EC-leasing, Moscow, Russian Federation

- *Corresponding Author:
- AA Berezin

EC-leasing, Moscow

Russian Federation

**Tel:**717531-8181

**Email:**[email protected]

**Received date:** November 27, 2017; **Accepted date:** January 27, 2018; **Published date:** March 07,
2018

**Citation:** Berezin AA , Shmid AV (2018) Theoretical Analysis of Blood Cell Concentration Oscillations Dynamics in Healthy Donors and Patients with Leukemia (*in vivo*). J Prev Med. Vol.3 No.2:9

A theoretical model has been proposed for acute leukemia developing mechanism (*in vivo*). The model describes the leukosis dynamics as an increase in the anomalous differentiation cycle energy of tumor blood cells (blasts) at the expense of the lymphocyte and segmented neutrophil cycle energy. The model has been formulated in terms of coupled Van der Pol equations with a time lag. The model solutions are in good agreement with experimental clinical data.

Acute leukemia; Anomalous differentiation cycle; Hematopoietic organs; Parametric oscillations

There are currently several theories of the leukemia development, such as the virus theory, virus-gene theory, clone theory, the theory of systemic development of this disease, etc. Leukemias are characterized by a systemic lesion of hematopoietic organs and a cyclic course of the disease. The theory, in which leukemia is treated as a systemic disease, holds that leukemia starts simultaneously throughout the entire hematopoietic system from the normal cells because of disturbances of the process of their differentiation (for example in some cases it triggers after a regular vaccination).

The research has been carried out in Hematological Department
(Head professor Machonova LA) of the Oncological Research
Center (director professor Blochin NN) by the cytologist professor
Peterson IS The main purpose of the research was to simulate the
process of sudden relapse happening in leukemic patients when
the concentration of leukocytes could sharply increase from lower
than 1000 cells per cubic mm to over 50000 cells during a day.
Therefore, the simulation could possibly predict the relapse and
help to modify the policy of chemotherapy in particular to reduce
its toxicity to normal cells. Since such a sharp increase of cell
concentration in case of relapse can be mathematically described within the dynamics or parametric oscillations, the authors
addressed the existing studies of biological rhythms in the blood.
The first researcher who described the oscillatory character of
leukocyte concentration during a day was American scientist
Franz Halberg [1]. He showed that leukocyte concentration in
peripheral blood of healthy humans could change by 50% during
a day. In cases of pathological states that variation maybe much
larger. Authors could not find in scientific literature the blood cell
concentration oscillations study. Accounting the Halberg results
the authors decided to investigate circadian blood cell rhythms in
healthy donors and in patients with acute leukemia. The results
of the study [2,3] showed that the circadian blood cell rhythms in
cases of leukemia had a broken phase of the cell concentration
oscillations to compare with a stable phase of the same
oscillations in the blood of healthy donors. The study of circadian
blood rhythms in healthy mice and leukemic mice confirmed
the phase breakage in the latter case [4]. The mentioned results
allowed suggesting in clinic the right time of the leukemic blood
rhythms phase correction through chemotherapy appropriate
time administration and transfusion of donor neutrophils [3].
Together with that, the blood hour rhythms (*in vitro*) in healthy
donors and patients with leukemia were studied [5]. All that
allowed elaborating a theoretical model for description blood
cell concentrations oscillatory dynamics in the blood of healthy
donors and in that of leukemia patients.

**Mathematical Model**

Earlier [6], we proposed a physical model for the segmentedneutrophil
and lymphocyte concentration oscillations (*in vitro*) in
the blood of a healthy donor. We showed that these oscillations
are described by coupled Van der Pol equations with a time lag:

(1)

where x and y are derivatives of respectively segmented
neutrophil and lymphocyte concentrations in the (*in vitro*) blood;
T_{x} and T_{y} are the delay times between an action and a change
in the concentrations, which are specified by the time of mutual
differentiation of cells; ω <1 reflect the nonlinearity coefficients
determined by the mechanism of protein synthesis by segmented
neutrophils and lymphocytes;ω is the oscillation frequency of
the derivatives of neutrophil and lymphocyte concentrations;
F_{1}(t),F_{2}(t) are random functions which account for the influence
of chaotic thermal fluctuations in the blood plasma electrolyte on
the derivatives of x and y concentrations. System (1) was studied
on a computer, and its solutions were shown to agree with the
experimental results.

