A modeling and optimization method for combined photothermal-photodynamic therapy

Abstract

The present application relates to the technical field of cancer treatment research, and discloses a kind of modeling and optimization method of photothermal-photodynamic combined synergistic therapy, comprising the following specific steps: constructing tumor growth model with time delay;On the basis of tumor growth model, respectively introduce photothermal and photodynamic therapy intensity term, tumor killing term, construct tumor treatment model.The present application solves the problem that the prior art lacks effective analysis of photothermal therapy and photodynamic therapy synergistic therapy, and it is difficult to predict the effect of photothermal and photodynamic synergistic therapy, and has the characteristics of being able to assist in improving the effect of cancer treatment.

Description

Technical Field

[0001] The present invention relates to the technical field of cancer treatment research, and more specifically, to a modeling and optimization method for photothermal-photodynamic combined synergistic therapy. Background Art

[0002] Cancer is one of the most serious threats to human health and life. In recent years, cancer incidence has continued to rise, with cancer increasingly affecting younger people, making it the leading cause of death and disease. This is not only due to the high mortality rate of cancer, but also to the difficulty and cost of diagnosing and treating cancer. Traditional treatments (including radiotherapy, chemotherapy, and surgery) have significant side effects, such as decreased immunity and damage to organs and normal tissues. Research is underway to develop new and effective tools to combat cancer.

[0003] In areas such as skin disease treatment, there are two new treatment methods: photothermal therapy and photodynamic therapy. They are simple to use, low-cost, and capable of targeted, precise treatment with minimal side effects. Researchers at home and abroad have applied these innovations to the diagnosis and treatment of cancer, leading to the development of photothermal and photodynamic therapy for cancer cells. These two methods have great potential for treating superficial skin cancers such as melanoma, nasopharyngeal carcinoma, breast cancer, and esophageal cancer.

[0004] Photothermal therapy (PTT) is a strategy in which nanoprobes convert absorbed light energy into heat energy under near-infrared light to kill tumor cells; photodynamic therapy (PDT) is a method that uses photosensitive molecules as photoactivators to promote oxygen source conversion under appropriate light excitation, generating cytotoxic singlet reactive oxygen free radicals to kill tumor cells. Both methods are excited by external light sources, so the treatment location and intensity can be precisely controlled, and the side effects are much smaller than radiotherapy and chemotherapy. Combining the two modes of photothermal and photodynamic therapy and supplemented by resonant energy conversion and near-infrared upconversion technology can achieve indirect light excitation of photosensitizers to kill deep-seated tumors.

[0005] In practice, photothermal therapy involves injecting nanoprobes with high photothermal conversion efficiency into the human body. The probes' targeting function allows them to accumulate at the tumor site. Under near-infrared light, the light energy is converted into heat, ultimately killing cancer cells through high temperatures. Photodynamic therapy involves injecting photosensitizing drugs into the human body. The drugs' targeting function allows them to accumulate at the tumor site. Under near-infrared light, the drugs are activated, triggering a photochemical reaction that kills the tumor.

[0006] Since the treatment process of photothermal therapy is similar to that of photodynamic therapy, photothermal and photodynamic therapy are suitable for combined treatment: the photosensitizer used in photodynamic therapy can be used as a photothermal conversion agent for photothermal therapy, such as indocyanine green (ICG). Gold nanorods as a photothermal conversion agent also have an enhancing effect on photodynamic therapy. Photosensitive molecules can obtain energy from the strong electromagnetic field generated by the surface plasmon resonance effect of gold nanorods and also become excited states, thereby producing singlet oxygen.

[0007] When light is irradiated onto the gold nanorod, a surface plasmon resonance effect is generated [4], and free electrons will undergo collective oscillation and couple with electromagnetic waves to form a near-field electromagnetic wave propagating along the metal surface. If the frequency of the incident light wave is consistent with the frequency of the electron oscillation, resonance will occur. In the resonance state, the energy of the electromagnetic field is efficiently converted into the collective vibration energy of the free electrons on the metal surface. Therefore, the nanogold rod can effectively absorb light of a specific wavelength and convert it into heat.

[0008] The rod-shaped gold nanoparticles have different degrees of electron polarization in different directions, generating two surface plasmon resonance modes corresponding to the two characteristic sizes of the long axis and the short axis. The transverse electron resonance produces a transverse absorption band, which has a small dependence on the aspect ratio and is located near 520 nm, and the intensity is weak; the axial electron resonance produces a longitudinal absorption band, which has a large dependence on the aspect ratio, and the axial intensity is large, which dominates the optical properties of the nanorod. Therefore, by adjusting the aspect ratio, the axial resonance absorption peak of the nanogold rod can be adjusted in the range of 520-1400 nm. Therefore, the axial resonance absorption peak of the nanogold rod can be adjusted to the near-infrared region, and the deep tissue tumor cell photothermal treatment can be carried out under the irradiation of near-infrared laser.

[0009] The enhanced local electric field of the gold nanorod is mainly distributed at both ends of the gold nanorod. If the photosensitizer is attached to the two sides of the gold nanorod with mesoporous silica as the medium, a nanodumbbell structure is formed, and the adsorption of the photosensitizer can significantly improve the performance of the photosensitizer. The photothermal effect of the gold nanorod itself can achieve the effect of photothermal-photodynamic synergistic therapy.

