Editorial board editor-in-chief

Δψ
= 
f (x, t)
The chemical reaction of medicine in a place of contagion, 
and even an external steering of the ield of reaction such as 
electromagnetic as well as mechanical or temperature, can be 
sources generating these forces.
The obtained equation is not an Laplace’s equation but 
a Poisson’s one. The solutions of equation are well-known, but 
inhomogeneity in equation 
f(x,t)
 
is unknown and there are only 
few works devoted to this subject.
The author’s work is the proposal of analyses of these pro-
cedures, which we can accelerate with analysis of inluence on 
medicine content. The medicine itself, of course, carries on a dei
-
nite potential, however, it is not known in fact, when reacting in 
chemically medicine compound in treated area. This potential can 
be stable if the strength of medicine does not undergo change, 
or is suitably labile if medicine decreases or causes a prolonged 
action when is surrounded with isolating substances. There are 
many possibilities for required solution.
Let us consider the case of one dimensional transportation 
of medicine into the treated medium, assuming the monotonous 
decrease of medicine on the way of treatment in following de-
scription: 
φ(x)
=
 A 
−
 Ax
We have got therefore a following problem:
δ
∫
°
T
[ x
°
²
+ 1 −
x] [
1 − 
x
°
²
− 1 −
x] dt 
= 0
that is the equation (we accepted 
A = 
1
for facilitation of cal-
culation), where:
H (x, x
°
) 
= 
x
° 4
 
− 2
xx
°
 + x
°
²
 + x
²
skipping coeficient 
½
also for omission of needless arithmetic 
dificulties.
Writing out an Euler’s equation for introduced problem we 
have got:
x
°°
[6 
x
°
² + 2
x
− 1] + 3 
x
°
²
− 
x 
= 0
It is a non-linear differential equation of the second order, 
which the optimum solution of a function
 
f(x,t)
 
which
 
creates
 
the best trajectory.
Equation is unusually dificult to solve. There is a necessity 
of the numeric approach for that. It is visible, however, that for 
weakening potential of medicine the shortest healing trajectory 
will be a compiled function of time.
Let us notice, however, that the found trajectory is the best 
for process of medicine treatment. Technician treatment operator 
has to take the care about the presence of the suitable portion 
of medicine on this trajectory. The different ways have been 
indicated already in this text. So just thw technician of treatment 
has to look for the way of an agreement of a treatment process 
with transportation of medicine on the best trajectory.
Summary and conclusions
The question of penetration of the xenobiotics into the structure 
of a biosystem such as the tissue of skin or osseous, in point of 
view of the pharmacological treatment as well as toxic interaction, 
can be described both by a diffusive model and the proposed 
new variation approach. 
The ield of density of diffusive medium or temperature is 
a solution in this irst model, however, the best trajectory of 
healing or toxic impulses is always obtained in variation model. 
In diffusive model we observe the retention of the chemical 
substance on both sides of an organic membrane, however, it 
is possible to observe the retention of penetrating substances in 
every point of this membrane in a new variation considerations.
It seems that the proposed new approach allows for better 
steering and controlling of resources of such optimal impulse 
and for obtaining the expected progress in treatment and it 
enriches the hitherto existing formula of diffusive analyses, 
because the analyzed processes by the aim of mathematical 
apparatus were not always diffusive. The condition is that it is 
important to know the properties of pharmacological substances 
with expected potentials of interaction, which are usually deined 
by pharmacologists for their practical applications in biosystem 
environment. 
Presented procedure of analysis of medicine penetration is 
not a hospital recipe, but it is a formula showing dificulties which 
should lead to obtaining the hospital procedure. 

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35
System biolog
y
Steering of the process of penetration of chemical substances into biosystem structure
This new proposal is also a challenge for pharmaceutical 
industry for undertaking the co-operation. 
Acklowledgements 
The work was supported by AGH University of Science and 
Technology in Cracow.
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BIO-ALGORITHMS AND MED-SYSTEMS
JOURNAL EDITED BY JAGIELLONIAN UNIVERSITY – MEDICAL COLLEGE
Vol. 7, No. 13, 2011, pp. 37-42