Furthermore, Ref. [6] showed that, along with the normal cells,
the tumor (leukemic) ones were also present in the (*in vitro*)
blood of acute-leukemia patients. **Figure 1A** present the plot of
oscillations in the lymphocyte and leukemic-cell concentrations
in the blood of a patient with acute lumphoblastic leukemia
under {*in vitro*} conditions. The plots of the oscillations for a
patient with myeloblastic leukemia are of the same character. In
this case, however, the concentrations of segmented neutrophils
and myeloblasts (tumor cells) oscillate in opposite phases.
These experimental results allow one to propose a physical and
mathematical model of both lymphoblastic and myeloblastic
acute leukemias. In addition to the oscillation processes in the blood of healthy humans, described by system (1), the model also
includes oscillations in the concentrations of leukemic (tumor)
cells. Then, along with the aforementioned mutual differentiation
of segmented neutrophils and lymphocytes (described by
system (1)), mutual differentiation between lymphocytes and
lymphoblasts shows up in the (*in vitro*) blood [5]. Mathematically,
mutual differentiation between lymphocytes and lymphoblasts is
expressed by the equations

(2)

where C_{B} and C_{B} are absolute concentrations of
lymphocytes and lymphoblasts, respectively, and z is the
derivative of the concentration of the protein synthesized
by the lymphoblasts in the blood. In this case, the equation

(3)

Which describes the dynamics of the rate of protein concentration
variations, should be added to system (2). As it is known normal
lymphocytes synthesize immunoglobulins and lymphoblasts
synthesize specific proteins. Here z is the derivative of the
concentration of the proteins synthesized by lymphoblasts in
the blood, T_{z} is the time of the lymphocyte differentiation to
the lymphoblasts. Such type of differentiation was found by
cytologist Peterson I.S. in the process of periodical blood analysis
of the thermostatic blood (*in vitro*). This study [7] showed the
existence of both processes – differentiation of blood cells and
dedifferentiation of them [7] lymphocytes into lymphoblasts and
opposite direction of differentiation. Such process occurs only
in relatively large amounts of incubated blood like 150–200 ml.
Moreover, the study in these conditions leukemic blood (both
lymphoblastic and myeloblastic leukemias) demonstrated the
opposite direction of differentiation but at an early stage of
malignization, as it was found by cytologist Peterson I.S. F_{3}(t)
is a random function which accounts for the influence of chaotic
thermal fluctuations in the blood plasma electrolyte on the rate
of protein diffusion into the lymphoblasts and lymphocytes, and
ω_{0} represents the frequency of (*in vitro*) cell free oscillations
corresponding to the oscillation period
2π/
ω_{0} =2 h 40 min found
experimentally [5]. With regard to (2), the equation describing
the concentration variation rate for the lymphocyte-synthesized
proteins will have the form

(4)

(OK) leukemic cells in the blood of a patient with acute
lymphoblastic leukemia (*in vitro*) Horiz.axis days, Vert axis
thousands of cells per 1 cubic millimeter.

**Figure 1B** shows the numerical solutions of coupled Equations
(1-4). Substitution of the x, y, and z values into the equations for
absolute concentrations yields the curves which are consistent
with the experimental data.

As the delay time T_{z} decreases by a factor of 3, the model solutions
show a fall of the lymphocyte absolute concentration and a
rise of the lymphoblast absolute concentration. This suggests
the conclusion that the leukemic process is the increase of the leukemic-cell differentiation cycle energy. The tumor cells have
a differentiation cycle of their own with parameters different
from those of the healthy-cell differentiation. In some instances,
this difference gives rise to an autoparametric excitation of
the amplitude of tumor cell concentration oscillations and the
tumor differentiation of cells in the buffer organs: bone marrow,
spleen, and lymph nodes. To put it differently, the initiation and
development of the leukemic process is the excitation of selfoscillations
with time parameters different from those of selfoscillations
of the healthy-blood cell differentiation and, hence,
insults in the energy take-off by the tumor cycle.

leukemic-cell differentiation cycle energy. The tumor cells have a differentiation cycle of their own with parameters different from those of the healthy-cell differentiation. In some instances, this difference gives rise to an autoparametric excitation of the amplitude of tumor cell concentration oscillations and the tumor differentiation of cells in the buffer organs: bone marrow, spleen, and lymph nodes. To put it differently, the initiation and development of the leukemic process is the excitation of selfoscillations with time parameters different from those of selfoscillations of the healthy-blood cell differentiation and, hence, insults in the energy take-off by the tumor cycle.

The experimentally observed synchronism in the oscillations of
segmented-neutrophil and lymphocyte absolute concentrations
throughout the entire volume of the (*in vitro*) blood under
study is a phenomenon similar to the formation of Benard
cells, synchronous contractions of the heart muscle, Belousov-
Zhabotinsky reaction, etc. At the same time, the synchronization
mechanism for blood cell concentration oscillations (*in vitro*)
requires the assumption of the existence of a medium with
properties inherent in a continuous dynamical system. The role of such a medium can be played by the strong blood plasma
electrolyte treated as dense physical plasma.

The previously developed mathematical model of oscillations in
the concentrations of segmented neutrophils, lymphocytes and
tumor cells in the (*in vitro*) blood is used as a basis for the model
of oscillations of the aforementioned concentrations (*in vivo*).