[0010] However, the prior art lacks effective analysis of the photothermal therapy and photodynamic therapy synergistic therapy, and there are problems that it is difficult to predict the effect of photothermal and photodynamic synergistic therapy, and therefore how to invent a model that can predict the effect of photothermal-photodynamic combined therapy is a technical problem that needs to be solved in the technical field. SUMMARY

[0011] In order to solve the problem that the prior art is difficult to predict the effect of photothermal and photodynamic synergistic therapy, a modeling and optimization method for photothermal-photodynamic combined synergistic therapy is provided, which has the characteristics of being able to assist in improving the effect of cancer treatment.

[0012] To achieve the above-mentioned object of the present application, the technical solutions adopted are as follows:

[0013] The beneficial effects of the present application are as follows:

[0014] A modeling method for combined photothermal-photodynamic therapy, comprising the following specific steps:

[0015] Constructing a tumor growth model with time delay;

[0016] On the basis of the tumor growth model, introducing a photothermal and photodynamic treatment intensity term and a tumor killing term, respectively, to construct a tumor treatment model.

[0017] Preferably, the tumor growth model with time delay is constructed, specifically:

[0018] The tumor growth model with time delay is constructed based on either a Logistic growth model with time delay or an Allee model with time delay;

[0019] If the Logistic growth model with time delay is used, the tumor growth model with time delay constructed is specifically:

[0020]

[0021] If the Allee model with time delay is used, the tumor growth model with time delay constructed is specifically:

[0022]

[0023] Wherein, x(t) represents the real-time tumor cell density, r0 represents the natural growth rate of tumor cells, K represents the maximum environmental capacity, τ represents the time delay caused by the cell cycle of tumor cells, and A represents the Allee coefficient.

[0024] Further, on the basis of the tumor growth model, a photothermal and photodynamic treatment intensity term and a tumor killing term are introduced, respectively, to construct a tumor treatment model, specifically:

[0025] If the Logistic growth model with time delay is used, the tumor treatment model constructed is specifically:

[0026]

[0027] If the Allee growth model with time delay is used, the tumor treatment model constructed is specifically:

[0028]

[0029] Wherein, H(t) is the real-time photothermal treatment intensity, is the time-delayed photodynamic treatment intensity; μH the killing rate of tumor cells by photothermal therapy, μ B the killing rate of tumor cells by photodynamic therapy; the lag time of photodynamic therapy relative to photothermal therapy, i.e., the phase difference between the two therapies in terms of effect.

[0030] Further, the change rate of the actual photothermal therapy intensity is represented as:

[0031]

[0032] the change rate of the actual photodynamic therapy intensity is represented as:

[0033]

[0034] wherein Q is a heat source term, representing constant heat generation of the photothermal conversion agent as a heat source under constant irradiation of a laser light source, a is the photothermal effect heat generation rate of the photosensitizer, C(t) is the photosensitizer concentration, β C is the active oxygen yield of the photosensitizer without enhancement, and β is the active oxygen yield of the photosensitizer increased under surface plasmon resonance enhancement, λ H is the non-therapeutic loss rate of heat, λ B is the non-therapeutic loss rate of active oxygen, γ H is the loss rate of heat during the therapy process, γ B is the loss rate of active oxygen during the therapy process.

[0035] Further, the change of C(t) is specifically represented as:

[0036]

[0037] Further, the initial conditions of the tumor therapy model are:

[0038] x(0) = K, H(0) = 0, B(0) = 0, C(0) = C0;

[0039] The model uses core-shell composite structure nanoparticles formed by coating noble metal nanoparticles with organic molecular photosensitizers as the therapeutic medium, all photosensitizers are enhanced by metal surface plasmon resonance, the nanoparticles are uniformly distributed in the tumor tissue, the external laser light source is constantly and uniformly irradiated, and the tumors are uniformly distributed in space.

[0040] An optimization method for combined photothermal-photodynamic therapy, based on the modeling method, comprising the following specific steps:

[0041] Substitute the initial parameters into the model, and initialize the tumor therapy model;

[0042] The parameters of φ, β and α are modified, the tumor treatment model is solved respectively, the solving results are compared, and the optimal parameter combination is found.

[0043] Based on the optimal parameter combination, an optimal synergistic treatment scheme is designed.

[0044] Preferably, the parameters of φ, β and α are modified, the tumor treatment model is solved respectively, the solving results are compared, and the main parameters affecting the tumor treatment effect are found, and the specific steps are as follows:

[0045] The parameters of φ, β and α are modified, the tumor treatment model is solved respectively, and the change of the tumor cell number density under different φ, β and α parameters is analyzed.

[0046] The influence of the change of the parameters of φ, β and α on the ratio N / K of the surviving cell number density N to the maximum environmental capacity K is analyzed.

[0047] The change of the tumor cell number density under different φ, β and α parameters and the influence of the change of the parameters of φ, β and α on N / K are comprehensively analyzed, and the parameter combination with the best tumor cell killing effect is obtained.

[0048] Further, the influence of the parameters of φ, β and α on the photothermal-photodynamic synergistic effect is analyzed, and the specific steps are as follows:

[0049] The influence of the change of the parameters of φ, β and α on Q is analyzed.

[0050] The photosensitivity C0 and Q are set as coordinate axes, the value of N / K is fixed, the simulated equivalent effect line graph is obtained based on the tumor treatment model through multiple simulations, the additive equivalent effect line when the photothermal treatment and the photodynamic treatment are in an additive relationship is compared with the simulated equivalent effect line graph, and the synergistic effect value is calculated.

[0051] The influence of the change of the parameters of φ, β and α on the heat production parameter Q and the synergistic effect value is analyzed, the key parameters in the parameters of φ, β and α are found, and the photothermal-photodynamic combined synergistic treatment is regulated based on the key parameters.

