Biological Disposition of Drugs and Inorganic Toxins

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Biological Disposition of Drugs and Inorganic
Toxins A number of metallic elements play a
crucial role in our nutrition, usually in small
quantities. Sodium is necessary to nerve
conduction, iron is essential part for
hemoglobin, magnesium is a component of chlorophy
II. Many other trace of metals are also required
for correct functioning of bio-molecules. On the
other hand, lead has toxic effect in our body.
There is a plenty of lead in our
environment. Most lead enters the biosphere
through human-made sources. It is available
through many sources, ie, paints, cable sheathing
and pipes, gasoline, batteries, ceramics,
plastics, etc. Following biological compartment
model shows the paths of lead into, around and
out of an organism
Environment
Blood
Soft-tissue
Bone
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Gas Exchange The lungs are gas exchange organs.
They can provide an efficient entry route into
our body for foreign substances such as lead. The
air we breathe may contain leaded gasoline or
from lead smelters. About 40 of the lead we
inhale is absorbed into our blood from
lungs. Blood is first compartment which absorbs
lead. Blood carries the lead to soft tissues -
second compartment ( all tissues which are not
bone ). All these tissues behave similarly and
absorbs lead similarly to about the same
degree. Third compartment is bone, in which lead
has very large half life. The arrows show the
direction of lead movement between various
compartments. Another pathway by which chemicals
can enter our body is through our skin. As such
skin is a sensor of, and a waterproof barrier to,
outside world. However, some chemicals can
penetrate the skin ( cell membrane). Such
compounds are tetra-methyl lead and tetraethyl
lead ( antiknock compounds found in leaded
gasoline). Cleaning hands with such liquids would
guarantee lead into our bodies.
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• Clinical Effect of Lead
• Lead poisoning is indicated by a variety of
  clinical symptoms, including gastrointestinal and
  mental disorders- vomiting and pain. In children
  there may be central nervous system disorders-
  drowsiness, irritability, and speech disturbances
  . Blue line along the gums may also be formed.
• It can also affect children IQ
• A Mathematical Model for Lead in Mammals
• As a first approximation we need only distinguish
  three tissue types bone, blood, and the soft
  tissues.
• Bone tends to take up slowly but retains it for
  very long periods of time in contrast to soft
  tissues, other than blood, where the turnover is
  much quicker.
• Blood is a transport agent of the metal.
• The disposition of the lead in the body can be
  followed as a three compartment system by
  tracking its movement into and out of these three
  tissues types .
• Lead enters the system by ingestion and through
  the skin and lungs.these paths usher the
  substance to the blood

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• From the blood, lead is taken up by the soft
  tissues and the bone. This uptake is reversible.
  Lead is also released by these organic reservoirs
  back to the blood.
• However,especially for the bone, leads
  half-life time is very long.
• Lead can be shed from the body via kidneys, from
  blood and some via hairs.
• Thus, blood is the main conduit through which
  lead moves among various compartments.
• Environment will be treated as another
  compartment. To account for the lead intake and
  elimination.
• Designate this as compartment 0
• Let xi, i 1, ,3 denotes the amount of lead in
  the compartment i ,and
• aij, i 0,..,3, j 1,..,3, denote the amount
  rate of movement of lead to compartment j from
  compartment i.
• The product aij . Xj is the rate at which the
  amount of lead increases in the compartment i
  due to lead in compartment j
• There is no reason that aij aji and, as noted
  above, the rate of movement from blood to bone is
  very different than the reverse rate. The units
  of a are per day.

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• As we will not keep track of the amount of lead
  in the environment, this is an open compartment
  model. Instead, we account for environmental
  intake by including a separate term IL(t),
  applied to compartment 1, blood.
• Compartment 1 is blood,
• compartment 2 is soft tissues
• and compartment 3 is bones.
• From the assumption , some of the rates are zero.
  Namely, a03 a23 a32 0
• There is no direct elimination to the environment
  from the bone and do direct exchange between
  bone and soft tissue. There is no aio, as there
  is no x0 term in our model.
• Finally there is no term aii, as compartment is
  our finest unit of resolution.
• With these basics, we can now prepare the model,
  which is driven from the simple fact the rate of
  change of lead in a compartment is equal to the
  difference between the rate of lead entering and
  lead leaving.
• Since there are three active compartments in the
  system, our model will have three differential
  equations, which will be written in matrix form.

