CN110428905B - Tumor growth trend prediction method - Google Patents

Tumor growth trend prediction method Download PDF

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CN110428905B
CN110428905B CN201910587445.7A CN201910587445A CN110428905B CN 110428905 B CN110428905 B CN 110428905B CN 201910587445 A CN201910587445 A CN 201910587445A CN 110428905 B CN110428905 B CN 110428905B
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tumor
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CN110428905A (en
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龚芳
蒋飞达
胡慧敏
李刚
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Affiliated Hospital of Jiangnan University
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Abstract

The invention provides a tumor growth trend prediction method, which can couple the tumor volume and the cancer cell density together through a velocity field u, can give the change process of the tumor volume, the radius and the cancer cell density along with time, and has better applicability. Which comprises the following steps: s1: acquiring the densities of T cells and cancer cells at the same time point in a subject to be predicted by the prior art; s2: the method comprises the following steps of obtaining data such as the killing rate of the T cells to the cancer cells and the density C of the cancer cells by the prior art: s3: substituting the acquired parameters into the mathematical model of the invention; s4: performing numerical simulation on the formula in the step S3, and predicting the trend of the density C of the cancer cells in the tumor according to the numerical simulation result; s5: the change in tumor radius R is calculated according to the formula of the invention for tumor radius R: s6: combining the density C of cancer cells in step S3 and the tumor radius R in step S5, the trend of change of the tumor according to the change of time t is derived.

Description

Tumor growth trend prediction method
Technical Field
The invention relates to the field of biotechnology, in particular to a tumor growth trend prediction method.
Background
Cancer is a major public health problem in the world and may occur in various age groups. In the course of cancer treatment, prediction of the growth trend of tumors is helpful to confirm whether the treatment method used is effective or whether the therapeutic drug used is effective. However, the growth of cancer cells is a highly complex process, and predicting the growth trend of tumors is a subject of the research process.
In the prior art, common methods for predicting the growth of tumors include: its growth is modeled by tumor volume (or radius), or by the density of cancer cells in the tumor, such as: a Logistic model of a growth model established by tumor volume (or radius) is taken as an example, and V is assumed to be volume, r is growth rate, and VmIs the maximum tumor volume, which is modeled as:
Figure BDA0002114925520000011
a growth model established by cancer cell density in tumors, assuming σ is cancer cell density, the model is:
Figure BDA0002114925520000012
the models of the two modes have a single prediction mode, and only the increase of one of the volume and the density is considered. In fact, changes in tumor volume and density should be considered simultaneously in the growth of the tumor. And due to the complexity of tumor growth, factors such as the convection diffusion effect of tumor cells, apoptosis, killing of immunized T cells, and the like, should also be considered. Therefore, the results predicted by the existing models have low applicability in actual work.
T cells are a type of lymphocyte (leukocyte subset) that plays an important role in cell-mediated immunity, and T helper cells, known as CD4+ T cells, can assist in the activation of other leukocytes, including cytotoxic T cells and macrophages, during the course of immunity. Cytotoxic T cells, called CD8+ T cells, can destroy virus-infected cells and cancer cells C and are also associated with transplant rejection, and there have been many studies on the extraction, measurement, analysis of T cells and cancer cells, with specific reference to: the published patent document CN200780014169.0T cellular assay, CN201810454480.7, rapidly detects T cell activation by RNA measurement using flow cytometry. During the research process, we found that the change of tumor volume is determined by the proliferation and death of cancer cells C, and the killing rate of cancer cells by T cells related to the proliferation and death of cancer cells, the growth rate of logic of cancer cells, the bearing capacity of environment and other data can be measured by the prior art, and the current common methods are as follows: stable isotope mass spectrometry, the specific measurement procedure is described in patent documents already published: CN201610362514.0 a method for quantifying carnitine compound holo-isotope internal standard mass spectrum, CN201510778281.8 a method for identifying olive oil production places based on isotope mass spectrum technology; in the case of radial symmetry, the tumor volume varies with the tumor radius r (t); therefore, the growth trend of the tumor can be predicted through the relationship among the T cells, the cancer cells and the normal healthy cells.