[0052] Further, the synergistic effect value R=S 阴影 / S Δ , wherein S 阴影 is the area between the simulated equivalent effect line and the additive equivalent effect line, S Δ is the triangular area surrounded by the additive effect straight line and the horizontal and vertical coordinate axes, R is a proportional coefficient, reflects the synergistic effect of the two means in the existing treatment scheme, and the greater the value of R is, the better the synergistic effect of the photothermal treatment and the photodynamic treatment is.

[0053] The beneficial effects of the present application are as follows:

[0054] The present invention discloses a modeling method for photothermal-photodynamic combined synergistic therapy. By simulating the spatiotemporal evolution of tumors during the treatment process, a complete photothermal-photodynamic theoretical model is established. The model analysis provides guidance for the preparation of probes and the optimization of treatment plans. This solves the problem that the existing technology lacks effective analysis of the synergistic treatment of photothermal therapy and photodynamic therapy, and it is difficult to predict the effects of the synergistic treatment of photothermal and photodynamic therapy. The method has the characteristic of being able to assist in improving the effect of cancer treatment. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 It is a flow chart of a modeling method for photothermal-photodynamic combined synergistic therapy of the present invention.

[0056] Figure 2 It is a flow chart of an optimization method for photothermal-photodynamic combined synergistic therapy of the present invention.

[0057] Figure 3 Schematic diagram of the curve showing the change of cell number density with treatment time at different φ values ​​in Example 3.

[0058] Figure 4 3 is a schematic diagram of the curve of cell number density changing with treatment time under different β values ​​in Example 3.

[0059] Figure 5 3 is a schematic diagram of the curve of cell number density changing with treatment time at different α values ​​in Example 3.

[0060] Figure 6 This is a cross-sectional diagram of N / K with variables φ, β, and α in Example 3.

[0061] Figure 7 This is a schematic diagram of the influence curve of φ on N / K in Example 3.

[0062] Figure 8 This is a schematic diagram of the effect of β on N / K in Example 3.

[0063] Figure 9 Schematic diagram of the effect of α on N / K in Example 3.

[0064] Figure 10 This is the isotropic curve diagram of the heat generated by the gold nanorods Q and the initial concentration of the photosensitizer C0 when the N / K value is 0.5 in Example 3.

[0065] Figure 11 The synergistic effect R=S based on the isobologram in Example 3 阴影 / S Δ Schematic diagram.

[0066] Figure 12 It is a schematic diagram of the iso-effect curve corresponding to the φ value in Example 3.

[0067] Figure 13 is a schematic diagram of the equivalent effect curve corresponding to the value of β in Example 3.

[0068] Figure 14 is a schematic diagram of the equivalent effect curve corresponding to the value of α in Example 3.

[0069] Figure 15 is a schematic diagram of the curve of the influence of φ on the synergistic effect in Example 3.

[0070] Figure 16 is a schematic diagram of the curve of the influence of β on the synergistic effect in Example 3.

[0071] Figure 17 is a schematic diagram of the curve of the influence of α on the synergistic effect in Example 3.

[0072] Figure 18 is a graph of the numerical solution of the model of the change of x with time t when the time lag τ = 1, Q = 1.5, and C0= 10 in Example 4.

[0073] Figure 19 is a graph of the numerical solution of the model of the change of x with time t when τ = 10, Q = 3, and C0= 20 in Example 4.

[0074] Figure 20 is a graph of the numerical solution of the model of the change of H with time t when τ = 1, Q = 1.5, and C0= 10 in Example 4.

[0075] Figure 21 is a graph of the numerical solution of the model of the change of B with time t when τ = 1, Q = 1.5, and C0= 10 in Example 4.

[0076] Figure 22 is a graph of the equivalent effect line when the value of x min is 4.91 in Example 4.

[0077] Figure 23 is a graph of the change of the value of x min with the value of under different Q and C0in Example 4.

[0078] Figure 24 is a graph of the equivalent effect line corresponding to different values of min when x = 4.91 in Example 4.

[0079] Figure 25 is a graph of the equivalent effect line under different values of β when x min = 4.91 in Example 4.

[0080] Figure 26 is a graph of the equivalent effect line when x min= 4.91, Q = 1.6, beta-C0 curve. DETAILED DESCRIPTION

[0081] The application will be described in detail below with reference to the drawings and specific embodiments.

[0082] Example 1

[0083] As shown in the figure, a modeling method of combined photothermal and photodynamic therapy includes the following specific steps: Figure 1

[0084] A tumor growth model with time delay is constructed.

[0085] On the basis of the tumor growth model, a tumor treatment model is constructed by introducing a photothermal and photodynamic treatment intensity term and a tumor killing term.

[0086] In one specific embodiment, a tumor growth model with time delay is constructed, specifically: a Logistic growth model with time delay is used, and the tumor growth model with time delay constructed is specifically:

[0087]

[0088] wherein x(t) represents the real-time tumor cell density, r0 represents the natural growth rate of tumor cells, K represents the maximum environmental capacity, and τ represents the time delay caused by the cell cycle of tumor cells.

[0089] In this embodiment, the most commonly used model in population dynamics modeling is the classic Logistic growth model. This model takes into account that the living space and environmental resources are limited and cannot provide the required space and energy basis for an unlimited number of population individuals, i.e., each population has a corresponding maximum environmental carrying capacity in its living ecological system. If the population density is too high, the average resources occupied by the individuals of the population will decrease, and the high population density will also cause the environment to deteriorate, leading to an increase in diseases and other phenomena. As a result, the population birth rate decreases and the mortality rate increases, thereby reducing the population density.

