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UDC 004:591.5:612:616-006
https://doi.org/10.15407/biotech11.01.039
SOME TRENDS IN MATHEMATICAL MODELING
FOR BIOTECHNOLOGY
O. M. KLYUCHKO1, Yu. M. ONOPCHUK2 1Kavetsky Institute of Experimental Pathology, Oncology and Radiobiology of the National Academy of Sciences of Ukraine, Kyiv 2Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Kyiv
E-mail: kelenaXX@ukr.net
Received 07.12.2017
The purpose of this publication was to show some trends of the development of modeling methods for biotechnology according to contemporary achievements in science and technique. Common approaches were outlined, some types of classification of modeling methods were considered, the role of methods of mathematical modeling in biotechnology in conditions of intensive development of information computer technologies was highlighted. The appropriate scheme of their interrelation was proposed. Several examples and mathematical models in three dimensions (1D, 2D, 3D models) were described respectively for processes in living objects at different levels of their hierarchical organization. In course of this research, the main attention was paid to some processes modeling in neurons, as well as their aggregates of different forms, including the cellular formation of glioma (1D, 2D, 3D brain processes models). Their consideration was begun with the models that had today only theoretical importance and was completed with a description of a model that might has practical application. The work was done after the analysis of approximately 250 current publications in fields of biotechnology, including the authors' original works.
Key words: 1D, 2D, 3D modeling, mathematical methods, information technologies.
Intensive development of computer facilities, information technologies and advanced training of scientists in the field of mathematical methods made it possible to apply increasingly complex methods for modeling in biotechnology; and this practice demonstrates a lot of advantages. Using modeling, the researcher can test well the structures of complex system and processes in it without invasion into this system [1]. Simulation can also be used in situations, when the system's behavior is not clear enough (models of "black box" or "grey box") [1]. Using some models, one can simulate the dynamic behavior of the system over a period of time [1]. Also, the methods of modeling permit to study the biotechnical system in cases, if invasion in it functioning is enough expensive or can damage the system by itself (in cases of biotechnical production, processes in living organism, and etc.) [1]. In our previous publications [1, 2] we analyzed a number of mathematical analytical methods for data processing in biotechnology, and outlined that some types of mathematic analysis have already become widely used: factor, dispersion, correlation, discriminant
ones; as well as methods of classification, other methods of statistical data processing [1, 2]. Special attention we paid to some powerful mathematical methods used in biotechnology, like neural networks [3], cluster analysis [4], image processing [5]. In the format of this article it was impossible to observe the great volume of publications devoted to task solutions using contemporary methods of modeling. So, at the beginning of this article some general approaches were outlined; some principles of classifications of mathematic modeling methods were suggested. Than we tried to demonstrate our understanding of the place of mathematic modeling methods among other mathematic methods from the point of view of information computer technologies (ICTs); we suggested also an appropriate scheme of their correlation. Further some case studies were represented: some suitable mathematic models are suggested for the description of different processes in living objects of different levels of organization. We would like to suggested a few models of the brain or brain elements of different level of hierarchical organization. Respectively,
these models we ordered as models of different number of dimensions (D): 1D model, 2D model and 3D model. We started from some models that have theoretical importance for today only — we outlined some examples of modeling techniques and software tools that can be used to simulate cellular networks and systems. Than we described a model of glioma with forecasting of patients life expectancy; we hope this will be important for the practice.
Mathematical models in biotechnology: basic concepts. There were a lot of contemporary publications where are stated, that mathematical models are tools that we can use to describe the past performance and predict the future performance of biotechnological processes [6-75]. Methods of mathematic modeling could be applied to processes operating at many different levels, from the action of an enzyme within a cell, to the growth of the cells within a commercial scale bioreactor; these methods were widely used for the development of bioinformatics methods, for the newest biomedical electronic information systems elaboration [1, 11, 74, 76-142]. Mathematical models could be powerful tools in both fundamental research and applied research; for their development the newest mathematic methods were used usually [1, 2-5, 41, 60-62, 66-74, 76-77, 116]. Mathematical models were important tools in the optimization of the performance of biotechnological processes. There were numerical excellent publications in this sphere with the description of analysis of mathematical models' structure, demonstrated how they contain state variables, independent variables, operating variables and parameters, and etc. [1, 7-10, 13-16, 18-28, 34, 63, 64, 75, 80]. Such publications contained usually some case studies, in which, for example, the differential equations were used to describe the operation of a simple enzyme reactor, and etc. Sure, the real experimental results and results of observations were put in base of such models [1, 12, 17, 2840, 72-74, 134-155]. These provided the basis for discussion of the various approaches that could be used in modeling, such as empirical versus mechanistic approaches, or structured versus unstructured approaches [7].
Classification of mathematic models. Great variability of the types of mathematical methods was developed during all period of this practice existence [1, 2-5, 41, 60-62, 66-74, 76, 77, 116]. Probably, according to the most general contemporary classification all models could be subdivided to 1 — physical, 2 — mathematical, and 3 — computer ones.
