Download Caffeine and the Cerebral Vasculature: Pharmacokinetics

Caffeine and the Cerebral Vasculature: Pharmacokinetics,
Vasoconstrictive Effects and Comparison to Hypertension
By Joel Buishas
Abstract:
Caffeine is a well-known and widely utilized
neuro-stimulant. It is also a vasoconstrictor and
has been shown to reduce Cerebral Blood Flow
(CBF) by an average of 27% [1]. Habitual
users potentially subject themselves to
hemodynamics similar to a patient with
hypertension. A model was developed to further
investigate the relationship between caffeine
consumption and cerebral blood flow. This was
done by determining the hemodynamics of the
cerebral vasculature at rest via the construction
of a system of conservation balance equations,
the construction of a convective-destruction
model to determine the concentration profile of
caffeine within the cerebral vasculature, and the
use of experimentally derived data to determine
the vessel constriction coefficient (VCC) a value
that relates the drug concentration in any given
vessel with the increase in resistance due to
caffeine induced constriction that the vessel will
undergo. The effects of two, three, and four
cups of coffee on the cerebral vasculature were
then compared with patients with varying
categories of hypertension. The model for the
system at rest was validated by comparison to
experimentally determined values for Cerebral
blood flow (CBF),mean arterial pressure
(MAP), and average blood flow rate through the
internal carotid arteries (ICA) and the jugular
veins (JV) (see table B for details). The results
of the model showed a CBF of 750ml/min, a
MAP of 84mmHg, and average blood flow rate
of 334ml/min (ICA) and 374ml/min (JV). This
suggests that the model accurately predicts the
steady state hemodynamics for the cerebral
vasculature at rest. The model for the system
under the influence of caffeine was validated by
comparison to the experimentally determined
27% decrease in CBF due to ingestion of 250mg
of caffeine.
The model yielded a CBF
reduction of 38% the model decreases CBF
by 11% more than expected. The system at
rest and the caffeinated system were also
validated by checking the overall mass
balance. The model showed that the CBF
of a person who habitually consumes two
cups of coffee was 592 ml/min and was
comparable to a patient with poorly
controlled hypertension with an average
CBF of 497 ml/min. This suggests that
habitual consumption of caffeine can lead to
a self-inflicted hypertensive state.
Introduction:
Hypertension is chronically elevated blood
pressure, this condition can lead to a host of
health problems including aneurism, stroke, and
has even been linked with cognitive decline and
dementia in elderly patients [2]. A prominent
effect of hypertension is a reduction in the
amount of blood supply to the brain.
Hypoperfusion impairs the flow of oxygen and
nutrients to brain tissue which can lead to brain
ischemia and neural degradation [2].
Caffeine mirrors the effects of
hypertension in that it increases pressure and
resistance and decreases blood flow. According
to the FDA the average American adult
consumes 300 mg of caffeine per day [3].
Caffeine causes cerebral vasoconstriction by
antagonizing adenosine receptors. It has been
demonstrated that Caffeine has the ability to
reduce CBF by an average of 27% [1]. Habitual
caffeine intake causes the vascular adenosine
receptor system to adapt to this vasoconstriction
but, it has been shown that the adenosine
receptors have a limited ability to compensate
for high amounts of caffeine use [1], meaning
that recurrent users of caffeine may have
chronically low CBF levels. The goal of this
series of network computations was to develop a
mathematical model to help gain insight into the
pharmacokinetics of caffeine in the human
cerebral vasculature, examine the effect of
caffeine on the hemodynamics of the system,
and to compare the effect of caffeine to the
effects of hypertension. A model was developed
to further investigate the relationship between
caffeine use and cerebral hemodynamics by
focusing on the increase in resistance to blood
flow due to the vasoconstrictive properties of
caffeine.
Methods:
Figure 1: Visualization of Human Cerebral Vasculature
The model for the human cerebral vasculature
(shown above) has 42 vessels and 36 nodes
which represent an input, an output, or a
junction. The goal of the network computations
was to determine the flow rate in all of the
vessels and the pressure drop across each vessel
in the system at rest.
In order to solve the system, the inlet
and outlet pressure are 100 mmHg and 5 mmHg
respectively [3]. The connection logic of the
vessels is given in the form of an .NWK file and
it allows for the creation of 29 momentum
balances equations for the system at steady state
(given below).