Disclosure of Invention
In order to solve the problem of low applicability caused by the fact that the change state of a tumor is not considered in the existing prediction method, the invention provides a tumor growth trend prediction method which can provide the change process of the tumor volume, the radius and the cancer cell density along with time and has better applicability. The technical scheme of the invention is as follows: a method for predicting tumor growth trend, comprising the steps of:
s1: obtaining a sample to be detected of cancer cells in a subject to be predicted;
the method is characterized in that:
the sample to be detected obtained in step S1 includes a sample to be detected including T cells at the same time point as cancer cells;
s2: the following data were obtained by performing analysis in conjunction with cancer cell, T cell samples:
eta: is the killing rate and normal number of the T cells to the cancer cells,
γ0: is the threshold value and normal number of the killing rate of the T cells to the cancer cells,
λC: is the logistic growth rate and normal number of cancer cells,
k: is the load bearing capacity of the environment;
s3: substituting the obtained parameters into the following mathematical model:
Figure BDA0002114925520000021
in the formula:
c is the density of the cancer cells,
t represents the time of day and t represents the time of day,
eta is the killing rate of the T cells to the cancer cells, normal number,
γ0is the threshold value of the killing rate of the T cells to the cancer cells, normal number,
Figure BDA0002114925520000022
xi is a normal number;
s4: performing numerical simulation on the formula in the step S3, and predicting the trend of the density C of the cancer cells in the tumor according to the numerical simulation result;
s5: the change in tumor radius R is calculated according to the following formula:
Figure BDA0002114925520000023
wherein u is the moving speed of the cell;
s6: combining the density C of cancer cells in step S3 and the tumor radius R in step S5, the trend of change of the tumor according to the change of time t is derived.
It is further characterized in that:
in step S3, γ0The expression of (a) is:
Figure BDA0002114925520000024
λC: the logistic growth rate, normal number of cancer cells,
k: is the load bearing capacity of the environment;
in step S3, λC、η、δCAnd gamma0Satisfies the equation: 2 (lambda)C-δC)＝η+γ0
In the formula: deltaCIs apoptosis or death or necrosis rate of cancer cells,
λC: cancer cell logistic growth rate, normal number
Eta: is the killing rate and normal number of the T cells to the cancer cells,
γ0: is the threshold value and normal number of the killing rate of the T cells to the cancer cells;
in step S5, the expression for u (r (t), t) is as follows:
Figure BDA0002114925520000031
in the formula:
s corresponds to r (0< r < R (t)),
(c) is the proliferative function of cancer cells, i.e. the net increase per unit volume of cancer cells per unit time, expressed by the formula:
Figure BDA0002114925520000032
in step S6, the trend of change of the tumor includes the following cases:
C0(r) represents the density of C at the initial moment,
Figure BDA0002114925520000033
when, there is the following relation:
(1-1)
Figure BDA0002114925520000034
when the temperature of the water is higher than the set temperature,
Figure BDA0002114925520000035
(1-2)0<γ0-η<when the number of the holes is 4, the number of the holes is four,
Figure BDA0002114925520000036
R↓R∞；
(1-3)-2<γ0-η<at the time of 0, the number of the first,
Figure BDA0002114925520000037
R↓R∞；
(1-4)-4<γ0-η<-2、0<C0<when the direction of the light beam is xi,
Figure BDA0002114925520000038
R↓R∞；
(1-5)-4<γ0-η<-2、
Figure BDA0002114925520000039
c → 0, R ↓ 0;
(1-6)γ0-η<at-4, C → 0, R ↓ 0;
Figure BDA00021149255200000310
when, there is the following relation:
(2-1)γ0-η>when the number of the holes is 4, the number of the holes is four,
Figure BDA00021149255200000311
R↓R∞；
(2-2)0<γ0-η<at the time of 2, the reaction kettle is opened,
Figure BDA00021149255200000312
R↑R∞；
(2-3)-4<γ0-η<at the time of 0, the number of the first,
Figure BDA0002114925520000041
R↑R∞
(2-4)γ0-η<-4、
Figure BDA0002114925520000042
c → ξ, R ≠ + ∞;
(2-5)γ0-η<-4、ξ<C0<1, C → xi, R ≠ infinity
Wherein:
R∞is a constant greater than R (0),
R∞is a constant less than R (0).