1 — Physical model, for example, engineers can elaborate like any device, or smaller copy of bioreactor for biomass cultures; in such a way its functional abilities might be checked.
2 — Mathematical models were developed as a set of mathematical expressions for description of biotechnological processes; the main abstractions and regularities of modeled processes have to be revealed carefully.
3 — Computer (or program) models meaned an algorithm and/or codes for the simulation (or imitation) of biotechnological processes.
Also, there were some attempts to group mathematical modeling methods for their classification. One such classification was done by the group of Ukrainian cybernetics who worked for biology in Kyiv, on 1970-th -1980-th: Piatigorsky B.Ya., Zaitman G.A., Cherkasky V.L., Chinarov V.A.(Kyiv, Ukraine, 1985) [1, 9, 10]. Groups of modeling methods are represented below according to their works as a number of alternative pairs (Fig. 1).
In one of more recent publication there were represented another classification, also based on contradictory pairs [7]. Types of models according to these authors were following (Fig. 2).
Because of fast progress in contemporary ICTs' spheres another types of classifications were possible as well. Below in this publication there are the results of authors' original research in modeling (mathematical and program ones as well) [1-5, 15, 16, 18-37, 6066] and mathematical processing of biological data [134-155], as well as a brief review of some other authors works in modeling [6-8, 11, 37-57, 67-133].
Contemporary information technologies: from models to information systems. Practice of mathematic modeling in biotechnology is seen as traditional one nowadays. From the other side, our époque is characterized by extremely intensive development of ICTs that obtain numerous forms and manifestations. Our task was to find the links between traditional mathematic modeling and modern ICTs; the results of such studying were described before in [1]; they are shown also at Fig. 3.
At the center of picture "Mathematical modeling" was placed as an extremely important compartment of methods and knowledge in computer disciplines, which is considered classic today. Until the early 1990s — the time of intensive development of information computer technologis — mathematical modeling occupied a priority position. This compartment produces the source data, models, methods, and etc.; all these had found an
Fig. 1. Classification of models for biological sciences: A, B, C, D, E are the different types — pairs of classification system groups according to [9]
A
Empirical models
B
Distributed models
Mechanistic models
Lumped models
G
Linear models Non-linear models
Fig. 2. Recent classification of models for biological sciences: A, B, C, D, E, F, G are the different types — pairs of classification system groups according to [7]
application in practice, which is presented at figure as compartment "Technical application". Such practical application meaned elaboration of various mechanisms or devices, for example robots, expert systems, automatic devices, other computer-controlled systems.
Process of development of notion "Information computer technologies" and its correlation with notions "Databases"
and "Mathematic modeling" might be demonstrated as few historical stages (Fig. 3).
1) Initially, for models that were created at the historic "Stage 1" not so much input data were needed; sometimes such data could be given even as a simple table representation on a sheet of A4 format.
2) At the "Stage 2" we showed the continuation of this historical process.
The amount of data that needed processing grew rapidly with time, and the input of "Mathematical modeling" began to present data structured in the form of simple databases (DB) (for example, using the first versions of DBase or FoxPro according to shown at (Fig. 3).
3) Due to the technical progress, increased power and speed of computers, the transition to personal computers (PCs) and the formation of networks, there were prerequisites that DB acquired their own increasing importance (Stage 3). With the time DB control systems have been so advanced and developed that many of the preparatory data operations became possible to be realized even before the data entered the part "Mathematical modeling". At the next step the level of development of input data operations became very high. This resulted in the "Information Technologies" developed with DB and included in them; and this structure could be isolated in a separate compartment as shown at figure for Stage 3. At Stage 3 of information systems (IS) development, the subset "Information technologies" included "databases".
Today we can state that at the current level of information computer technologies development the different methods and technologies coexist successfully: older in time of their invention as well as the latest ones. Their significance and role in the modern world is determined not by the time of their creation, but how much they are needed in practical human activity.
Description of some mathematic models. So, the methods of mathematic modeling could
be used nowadays for the solution of different tasks in biotechnology. Such practice began even from the first steps of computer use; in Kyiv, Ukraine there were the works of 1950-60-th years of cybernetics who worked for biology: Amosov N.M., Hlushkov V.M., Kostyuk P.G., and later — representatives of their scientific schools Shkabara E.A., Ivanov-Muromsky, Piatigorsky B.Ya., Onop-chuk Yu.M. and many others [1, 10, 11]. It was not possible to list all tasks solved by these representatives of Kyiv scientific schools of cybernetics during few decades until XXI century. The numerical models of different types of processes in nervous systems, in cells and cell aggregates, biological objects, biological organism functioning, models of experimental biotechnological equipment, and many others were elaborared.