∑ 𝐹𝑙𝑜𝑤𝑠 𝐼𝑁 + ∑ 𝐹𝑙𝑜𝑤𝑠 𝑂𝑈𝑇 = 0 (1)
The length and vessel geometries are also
known. This allows for the creation of 42
constitutive equations (given below).
∆P = F α
(2)
128∗𝑢∗𝑙
𝜋∗𝐷 4
(3)
𝛼=
(See appendix A for a complete list of balance
and constitutive equations used in the model).
The equations were put into the form of the
standard linear algebra problem (3). Where A
was the coefficient matrix, x was the vector of
unknowns and b was the target vector. The
equation was solved with LU decomposition in
Matlab.
Ax − b = 0 (4)
.
The complexity of the vessels between the
Circle of Willis and the Venous system
including the main branches of the cerebral
artery, Pial network, Intracerebral arteries,
Microcirculation, Intracerebral veins, and Pial
veins necessitated an approach utilizing
constrained optimization to obtain relevant
resistance values in the vessels mentioned
above. The optimal hydraulic resistance in these
complex series of vessels was calculated by
solving the system for an initial resistance value
of 0.3 Pa*s/mm^3. The blood flow rate
calculated was compared to the ideal blood flow
rate of 750 ml/min the difference was considered
the error. The process was repeated 1000 times
incrementing the initial resistance value by
0.015% each time. The error surface was
constructed by plotting the error against the
incremented resistance values. The resistance
with the lowest error was chosen as the ideal
resistance (See table A for ideal resistances).
To determine the concentration profile
of caffeine in the cerebral vasculature for any
dose steady state mass balances with convection
and destruction terms were constructed for each
of the 36 nodes in the system using equations 48 below.
∑ 𝐼𝑁 − ∑ 𝑂𝑈𝑇 = 𝐷𝑒𝑠𝑡𝑟𝑜𝑦𝑒𝑑
(5)
Equation 4 above describes the mass balance for
molar flux of caffeine for each vessel in the
cerebral vasculature.
∑𝑛𝑖=1 Φ𝑄𝑛 𝑄𝑛
(6)
Equation 5 above describes the molar flux into
the node. Where n is the number of vessels
going into each node, Q is the flow rate obtained
from the system at rest, and Φ𝑄𝑛 is the unknown
molar concentration of caffeine in each inflow.
∑𝑚
𝑗=1 Φ𝑗 𝑄𝑗
(7)
Equation 6 above describes the molar flux out of
the node. Where m is the number of vessels
leaving the node, 𝑄𝑗 is the flow rate out of the
node obtained from the system at rest and Φ𝑗 is
the unknown molar concentration of caffeine in
the outflow.
∑36
𝑠=1 Φ𝑠 ∗ 𝑉𝑠 ∗ 𝑘𝑠
(8)
Equation 7 above describes the destruction term
for the each mass balance equation. Where Φ𝑠
is the unknown molar concentration of caffeine
in the blood leaving the node, 𝑉𝑠 is the known
volume of blood leaving the node, and 𝑘𝑠 is the
unknown destruction constant and represents the
removal of caffeine from the system as it is
absorbed by the vessels in (see appendix B for
an example network to demonstrate construction
of necessary mole balances). The destruction
constant k was determined by Newton et. al. to
be a linear function of caffeine concentration
(Φ). A linear regression was performed on the
data obtained from Newton et. al. to obtain an
equation for k given below.
𝑘𝑠 (𝛷) = -0.017* Φ𝑠 + 0.1607 (8)
Both k and Φare unknowns, Figure 2 below is a
schematic representation of the method used to
solve the system of two unknowns and two
equations. An initial guess was made to solve
the system for Φ, k’ was then calculated using
equation 8 above. Then k’ was compared to the
previously determined k by finding the infinity
norm for the difference between k and k’. k was
then updated to k’ and the system was solved
again for Φ. Once the difference between k and
k’ was smaller than 0.001 the model obtained
the desired concentration profile and k vector.
Figure 2: Schematic for finding the solution of the system of two equations and two unknowns consisting of the
concentration vector and destruction constant using an initial guess of k to solve the system for 𝛷 then iterating and
updating k until the infinity norm is sufficiently small
Experiments by Lunt et. al. determined that a
250mg dose of caffeine results in a 2% reduction
in middle cerebral artery (MCA) diameter.