[0090] In one specific embodiment, a Logistic growth model with time delay is used, specifically:

[0091]

[0092] wherein H(t) is the real-time photothermal treatment intensity, is the time-delayed photodynamic treatment intensity; μ H is the tumor cell killing rate of photothermal therapy, μ B is the tumor cell killing rate of photodynamic therapy; ​It is the lag time of the photodynamic therapy process relative to the photothermal therapy process, that is, the phase difference between the effects of the two treatments.

[0093] In a specific embodiment, the actual rate of change of photothermal therapy intensity is expressed as:

[0094]

[0095] The actual change rate of photodynamic therapy intensity is expressed as:

[0096]

[0097] Where Q is the heat source term, representing the constant heat generation of the photothermal converter as a heat source under constant laser light irradiation, α is the heat generation rate of the photothermal effect of the photosensitizer, C(t) is the photosensitizer concentration, β c is the ROS production rate of the photosensitizer without enhancement, β is the ROS production rate of the photosensitizer enhanced by surface plasmon resonance, and λ H is the non-therapeutic heat loss rate, λ B is the non-therapeutic loss rate of reactive oxygen species, γ H is the heat treatment process loss rate, γ B is the loss rate during the active oxygen treatment process.

[0098] In this embodiment, since the photothermal effect of the metal nanoparticles is directly related to the surface plasmon, βQ can actually be used to indirectly describe the increased active oxygen production rate of the photosensitizer under the surface plasmon enhancement.

[0099] In a specific embodiment, an unstable organic photosensitizer such as ICG actually decomposes continuously in a negative exponential relationship with time under light, and the change of C(t) is specifically expressed as:

[0100]

[0101] In a specific embodiment, the initial conditions of the tumor treatment model are:

[0102] x(0)=K, H(0)=0, B(0)=0, C(0)=C0.

[0103] Example 2

[0104] More specifically, in this embodiment, factors such as the uneven distribution of tumors in space and the differences between similar tumor cells are ignored, and a tumor growth model with time lag is constructed, specifically:

[0105] The Allee model with time lag is used to construct a tumor growth model with time lag as follows:

[0106]

[0107] Among them, x(t) represents the real-time tumor cell density, r0 represents the natural growth rate of tumor cells, K represents the maximum capacity of the environment, τ represents the time lag caused by the cell cycle of tumor cells, and A represents the Allee coefficient.

[0108] In this example, the Allee model addresses the shortcomings of the logistic model. A population with the Allee effect takes into account the cooperative behavior between individuals. This means that a minimum population density sufficient for survival is required to ensure that individuals within the population have sufficient conditions and opportunities for cooperation. If the density falls below a certain threshold, the population density will be too low, resulting in a lack of cooperation between populations, reduced individual fitness, and the population will become extinct.

[0109] In a specific embodiment, based on the tumor growth model, the treatment intensity terms of photothermal and photodynamic therapy and the tumor killing term are introduced respectively to construct a tumor treatment model, specifically:

[0110] Assuming that the dose-effect relationship curve is linear, using the Allee growth model with time lag, the constructed tumor treatment model is as follows:

[0111]

[0112] Where H(t) is the real-time photothermal therapy intensity, is the intensity of time-delay photodynamic therapy; μ H is the killing rate of tumor cells by photothermal therapy, μ B is the killing rate of tumor cells by photodynamic therapy; It is the lag time of the photodynamic therapy process relative to the photothermal therapy process, that is, the phase difference between the effects of the two treatments.

[0113] In a specific embodiment, considering that the heat conduction equation and the diffusion equation describe the transfer of heat and the diffusion of substances respectively, their forms are actually the same, and first-order pharmacokinetics is based on the results obtained from the diffusion equation, the actual rate of change of photothermal therapy intensity is expressed as:

[0114]

[0115] The actual change rate of photodynamic therapy intensity is expressed as:

[0116]

[0117] Where Q is the heat source term, representing the constant heat generation of the photothermal converter as a heat source under constant laser light irradiation, α is the heat generation rate of the photothermal effect of the photosensitizer, C(t) is the photosensitizer concentration, β Cis the ROS production rate of the photosensitizer without enhancement, and β is the ROS production rate of the photosensitizer enhanced by surface plasmon resonance, λ H is the non-therapeutic heat loss rate, λ B is the non-therapeutic loss rate of reactive oxygen species, γ H is the heat treatment process loss rate, γ B is the loss rate during the active oxygen treatment process.

[0118] In a specific embodiment, the change of C(t) is specifically expressed as:

[0119]

[0120] In a specific embodiment, the initial conditions of the tumor treatment model are:

[0121] x(0)=K, H(0)=0, B(0)=0, C(0)=C0;

[0122] In this embodiment, the model uses core-shell composite structure nanoparticles formed by precious metal nanoparticles coated with silica shells containing organic molecular photosensitizers as the treatment medium. All photosensitizers are enhanced by the metal surface plasmon resonance effect. The nanoparticles are evenly distributed in the tumor tissue, the external laser light source is constant and uniformly irradiated, and the tumor is evenly distributed in space.

[0123] Example 3

[0124] like Figure 2 As shown, an optimization method for photothermal and photodynamic combined synergistic therapy, based on the modeling method, includes the following specific steps:

[0125] Substituting initial parameters into the model to initialize the tumor treatment model;

[0126] Modify the φ, β, and α parameters and solve the tumor treatment model separately, compare the solution results, and find the optimal parameter combination;

[0127] Based on the optimal parameter combination, the optimal synergistic treatment plan is designed.