Intensive works in this direction were carried out through over the world. For example, in publications [1, 6-9] there were a lot of other models' descriptions. In work of [7] there was a mathematical model developed for enzyme reactor consisted on a reservoir containing a substrate solution which circulated through the column with immobilized enzyme; the use of this model for the estimation of enzyme kinetic constants was demonstrated. Also there was a structured metabolic model developed in 1984 for the growth studying of a single Esherichia coli cell and etc. Authors in [6] outlined modeling techniques and software tools that could be used to simulate cellular networks and systems. First, they presented
Fig. 3. Development of notion "Information computer technologies" and their correlation with notions
"Databases" and "Mathematic modeling" [1]
a review of stoichiometric models and Flux Balance Analysis and then discuss continuous deterministic models composed of ordinary differential equations. Next, they presented the continuum descriptions, often used for the simulation of model signal transduction and gene expression programs with stochastic models. Lastly, they discussed the application of qualitative modeling tools and techniques such as fuzzy-logic and petri nets to the problem of modeling complex intracellular networks. For the illustration of abovementioned basic concepts a few case descriptions were suggested.
1D mathematic modeling of electrical currents in biocells. Electrical signal transmission through myelinized fiber. The excitation spreading in myelinized fibers is seen as classic example of 1D mathematic modeling of electrical signals transmission in nervous system. If the maximal action potential significantly exceeded the threshold and if the front of potential rise is sufficiently narrow, the potential increase was very fast. It could be assumed that the excitation transfer occurs from one Ranvier node to the nearby node and subsequent nodes influenced only slightly one on another. This task solution leaded to a problem with distributed and lumped parameters [13, 14]. In case suggested below it was supposed that the action potential is many times greater than the threshold, and the front rise of potential was equal to 1.5-2 intervals between the Ranvier nodes [13, 14]. So, the intervals between the nodes formed the distributed system and the potential change in it could be described by the equation
if i = 0. For simplicity there were used dimensionless variables, namely, the coordinates are measured in intervals L, and the time t — in units of E1C1. For such cases the equation for the potential at intervals looks like much simpler:
= - (1)
Lets also suppose that at time T = 0 the excitation started at the node located in point X = 0 due to the mechanism of membrane potential changes in it, and according to (2).
Electrical current generated by this node initiated processes in the next node. So, charging capacity of myelinized interval and capacity of the next node located at X = 1. Then there were assumed that processes at the next node (charging node X = 2) did not occur until the time of stimulus "hopping" at excitation node X = 1. Therefore for boundary conditions at X = 2 it was possible to assume that
9 (2, T) = 0
(3)
Presence of lumped capacitance C2 at Ranvier node and at the point X = 1 was reflected by the simple expression
(4)
It was assumed that at point of process start
9 (X, 0) = 0.
(5)
This may be in case of significant difference between the threshold and action potentials, and the presence of small initial charge on the fiber had not significant influense on results.
The task was solved under the condition 0< T < TL , where TL — magnitudeless dimensionless time of excitation "hopping" from one node to another. The purpose of task was to calculate the time of excitation spreading for the pulse velocity, proportional to Tl.
Because of the lumped capacitance at the point X = 1 task should be divided into two parts: in the domain 0 < X < 1, and in 1 < X < 2.
Lets set that
9 (1, T) = ¥ (T). (6)
Then the solutions should be substituted into equation (4) and from it was possible to determine the unknown function ^ (T). Solution of equation (1) in the domain 0 < X < 1 with the boundary conditions (2) and (6) and the initial condition (5)
<p(X,T) = 2n^nsm(nnX)e~n2j'2T j t
(7)
In domain 1 < X < 2 with boundary conditions (6) and (3) and the same initial condition we get
9 (2, T) = 9o (T)
(2)
<p(X,T) = 2n^nsm(nn(X - l))e""2,r3r j
(8)
For further calculations one needed to know the type of function %(T), i.e. the change of working node potential in time. We approximated this dependence by a function
(9)
where — maximum capacity; T1 — rise time capacity. Because of abovementioned reasons, the initial value of this function can be set equal to zero.
As follows from the experimental data according to [9], such approximation is not far from the truth. Substituting (7), (8) and (9) into (4), we obtain the equation for unknown function y(T). For its solution it is convenient to use the Laplace transform. Omitting the intermediate steps, we give the final expression for Laplace transform
(10)
Function y(T) had a pole of second order at point p = 0, stipulated by the multiplier 1/p2, and a number of poles of the first order, determined by the roots of equation
(11)
This equation had real roots. Roots of the equation are defined as the point of intersection of some relative curves with the horizontal line drawn at the level -C2/C1L. All the roots of (11) are negative and layed to the left of n2n2. We denoted the found roots p0, p1,
Knowing the poles of ip{p) function, it is easy to do the inverse Laplace transform:
1 C,L + C2
2 4rCji
CO £
Vk2 +n2Ti2
SCO
Tl=—o
(pk +n2TT2)2 (12)
The time of excitation "hopping" TL from one node to another is determined by condition that at time T = TL the node potential at point X = 1 should reach the threshold
<P*Ti_1T ciL <p2 2 4 CXL
r.j 1 Y oj (
C2 1 V Pk + n2JI2
2Aj 1
h=0 A"=—+n2n2y
Because the true dimensional hopping time from one node to another is R1C1L2Tl, and the distance between nodes L, then velocity of the pulse is
As an example, lets compare with physical processes at the frog myelinized nerve fiber described in work [14]. In cases of some known spread values, we took the average value. So, we assumed that
At the end of calculations lets study two cases: the front widths of the intervals were 1.5 and 2, i.e. when T1 = 1,5TL and T1 = 2TL. For the case T1 = 1,5TL the calculation gave the speed v = 29 m/s. For T1 = 2TL we obtained v = 21 m/s. From other side, the experimentally measured value of velocity is 23 m/s. So, close values in all these cases demonstrated good adequacy of performed and simulations as well.