Mathematical model of daily and hourly oscillations in
the segmented neutrophil, lymphocyte and leukemic cell
concentrations (*in vivo*). The experimental data obtained and
the results of mathematical simulation of (*in vitro*) oscillations of
segmented neutrophils, lymphocytes, and leukemic cells allowed
developing a mathematical model to describe the daily and
hourly oscillations in the concentrations of these cells (*in vivo*).
The basic system of equations has the form:

(5)

Where N is segmented-neutrophil concentration in the bone
marrow and L is lymphocyte concentration in the lymph nodes
(both normalized to the statistically averaged concentrations for
a given age group). C_{N} and C_{L} are the departures of respectively
the segmented-neutrophil and lymphocyte concentrations
from their averaged values in the peripheral blood. The righthand
sides of Eqs (5) describe the external action of the daily
and monthly variations in the cosmic ray intensity and of earth
magnetic field and being initial amplitudes of
these agents. , are the phase shifts equal to 4 and 2 hours, are the phase
shifts equal to 4 and 2 days, and are the
coupling coefficients.

(6)

These two equations reflect the lagging influence of the
dampers of neutrophil and lymphocyte concentration oscillation
amplitudes due to the damping effect of the bone marrow (for
C_{L} ) and the lymph nodes and spleen (for C_{L} ). The delay time T_{N} is

(7)

Where c_{N} is the bone marrow volume, c_{N} is the specific rate of
neutrophil absorption from the peripheral blood by the bone
marrow, p_{N} is the surface area of the bone marrow, p_{N} is the
neutrophil percentage in the bone marrow and K_{N} is the specific
rate of the neutrophil outflow from the bone marrow to the
peripheral blood proportional to the rate of the bone marrow
myeloblast differentiation to the neutrophil stage, and

(8)

where c_{L} is the spleen and lymph node volume, c_{L} is the specific rate of the lymphocyte absorption from the peripheral blood by
the spleen and lymph nodes, S_{L} is the lymphocyte percentage in
the spleen and lymph nodes, S_{L} is the surface area of the spleen
and lymph nodes, and K_{L} is the specific rate of the lymphocyte
outflow to the peripheral blood is proportional to the rate of the
lymphoblast differentiation in the spleen and lymph nodes to the
lymphocyte stage.

(9)

Here, P is the myeloblast concentration in the spleen and lymph
nodes and R is the lymphocyte concentration in the bone marrow.
Two Equations (9) Describe the mutual synchronization of
oscillations in the c_{L}and c_{L}concentrations due to the neutrophil
differentiation from the myeloblasts in the spleen and lymph
nodes and the lymphocyte differentiation from the lymphoblasts
in the bone marrow. Then

(10)

where c_{b} is the specific rate of lymphocyte absorption from
the peripheral blood by the bone marrow, K_{b} is the lymphocyte
percentage in the bone marrow and K_{b} is the specific rate of
the lymphocyte outflow to the peripheral blood from the bone
marrow proportional to the rate of the bone marrow lymphoblast
differentiation to the lymphocyte stage. The delay time n T (equal
to the time of this differentiation) is

(11)

Where p_{S} is the specific rate neutrophil absorption from the
peripheral blood by the spleen and lymph nodes p_{S} is the
neutrophil percentage in the spleen and lymph nodes and K_{S} is
the specific rate of the neutrophil outflow to the peripheral blood
proportional to the rate of the lymphoblast differentiation in the
spleen and lymph nodes to the neutrophil stage.

Depending on the delay times T_{N} ,T_{L},T_{x},T_{y} the systems
described by the Van der Pol equations with a time lag were
shown to have different oscillation modes [9]. So, synchronous
stable oscillations with period T_{0} were observed for oscillations switch abruptly
from the in-phase to the antiphase mode. The in-phase mode of
daily oscillations in the segmented-neutrophil and lymphocyte
concentrations is observed (as shown in **Figures 1A** and **1B** in Ref.
[6]) in healthy humans and animals (mice), while the antiphase
one is observed in patients with infectious and inflammatory
diseases (**Figure 3** in Reference [6]).

Computer analysis of the joint solution of system (5, 6, 9)
and system (4) of Reference [6] showed both qualitative and
quantitative agreement of the mathematical model with the
experimental data for healthy donors (**Figure 2**) and patients
with infectious diseases (**Figure 3**) and the development of
acute leukemia (**Figure 4**). In addition, we studied the model solutions for an impulse action on the system (which models e.g.,
vaccination and the effects of various chemical and other stressor
agents on the blood). If the blood was affected at the time instant
corresponding to the inflection point of the function describing
the concentration variation for the lymphocyte-lymphoblast
differentiation cycle, we found a high probability of leukemia
development (that is, an autoparametric excitation of this
normally very small cycle, and a predominant redifferentiation
of buffer organ cells to lymphoblasts). Sometimes this is actually
observed upon vaccinations and chemical poisoning. In order to
simulate this kind of effects, an impulse action (shown by the
arrow in **Figure 1B**) was exerted on the model within the time
interval in which the regular antiphase oscillations develop in the
model solutions. As is evident from the plots obtained (**Figure 1B**), the oscillation character changes drastically upon an impulse
action. In particular, a violation of the phase and amplitude
stability is observed. In addition, both the experimental system
and its mathematical model possess "memory" with respect to
external influences, which is also a characteristic feature of open
thermodynamic systems.