[0128] In this example, to investigate through simulation the impact of various parameters on the efficacy of combined photothermal and photodynamic therapy, as well as the synergistic effects between metal nanoparticle photothermal therapy and molecular photosensitizer photodynamic therapy, baseline values ​​were set for each parameter in the model. In this example, the maximum environmental capacity, K, was set to 10, and the natural tumor growth rate, r0, was set to 1. Based on these values, reasonable numerical ranges for the other parameters were determined. The final parameter baseline values ​​are shown in Table 2-2.

[0129] Table 2-2 Parameter names, meanings and reference values

[0130]

[0131] Where tumor cell natural growth rate r0, the maximum capacity of the environment K, Allee coefficient A is the cell itself influence parameters. The light-heat effect of photosensitizer heat production rate α, the surface plasmon enhanced photosensitizer activity oxygen production rate β is the performance of the probe itself parameters. Photosensitizer initial concentration C0, the relative lag of photodynamic therapy is the use of probes in the treatment process of human-controlled factors.

[0132] In one embodiment, the tumor treatment model is a nonlinear ordinary differential equations. Such an equation is difficult to find the analytical solution. Therefore, the embodiment of the present application selects the use of 4 order Runge-Kutta method to find its numerical solution, the tumor treatment model using Runge-Kutta RK4 method, o(h 5 ) of the truncation error, the specific steps are:

[0133] Given the initial conditions y ′ =f(x,y),y(x0)=y0

[0134] Where y represents all dependent variables, x represents all independent variables;

[0135] Using RK4 method iteration:

[0136]

[0137] Where

[0138] k1=f(x n ,y n )

[0139]

[0140]

[0141] k4=f(x n +h,y n +hk3)

[0142] Where h is the time interval.

[0143] In this embodiment, the Mathematica software and Matlab are used as calculation tools, when solving, the time t range is selected as 0-5, the solving step is 0.1, and the precision is 10 -5 .

[0144] In one embodiment, the parameters of φ, β, α are modified and the tumor treatment model is solved respectively, the main parameters affecting the tumor treatment effect are found by comparing the solving results, and the specific steps are:

[0145] Modify the φ, β, and α parameters and solve the tumor treatment model separately to analyze the changes in tumor cell density under different φ, β, and α parameters;

[0146] In this example, to measure the impact of different φ, β, and α parameters on the treatment effect, the ratio (N / K) of the number of surviving cells (N) after one unit of time after the start of photothermal-photodynamic therapy to the maximum environmental capacity K was used as an indicator. The smaller the N / K value, the more tumors were killed by the photothermal-photodynamic synergistic therapy, and the better the killing effect.

[0147] In this embodiment, the curve of cell number density changing with photothermal-photodynamic therapy time is shown in the figure below: Figure 3 、 4 , 5. It can be seen that β has a greater impact on the treatment effect.

[0148] Analyze the effects of changes in φ, β, and α parameters on the ratio N / K of the number density of surviving cells to the maximum environmental capacity K;

[0149] In this embodiment, Figure 6 As shown in the figure, to analyze the effects of varying the parameters φ, β, and α on the ratio N / K (the ratio of the number density of viable cells N to the maximum environmental capacity K), a cross-sectional plot of N / K was constructed using φ, β, and α as variables. The cross-sectional plot shows the N / K values ​​obtained when varying the three parameters φ, β, and α within a reasonable range. This plot roughly illustrates that N / K decreases with increasing β and α, indicating that larger β and α values ​​result in greater tumor cell killing. However, the influence of the treatment phase difference φ on treatment efficacy is more complex.

[0150] A comprehensive analysis was conducted on the changes in tumor cell density under different φ, β, and α parameters, and the impact of changes in φ, β, and α parameters on N / K, to obtain the parameter combination with the best tumor cell killing effect.

[0151] In this embodiment, in order to comprehensively analyze the changes in tumor cell density under different φ, β, and α parameters, as shown in FIG. Figure 7 、 8 As shown in Figures 9, based on the effects of changes in φ, β, and α parameters on N / K, a schematic diagram of the influence curve of φ, β, and α on N / K was also constructed. When φ = 0, that is, when there is no phase difference between photothermal and photodynamic, N / K has the best killing effect on tumor cells. N / K decreases sharply with the increase of β and gradually decreases with the increase of α. However, it is obvious that the impact of changing parameter α on N / K is much smaller than the impact of changing parameter β on N / K.

[0152] In this embodiment, based on the above analysis, it is concluded that in actual treatment, the time delay φ should be reduced to close to 0, and β should be increased to at least 0.08.

[0153] In one specific embodiment, in order to verify that the model can describe the synergistic effect between photothermal therapy and photodynamic therapy, the Loewe isobologram about photothermal and photodynamic is needed. In the Loewe isobologram, each curve represents the curve of the same therapeutic effect obtained by two drugs or treatment methods at different doses. The upper convex isobologram represents the antagonistic effect of the two drugs or treatment methods, the lower concave represents the synergistic effect of the two drugs or treatment methods, and the straight line represents the additive effect of the two drugs or treatment methods.

[0154] In this embodiment, the effects of φ, β, and α parameters on the photothermal-photodynamic synergistic effect are analyzed, and the specific steps are as follows:

[0155] The effects of φ, β, and α parameter changes on Q are analyzed.

[0156] The photosensitivity C0 and Q are set as coordinate axes, the value of N / K is fixed, and the simulated isobologram is obtained by multiple simulations based on the tumor treatment model. The additive isobologram and the simulated isobologram are compared when the photothermal and photodynamic therapies have an additive relationship, and the synergistic effect value is calculated.