2D modeling of matrix of bionic element in biotechnical system. Experimental study of the structure and function of neurons in the brain and other cells is an important area of brain research [15-17] including researches with fluorescent markers as well as biotechnology during the last decades. One interesting way of further carrying of such investigations is a construction of biotechnical systems where the technical part and biological one (implants, cultured samples etc.) were combined. Such combination have good prospects for their use for organism damage corrections, for disabled peoples cure, and etc.
Despite the fact that the experiments in this area of science are extremely laborintensive, high-tech and expensive, the number of Ukrainian researchers still keep them now in collaboration with foreign scientists [3740]. Among Ukrainian researchers several research groups perform optical registration on the cells of various tissues and single cells [38-40]. All such studies have to be carried out under the control of luminescent, confocal microscopes using fluorescent markers, in our experiments — primulin and bis benzimid [1517]. Tracking the processes during experiments performed the sets of digital pictures recorded in the computer memory. The sources of experimental data that were taken into account for modelings — electrophysiological experiments (patch-clamp, voltage clamp, and others [13, 14]. Results of such experiments
were combined with optical ones in attempts to construct suggested 2D model — our observed biotechnical system. Briefly, the primary relative 2D physical model for this task was following. The samples of brain neurons were inquired into the observed biotechnical system as rectangular 2D plate with cultured brain neurons constructed as 2D matrix. Taking as the base the results of electrical and optical properties of natural studied brain neurons, we developed computer models for 2D neuronal matrix (biotechnical system) with properties of memory and symbols coding.
Physical and biochemical nature of 2D matrix from bionic elements. In base of original model (version of 2-dimentional network from brain neurons) experimental results described in our previous articles are put [15-17]. The sense of registered phenomena [17] was in the fact, that input signal changes of optical properties of the neuron (luminescence of neuron) become more intensive in case of neuron excitation. Other results that were taken into account during modeling were the results of transmembrane electric currents studying in experiments with patch-clamp, voltage-clamp, and etc. [13, 14]. Basing on these facts the 2D matrix as our model was elaborated. This matrix may be imagined as an abstraction of one-layer cell culture where neurons are ordered as it was shown on Fig. 4; the signal could be transmitted from cell to cell. This abstract construction has to be incorporated into the electric circuit with measuring and information storage devices described in numerical manuals (Fig. 4). Such system was biotechnical-one, constructed using principles of bionics; some laboratories of the world reported about such constructions [1]. Developing program model, we based on abstract supposion of 2D neuron matrix existing. Schematically it is suggested below.
At the next step we would like to elaborate a program model of this neuronal matrix that was called "quasi screen" (Fig. 4) this abstract
1.b
1,3
m
A
B
_/7v_ ~VJ2
Fig. 4. Biotechnical system as union of bionic element (1) and electronic element (2):
1, A — for one neuron or its fragment; 1, B — for 2D neuron matrix
neurons there were ordered on 2D matrix. Simultaneously the screen pixels have been brought into compliance with these abstract neurons. All these stages of our investigations were done, and results were already published in different scientific and technical journals [15, 16]. Further stages according to our plans were following: 1- simulation of 2D "quasi-screen" functioning (model 1), and elaboration of program for the simplest symbols coding at such screen (model 2). Obtained results were suggested below.
Description of a "quasi screen" with bionic elements for biotechnical system. Probable "quasi screen" with bionic elements for biotechnical system may be imagined as 2D neuron matrix on plastic plate with cultural brain neurons in dissociated culture incorporated into electric circle with standard measuring and data stored devices. 2D neuron matrix may be imagined as a model like chessboard with alternating rows and columns that separate the cells which contain optically active elements. The structure of such a screen we described previously in details [15]. For our model we assumed that in some cells of this screen there are bionic photosensitive elements [15].
Assumptions of the model. When designing a model, we made the following assumptions.
1) Each screen pixel can be in 3 states, corresponding to 3 states of natural neuron (quiet, excited and refracter state of neuron). These 3 states were coded by "red", "yellow", "achromatic" colors respectively.
2) Also, a pixel could be described by three-phase model. Nature of these phases also corresponded to quiet, excited and refracter state of neuron.
3) The demonstrational velocity of the model has been increased 60 times: one minute of real processes in nature (as recorded in the experiment), corresponds to 1 second model.
4) Image coding it was possible to realize by establishing of different color points pattern in any set time interval.
As it was mentioned above, the lightsensitive elements of "quasi screen" were bionic elements with variable optical properties. According to our experiments these bionic elements may be following.
