Download Modeling Drug Delivery in the Rhesus Monkey - LPPD

Steady-State Blood Flow Model
of the Rhesus Monkey
By
Koula Quirk
RET Fellow 2010
Advisor: Dr. Andreas A. Linninger
Laboratory for Product and Process Design
Department of Bioengineering
University of Illinois at Chicago
RET Program NSF EEC-0743068
(A. Linninger, PI)
Abstract
A static one-dimensional whole body model of the rhesus monkey was designed with the
organs represented as compartments. Blood flow was simulated through these
compartments as blood pressure drops and was distributed throughout the body. A
steady-state computational model of drug delivery in the rhesus monkey was constructed
from physiological values derived using cardiac output, organ resistance and blood
pressure experimental data from the literature. The model predicts a steady-state view of
regional organ blood flow and blood pressure.
Background
Pharmacokinetics is a branch of pharmacology that studies, in living organisms, the way
a drug is absorbed, metabolized, distributed, and eliminated by the body. It is a complex
process that is not well understood- thus giving rise to dangerous side effects and
increasing risks of drug treatments. A steady-state analog has been completed modeling
pharmacokinetics in the rat and the human model is in progress, both in LPPD. For
allometric scaling, an intermediate animal model is desirable to gain a more precise
understanding of drug delivery. The number of animals used in clinical drug trials is
enormous and can be vastly reduced with the design of computational models to simulate
drug distribution inside the body. The rhesus (Macaca mulatta) monkey is anatomically
and physiologically close to humans, sharing about 93% of our genes (Ottesen, 2006). It
breeds well in captivity, is widely available, and is the non-human primate of choice for
conducting research on health-related topics. For these reasons we began work on
modeling drug delivery in the rhesus monkey.
Introduction
Physiological data obtained from both invasive and non-invasive methods has grown
exponentially in the past several years. The field of applied mathematical modeling in
human and animal physiology continues to develop along with the increasing amount of
available data. Large amounts of data can be better understood by using mathematical
models to help researchers and doctors develop new insights into physiological functions
and relationships.
The model developed for the rhesus monkey was derived using quantitative results based
on experimental measurements.
Methods
The main source of data was drawn from the Forsyth paper (Forsyth, 1968). They used
the radioactive microsphere technique to measure the distribution of cardiac output and
regional organ blood flow. This method makes repeated measurements of blood flow in
2
multiple organs by using plastic microspheres labeled with different radioactive nuclides.
The male rhesus monkeys weighed between 3.1 and 6.8 kg.
Using Kirchoff’s conservation (current and voltage) laws, Darcy’s law, and the HagenPoiseuille equation to derive the calculations, we modeled blood flow through
compartments and constructed a static one-dimensional model to simulate blood
distribution through the organs, or tubes. The appendix contains calculated values.
Kirchoff’s law states, in essence, that the sum of voltages in a closed circuit is zero. That
is, input must equal output, i.e., in our case, blood is conserved. Darcy’s law, analogous
to Ohm’s law, describes flow of fluid (blood) through a porous medium. That is, the flow
rate is directly proportional to the drop in vertical elevation between two places in a
medium and indirectly proportional to the distance between them. In our case, the flow
rate is a function of the pressure change divided by the resistance of the body organ. The
Hagen-Poiseuille equation is a physical law that gives the resistance when you know the
pressure drop in fluid flowing through a long cylindrical pipe (in our case arteries and
veins).
Equation 1 below is Darcy's law; Equation 2 is the Hagen-Poiseuille equation:
(1)
(2)
where:
F = blood flow (m*s-1)
P = pressure (Pa)
R = resistance (m-1)
ν = fluid viscosity (Pa·s)
L = length of tube (m)
r = radius of tube (m)
3
In standard fluid dynamics notation:
or
(3)
(4)
The change in pressure is equal to the pressure leaving the organ subtracted from
the pressure at the input to the organ.
where: ΔP is the pressure drop
L is the length of pipe
μ is the dynamic viscosity
Q is the volumetric flow rate
r is the radius
d is the diameter
π is the mathematical constant
Results
Figure 1 shows our predicted results for the rhesus monkey. Blood pressure drops from
110 mmHg as it leaves the heart and drops according to the resistance of each organ.
Table 1 and 2 also show that most of the percentages of cardiac output values in the
rhesus monkey are in the same broad range as those found in humans and rats (Forsyth,
1968). The main difference is the lower fraction of output to the large and small
intestines found in the monkey (8%) compared to the rat (19%). The values for the
rhesus brain (7%) are midway between man (14%) and rat (1%). Blood pressure for the
rhesus is 110mmHg, much lower than humans at 120 or even rats at 116. Flow rates to
the kidneys were similar between man, rat, and rhesus monkey. Percentage of cardiac
output to the liver was the lowest in the rat.