[0157] In this embodiment, the values of C0 and Q are set so that the value of N / K is exactly 0.5 under a series of C0 and Q values, thereby obtaining the same therapeutic effect. By the above method, within the reasonable range of C0 and Q, the values of C0 and Q are traversed at a specific step size, the N / K value corresponding to each pair of C0 and Q values is calculated, and a series of C0 and Q values corresponding to N / K=0.5 are selected. Finally, taking Q and C0 as the horizontal and vertical axes of the coordinate graph, respectively, the C0 curve about Q is drawn, that is, the simulated isobologram as shown in Figure 10 The intersection of the synergistic effect curve and the horizontal and vertical coordinate axes can obtain the corresponding additive isobologram.

[0158] In this embodiment, as shown in Figure 11 The synergistic effect value R=S 阴影 / S Δ , wherein S 阴影 is the area bounded by the simulated isobologram and the additive isobologram, S Δ is the triangular area surrounded by the additive effect straight line and the horizontal and vertical coordinate axes, and R is a proportionality coefficient reflecting the synergistic effect of the two means in the existing treatment scheme. The larger the value of R is, the better the synergistic effect of photothermal and photodynamic is.

[0159] Analyze the impact of changes in φ, β, and α parameters on the heat production parameter Q and the synergistic effect value, find the key parameters among φ, β, and α parameters, and regulate the combined photothermal and photodynamic synergistic therapy based on the key parameters.

[0160] In this embodiment, in order to analyze the influence of the changes of φ, β, and α parameters on the heat generation parameter Q and the synergistic effect value, as shown in FIG. Figure 12 、 13 As shown in Figures 14 and 15, the iso-effect curves corresponding to the changes in the parameters φ, β, and α when the value of N / K is 0.5 are constructed by the tumor treatment model; from these iso-effect curves, it can be seen that in the process of killing tumor cells by nanoprobes, there is an obvious synergistic effect between photothermal therapy and sensitizer photodynamic therapy in the model established by the present invention. Therefore, the ability of the model proposed by the present invention to describe the synergistic effect of photothermal and photodynamic therapy is verified. It can be seen that in different The isoeffective lines obtained under the parameters of β and α are The β and α parameters increase and gradually shift downward. However, the degree of downward shift varies greatly among the three parameters, and the change of the β parameter has the greatest impact on the iso-effect line.

[0161] In this embodiment, the synergistic effect is measured using the R value, such as Figure 15 、 16 As shown in Figures 17, further, a schematic diagram of the curve of the influence of changes in φ, β, and α parameters on the synergistic effect was constructed. It can be seen that the change in the active oxygen production rate β of the photosensitizer increased under surface plasmon enhancement has a dramatic effect on the synergistic effect R. As β gradually increases from 0.01 to 0.15, the synergistic effect R gradually increases from 0.25 to about 0.66, but the rate of increase gradually decreases, making the change in the synergistic effect gradually stabilized.

[0162] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included in the scope of protection of the claims of the present invention. It can also be seen that the synergistic effect R decreases linearly with the increase of the photothermal heat production rate α of the photosensitizer, but the rate of decrease is slow. As α gradually increases from 0.01 to 0.15, the synergistic effect R gradually increases from 0.57 to about 0.51, that is, the synergistic effect R is not sensitive to changes in α.

[0163] In this embodiment, in summary, the increased photosensitizer reactive oxygen species yield β under surface plasmon enhancement is a key factor affecting the synergistic effect of probe photothermal-photodynamic therapy. Therefore, the photosensitizer reactive oxygen species yield parameter should be adjusted as high as possible during the probe preparation process to improve the synergistic effect and thus enhance the therapeutic effect.

[0164] The present invention simulates the combined photothermal and photodynamic therapy of probes, prepares nano-dumbbell structure probes, characterizes the probe performance, and studies the photothermal and photodynamic synergistic behavior of the nano-probes obtained under different preparation parameters.

[0165] Based on the present invention's analysis, this example prepared a large number of gold nanorods and measured their absorption spectra. Subsequently, the ends of the gold nanorods with absorption peaks around 800 nm were coated with SiO2 to create nano-dumbbell-shaped probes. The probe morphology was then observed using a transmission electron microscope, and the ideal probes were selected for indocyanine green (ICG) attachment, yielding the final sample.

[0166] The analysis of the present invention shows that the therapeutic effect is best when photothermal and photodynamic therapy start to work at the same time. The most important factor for the synergistic effect of photothermal and photodynamic therapy is the increased photosensitizer reactive oxygen production rate β under surface plasmon enhancement. This result has guiding significance for parameter regulation in the probe preparation process and optimization of the treatment process. The experiment successfully controlled the aspect ratio of the gold nanorods by changing the experimental parameters, thereby controlling the resonance absorption peak of the gold nanorods. The experiment successfully prepared a nano dumbbell-shaped probe and determined the appropriate parameter range for the preparation. Preliminary biological experiments also proved that after the nano dumbbell-shaped probe was attached to the photosensitizer ICG, photothermal and photodynamic therapy had a synergistic effect.

[0167] The resulting dumbbell-shaped A@mSiO2-ICG nanocomposite exhibits more efficient photothermal-photodynamic synergy than fully encapsulated Au@mSiO2-ICG nanoprobes. Once mature, this technology could be even more effective than photothermal or photodynamic therapy alone, potentially helping to eliminate cancer cells more efficiently, precisely, and with fewer side effects.

[0168] Example 4

[0169] In this embodiment, the Runge-Kutta method program written in Mathematica is used to calculate the numerical solution of the model equations, and directly provide an image of the change of the tumor cell number density x with time t.