1) complex protein molecule and molecule optically active substance (fluorochrome);
2) "luminescent" neurons described in our article [17].
Our model was designed for the "screen" where optically active elements are the neurons with fluorochrome molecules inside. We have
developed a model based on dynamic changes in the optical characteristics of the neurons that we described previously in [17].
Dynamics of optical neuron characteristics changes in screen matrices. According to the results of the experimental neuron reaction studying, the receipt of the excitation signal can be represented as three consecutive phases that have their physical, biochemical and physiological sense (Fig. 5).
Vectors mean transition of the system from one state to another. Vertices of the triangle are three states in which the system can be (Fig. 5). These three states on the model were colored with white, red and yellow colors. Respectively,
- Top 1 corresponds to described phase 1 and 4 (achromatic).
- Top 2 corresponds to phase 2 (red).
- Points at vector 2-3 means transition (yellow, orange).
- Top 3 corresponds to phase 3 (achromatic).
This model 1 demonstrated the activity of "quasi screen" with neurons for the case when activated signal were inputted to active element at the left top of neuronal matrix and than moved to the right bottom angle. We supposed that each active element of the screen may be activated with further relaxation. Below we described these phases of activation and deactivation of each element.
1) The first phase of bionic element activation in time moment t0, before activated signal was inputted (Fig. 5). Moment t0 is characterized by signal amplitude A=0, fluorescence intensity was going to zero (E0). Such pixels were achromatic at our model (Figs. 4, 6-8).
2) The second phase of bionic element excitation was at time moment (t1). The phase was depicted at Fig. 5 as point 2. Interval t1-t0 =20 seconds. During system transition
from the phase 1 to phase 2 the amplitude of activated signal — electrical current growed to the maximum A = Amax. Pixel luminescence was maximal (Emax). At model the neurons of this phase were marked by red color (Fig. 4, 7, 8).
3) Additional points were characteristic (middle points) for the excitation of bionic elements at time moment ((t2 t3) (depicted on Fig. 5 as point at vector between 2 and 3). Interval t2-t0 = 30 seconds and time points near it. At these time points the amplitudes of activated signal (electrical currents) are going to decrease (bionic element luminescence is decreased) A = Amin. Pixel luminescence decreased to minimal values (Emin). At model elements of this phase were marked by yellow and orange depending on current amplitudes and intensity of luminescense (Figs. 4, 7, 8).
4) The fourth phase of bionic element activation was at time moment t4. Interval t4-t0 = 180 seconds. The phase was characterized by signal amplitude (electric current) A = 0, luminescence intensity was going to zero (E0). These elements were achromatic at model too (Fig. 6).
At the "screen" at Figs. 6-8 the bionic elements were depicted by the circles at defined positions on screen matrix. This arrangement of bionic screen elements was determined by conditions in which the existence of bionic elements is possible under the experimental conditions. The initial view of the model and the initial step of the program was shown at Fig. 6.
The next figure demonstrated the model at 101th second from process start. It was possible to see that one element might be at the second phase; some other elements were at phase number three, and so on.
At the last figure there were demonstrated the program work at 120th second from process start. One can see here that all elements moved gradually, after specified period of time, from
Fig. 5. Each pixel of the screen works by three-member cycle
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o o o
o 0 o
0 o 0
o 0
o o
Fig. 6. Initial phase
□□□□□□□□□□
□BDDBDDDEL □□□□□□□□□□
□□□□□□□HDD □mnPDDHDD
Fig. 7. Working model at 101th second from process start
one phase to another. It means that activated signal moved from the upper left angle to the bott.
In developed original model of this biotechnical system two figures: triangle and circle can be coded. Systems which may be described in such ways potentially are the systems with "memory", if system can stay in some of states longer than in other states. Thus, the proposed model could be used for modeling the "screen" with elements that have properties of memory.
Model algorithm. Model described above was abstract and basic. Its algorithm was given on Fig. 9.
So, in such a way we tried to make a model of coupling between transmembrane electrical currents and their visualization using fluorescent drugs. Taking as the base the results of electrical and optical properties of natural brain neurons studying we develop computer models for 2D neuronal matrix (biotechnical system, "quasi screen") with properties of memory and possibilities of symbols coding. Program model of electrical signal propagation (and, respectively, information signal propagation) in this biotechnical system was elaborated.
3D model: mathematical model of estimation of hypoxia development in brain gliomas. As it was noted in our previous publication [1], the state of hypoxia in tissues (the brain in particular) can occur dye to the low oxygen concentration at altitudes, due to diseases or metabolic disorders, and etc. The development of tumors of the brain — glioma — also leads to the development of hypoxic phenomena. The study of these phenomena were done by Prof. Onopchuk Yu.M. (Glushkov Instituite of Cybernetics, NAS of Ukraine, Kyiv) and researcher Guzhavskaya N.V. (Institute of Neurosurgery, NAS of Ukraine, Kyiv) [31].
