To determine the necessary conditions for creating a safe environment for the embryo exposed to a laser beam and increasing cell viability, we researched to calculate the optimal technical parameters and temperature indicators of laser exposure to the embryo. For this purpose, we controlled the temperature conditions of laser action on the embryo. The calculated mathematical model of laser action is presented, and the temperatures of laser action on the layers of the embryo are calculated by using the methods of separated variables and uncertain coefficients. Using the surface-volume ratio of the volume of thermally injured cells to the volume of the embryo cell layer, an applied optimization mathematical model for minimizing the volume of thermally injured embryos was implemented. This made it possible to calculate the optimal power and time of laser exposure to the embryo. Thus, the application of the results of this study made it possible to determine the parameters of the safety environment for a bioobject (embryo) in which the latter is unevenly heated. The results of the above studies can be used in solving general problems aimed at improving the quality of functioning of several technical systems with distributed parameters containing concentrated, discrete sources of scanned laser radiation.
From the point of view of innovative biotechnology projects, the security environment is a set of external and internal relations related to certain biotechnological manipulations with a bioobject, as well as conditions, factors, and circumstances that have influenced and may influence these relations in a certain way. The stimulators of the development of the security environment are known to be the interests of forces, and all actions of such forces are aimed at achieving their own or group interests. In the context of our study, the interests of the forces mean, on the one hand, the interests of the project implementers, and on the other hand, the interests of competitors. It is the competitors who are interested in causing damage, liquidation of the project, or destruction of its main objects at the first stages of implementation of an innovation project, and in discrediting the results among the scientific or business community at later stages. Therefore, understanding the true interests of competitors is a priority task in predicting and implementing the conditions of the project's security environment [1–2]. The interests of the forces are manifested and specified in the problems of relations that arise during interaction and are hidden. According to innovative projects, the interests are aimed at obtaining/preserving information that constitutes scientific and commercial secrets and provides advantages in the competitive struggle. Therefore, understanding the true interests of the actors is a priority task for creating conditions for project information security.
For this study, we have chosen an innovative project that is intended to be implemented in the activities of a biotechnology laboratory, institute, veterinary medicine center, etc. [3–4]. Its essence is to calculate the optimal parameters and improve the quality of the safe environment where the embryo is exposed to a laser beam. The purpose of our study is to apply mathematical programming methods to determine the optimal conditions of the safe environment for the embryo. Given that a bioobject is a living organism, such conditions can be considered a safe environment for it, because in this environment it remains safe and can maintain its own life, develop, and reproduce.
An increase in the quality of laser division of the embryo is achieved by creating conditions of a pectic-free medium where the embryo is placed, by increasing its viability in the biotechnological process. It is necessary to provide control over the temperature conditions of laser heating and the consumption of technical resources (power and time) of laser emitters. Control over temperature regimes can be ensured through the implementation of restrictions not to exceed the temperature of heating of the embryo for 370C during multiple iterative processes of realization of computational mathematical models (boundary value problems) for multidimensional, nonstationary differential equations that describe the state of the embryo under laser influence. Due to the specificity of boundary value problems and specific features of the temperature function of the embryo, it is impossible to guarantee the existence and uniqueness of their solutions (multidimensionality, nonlinearity). To justify the correctness of mathematical models, the authors propose to use specialized methods and estimates from above on the solving functions of solutions of differential equations in the space of smoothly growing distributions of slow degree growth. This will allow us to obtain necessary and sufficient conditions for guaranteeing the correctness not only of the above mathematical model but also of applied optimization mathematical models of the process of laser treatment of the embryo, as well as of the general problem of creating an ash-free environment for the bioobject by increasing the viability of the separable parts of the embryo.
In [5–6], computational mathematical models
were developed for thermophysical and
mechanical systems containing sources of
physical field loading. Necessary and sufficient
conditions for the stability of solutions for
systems of nonlinear differential equations
with a small external perturbation in the
righthand side of the main differential equation of
the boundary value problem were determined
and proved [7–8]. The authors of [9] solve the
problem of determining the correctness
conditions for boundary value problems with
systems of linear, inhomogeneous partial
differential equations. The authors derive
hyperbolicity conditions for solutions of
homogeneous and inhomogeneous impulse
systems with distributed parameters. Paper
[
Methods for improving the accuracy of
solving applied problems of cutting a fixed-size
area [
Our publication is in the context of previous
scientific research, which the authors were
engaged in in different years to increase the
effectiveness of the implementation of
innovative developments in various branches
of agriculture and agrarian business, in
particular [
In the articles [
Some aspects of solving applied
optimization problems to improve the design
quality of biotechnological systems in
agriculture are studied in publications [
However, the parameters of the safety environment for such biological objects, which are under laser radiation and allow them to be viable, have not been determined. Therefore, this publication is relevant and allows you to solve exactly these questions.
Determine the parameters of the safety environment where the embryo is located, which should be created for a specific bioobject.
To calculate the temperature regimes of laser heating of the embryo, it is necessary to solve the computational mathematical model (boundary value problem) of the process of laser action on the embryo. Boundary value problem of the system of differential equations of heat conduction describing the state of an embryo under laser influence:
We got it: e−1097c 16437563600 11++1706495,,27cc ,
16437563600 11++1706495,,27cc = 1 , and therefore c1 264708,3.