[0170] Let Q be 1.5, C0 be 10, and other parameters be baseline values, and solve the model equations for the time t range of 0 to 5. The obtained image of the tumor cell number density x changing with time t is as follows: Figure 18 As shown in the figure, we can see that x reaches a minimum value and then rises again. This is because the equation The concentration of organic dye photosensitizer C decreases exponentially with time, which makes the actual photodynamic therapy intensity B as Figure 19At the same time, due to the decomposition of the photosensitizer, the heat generated by the photosensitizer photothermal effect decreases, and the actual photothermal treatment intensity H will also decrease over time until it reaches a certain value, such as Figure 20 As shown, therefore, due to the decrease in treatment intensity, x will rise again to an equilibrium point less than 10.

[0171] However, in actual treatment, this situation where x rises again will not occur. This is because in actual treatment, the tumor will be treated more intensively within a time period shorter than the tumor cell cycle to ensure that all tumor cells are killed. To verify this, in this example, the τ value representing the tumor cell cycle is adjusted to 10, and the resulting xt curve is as follows: Figure 21 As shown, at this time, the tumor cells are almost completely killed at t = 3. This is consistent with the actual situation. However, in this embodiment, in order to facilitate the study of the impact of different parameters on efficacy and synergistic effects, the time lag τ is still taken as the baseline value of 1 in the following sections.

[0172] In this embodiment, when constructing the simulated isobologram, the minimum value of x in a single simulation is selected as the indicator of therapeutic effect. Using the baseline values ​​of the parameters, and setting the value of C0 to 20 and the value of Q to 0, the minimum value of x at this time is calculated. min Then set the value of C0 to 0, and calculate Q = 3.51 to make x min The value of is the same. Figure 22 As shown in the figure, the indicative dose values ​​required to achieve the same therapeutic effect using metal nanoparticles alone for photothermal therapy and organic dye photosensitizers alone for photodynamic therapy were obtained. With Q and C0 as the horizontal and vertical axes of the coordinate graph, respectively, the two sets of values ​​obtained previously represent the two points on the two axes. Connecting these two points with a straight line yields the line corresponding to the additive effect in the isobologram.

[0173] With a fixed step size of 0.4 for Q, calculate the C0 value corresponding to each Q value in the range of 0 to 3.51 so that each set of (Q, C0) data can make x min Using the obtained data points, plot x min The simulated isoelectric curve when the value of is 4.91.

[0174] It can be clearly seen that x min The isotropic effect line for a value of 4.91 lies below the additive isotropic line. This indicates a significant synergistic effect between metal nanoparticle photothermal therapy and organic dye photosensitizer photodynamic therapy. This validates the proposed model's ability to describe the synergistic effects of photothermal and photodynamic therapy.

[0175] In this embodiment, when analyzing the influence of the changes of φ, β, and α parameters on the heat generation parameter Q and the synergistic effect value, four groups of parameters are selected: Q = 2, C0 = 7; Q = 2, C0 = 5; Q = 1, C0 = 5; Q = 1, C0 = 7. The parameters other than the reference value are taken as the reference value. The obtained curve is as follows Figure 23 As shown in the figure, x min Value The increase in the value increases almost linearly. This means that the smaller the delay between the photodynamic therapy phase and the time when photothermal therapy produces an effect, the better the effect of the combined treatment. The slope of the straight line when Q = 2, C0 = 7 is almost the same as the slope of the straight line when Q = 1, C0 = 7, the slope of the straight line when Q = 2, C0 = 5 is almost the same as the slope of the straight line when Q = 1, C0 = 5, and the slope of the straight line when C0 = 7 is greater than the slope of the straight line when C0 = 5. This shows that the relative lag of photodynamic therapy has a more significant impact on the effect of photodynamic therapy.

[0176] When the three iso-effect lines from top to bottom correspond to x min =4.91 isoeffective line as Figure 24 As shown in the figure, it is clear that the smaller the relative hysteresis of photodynamic therapy, the lower the photosensitizer dose required to achieve the same therapeutic effect. The shapes of the three isoelectric lines remain basically unchanged, indicating that the relative hysteresis of photodynamic therapy does not significantly affect the intensity of the synergistic effect.

[0177] β=0、β=0.05、β=0.1, under three parameter values, x min =4.91 when the isoeffective line is as follows Figure 25 As shown in the figure, all other parameters were taken as baseline values. It can be seen that as the β value increases, the downward curvature of the curve increases. This indicates that under a stronger surface plasmon-enhanced photodynamic effect, the synergy between photothermal and photodynamic therapy is stronger, and a lower dose of combined therapy can achieve the same therapeutic effect.

[0178] It is important to note that the isotropic effect line when β = 0 is close to a straight line, that is, close to the isotropic effect line of the additive effect. This indicates that when β = 0, the synergy between photothermal and photodynamic therapy is so weak that it can be ignored. This indicates that in the situation simulated by this model, the synergistic effect of photothermal and photodynamic therapy in combined therapy is dominated by the surface plasmon-enhanced photodynamic effect, while the photothermal effect of the photosensitizer has a weak impact on the synergistic effect.

[0179] The dose used in combination therapy can be used as an indicator of the strength of the synergistic effect under different β conditions. For the convenience of selection, the Q value can be fixed to 1.6, and the x value under different β conditions can be calculated. min=4.91 is used as the indicator of synergistic effect. The C0 values ​​when β is 0, 0.025, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, and 0.35 are calculated respectively. The β-C0 is plotted as Figure 26 As shown in .