The invention relates to a tumor growth trend prediction method, which simultaneously predicts the change condition of the density of cancer cells and the volume of tumor by coupling the volume (or radius) of tumor and the density of cancer cells; because the change in tumor volume is determined by the proliferation and death of cancer cells, and in the case of radial symmetry, the tumor volume changes with the change in radius r (t), the growth tendency of cancer cells is marked by their density C; the size change trend of the tumor itself is determined by the tumor radius R. Compared with the prior art, the technical scheme also considers the convective diffusion effect, apoptosis, competition condition and free boundary change of the cancer cells on the aspect of logistic growth, so that a cancer cell model is more comprehensive than before, the relation between the asymptotic property and radius of the cancer cells and time is given according to the multiplication characteristics of the cancer cells and the application of a derivative, the comparison principle of a differential equation and the free boundary condition, the change process of the cancer cells along with the time is analyzed based on the convective diffusion model, and the obtained prediction result is more applicable.
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FIG. 1 shows a view of the embodiment C of the present invention0(r)∈(0，1/2)、γ0-η>A tumor prediction trend plot at 0;
FIG. 2 shows a view of a view point of a liquid crystal display device C according to an embodiment of the present invention0(r)∈(0，1/2)、γ0-η<-a tumor predictive trend map at 4;
FIG. 3 shows a block diagram of a block diagram C according to an embodiment of the present invention0(r)∈(0，1/2)、-4<γ0-η<-a tumor predictive trend map at time 2;
FIG. 4 shows a view of a cross section C in an embodiment of the present invention0(r)∈(1/2，1)、γ0-η>A tumor prediction trend plot at 0;
FIG. 5 shows a view of a view point of C in an embodiment of the present invention0(r)∈(1/2，1)、-4<γ0-η<-a tumor predictive trend map at time 2;
FIG. 6 shows a view of a cross section C in an embodiment of the present invention0(r)∈(1/2，1)、γ0-η<-a tumor predictive trend plot at 4.
Detailed Description
As shown in fig. 1 to 4, the method for predicting tumor growth trend of the present invention predicts tumor production trend based on the convection diffusion equation.
To model cancer expressed by cancer cells C and normal healthy cells N, T T, C (x, T), N (x, T), and T (x, T) were used to represent points (x, T) ∈ R3Density of cancer cells C and normal healthy cells N, T cells T at X [0, + ∞ ], assuming all spatial intervalsThe amount is radially symmetric and the tumor is spherical r ═ x uti<R (t), wherein the radius R (t) varies over time. The tumor boundary r ═ r (t) is a free boundary. The density of cells is assumed to be uniform throughout the tumor, i.e.:
C(r,t)+N(r,t)+T(r,t)＝θ， (1.1)
wherein: r is more than or equal to 0 and less than or equal to R (t), t is more than 0
θ is a constant, and C (r, T), N (r, T), and T (r, T) represent the densities of cancer cells, normal healthy cells, and T cells at point (r, T), respectively.
Since tumor cells grow abnormally fast, equation (1.1) implies that there is an internal pressure between the cells, thereby creating a velocity field
Figure BDA0002114925520000057
Suppose that:
Figure BDA0002114925520000051
wherein
Figure BDA0002114925520000052
Is the unit radial vector.
In formula (1.1), T is constant, and θ -T is 1; assuming that N is 0 and θ is 1, equation (1.1) is transformed into:
C(r,t)+T(r,t)＝1 (1.3)
wherein r is not less than 0 and not more than R (t), and t is greater than 0.
Cancer cells are assumed to undergo logistic growth "constant × C (1-C/K)" and to be killed by T cells at a rate "constant × TC", where K is the environment-bearing capacity, and cancer cells are assumed to die with apoptosis or necrosis at a rate δC. Then, the density of cancer cells C satisfies the following convection-diffusion equation:
Figure BDA0002114925520000053
u is the velocity, K is the bearing capacity of the environment, λCIs the logistic growth rate, normal number of cancer cellsEta is the killing rate and normal number of the T cells to the cancer cells, deltaCCancer cells die at the rate of apoptosis or necrosis.