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In words, first equation says that lead leaves
blood for the environment, soft tissue and bone
at a rate proportion to the amount in the blood
lead enters blood from soft tissues and bone in
proportion to their respective amounts and lead
enters blood from the environment according to
IL(t). The algebraic sum of these effects is the
rate of change of lead in blood. In matrix
formulation, this can be written as X / AX
f, where X/ is first derivative of X wrt time,
and f IL(t) 0, 0T
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A is a 3x3 matrix -(a01 a21 a 31) a12 a
13 A a21 -(a02a12) 0 a31
0 -a13 The solution will be given
by X(t) We assume that the intake of lead
is constant. This is a reasonable assumption.
Then f is also constant and we may carry out the
integration on the right hand side.
e
At-A-1e-As 0 t f - e Ate-At - I A-1f
- I- e-At A-1f.
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The complete solution becomes X e At x0 -
I- e-At .A-1f e At x0 A-1f -
A-1f This is the solution of the system provided
A-1 exist and that the exp(At) is
computable. The long term prediction of the
model The long term solution is predicted by the
eigenvalues of the matrix A. It is easily seen
that this is compartment matrix. The diagonal
terms are all negative. The eigenvalues will be
all negative. In such case e At tends to zero
matrix as t tends to infinity. As a result, long
term result of lead in the body is given by X
--gt - A-1f as t tends to infinity
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Example Lead is measures in Micrograms and the
time unit is per day. Rate of 49.3 is given is
the rate of ingestion micrograms per day. Other
coefficients are a01 a12 a13 a21 a02 a31 IL
0.0211 0.0124 0.000035 0.0111 0.0162 0.0039 49.3
The graphs of the outputs are shown in the
figure The figure shows graphs of the total leads
in compartments 1.2 and 3 over a period of 265
days. The horizontal axis is in days and the
vertical axis is micrograms of Lead. We have
taken the initial value of X0 to be 0 ( all
components are zero) One observation is that the
level of lead in blood and soft tissue approaches
a steady state quickly. Lead achieves a steady
state in blood of about 1800 units and about 700
units in soft tissue. The level of lead in bones
continue to increase
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• after one year.
• In this model, the bones continue to absorb lead
  because of constant input rate.
• On the other hand, bones release lead very
  slowly. This high level of lead in bones has
  implications for severe chronic biological
  problems.
• The level of lead in steady state is the next
  subject for discussion.
• For this model, the long term behavior of the
  solution is computed based on the eigenvalues of
  the matrix A.
• Which will be - 0.0447, -0.02, and -0.00003.
  All the values are negative.
• Computation of -A-1 f yields ( 1800, 698,
  200582 ), f (439/10, 0, 0 ).
• This model predicts the long-range forecast as
  follows
• The level of lead in blood rise to 1800
  micrograms.
• The level of lead in soft tissue will be 698
  micrograms.
• The level of lead in bones will be 200582
  micrograms.
• Various studies show that the approximately 95
  of the body lead is lodged in bones and continues
  to accumulate until the age of 60 years.

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• Pharmacokinetics
• the routes for dispersion of drugs through the
  body follow the same pattern as that of lead. We
  will examine how body handles the ingestion of a
  drug taken periodically ( say, every six hours
  ,ie decongestant ).
• We keep track of drug in two compartments the
  gestro-intestinal tract and circulatory system.
• The mathematical importance of this model is that
  the limit for this system is periodic.
• A two compartment pharmaco-kinetic model is used
  to construct a drug utilization scenario.
• The drug is taken orally on a periodic basis
  resulting a pulse of dosage delivered to the GI
  tract.
• From here the drug moves into the blood stream,
  without a delay, at a rate proportional to its
  concentration in the GI tract and independent of
  its concentration in the blood.
• Finally the drug is metabolized and cleared from
  the body at a rate proportional to its
  concentration there.

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Model has two compartments. Let x(t) denotes the
concentration in the GI tract and y(t) denotes
its concentration in the blood. D(t) is the
amount of drug. It may be constant or a function
of time. The governing equations are System
is a linear with the forcing function D(t), its
solution in the matrix form is given by Where
Y and P are vectors given by Y and M is the
coefficient matrix given by Note that it
is a compartment model , diagonal terms are
negative and column sums are zero. Consequently,
the first term exp(Mt)Y0 is transient,that
is it tends to 0 with time.
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Therefore, asymptotically, the solution tends to
the second term Y --gt
, which is periodically driven by
D. Periodic solutions predict serum concentration
cycles. Exact shape of the dosage profile, D(t),
depends on how the pharmaceutical company has
buffered the drug. We assume that the drug is
taken every six hours and dissolves within six
hours, providing a unit pulse dosage with height
2 and pulse width of 1/2 on a interval of
0,6. The rate parameters a and b are given in
half lives, a a ln 2, b (1/5) ln2. For a
numerical solution, assume zero initial
conditions. X(0) 0 and y(0) 0, which means
that no drug is initially present in the
system. ( Read book to check the solution in
Maple language , page 228). The solution is shown
in the figure. Solution of x(t) and y(t) are
shown on the same graph. Both are oscillatory.
y(t) increases in time with oscillatory
behavior. Phase plot between the variables x(t)
and y(t) is also shown. It shows that the
asymptotically, the solution tends to be periodic
( non sinusoidal), called limit cycle.
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Input is taken as periodic
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From the figure, the high concentration level in
the blood is 1.5 while the low is about 0.9 on
the limit cycle. In designing a drug, it is
desirable to keep the concentration as uniform as
possible and to reach limit cycle as quickly as
possible. The parameters a and b tends to be
characteristics of the drug itself. The
asymptotic solution is found from the second term
of the solution equation Y We can also
solve the equations explicitly as two independent
equations x(t) -ax(t) 2 with x(0) x0
which is given as x1(t) 1/ln2 (x0 -1/ln2)
2 -2t Follow this part of the solution from x 0
to1/2. For t 1/2 to 6, there is no input D(t)
0 the equation becomes x(t) - ax(t) with
initial condition, x(1/2) x1(1/2).
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The solution for this part will be X2(t) 2
-2t ( x0 1/ln2 )) For the solution to be
periodic, we have to put another condition that
x0 x(6), and solve for x0. See the graph
for the value of x0 Similarly, the solution for
y(t) can be calculated. y(t) b y(t) a
x(t). The solution of y(t) will also be in two
parts, as the solution of x(t) is in two parts.
One part will be valid from y from 0 to 1/2 and
other from 1/2 to 6. The solution of x(t) and
y(t) are shown in the graphs. Both of them are
periodic. Note that the y(t) level should be high
enough to be effective, but not so high as to
cause side effects. It is up to the drug
companies to adjust the constants a and b so that
the periodic solutions are maintained.The company
has to determine the appropriate level of y, and
adjust a and b to maintain that level.
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