[0180] As can be seen, as the β value increases, the required C0 value decreases, but at a slower rate. This indicates that the photothermal and photodynamic synergistic effects increase with the strengthening of the surface plasmon phenomenon. When the surface plasmon phenomenon's contribution to photodynamics is weak, its enhancement has a strong impact on the synergistic effect. However, as the surface plasmon phenomenon's ability to promote photodynamics becomes stronger, the enhanced synergistic effect becomes less pronounced. This suggests that there may be an optimal cost-effectiveness ratio when enhancing the synergistic effect caused by surface plasmons to increase therapeutic efficacy.

[0181] Obviously, the above embodiments of the present invention are merely examples for the purpose of illustrating the present invention, and are not intended to limit the embodiments of the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the claims of the present invention.

Claims

1. A modeling method for photothermal-photodynamic combined synergistic therapy, characterized by: The specific steps include: A tumor growth model with time lag was constructed, specifically: A tumor growth model with time lag is constructed based on either the Logistic growth model with time lag or the Allee model with time lag; If the Logistic growth model with time lag is used, the tumor growth model with time lag is constructed as follows: If the Allee model with time lag is used, the tumor growth model with time lag is constructed as follows: in, Represents the real-time tumor cell density, represents the natural growth rate of tumor cells, Represents the maximum capacity of the environment, represents the time lag caused by the cell cycle of tumor cells, represents the Allee coefficient; Based on the tumor growth model, the treatment intensity terms of photothermal and photodynamic therapy and the tumor killing terms are introduced to construct a tumor treatment model. Specifically: If the Logistic growth model with time lag is used, the constructed tumor treatment model is as follows: If the Allee growth model with time lag is used, the constructed tumor treatment model is specifically: in, is the real-time photothermal therapy intensity, is the intensity of time-delay photodynamic therapy; is the tumor cell killing rate of photothermal therapy, is the killing rate of tumor cells by photodynamic therapy; It is the lag time of the photodynamic therapy process relative to the photothermal therapy process, that is, the phase difference between the effects of the two treatments.

2. The modeling method for photothermal-photodynamic combined synergistic therapy according to claim 1, characterized in that: The actual rate of change of photothermal therapy intensity is expressed as: The actual change rate of photodynamic therapy intensity is expressed as: Where Q is the heat source term, which represents the constant heat generated by the photothermal converter as a heat source under constant laser light irradiation. is the heat generation rate of the photosensitizer's photothermal effect, is the photosensitizer concentration, is the active oxygen production rate of the photosensitizer without enhancement, and is the active oxygen production rate of the photosensitizer increased by surface plasmon resonance enhancement, is the non-therapeutic heat loss rate, is the non-therapeutic loss rate of active oxygen, is the heat loss rate during the treatment process, is the loss rate during the active oxygen treatment process.

3. The modeling method for photothermal-photodynamic combined synergistic therapy according to claim 2, characterized in that: The changes are specifically expressed as follows: 。 4. The modeling method for photothermal-photodynamic combined synergistic therapy according to claim 3, characterized in that: The initial conditions of the tumor treatment model are: ； The model uses core-shell composite structure nanoparticles formed by shell-coated precious metal nanoparticles containing organic molecular photosensitizers as the treatment medium. All photosensitizers are enhanced by the metal surface plasmon resonance. The nanoparticles are evenly distributed in the tumor tissue, the external laser light source is constant and uniformly irradiated, and the tumor is evenly distributed in space.

5. A method for optimizing photothermal and photodynamic combined synergistic therapy, characterized by: The modeling method according to any one of claims 2 to 4 comprises the following specific steps: Substituting initial parameters into the model to initialize the tumor treatment model; Revise 、 、 Parameters and solve the tumor treatment model separately, compare the solution results, and find the optimal parameter combination; Based on the optimal parameter combination, the optimal synergistic treatment plan is designed.

6. The optimization method for photothermal and photodynamic combined synergistic therapy according to claim 5, characterized in that: Revise 、 、 Parameters and solve the tumor treatment model respectively, compare the solution results, and find the main parameters that affect the tumor treatment effect. The specific steps are: Revise 、 、 Parameters and solve the tumor treatment model separately, analyze the different 、 、 Changes in tumor cell number density under parameter conditions; analyze 、 、 The effect of parameter changes on the ratio N / K of the number density of surviving cells to the maximum environmental capacity K; Comprehensive analysis of different 、 、 Changes in tumor cell density under parameter conditions, based on 、 、 The effect of parameter changes on N / K was used to obtain the parameter combination with the best tumor cell killing effect.

7. The optimization method for photothermal and photodynamic combined synergistic therapy according to claim 6, characterized in that: We further analyzed 、 、 The influence of parameters on the photothermal-photodynamic synergistic effect, the specific steps are: analyze 、 、 The impact of parameter changes on Q; Setting light sensitivity Using and Q as coordinate axes and fixing the value of N / K, multiple simulations were performed based on the tumor treatment model to obtain simulated isobolograms. The additive isobolograms and simulated isobolograms when photothermal and photodynamic therapy are in an additive relationship were compared to calculate the synergistic effect value. analyze 、 、 The effect of parameter changes on the heat production parameter Q and the synergistic effect value is found 、 、 Key parameters among the parameters, and the preparation plan of the probe used in photothermal-photodynamic therapy is regulated based on the key parameters.

8. The optimization method for photothermal and photodynamic combined synergistic therapy according to claim 7, characterized in that: The synergistic effect value ,in is the area bounded by the simulated iso-effect line and the summed iso-effect line, is the area of ​​the triangle enclosed by the summing effect straight line and the horizontal and vertical coordinate axes, is a proportional coefficient that reflects the synergistic effect of the two methods in the existing treatment plan. The larger the value, the better the synergistic effect of photothermal and photodynamic.

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