Cancer cells are recognized by dendritic cells, which can activate T cells by introducing an increase term λ in the T cell's convection diffusion equationTC to indicate such activation:
Figure BDA0002114925520000054
by scaling, take λ in equation (1.5)T＝δT＝1。
In the tumor region, the cancer cells C, T cells T ideally eventually reach equilibrium, i.e., stabilize. While generally finding a stable point, let the right-end term of the equation be 0, and then the formula (1.3) has:
Figure BDA0002114925520000055
the stable point is obtained by calculating the stable point,
Figure BDA0002114925520000056
let the right-end term of equation (1.4) be 0, the stable point is substituted to obtain:
Figure BDA0002114925520000061
simplifying to obtain:
Figure BDA0002114925520000062
get
Figure BDA0002114925520000063
Thus obtaining: 2 (lambda)C-δC)＝η+γ0 (1.7)
Under the assumed conditions (1.6), (1.7) it is readily seen that C.ident.T.ident. 1/2 is a solution of the equations (1.4), (1.5), since the values to the right of (1.4) and (1.5) are both 0 when C.ident.T.ident. 1/2. By the formula (1.7), the nonlinear proliferation function of cancer cells and T cells was obtained.
Applying a no-flux boundary condition:
Figure BDA0002114925520000064
equation (1.8) is the homogeneous Neumann boundary condition r ═ r (t) of C, T on the free boundary, assuming that the free boundary r ═ r (t) moves with cell velocity u, i.e.:
Figure BDA0002114925520000065
i.e. the formula for the change in tumor volume.
Changes in tumor volume are determined by the proliferation and death of cancer cells. In the case of radial symmetry, the volume varies with the variation of the radius r (t). This is obtained by equation (1.4) and equation (1.5):
Figure BDA0002114925520000066
as can be seen from equation (1.3), r ≦ 0 ≦ R (T), T >0, and C + T ≡ 1, then equation (2.1) changes:
Figure BDA0002114925520000067
wherein
Figure BDA0002114925520000068
This can be obtained by equation (2.2):
Figure BDA0002114925520000069
as can be seen from equation (1.9) and equation (2.4):
Figure BDA00021149255200000610
i.e. to obtain a formula for the change in tumor radius R.
The growth rate of cancer cells depends on the number of cells contained in them. The function f (c) in equation (2.3) is referred to as the proliferative function of cancer cells, i.e. the net increase per unit volume of cancer cells per unit time. Then, combining equation (1.7) and equation (2.3), we can see:
Figure BDA0002114925520000071
if η ≠ γ0The function f (c) is non-linear in equation (2.6). Otherwise, f (C) is a linear function in equation (2.6).
From equation (2.4) and equation (2.6), we immediately obtained u ≡ 0 when C ≡ 1/2 holds true. That is, the free boundary r ≡ r (T) remains unchanged for this particular solution C ≡ T ≡ 1/2.
The equation for C is derived by calculating the advection term from equation (1.4), namely:
Figure BDA0002114925520000072
substituting equation (2.2) into equation (2.7) can result in:
Figure BDA0002114925520000073
wherein:
Figure BDA0002114925520000074
wherein:
Figure BDA0002114925520000075
let η ≠ γ0ξ is made to hold.
Assuming C and T are positive numbers:
c (r, T), T (r, T) epsilon (0,1), and for r ≦ 0 ≦ R (T), T > 0.
The change of T cells predicts the formula T (r, T), and the advection term of the formula (1.5) is calculated by the formula (2.2) and the formula (1.3), namely:
Figure BDA0002114925520000076
wherein:
Figure BDA0002114925520000077
indicating the proliferative function of T cells.
Rewrite equation (1.5) by equation (2.10) as follows:
Figure BDA0002114925520000081
wherein:
Figure BDA0002114925520000082
then:
Figure BDA0002114925520000083
let η ≠ γ0So that
Figure BDA0002114925520000084
This is true.
Experimental data for tumor growth were well fitted by solution of the ODE model. Then, the formula (2.8) and the formula (2.11) are simplified by the boundary condition formula (1.8) and substituted into the ordinary differential equation.