Discussion
According to Forsyth the majority of cardiac output values found in the rhesus monkey
are within the same range as those (culled from other reports) in man and rat. The main
difference is the lower percentage of output to the large and small intestines in the
monkey (8%) compared to the rat (19%). Additionally, there is a greater total cardiac
output per body weight in monkeys than is reported in other species (Forsyth, 1968). This
is reflected in the greater flows seen in most of the organs for the monkey compared to
other species. The data also show a much higher total cardiac output per body weight in
rhesus monkeys than is reported for other species.
4
Figure 1 shows the analogous computational model derived from experimental data in
the literature. Figure 2 shows a general anatomical view of a monkey for reference
purposes. The colors of the diagram correspond to changing pressure values through the
whole system. Pressure is highest where the blood leaves the heart and enters each organ
and drops according to organ resistance; pressure eventually drops to zero as it returns to
the heart at the output of all the organs. Table 1 shows the pressure drops predicted in the
model simulation. Numbers 1-45 represent the points of blood output from major organs
(1=heart). Values are the predicted blood pressures of the rhesus monkey. These were
calculated based on the data in the Forsyth article. Figure 3 is included as reference to
show the general human circulatory system.
Conclusion
New and continuing developments in computer technology aid mathematical modeling.
The improvement of existing mathematical models and construction of new ones can lead
to better simulation environments which are necessary for education and research.
Mathematical models can help avoid misunderstandings and wasted effort. Most
concepts can be clearly defined only by using mathematics. Mathematical models often
give rise to new and important questions that could not be asked without the use of these
models. For example, under what conditions is the cardiovascular system stable? How
does the topology of the vascular system influence the function of the system? This, in
turn, deepens our understanding of human and animal physiology.
It is hoped that the pharmacokinetic model of drug delivery in the rhesus monkey
contributes to this end.
5
Figure 1: Analogous blood flow
computational model (based on previous
work on the rat model by Cierra M. Hall at
LPPD/UIC.
Figure 2: Monkey anatomy, for reference.
(From www.infovisual.info, permission
granted.)
6
Table 1: Simulated blood pressure values in the rhesus monkey (mmHg).
% Solution from Delphi:
% Blood Pressures:
% x(1) = 110;
% x(2) = 109.000000308445;
% x(3) = 105.000000266026;
% x(4) = 104.999998536621;
% x(5) = 13.0000010838241;
% x(6) = 108.999997482744;
% x(7) = 104.999997672403;
% x(8) = 13.0000020063037;
% x(9) = 12.0000020490089;
% x(10) = 11.9999992233083;
% x(11) = 5.9999993852343;
% x(12) = 19.9999974855523;
% x(13) = 12.999997268887;
% x(14) = 11.9999972403551;
% x(15) = 103.999988470877;
% x(16) = 101.239988934643;
% x(17) = 12.920003674989;
% x(18) = 12.0000038273516;
% x(19) = 103.99998255177;
% x(20) = 101.239983370683;
% x(21) = 12.9200094757135;
% x(22) = 12.0000097464582;
% x(23) = 103.999977300554;
% x(24) = 101.239978432723;
% x(25) = 12.9200146244713;
% x(26) = 12.000014997675;
% x(27) = 103.999973305203;
% x(28) = 101.239974677094;
% x(29) = 12.9200185399147;
% x(30) = 12.0000189930253;
% x(31) = 103.999970565719;
% x(32) = 101.239972101602;
% x(33) = 12.9200212184552;
% x(34) = 12.0000217325093;
% x(35) = 103.999968454168;
% x(36) = 101.239970109742;
% x(37) = 12.9200232992409;
% x(38) = 12.0000238440601;
% x(39) = 103.999995057873;
% x(40) = 77.0000006413871;
% x(41) = 104.00000214443;
% x(42) = 102.000002032651;
% x(43) = 108.999998424644;
% x(44) = 12.0000011071087;
% x(45) = 4;
Numbers 1-45 represent the points of blood output from major organs (1=heart).
Values are the predicted blood pressures of the rhesus monkey.
These were calculated based on the data in the Forsyth article.
This corresponds to Figure 1 diagram: That is, #1 represents the blood pressure output
from the heart, and #45 represents the pressure at the input to the heart.
7
Figure 3: Main circulatory routes, arterial and venous
systems, showing blood flow to major organs.
