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Modeling Part 3

Targeting Ability, Effectiveness and Safety

Targeting Ability

To evaluate the in vivo antitumor potential of M1, we established a preclinical tumor model.

We selected several groups of male and female mice and gave them a rapid intravenous injection of M1.The concentration of M1 was collected in different parts of the mouse body, and the mean value was taken to plot the results, as shown in the figure below.

The concentration of M1 was collected in different parts of the male mouse body

The concentration of M1 was collected in different parts of the female mouse body

From the experimental results, we can see that M1 is enriched in the tumor area in mice.In other parts of the mice, the concentration level of M1 was low, which was close to the level of normal mice, which reflected the targeted effect of M1 and thus ensured the targeted effect of combined therapy.

Effectiveness

Fitness

Meanwhile, the mean value of Tumor Volumn in both control and experimental mice was obtained, as shown in the table below.^[1]

Tumor Volumn in both control and experimental mice
	3	6	9	12	15	18	21
M1	0.12	0.20	0.28	0.40	0.68	0.88	0.95
Control	0.10	0.11	0.14	0.15	0.18	0.28	0.42

We used Fourier and Gaussian functions to fit the images of M1 and the control group, the table is as below:

Tumor Volumn in both control and experWe know this from experiments on ten primate cynomolgus monkeysimental mice

Consistent with the in vitro experiments, evident antitumor effects were observed in M1-treated animals.

Improved ODE Model

Considering the different environment in human body because of the immune effect and the limitation of the number of uninfected tumour cells U(t), infected tumour cells I(t), we decided to change our model:^[2]
$$\left\{ \begin{aligned} \frac{dU(t)}{dt}&=r*U(t)*\frac{1}{\varphi*U(t)+\psi*I(t)}-\beta U(t)M(t)\\ \frac{dI(t)}{dt}&=r*I(t)*\frac{1}{\varphi*U(t)+\psi*I(t)}+\beta U(t)M(t)-\alpha I(t)-\frac{I(t)}{I(t)+U(t)}\\ \frac{dM(t)}{dt}&=\sigma I(t)-uM(t)-M(t)*\frac{I(t)}{I(t)+U(t)}\\ \end{aligned} \right.$$
In the equations above, M(t)is the titer of the M1. The proportion of infected tumour cells among all the tumor cells as $\frac{I(t)}{I(t)+U(t)}$ means the immune effect on the titer of the infected tumour, meanwhile, in order to take the limitation of tumour cells to themselves into consideration, we add $\frac{1}{\varphi*U(t)+\psi*I(t)}$ to show the counteraction to its own growth with a large amount of tumour.

The solution of the equations is showed below:

This solution is more reasonable. It proves the effectiveness of M1 with the reduction of the uninfected tumour cells. Obviously the solution tends to be stable and the infected tumour cells will gradually become zero. In addition, the remain of the tumour cells also prove that the second dose is necessary.

Safety

As for the safety of our system, we know this from experiments on ten primate cynomolgus monkeys. In this experiment, Ten captive-bred 4 to 6-year-old male- or female- specific pathogen-free cynomolgus macaques (M. fascicularis) weighing between 3.5 and 6 kg were obtained and Five randomly selected macaques were treated with oncolytic virus M1; the left five ones were subjected to the control group–receiving vehicle after the same schedule as the virus injections in the test group. All animals were bled 2 days before viral injection to confirm the absence of neutralizing antibodies.^[3]

During the experiment, the body weight and body temperature of experimental macaques and control macaques were measured weekly. In addtional, the count of white blood cells, lymphocyte,monocyte, albumin(ALB), globulin(GLB), aspartate amino-transferase (AST) , alanine transaminase (ALT), alkaline phosphatase (ALP) Were examined and counted. The results are listed as below:

Tumor Volumn in both control and experimental mice
Diagnosis	Control	Test	p value
Body weight	5.26 ± 1.39	5.81 ± 0.8	0.2099752 ≥ 0.05
Body temperature	38.42 ± 0.48	38.22 ± 0.52	0.2536536 ≥ 0.05
WBC,cells/nl	11.12 ± 4.31	9.54 ± 4.01	0.2640504 ≥ 0.05
LY,%	34.72 ± 9.62	51.16 ± 12.14	0.006704666
MONO,%	4.86 ± 2.32	5.2 ± 3.17	0.3722814 ≥ 0.05
AST,U/liter	28 ± 5.24	28.2 ± 4.55	0.3862507 ≥ 0.05
ALT,U/liter	32.4 ± 16.77	26.4 ± 3.78	0.2056938 ≥ 0.05
ALB,g/liter	44.66 ± 2.79	46.22 ± 1.81	0.1300045 ≥ 0.05
GLB,g/liter	32 ± 3.50	30.26 ± 4.11	0.2247176 ≥ 0.05
ALP,U/liter	175 ± 76	181.8 ± 70.48	0.3788848 ≥ 0.05

We run hypothesis tests on the data:

H₀: No statistically significant differences

H₁: There are statistically significant differences

All the examinations mentioned above were performed to evaluate the potential comprehensive long-term effects of multiple rounds of injections on macaques. As shown in Table, no significant differences can be found in the test group compared withthe control, except for the increased percentage of lymphocytes in the white blood cells, which is a common phenomenon during virus infection.

That is to say we should accept the H₀, and our system is safe and dependable in a way.

The details of this experiment can be seen in the reference.

*The Contact Ratio β and Spatial Distribution

Actually, in the research, the contact ratio β is different due to the position. The uninfected tumor cells tend to have a higher contact ratio when they are around an infected tumor cell. And as we can see, β decreases as the cells move away from the infected one. Thus, we should consider the distribution of infected tumor cells.

We could collect the fluorescent figure of different time, and regard the fluorescent points as points in the plot. Then we could use k-means to cluster. And we can get the mean and variance of each cluster. Thus we can use Gaussian Mixture Model to make an approximate distribution based on the clustering. That is because the gaussian mixture distribution can approximate the non - negative integrable function with arbitrary precision. Then we could get the figure below.

Basee on the GMM, we could finally get the distribution of infected tumor cells. It is the distribution of the contact ratio β as well.

The similar method can be applied into 3-dimensional human body. There is a little difficult in adapting the time-varying distribution to the ODE. But we will try to finish it in the future.

Renference

[1]Rafael Sanjuán,Valery Z Grdzelishvili. Evolution of oncolytic viruses[J]. Current Opinion in Virology,2015,13.

[2]Zizi Wang,Zhiming Guo,Huaqin Peng. A mathematical model verifying potent oncolytic efficacy of M1 virus[J]. Mathematical Biosciences,2016,276.

[3]Haipeng Zhang, Yuan Lin, Kai Li, Jiankai Liang, Xiao Xiao, Jing Cai, Yaqian Tan, Fan Xing, Jialuo Mai, Yuan Li, Wenli Chen, Longxiang Sheng, Jiayu Gu, Wenbo Zhu, Wei Yin, Pengxin Qiu, Xingwen Su, Bingzheng Lu, Xuyan Tian, Jinhui Liu, Wanjun Lu, Yunling Dou, Yijun Huang, Bing Hu, Zhuang Kang, Guangping Gao, Zixu Mao, Shi-Yuan Cheng, Ling Lu, Xue-Tao Bai, Shoufang Gong, Guangmei Yan, and Jun Hu. Naturally Existing Oncolytic Virus M1 Is Nonpathogenic for the Nonhuman Primates After Multiple Rounds of Repeated Intravenous Injections. Human Gene Therapy 2016 27:9, 700-711

Modeling Part 2

Modeling Part 4

Team:SYSU-Medicine/TES