The no-flux boundary condition (1.8) on r ═ r (T) is compared with the solutions of C (r, T) and T (r, T) in the equations (2.8 and (2.11), respectively, and the solution of the ordinary differential equation
Figure BDA0002114925520000085
Figure BDA0002114925520000086
And
Figure BDA0002114925520000087
suppose that:
a<C(r,0)<b，0≤r≤R(0) (2.14)
for constants a and b, the principle of comparison using the parabolic equation infers if:
a<C10≤C(r,0)≤C20<b
for R ≦ 0 ≦ R (0)
Figure BDA0002114925520000088
Is the initial value of the first equation of the formula (2.13)
Figure BDA0002114925520000089
When j is 1,2, then:
Figure BDA00021149255200000810
wherein: r is more than or equal to 0 and less than or equal to R (t), t is more than or equal to 0 (2.15)
Similarly, assume a<T10≤T(r,0)≤T20<b，0≤r≤R(t)，t≥0，
Figure BDA00021149255200000811
The second equation of equation (2.13) has an initial value of
Figure BDA00021149255200000812
When j is 1,2, then:
Figure BDA00021149255200000813
wherein: r is more than or equal to 0 and less than or equal to R (t), and t is more than or equal to t0 (2.16)。
In summary, it can be found that:
the formula for the change in tumor volume is:
Figure BDA0002114925520000091
the formula for the change in cancer cells C is:
Figure BDA0002114925520000092
combining the density C of cancer cells and the tumor radius R, the trend of the change of the tumor according to the change of time t includes the following cases:
C0(r) represents the density of C at the initial moment, C0(r) ∈ (0,1/2), the following relationship exists:
(1-1)
Figure BDA0002114925520000093
when the temperature of the water is higher than the set temperature,
Figure BDA0002114925520000094
R↑R∞；
(1-2)0<γ0-η<when the number of the holes is 4, the number of the holes is four,
Figure BDA0002114925520000095
R↓R∞；
(1-3)-2<γ0-η<at the time of 0, the number of the first,
Figure BDA0002114925520000096
R↓R∞；
(1-4)-4<γ0-η<-2、0<C0<when the direction of the light beam is xi,
Figure BDA0002114925520000097
R↓R∞；
(1-5)-4<γ0-η<-2、
Figure BDA0002114925520000098
c → 0, R ↓ 0;
(1-6)γ0-η<at-4, C → 0, R → 0;
C0(r) ∈ (1/2,1), the following relationship is present:
(2-1)γ0-η>when the number of the holes is 4, the number of the holes is four,
Figure BDA0002114925520000099
R↓R∞；
(2-2)0<γ0-η<at the time of 2, the reaction kettle is opened,
Figure BDA00021149255200000910
R↑R∞；
(2-3)-4<γ0-η<at the time of 0, the number of the first,
Figure BDA00021149255200000911
R↑R∞
(2-4)γ0-η<-4、
Figure BDA00021149255200000912
c → ξ, R ≠ + ∞;
(2-5)γ0-η<-4、ξ<C0<1, C → xi, R ≠ infinity
Wherein:
R∞is a constant greater than R (0),
R∞is a constant less than R (0).
In the present embodiment, the invasiveness of the growth of the cancer cells is measured by the values of C and R. In the early stages of the tumor, namely: c0(r) e (0,1/2), tumor volume will increase or decrease if T cells kill cancer cells at a low rate; if the kill rate eta is greater than gamma0The tumor volume will continue to decrease. Furthermore, if the killing rate η is large enough, the T cells can kill cancer cells very effectively, even successfully. In the late stages of the tumor, i.e.: c0(r) epsilon (1/2,1), if T cells kill cancer cells at a low rate, the tumor volume will continue to increase or decrease, and while T cells kill cancer cells at a higher rate, the tumorThe volume will increase. In fact, in the advanced stages of the tumor, cancer cells dominate and can induce T cells to be assimilated. The interferon secreted by cancer cells can inhibit the function of T cells and promote the growth of cancer cells. Once T cells are inhibited by cancer cells, they no longer function.