Thus, the temperature of heating of the The Dirichlet boundary conditions at the beginning and the end of laser heating: T (0;0) = 100 0C; (2) T (53;2550) = 37 0C.
Boundary conditions of the 3rd kind, specifying the specific heat flux from the embryo to the environment where this microbiological object is placed: −0, 67 T1 (0,t) = 4, 4 . (3)
r
Using the methods of separated variables and indeterminate coefficients, we obtained the solution of the differential equation (1), which we will write based on the general form of the boundary value problem (1)-(3): c1ect1 1+ c1ect2 1+ cr2
1 + = T (r1,t1 ) + 9a qr2
1 ; 6a cr22 + = T (r2 ,t2 ) + qr22 , 9a 6a where T (r,t ) is laser exposure temperature; q is the power density of heat loads in the layers of the embryo; a is diffusivity coefficient.
We divide the first equation by the second equation to find the unknown constants: T (r1,t1 ) + q6ra12 1+ ec(t1−t2 ) = T (r2 ,t2 ) + qr22 1+ 6a cr22 c9ra12 . (5) 9a Let's find the constant с1 : с1 =
T (r1,t1 ) + qr2
1 6a . ect1 1+ cr12
9a (4) (6) perivitelline space:
The temperature that is required to maintain the viability of the blastomere cell layer is 370C. When exposed to a laser beam on an embryo with a temperature greater than the 370C the microbiological object is injured. To calculate the volume Vsc of a heat-injured segment of embryonic embryos, it is possible to use the following formula:
Vsc(T) = r1h12 − h313 + Rh22 − h323 (7) where r1 is depth of laser beam penetration into the blastomere cell layer; h1 = C1B , h2 = C2B is segments of the segment connecting the geometric center of the embryo and the laser beam source of action in the form of a spot; R is radius of the cell layer.
Note that the volume Vzp+ pp (T ) segment of the layers of the pellucid zone and perivitellar space of the embryo is calculated by the formula (7). In this case r1 = O1A – is the depth of penetration of the laser beam into the perivitellated space.
To find the values h1 , h2 from the formula (7) consider a triangle О1О2А, formed by the segments r1, R and the segment connecting the geometric center of the blastomere cell layer and the laser beam source of action in the form of a spot (Fig. 1).
O1 isotherm r1 C1 А
R В С2
O2 embryo triangle О1О2А, and then divide the found area by the length of the side О1О2. Thus, the formula for determining the length of the height АВ has the following form:
AB = 2 p ( p − r1 )( p − R)( p − O1O2 ) ,
O1O2 where p = O1A + O2 A + O1O2 is half-perimeter.
2
To find the segments h1 , h2 parties O1O2 calculate the length of the segments O1B , O2 B : (8) (9) O1B = r12 − AB2 ; O2B = R2 − AB2 . h1 = C1B = r1 − O1B; h2 = C2 B = R − O2B.
Knowing the length of the segments O1B , O2 B , calculate the lengths of the segments we are looking for h1 = C1B , h2 = C2B :
Substituting the found values h1 , h2 in formula (7), let us calculate the volume of the thermally traumatized segment of the embryo blastomere cell layer, taking into account the volume of Vzp+ pp (T ) a segment of the irradiated layers of the pellucid zone and perivitellar space.
To determine the volume Vsc (T ) of the heattraumatized segment of the embryo blastomere cell layer, it is necessary to subtract the volume of the irradiated segment of the embryo blastomere cell layers, perivitellar space, and pellucid zone from the volume of the segment of the irradiated embryo blastomere cell layers, perivitellar space and the pellucid zone Vzp+ pp (T ) segment of the irradiated layers of the pellucid zone and perivitellar space of the embryo.
To find the volume of the blastomere cell layer, it is possible to use the following expression:
V Vc n , 2 (10) where V is the volume of the blastomere cell layer; Vc is the volume of one cell; n is cell count.
Using the data on the radius and number of
blastomere cells characteristic of the embryo at
the late morula stage of development, we
obtained that the volume of the blastomere
layer is equal to 138430,7 the micrometer. To
calculate the optimal values of power and time
of laser exposure, it is necessary to implement
an applied optimization mathematical model of
minimizing the volume of thermally injured
cells at the selected parameters of laser
exposure [
K = Vetc → min ,
Vc (11) where Vetc is the volume of the thermally traumatized segment of the cell layer, which is calculated by the formula (7).
Thus, applying the surface-volume relation from the above-applied optimization mathematical model, for a countable finite number of iterations from the solution of boundary value problems (1)–(3), we obtained the optimal parameters of the safe environment (power in the 124 milliwatts and the duration of laser exposure in 3 microseconds).
The article proposes mathematical models and methods of their realization to control the temperature modes of laser heating of embryo, to search for optimal parameters of laser exposure on embryo. An applied optimization mathematical model for minimizing the volume of thermally-injured blastomere cells is proposed. It is based on the surface-volume ratio of the volumes of traumatized cells of embryos to the volume of the whole cell layer. Having applied specialized methods from the theory of nonlinear mathematical programming in the space of generalized functions of slow step growth, the correctness of the given computational and applied optimization mathematical model is proved.
The results of the researches carried out in this work are expediently applied to increase the viability of microbiological objects under the influence of physical field sources by improving the quality of the safe environment where they are placed.