Biology Reference
In-Depth Information
1. All doses must be given at equal time intervals.
2. All doses, except possibly for the first dose, must be equal.
3. The MEC must be achieved as quickly as possible.
4. The concentration of the drug should remain between the MEC and
MTC at all times.
E
XERCISE
1-23
It is not common practice for pharmaceutical companies to make the
MEC and MTC figures for their drugs available. Instead, users are
provided with recommended doses and time intervals. It may be
interesting to consider the following question: Assuming that
pharmaceutical companies follow objectives 1-4 from Exercise 1-22
when determining the dose regimens for their drugs and that the
half-life is known, could you estimate the drug's MEC and MTC?
XI. USING COMPUTER SOFTWARE FOR SOLVING
THE MODELS
For most models developed thus far, we have presented analytical
solutions. Knowing the analytical form of a solution allows for direct
calculation of the predicted value. For example, knowing that the
solution of Eq. (1-2) is given by P(t)
P(0)e
rt
, where P(0)
¼
¼
5.3 and
r
¼
0.297, we can calculate that for t
¼
2.5, the model predicts a
t (DT ¼ 0.5)
P (t)
5.3e
(0.297)(2.5)
population size of P(2.5)
11.1 million for the United
States for the year 1825. In the same way, using the solution of the
discrete model p
n
¼
¼
¼
0.0
5.300
k)
n
p
0
from Exercise 1-1, we can calculate that if
(1
þ
0.5
6.148
p
0
¼
0.345, according to the discrete model (1-1), the U.S.
population in 1880 will be p
8
¼
5.3 and k
¼
1.0
7.133
k)
8
p
0
¼
(1.345)
8
(5.3)
(1
þ
¼
56.8 million.
1.5
8.275
2.0
9.599
It is not always easy to solve a model analytically, and, as the
sophistication of the models increases, the mathematics for solving the
equations become increasingly more challenging. When it is difficult (or
sometimes impossible!) to obtain the actual analytic solution, numerical
solutions are used instead. A numerical solution does not give us a function
as the analytical solution does, but instead provides us with a table of
values for the unknown function. For example, a numerical solution for
the problem
dP
2.5
11.136
3.0
12.919
3.5
14.987
4.0
17.387
4.5
20.170
5.0
23.399
ð
t
Þ
¼
rP
ð
t
Þ;
P
ð
0
Þ¼
5
:
3, for r
¼
0.297 is presented in Table 1-7.
5.5
27.145
dt
6.0
31.491
The left column contains a list of values for t, and the right column
contains the values of the numerical solution P(t) at these points. The
TABLE 1-7.
A numerical solution for dP(t)/dt
¼
rP(t).