According to the technical scheme, firstly, the proliferation of cells is measured by using a stable isotope mass spectrometry method, and the proliferation rate of the cells is obtained by taking C or T cells as an example, so that the estimation of other parameters is obtained; the following specific data were obtained:
λC＝1.86/day,δC＝0.18/day，K＝0.75g/cm3，γ0＝λC/K＝2.48cm3/g·day；
then, the formula (1.7)2 (lambda)C-δC)＝η+γ0Eta is 0.88cm3/g·day
γ0-η＝2.48-0.88＝1.6cm3/g·day
Because: gamma ray0-η∈(0,2)
When the temperature of the molten metal is higher than the set temperature,
Figure BDA0002114925520000101
then:
Figure BDA0002114925520000102
R(t)↓R∞，t→+∞；
when the temperature of the molten metal is higher than the set temperature,
Figure BDA0002114925520000103
then:
Figure BDA0002114925520000104
R(t)↑R∞，t→+∞。
referring to fig. 1 to 6 of the drawings, the abscissa of fig. 1 to 6 represents time t, and the ordinate represents the density C of cancer cells.
FIG. 1 to FIG. 3 are C0In the case of (r) ∈ (0,1/2), γ0-eta growth prediction of tumors at different ranges. FIG. 1 is γ0-η>In the case of 0, the curve in the graph shows an increasing trend, indicating that at C0(r) ∈ (0,1/2), the density of cancer cells will increase with time, but finally it will tend to a stable state, and the density of cancer cells will not increase without limit; FIG. 2 is γ0-η<-4, the curve in the graph shows a decreasing trend, indicating a decrease at C0(r) ∈ (0,1/2), the density of cancer cells decreases with time, because the initial density of cancer cells is still small, the killing rate of cancer cells by T cells increases, and finally the density of cancer cells approaches 0, i.e. the tumor is cured; FIG. 3 is-4<γ0-η<-2, in which case two curves appear in the graph: one increment and one decrement. When 0 is present<C0<ξ, the density of the cancer cell increases with time, but its radius shrinks; conversely, when xi<C0<1/2, the density of cancer cells decreases with time and finally becomes 0.
FIG. 4 to FIG. 6 are C0In the case of (r) ∈ (1/2,1), γ0-eta growth prediction of tumors at different ranges. FIG. 4 is γ0-η>0, the curve in the graph shows a decreasing trend, indicating that at C0(r) epsilon (1/2,1), the density of the cancer cells is reduced along with the increase of time, namely the spread of the cancer cells is controlled, but the radius of the cancer cells is increased, and finally, the density of the cancer cells tends to be in a stable state and does not shrink without limit; FIG. 5 is-4<γ0-η<The case of-2, similar to the meaning of fig. 4, but the change in cancer cell density is faster in fig. 5 than in fig. 2 because of the increase in killing rate; FIG. 6 is γ0-η<-4, in which case two curves appear in the graph: one increment and one decrement. When 0 is present<C0<ξ, the density of cancer cells increases with time; conversely, when xi<C0<1/2, the density of cancer cells decreases with time, but finally only the curve of cancer cell growth after T cell failure.
Two existing predictive models described in the background, one for the growth of the tumor volume (or radius) and one for the density of cancer cells in the tumor, consider only a certain amount of increase in volume and density, respectively, and if these two cases are combined, constitute the following models:
Figure BDA0002114925520000111
this combined model also cannot account for both the change in cancer cell density and tumor volume, but only by fixing the conditions on one side to obtain another variable, e.g., accounting for the change in volume assuming the density of cancer cells C is unchanged; or to account for changes in cancer cell density, assuming that the volume V is constant (fixed boundary).
The unknown function of the model in the technical scheme of the invention is the density of T cells and C cells, and the change of the regional boundary shows the change of the tumor size:
Figure BDA0002114925520000112
the tumor model of the present invention is a coupling of the two, with the tumor volume and the cancer cell density coupled together by the velocity field u. For example, when u is 0, the model herein is a model of cancer cell density, which corresponds to a particular case where the former model is a tumor model in the present invention. The tumor model of the present invention can simultaneously account for changes in tumor volume and cancer cell density.