Estimating the degree of tumor influence on organism, researches were faced a number
120500 Clear
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□□□□□□□□ED □□□□□□□□□□
□□□□□□□□□□
□■□□□□□□□□
Fig. 8. The final phase
j™* * slep_period > limar ||r* step_peri<Mi - tfiaíi0e_period s\ timar)
CD
Fig. 9. Algorithm of basic model for electrical signal linear propagation in 2D neuron matrix
of problems that they tried to solve by analyzing the clinical and laboratory data obtained during the patient examination. Based on these data, they would determine how much the initial loading caused by the tumor affects the patient life expectancy in the catamnesis, how to predict the possible time intervals of the tumor recurrence, the possibility of complications during the operation and in the early postoperative period. The process of tumor development in organism was considered as an stochastic one, although it is often preceded by functional changes in the organism, caused by the influence of adverse and extreme external environmental conditions, in which the vital activity of the organism was carried out. Therefore, the methods of the theory of reliability of complex systems were used to estimate the tumor pressure on the organism and the degree of risk
to patient's health. Consequently, the refuse of the system from functioning is associated with the achievement of the load function S (t), which reflects the degree of tumor influence on organism, a certain boundary level S (t). Naturally, the effectiveness of conservative or surgical treatment can be estimated using the scale [0, S(t)].
First of all, let's observe the model of clinical data of patients with brain glioma. When a patient arrived to hospital for treatment, a number of clinical indices of his condition were determined. Among the most important were: the degree of malignancy of the tumor (indicator f1**), tumor localization (f2**), the presence of mediastinal structures of the brain displacement (f3**), the presence of perifocal edema (f4**), tumor density (f5**), the presence of cysts formation (f6**), the type of disease course (f7**), the severity of general brain symptomatic (f8**), the severity of the focal symptomatic (f9**), the severity of stagnation in fundus (f10**), the tumor's consistency (f11**), the connection of the tumor with main vessels (f12**). Most of these indices were determined by tomographic, laboratory and exclusive research.
The values of each of these functions (fj**, i=1,12), which significantly affected the condition of patient with brain gliomas of different degrees of malignancy. They determined the functional capabilities of organism in life processes realisation, and they can be called a discrete set. We can assume that this value ws equal to:
1, if gliomas of the first degree were detected,
2, with gliomas of the second degree, M (f1**) = 3, with ependymomes, gliomas of the
third degree,
4, with gliomas of the fourth degree.
Location of the tumor was enough important. Therefore, we estimated M (f2**) as follows:
1 — convectional localization, forehead, crown, temporal, M (f2**) = 3 — damage of medial structures,
4 — subcortical localization.
Tomographic studies made it possible to estimate the region of values M (f3**):
0 — shifts were absent,
1 — presence of brain median
structures shifts, up to 4 mm, M (f3**) = 2 — brain median structures shifts from 4 mm to 7 mm,
3 — brain median structures shifts
from 7 mm to 10 mm,
4 — shifts were more than 10 mm
In the same way the area of values of function M (f4**) and M (f5**) was estimated:
0 — if perifocal cerebral edema is
1-1.5 cm or it was absent,
1 — diameter of perifocal edema was
1.5-2 cm;
M (f4**) = 2 — perifocal edema was 2-3 cm, 3 — edema was 3-4 cm.
1 — tumor density was 15-20 conventional units (c.u.) M (f5**) = 2 — tumor density was 20-25 c.u.
3 — tumor density was more than 25 c.u. Indicators of functions f6-f11** were important for the evaluation of brain tumor loading on organism. Clinicians defined the set of their values as follows:
0 — cysts were absent,
M f6** = 1 — overwhelming was present,
2 — necrosis, cysts were present,
1 — the consistency of the tumor was
soft,
M f7** = 2 — consistency of the tumor was dense,
3 — tumor was chalked.
A number of indicators used for the estimation of the quantitative value of tumor loading were determined by doctor indirectly, exclusively, in the process of patient examination. It was stated that functions f8**, f9** and f10** may have two values "0" or "1".
M (f8**) = 0 — there was no general brain symptomatology, 1— symptomatology was followed. M (f9**) = 0 — focal symptomatology was not observed, 1 — symptomatics was present? M (f10**) = 0 — stagnation on the fundus was absent or there was a moderate stagnation, 1 — stagnation on the fundus was clearly expressed. The indices f7** and f12** were linked with possible influence on functional changes of the whole organism:
1 — epileptic type of disease, M (f11**) = 2 — hypertensive type of disease, 3 — vascular type of disease.
0 — vessels were immured in a tumor, M (f12**) = 1 — main vessels did not belong to the tumor, 3 — main vessels were loosely soldered.