(From Tortora, 1990, permission granted)
Appendix
Figure 4: Cardiac output values at start of radioactivemicrosphere method. (From Forsyth, 1968)
Figure 5: Organ blood flow and resistance in the rhesus monkey
showing significantly higher percentage of CO to liver.
(From Forsyth, 1968)
9
Figure 6: Comparison of CO values in organs of man,
dog, rat, baboon, and rhesus monkey, showing lower
fraction of output to the large and small intestine in the
rhesus monkey.
(From Forsyth, 1968)
10
Cardiac Output (L/min)
Mean Arterial Pressure (mmHg)
Average Weight (kg)
Cardiac Output/wt (L/(kg*min))
Rat
0.1060
116.0000
0.3790
0.2797
Rat
1.0068
0.0528
5.1011
1.5156
2.7000
0.9000
13.8000
22.4000
0.7600
4.4000
0.7500
4.3000
Rhesus
1.0860
110.0000
4.1000
0.2649
0.5
Rhesus
0.6272
0.2827
4.8084
0.7496
6.5000
4.6000
12.3000
11.0000
2.8000
4.4000
0.7000
4.1000
(Delp, 1998)
(Forsyth, 1968)
0
Organ Flow Rates
Brain (L/(kg*min))
Liver (L/(kg*min))
Kidneys (L/(kg*min))
GI Tract (L/(kg*min))
Brain (% CO)
Liver (% CO)
Kidneys (% CO)
GI Tract (% CO)
Brain (% wt)
Liver (% wt)
Kidneys (% wt)
GI Tract (% wt)
Man
5.2000
120.0000
70.0000
0.0743
1.0000
Man
0.5600
0.2500
3.7500
2.9000
11.4000
4.6000
17.5000
18.1000
2.0000
2.6000
0.4000
1.4000
Average
2.1307
115.3333
24.8263
0.2063
SD
2.702907
5.033223
39.16576
0.114552
Average
0.7313
0.1952
4.5532
1.7217
6.8667
3.3667
14.5333
17.1667
1.8533
3.8000
0.6167
3.2667
SD
0.240923
0.124391
0.710802
1.089938
4.361575
2.136196
2.67644
5.757025
1.027878
1.03923
0.189297
1.619671
(Brown, 1994) t23
*GI
Tract:
large
intestine,
smallsmall
intestine,
cecum,cecum,
pancreas,
spleen,spleen,
stomachstomach,
and sometimes
the colon, the
rectum and eso
*GI
Tract:
large
intestine,
intestine,
pancreas,
and sometimes
colon, rectum and esophagus.
*GI Tract: large intestine, small intestine, cecum, pancreas, spleen, stomach, and
Table 2: the
Comparison
of values
for cardiac output and organ blood flow for rat, rhesus monkey,
sometimes
colon, rectum
and esophagus.
and human.
(Compiled byof
Cierra
M. Hall,
REU 2010
at LPPD/UIC)
Comparison
values
for Cardiac
output
and organ blood flow for Rat, hum
Acknowledgements
Financial support by NSF RET Grant – EEC 0743068 (Andreas Linninger, PI) is
gratefully acknowledged. For their generosity in patiently sharing their skills and
knowledge I also thank:
Prof. Andreas A. Linninger
Cierra M. Hall, REU 2010
Sukhi Basati, PhD. Candidate
The Laboratory for Product and Process Design staff
11
References
Normal distribution of cardiac output in the unanesthetized, restrained rhesus monkey.
Am. J. Physiol. 25: 736-741, 1968, Forsyth, Nies, Wyler, Neutze, and Melmon
Physiological parameters in laboratory animals and humans. Pharm Res 10(7): 10931095, 1993, Davies, B. and T. Morris
Dynamic pressure-flow relationships of brain blood flow in the monkey. J Neurosurg
41(5): 590-596, 1974, Early, C. B., R. C. Dewey, et al.
Redistribution of cardiac output during hemorrhage in the unanesthetized monkey. Circ
Res 27(3): 311-320, 1970, Forsyth, R. P., B. I. Hoffbrand, et al
Modelling of individual pharmacokinetics for computer-aided drug dosage.
Comput Biomed Res 5(5): 411-459, 1972, Sheiner, L. B.,
B. Rosenberg, et al.
Applied Mathematical Models in Human Physiology. Series: Siam Monographs on
Mathematical Modeling and Computation 9, 2006, J. T. Ottesen, M. S. Olufsen,
and J. K. Larsen
Principles of Anatomy and Physiology, 12th ed. 2009, G. J. Tortora and B. H. Derrickson
biosystems.okstate.edu
www.wikipedia.org
www.thefreedictionary.com
12