According to equation 3 this corresponds to an
increased resistance value. With this
information and the concentration profile from
the model, the vessel constriction coefficient
(VCC) was calculated using equation 9a and 9b
below.
∆∅𝑀𝐶𝐴 𝑉𝐶𝐶 + 𝛼𝑜𝑙𝑑 𝑀𝐶𝐴 = 𝛼𝑛𝑒𝑤 𝑀𝐶𝐴 (9a)
𝑉𝐶𝐶 =
𝛼𝑛𝑒𝑤 𝑀𝐶𝐴− 𝛼𝑜𝑙𝑑 𝑀𝐶𝐴
∆∅𝑀𝐶𝐴
(9b)
The VCC relates the drug concentration in any
given vessel with the increase in resistance due
to caffeine induced constriction that the vessel
will undergo. Given that the diameter of the
middle cerebral artery is 3.9mm the reduced
MCA diameter is 3.8mm. The new resistance
value was calculated to be 0.53PA*s. The VCC
was calculated to be 1.74(PA*s)/mole.
The concentration profile was then
determined for two, three, and four cups of
coffee, the resistances in each vessel were
altered using the VCC, and the pressure drop
and blood flow rates were recalculated using the
method for the system at rest.
Results:
Pressure drops, resistances, and flow rates in the
right and left ICA, Microcirculation, right and
left IJV calculated from the network equations
for the system at rest are shown to the right in
table A, B, and C. The model performed as
expected; an increase in the caffeine dosage
correlates to an increase in resistance to flow
and pressure drop across the vessels along with a
decrease in the flow rate through the vessels.
The results for the concentration profiles
obtained for 2, 3, and 4 cups of coffee are
illustrated by Figure 3. The data obtained
conforms to expectations; the caffeine
concentration exponentially and asymptotically
approaches zero as distance from the inlet
increases. Knowledge of the local concentration
of caffeine in each vessel allows for a targeted
approach to modeling the vasoconstrictive
effects of caffeine on the cerebral vasculature
Blood Flow Rates [ml/min] for System at
Rest and Insulted System
RICA
LICA
Micro
RIJV
LIJV
CBF
0 oz
coffee
333.9
343.3
749.9
364.8
385.1
749.9
8 oz
coffee
247.6
251.2
592.4
293.4
298.9
592.4
16 oz
coffee
170.3
172.4
413.4
205.3
208.1
413.4
24 oz
coffee
129.0
130.5
316.2
157.3
158.9
316.2
Table A: Flow rates for key vessels in human
cerebral vasculature for 0, 2, 3, and 4 cups of coffee.
Pressure Drops [Pa] for System at Rest
and Insulted System
Rest
8 oz
16 oz
24 oz
coffee coffee coffee
1854
1578
1396
RICA 1114
1870
1590
1406
LICA 1130
10820
922.4
994.7
1045
Micro
282.3
273.3
274.5
RIJV 120.9
282.3
273.3
274.5
LIJV 120.9
Table B: Summary of pressure drops across key
vessels in human cerebral vasculature for 0, 2, 3, and
4 cups of coffee.
Resistances [Pa] for System at Rest and
Insulted System
Rest
RICA
LICA
Micro
RIJV
LIJV
0.2001
0.1975
0.0001
0.0198
0.0188
8 oz
coffee
0.4492
0.4466
0.0934
0.0577
0.0566
16 oz
coffee
0.5560
0.5534
0.1443
0.0798
0.0788
24 oz
coffee
0.6492
0.6465
0.1984
0.1047
0.1036
Table B: Summary of pressure drops across key
vessels in human cerebral vasculature for 0, 2, 3, and
4 cups of coffee.
Caffeine Concentration
[mMole]
0.35
Concentration Profile of Caffeine
0.3
0.25
0.2
2 Cups Coffee
0.15
3 Cups Coffee
4 Cups Coffee
0.1
0.05
0
Distance from Inlet
Inlet
Veins
COW
Figure 3: The concentration profile of caffeine for 2, 3, amd 4 cups of coffee in the cerebral vasculature. The
concentration is higher in the Circle of Willis due to the convergence of the inlets.