The influence of tumor on human organism studied only through clinical parameters, could be estimated by a linearly weighted function:
F cl = Z 81*f1, (1) where 8i were coefficients of importance of some indices in the system of organism estimation, which could be determined as a result of factor analysis of the data. In general, Si was taken into account during the construction of the sets M (fi, i=1,12), and therefore in the simplest case we could assume that
F cl = Z ficl (2)
Models of laboratory data of patients with brain glioma. The development of a malignant tumor of different nosology required the strengthening of the circulatory system functions in places of tumor localization, which supplied oxygen to the cells of the brain, enzymes and other essential substances for vital activity. Therefore, for the estimation of the degree of tumor influence on the systemic states of the organism, it was necessary to take into account the parameters of laboratory examinations of patients. In the practice of neurosurgical clinics among the most important laboratory indices were: f1* — severity of hemoglobin content in blood, f2* — severity of leukocyte contents in blood, f3* — severity of sugar abnormalities in blood, f4* — violation of the level of the prothrombin index,
f5* — violation of fibrinogen content.
The weight of these violations was estimated by the ball system, which forms the values of functions fi, i = 1,5.
M (f1*) = 1, if Hb <= 130 g/l,
2, if Hb > 130 g/l;
M (f2*) = 1, if 0 <= leukocytes <= 4 (*109 /l),
2, if leukocytes >4 (*109 /l);
M (f3*) = 1, if 0 < sugar <= 4 mmol/l
2, if sugar >= 6.6 mmol/l.
The authors denoted the prothrombin index as PI
M (f4*) = 1, if PI <= 80 (%),
2, if 80 <= PI <= 100,
3, if PI >100.
The authors denoted the fibrinogen through F.
M (f5**) = 1, if F<= 3 g/l,
2, if 3 <= F <= 4 g/l,
3, if F > 4 g/l.
The data of laboratory tests of patient were used for the estimation of tumor influence on organism state according to laboratory indices:
F1 = Z f.1
(3)
and complex influence of clinical and laboratory indices:
Thus, when determining the limiting value of the tumor load on human organism using the linear regression equation (4), we used the clinical and laboratory data of the patients of the neurosurgical clinic. The analysis of estimations of loading demonstrated that after the reaching of the limit value S, equal to 18 points, 421 patients (58%) had a life expectancy of about 3 years in catamnesis. 300 people have lived less than 3 years — the loading of their tumor was less than 18 points. Statistical data analysis demonstrates that obtained data were reliable (P < 0.01). For people who lived in catamnesis up to 15 months, the average tumor loading was 18.03 ± 0.05 points, from 15 months to 3 years — 18.15 ± 0.12 points, from 3 years to 5 years — 17.07 ± 0.03 points, and more than 5 years — 16.04 ± 0.02 points. Thus, an essential correlation of value of tumor loading with the duration of life in the catamnesis was found (Fig. 10).
F = Fc1 + F1 = f1 + Zf
(4)
Fig. 10. Life expectancy depends on the size of the tumor load
For patients with relapses of disease in catamnesis up to 15 months, the average tumor load was set at 18,18 ± 0,1 point (IV), from 15 months to 3 years — 18,17 ± 0.06 points (III), from 3 years up to 5 years — 16.17 ± 0.3 points, more than 5 years — 17.6 ± 0.42 points. For patients with relapses of the disease in catamnesis, 179 people (56%) had a load of more than 18 points, 139 people (44%) — below 18 points. The differences were statistically significant (P < 0.05). Thus, it can be argued that value of tumor load on organism, found basing on clinical and laboratory data of patient in hospital, substantially influences on the possibility of tumor relapses. The calculation of tumor load on organism according to formulas above demonstrates that patients with a load of 19.48 ± 0.2 points had complications during the surgical operation; the load of patients
without operational complications was 16.86 ± 0.43 points.
Clinical and laboratory data of patients examination in the early postoperative period gave a possibility to found that complications in this period had patients whose load in the preoperative period was 18.74 ± 0.06 points, while for patients without complications — 17.35 ± 0.04 points . It should be noted that among patients with a load above 18 points, 39% had no complications in the postoperative period. The average load of the tumor on organism depending on the group of glioma demonstrated that at stage I of malignancy the average load was set at the level of 14.9 points, at the stage II — 16.14 points, at stage III — 18.75 points, and at stage IV — 19.08 points (Fig. 11).
The analysis of the data obtained using linear regression about the value of tumor load on organism demonstrated a high correlation with the Karnovsky index, which is used for the estimation of disease severity (Table).
Thus, the use of linear regression method for the processing of clinical and laboratory data of patients with brain gliomas allowed us to determine the limit value (18 points) of the tumor load on organism; and in each case of the disease permitted to plan an individual strategy for patient treatment.
Thus, in present publication the analysis of some ways of modeling methods development for biotechnology according to contemporary achievements in science and technique was done. At the beginning of this article the general approaches were demonstrated, some types of classifications of modeling methods were observed. Previously we outlined some types of mathematic analysis widely used for modeling purposes: factor, dispersion, correlation, discriminant ones; as well as methods of classification, other methods of statistical data processing. Special attention it was necessary to pay to such powerful mathematical methods used in biotechnology as neural networks, cluster analysis, and
2b -1 20 -
15 ■
0 -
I II III IV
Fig. 11. Average load of the tumor on organism depending on the group of astrocytoma (in scores)
image processing. The role of mathematic methods modeling for biotechnology in present époque of information computer technologies intensive development was studied and appropriate scheme of interrelation of all these spheres was suggested. Further in our article some case studies are given; as well as some mathematic models developed for processes in living objects of different levels of hierarchic organization. Actually, the models for objects of 1D, 2D, 3D cases were suggested. In course of this the main attention was paid to some processes modeling in neurons as well as in their aggregates of different forms, including glioma of the brain. Starting from the models that have only theoretical importance for today, we suggested 1D and 2D models for some processes at neurons. In 2D model of neuronal matrix we tried to make a model of coupling between chemosensitive transmembrane electrical currents and their visualization using fluorescent drugs taking as the base the results of electrical and optical properties of natural brain neurons studying. Program model of signal propagation and symbols coding in this biotechnical system was done as well.
