is shown in Fig. 4. At a glance, there does not seem to be any difference 3 group
of contour. With a more meticulous observation, one can see a subtle difference between them.
However, it is safe to say that the steady state concentration profile is not sensitive to changes in
nanoparticle size.
As nanoparticle size become larger, the steady state concentration profile encompasses a larger
area (or volume in 3-D). It agrees with the previous quantitative observation that steady state
volume averaged concentration increases as nanoparticle size become larger. Convection plays a
larger role as nanoparticle size decreases because the diffusivity of nanoparticle decreases. The
more significant role of convective flows can explain the larger concentration profile surrounding
the inlet.
7

8. (a) (b) (c)
Figure 4. Steady state nanoparticle concentration profile of: (a) group 1 (d = 19.8 nm) (b) group
3 (d = 21.4 nm) (c) group 5 (d = 24.2 nm).
Maximum Concentration of Nanoparticle
The simulation result shows that the maximum nanoparticle concentration in tumor region is
6.022 kg m−3
, 6.036 kg m−3
and 6.054 kg m−3
for group 1, 3 and 5 respectively. On the other hand,
the maximum nanoparticle concentration in the brain region (non-tumor region) is 0.681 kg m−3
,
0.761 kg m−3
and 0.811 kg m−3
for group 1, 3 and 5 respectively.
As nanoparticle size become larger, the maximum nanoparticle concentration, both in brain
and tumor region, increases. While one might want to reach a specific concentration threshold to
achieve a therapeutic effect, a lower concentration of nanoparticle in the brain region (non-tumor
region) might be desirable if we want to minimize the toxicity of the treatment. It should be noted
that the dependency of maximum nanoparticle concentration to nanoparticle size in brain region
is much higher than those in tumor region; maximum nanoparticle concentration in brain region
decreases at a much faster rate for a smaller nanoparticle. This might be another parameter that
need to be considered and controlled when designing the drug.
CONCLUSION
This computational modeling effort enables the prediction and understanding of nanoparticle
drug distribution in human brain. However, models are not without assumptions and simplifica-
tion. In reality, the brain tissue in not homogeneous in nature; therefore, the transport means, such
as diffusion and convective flow, will be more likely to be anisotropic and spatially varied (Lin-
niger , 2008). Idealized geometry is also used in this current model, a more realistic geometry of
human brain is desired for a more accurate simulation. We also make a number of assumptions in
the mathematical model used to simplify the simulation, such as the first-order law for the release
kinetic. A research on a more accurate model can lead to a more realistic simulation result. Fi-
nally, a more thorough independent parameter study of drug release rate and diffusivity coefficient
might need to be conducted in order to gain a better understanding of the effect of each individual
transport properties.
This simulation study has provided an insight toward how nanoparticle size effect its temporal
and spatial distribution when infused in brain. Nanoparticle size can be an important design
parameter to control if we want to fabricate a new drug delivery method to treat brain tumor using
nanoparticle carriers. On the other hand, nanoparticle concentration profile can give an insight
8

9. on how drug is distributed throughout the brain and tumor. The predicated drug concentrations
can be used to evaluate the treatment effectiveness. With this understanding, a more efficient drug
transport can be achieved.
REFERENCES
American Brain Tumor Association. (2015). Brain Tumor Statistics. Retrieved July 01, 2016, from
http://www.abta.org/about-us/news/brain-tumor-statistics/
Arifin, D. Y., Lee, K. Y., & Wang, C. (2009). Chemotherapeutic drug transport to brain tumor.
Journal of Controlled Release, 137(3), 203-210. doi:10.1016/j.jconrel.2009.04.013
Arifin, D. Y., Lee, K. Y., Wang, C., & Smith, K. A. (2009). Role of Convective Flow in Car-
mustine Delivery to a Brain Tumor. Pharm Res Pharmaceutical Research, 26(10), 2289-2302.
doi:10.1007/s11095-009-9945-8
Baxter L.T., Jain R.K. (1989). Transport of fluid and macromolecules in tumors. I. Role of inter-
stitital pressure and convection. Microvascular Research, 37, 77-104.
Cu, Y., & Saltzman, W. M. (2009). Controlled Surface Modification with Poly(ethylene)glycol En-
hances Diffusion of PLGA Nanoparticles in Human Cervical Mucus. Molecular Pharmaceutics,
6(1), 173-181. doi:10.1021/mp8001254
Kimelberg H.K. (2004). Water homeostasis in the brain: basic concepts. Neuroscience, 120, 852-
860.
Linninger, A. A., Somayaji, M. R., Mekarski, M., & Zhang, L. (2008). Prediction of convection-
enhanced drug delivery to the human brain. Journal of Theoretical Biology, 250(1), 125-138.
doi:10.1016/j.jtbi.2007.09.009
Pandey, S. K., Patel, D. K., Maurya, A. K., Thakur, R., Mishra, D. P., Vinayak, M., . . .
Maiti, P. (2016). Controlled release of drug and better bioavailability using poly(lactic acid-co-
glycolic acid) nanoparticles. International Journal of Biological Macromolecules, 89, 99-110.
doi:10.1016/j.ijbiomac.2016.04.065
Perry R.H. (1996) Perry’s chemical engineers’ handbook. New York: McGraw-Hill. p. 59-71.
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