Validation:
The model for the cerebral vasculature at rest
was validated by comparing CBF, MAP, and
flows in the ICA, right and left IJV.
The overall CBF was 750.14 ml/min and it was
calculated by adding the blood flow of the inputs
in the system, the Internal Carotid, Vertebral,
and Basilar arteries. The Mean Arterial Pressure
was computed to be 84.3mmhg and it was
calculated by taking the average pressure at each
node of all the arteries in the system up to the
microcirculation. The flow in the Internal
Carotid artery was 334.2ml/min and the flow in
the right and left jugular vein was 385.3ml/min
and 364.8ml/min respectively. The model for
the caffeinated system was validated by
comparison to the experimentally determined
CBF reduction due to ingestion of 250mg of
caffeine by Lunt et. al. See table D below for
summary and comparison to literature values).
The computations for the system at rest were
validated by checking the overall mass balance
using equation 10 below. The model for the
caffeinated system was validated by hand checks
of randomly selected mole flux equations.
∑ 𝐵𝑙𝑜𝑜𝑑 𝐹𝑙𝑜𝑤 𝐼𝑁 = ∑ 𝐵𝑙𝑜𝑜𝑑 𝐹𝑙𝑜𝑤 𝑂𝑈𝑇 (10)
The overall mass balance equation validated
the computations for the system at rest and
the randomly selected mole flux equations
validated the model for the concentration
profile of caffeine. Table D shows that the
model reliably predicts the steady state and
caffeinated hemodynamics of the system at
rest.
Literature
Model
CBF
CBF250mg
MAP
ICA
750.0
547.5
87.00
239.0
749.9
465.4
84.30
334.2
LIJV
357.0
385.3
RIJV
309.1
364.8
Table D: Comparison of experimentally determined
values for hemodynamics of cerebral vasculature and
the values calculated from the model. CBF[ml/min],
MAP[mmHg], flow rates[ml/min].
Discussion:
The results of the series of network computation
suggest that caffeine has a significant effect on
the hemodynamics of the cerebral vasculature.
Two cups of coffee (16oz) correspond to a 21%
reduction CBF. Two cups also increases MAP
to 85.9mmHg. Three cups significantly reduces
CBF by 45% and increases MAP to 87.5mmHg.
Four cups reduces CBF of 58% and increases
MAP to 88.7. This is significant because the
average person suffering from poorly controlled
hypertension has a CBF of 597ml/min which is a
120
20% decrease [3] and a MAP of 110mmHg.
Caffeine is the most widely consumed neurostimulant in the world and the average American
consumes about 300mg of the drug daily. The
data from the model suggests that this has
serious implications for possible caffeine
induced hypoperfusion because the model shows
significant link between caffeine use and CBF
reduction even though even larger doses of
caffeine do not seem to affect MAP as much as
CBF (see Figure 4-5 for summary).
MAP in Cerebral Vasculature
MAP [mmHg]
110
100
90
80
70
60
Rest
16oz
32 oz
64 oz
Hypertension
Figure 4: MAP for the cerebral vasculature at rest compared to MAP with 2, 3, and 4 cups of coffee and MAP of
average patient with poorly treated hypertension.
800
CBF Levels in Caffeinated Vasculature
CBF [ml/min]
700
600
500
400
300
200
100
0
Rest
16oz
32 oz
64 oz
Hypertension
Figure 5: CBF for 2, 3, and 4 cups of coffee compared to normal CBF and average CBF of patients with poorly
controlled hypertension.
Conclusion
The model demonstrated that for steady state
blood flow through the human cerebral
vasculature physiologically meaningful values
can be obtained for blood flow rates and
pressure drops with knowledge of the logical
connectivity of the network, the physical
geometry of the blood vessels, and the input and
output pressures. The values calculated for
overall CBF and MAP as well as localized blood
flow rates through key arteries and veins were
reasonably similar to the values found in the
literature.
The model further demonstrated that a
convective destruction model for the flow of
caffeine through the cerebral vasculature can
reasonably approximate the effects that caffeine
consumption has on CBF. Ingestion of common
amounts of caffeine can have profound effects
on the hemodynamics of the cerebral
vasculature. Most notably the vasoconstrictive
effects of caffeine could lead to a self-induced
hypoperfusive state.
Acknowledgements:
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