In present article we outlined modeling techniques to simulate neuronal processes not for single neurons or their 2D networks and systems only. As an example of 3D modeling we suggested the model of glioma with its analysis; this work we see as extremely important for the practice. Such analysis of the data obtained using linear regression about the value of tumor load on organism demonstrated a high correlation with the Karnovsky index, which is used for the estimation of disease severity. So, the use of linear regression method for the processing of clinical and laboratory data of patients with brain gliomas allowed the authors of this model to determine the limit value (18 points) of the tumor load on organism and in each case of the disease permitted to plan individual strategy for patient treatment.
Comparative analysis of the Karnovsky index and estimation of load weight by the linear regression method
Tumor loading before surgery Tumor loading after surgery
17.9 18.15
18.15 18.09
7.23 17.87
15.28 17.15
15.3 16.7
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ДЕЯК1 ТЕНДЕНЦП В МАТЕМАТИЧНОМУ МОДЕЛЮВАНН1 ДЛЯ БЮТЕХНОЛОГП
О. М. Ключко1 |Ю. М. ОнопчуК
Институт експериментально! патологи, онкологи та радмбмлоги iM. Р.в.Кавецького НАН Украши, Ки1в 21нститут шбернетики iM. B.M. Глушкова НАН Украши, Ки1в
E-mail: kelenaXX@ukr.net
Мета роботи — показати деяк тенден-ц^ в розробленн методiв моделювання для бiотехнологii в^пов^но до сучасних досяг-нень науки i технiки. Окреслено загальнi пщходи, розглянуто деякi типи класифь кацп методiв моделювання, висвiтлено роль математичних методiв моделювання у бiотехнологii за сучасного штенсивного розвитку iнформацiйних комп'ютерних технологiй. Запропоновано вiдповiдну схему ix взаемозв'язку. Наведено кiлька при-кладiв i вiдповiдно кiлька математичних моделей у трьох вимiраx (1D, 2D, 3D моде-лi) для проце^в у живих об'ектах на рiз-них рiвняx ix ieрарxiчноi органiзацii. Пiд час цього дослщження основну увагу було придiлено моделюванню деяких процесiв у нейронах, ixrnx агрегатах декiлькоx рiз-них форм, включаючи клiтинне утворення глiоми (1D, 2D, 3D моделi процесiв у моз-ку). Ix розгляд розпочато з моделей, що ма-ють на сьогодш лише теоретичне значення, а завершено описом модел^ яка може набути практичного застосування. Роботу вико-нано шсля аналiзу близько 250 публiкацiй з бштехнологи, включаючи оригiнальнi ав-торськi розробки.
Ключовi слова: 1D, 2D, 3D моделювання, математичш методи, шформацшш технологи.
НЕКОТОРЫЕ ТЕНДЕНЦИИ В МАТЕМАТИЧЕСКОМ МОДЕЛИРОВАНИИ ДЛЯ БИОТЕХНОЛОГИИ
Е. М. Ключко1 \Ю. Н. Онопчук2
1Институт экспериментальной патологии,
онкологии и радиобиологии им. Р.Е. Кавецкого НАН Украины, Киев 2Институт кибернетики им. B.H. Глушкова НАН Украины, Киев
E-mail: kelenaXX@ukr.net
Цель работы — показать некоторые тенденции в разработке методов моделирования для биотехнологии в соответствии с современными достижениями в области науки и техники. Изложены общие подходы, рассмотрены некоторые типы классификации методов моделирования, освещена роль методов математического моделирования в биотехнологии в условиях интенсивного развития информационных компьютерных технологий. Предложена соответствующая схема их взаимосвязи. Приведено несколько примеров и соответственно несколько математических моделей в трех измерениях (1D, 2D, 3D модели) для процессов в живых объектах на разных уровнях их иерархической организации. Во время этого исследования основное внимание уделялось моделированию некоторых процессов в нейронах, а также их агрегатах нескольких различных форм, в том числе и клеточном образовании глиомы (1D, 2D, 3D модели процессов в мозге). Их рассмотрение начато с моделей, имеющих сегодня только теоретическое значение, а завершено описанием модели, которая может иметь практическое применение. Работа выполнена после анализа около 250 публикаций в области биотехнологии,включая оригинальные авторские разработки.
Ключевые слова: 1D, 2D, 3D моделирование, математические методы, информационные технологии.