These studies of the model also showed that the introductions of additional buffers (e.g., segmented neutrophils from a donor) into the blood are among simple effective methods to affect a leukemic process so that the leukemic cells redifferentiate to lymphocytes. The introduction results in a rapid decrease in the amplitude of the leukemic cycle of the differentiation. This method was clinically tested and proved to be highly effective.

Now consider the reasons for the change in the blood cell
differentiation cycle parameters. We believe that the delay times
T_{N} ,T_{L},T_{x},T_{y} are genetically determined and affected by various
changes in a DNA molecule. These changes, as a consequence,
violate the amplitudes and phases of daily oscillations in the
concentrations of these types of leukocytes. In fact, the reasons
for the change in the differentiation cycle parameters lie at a micro
level; namely, in the changes of the parameters of the vibrational dynamics of a DNA molecule. The latter changes can be produced,
e.g., by an incorporation of viruses that affect the molecular
structure. Moreover, owing to a number of genetic differences
in the inbred mice of the AKM strain, the differentiation delay
periods can change at a level of the characteristic differences
between the DNA molecule of these mice and e.g., the statistically
averaged DNA molecule of mongrel mice. It is these differences
that lead to inborn changes in the experimentally observed (see **Figure 2** in [6]) daily oscillations in the segmented-neutrophil and
lymphocyte concentrations.

Thus, on the basis of mathematical simulation of the
hematopoietic system both (*in vitro*) and (*in vivo*) and comparison
between the results of simulation and experimental data, one
may conclude that the blood system is an open thermodynamic
system with a number of attractor cycles of the differentiation
with genetically determined cycle parameters. In view of
various genetic disturbances and inborn differences in the DNA
molecule structure, the cycle parameters can change either
spontaneously or under external factors. The presence of two
small (lymphoblastic and myeloblastic) cycles in such an open
thermodynamic system, the blood, is always observed in healthy
humans. As it was shown by the computer study of the model
an autoparametric excitation of one of these cycles and the
development of lymphoblastic and myeloblastic leukemias as well
as the relapse of the disease (for the possible aforementioned reasons) represent a typical physical scenario of quick energy
transfer between the cycles.

Numerical analysis of the model was followed by clinical studies aimed at a redistribution of the cell cycle energy with the aid of segmented-neutrophil suspension and traditional cytostatics. A preliminary scenario of the treatment was first worked up using computer model in order to determine the optimum point in time to affect the blood system. Despite an obvious complexity of this method, (it needs up to 8 blood samplings a day), the leukemic process disappeared completely in some cases. The utmost difficulty lies in the «lucky» coincidence between the model dynamics and the dynamics of the oscillation processes in the blood of a particular patient with acute leukemia. Furthermore, the symptoms of allergic diseases (manifested over a long period of time) were found to disappear upon medical treatment with intravenous neutrophil suspension injection.

Based on the results of clinical studies and mathematical simulations, we can formulate the following conclusions:

- in treatment of acute leukemic patients, the chemotherapy courses should be administrated at a certain time of day because the temporal variation in the cell sensitivity to cytostatics in an organism is of oscillatory character (similar to the blood cell concentration oscillations). Specifically, in continuous drop infusion of chemotherapy drugs during a day, their concentration should normally approach its minimum and maximum values within the time intervals from 6 a.m. to 8 a.m. and from 6 p.m. to 8 p.m., respectively. The maximum and minimum drug concentrations should differ by not less than an order of magnitude. Similar recommendations hold true for onetime infusions.

- it is recommended that traditional of vaccinations children be made within the period from 2 p.m. to 6 p. m. and not less than one month before or after the birthday of a child.

- Contact with strong chemicals should be avoided from 2 a.m. to 12 a.m.

- other pathological processes such as autoimmune diseases (bronchial asthma), some kinds of tumors, as well as the dynamics of a number of viral diseases can be described similarly to the above model. The approach given above was elaborated for improving the treatment policy whereas general theoretical model was carried out for general description of leukemia origin [5]. In blessed memory of Lenochka Berezina, who died at 20 of leukemia.

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- Berezin AA, Pasta F (2017) Ulam recurrence and coupled discrete ginzburg-landau chains in the description of various
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