Claims (5)

1. A method for predicting tumor growth trend, comprising the steps of:
s1: obtaining a sample to be detected of cancer cells in a subject to be predicted;
the method is characterized in that:
the sample to be detected obtained in step S1 includes a sample to be detected including T cells at the same time point as cancer cells;
s2: the following data were obtained by performing analysis in conjunction with cancer cell, T cell samples:
eta: is the killing rate and normal number of the T cells to the cancer cells,
γ0: is the threshold value and normal number of the killing rate of the T cells to the cancer cells,
c: is the density of the cancer cells and is,
λC: is the logistic growth rate and normal number of cancer cells,
k: is the load bearing capacity of the environment;
s3: substituting the obtained parameters into the following mathematical model:
Figure FDA0003437899000000011
in the formula:
c is the density of the cancer cells,
t represents the time of day and t represents the time of day,
eta is the killing rate of the T cells to the cancer cells, normal number,
γ0is the threshold value of the killing rate of the T cells to the cancer cells, normal number,
Figure FDA0003437899000000012
xi is a normal number;
s4: performing numerical simulation on the formula in the step S3, and predicting the trend of the density C of the cancer cells in the tumor according to the numerical simulation result;
s5: the change in tumor radius R is calculated according to the following formula:
Figure FDA0003437899000000013
wherein u is the moving speed of the cell;
s6: combining the density C of cancer cells in step S3 and the tumor radius R in step S5, the trend of change of the tumor according to the change of time t is derived.
2. A method according to claim 1, wherein the method comprises the steps of: in step S3, γ0The expression of (a) is:
Figure FDA0003437899000000014
λC: the logistic growth rate, normal number of cancer cells,
k: is the load bearing capacity of the environment.
3. A method according to claim 2, wherein the method comprises the steps of: in step S3, λC、η、δCAnd gamma0Satisfies the equation: 2 (lambda)C-δC)＝η+γ0
In the formula: deltaCIs apoptosis or death or necrosis rate of cancer cells,
λC: the logistic growth rate, normal number of cancer cells,
eta: is the killing rate and normal number of the T cells to the cancer cells,
γ0: is the threshold value and normal number of the killing rate of the T cells to the cancer cells.
4. A method according to claim 3, wherein the method comprises the steps of: in step S5, the expression for u (r (t), t) is as follows:
Figure FDA0003437899000000021
in the formula:
s corresponds to r (0< r < R (t)),
(c) is the proliferative function of cancer cells, i.e. the net increase per unit volume of cancer cells per unit time, expressed by the formula:
Figure FDA0003437899000000022
5. the method of claim 4, wherein the method comprises: in step S6, the trend of change of the tumor includes the following cases:
C0(r) represents the density of C at the initial moment,
Figure FDA0003437899000000023
when, there is the following relation:
(1-1)
Figure FDA0003437899000000024
when the temperature of the water is higher than the set temperature,
Figure FDA0003437899000000025
R↑R∞；
(1-2)0<γ0-η<when the number of the holes is 4, the number of the holes is four,
Figure FDA0003437899000000026
R↓R∞；
(1-3)-2<γ0-η<at the time of 0, the number of the first,
Figure FDA0003437899000000027
R↓R∞；
(1-4)-4<γ0-η<-2、0<C0<when the direction of the light beam is xi,
Figure FDA0003437899000000028
R↓R∞；
(1-5)-4<γ0-η<-2、
Figure FDA0003437899000000029
c → 0, R ↓ 0;
(1-6)γ0-η<at-4, C → 0, R ↓ 0;
Figure FDA0003437899000000031
when, there is the following relation:
(2-1)γ0-η>when the number of the holes is 4, the number of the holes is four,
Figure FDA0003437899000000032
R↓R∞；
(2-2)0<γ0-η<at the time of 2, the reaction kettle is opened,
Figure FDA0003437899000000033
R↑R∞；
(2-3)-4<γ0-η<at the time of 0, the number of the first,
Figure FDA0003437899000000034
R↑R∞
(2-4)γ0-η<-4、
Figure FDA0003437899000000035
c → ξ, R ≠ + ∞;
(2-5)γ0-η<-4、ξ<C0<1, C → xi, R ≠ infinity
Wherein:
R∞is a constant greater than R (0),
R∞is a constant less than R (0).
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