Dynamic Equations




Dynamics of Oncolytic viruses


Oncolytic viruses have the ability to infect and lyse cancer cells, while aiming to minimize harm or reduce damage to normal cells. This feature endows them with tremendous potential in cancer treatment, and various viruses have shown promising results in clinical trials. However, their feasibility remains elusive, and successful modeling predictions remain a subject of study from experimental and mathematical perspectives. Drawing inspiration from the dynamics handling methods of many biological systems, an attempt is made here to establish ordinary differential equations for a mathematical description of the infection process of oncolytic viruses.


After extensive literature research, we have identified two core issues in the modeling process. Firstly, the direct protein interactions play a crucial role, such as the transferrin receptor 1 (TfR1) and the surface proteins NA (neuraminidase) and HA (hemagglutinin). These protein interactions directly mediate the infection process and subsequently affect the infection rate, targeting specificity, and other relevant factors of interest. Secondly, the overall dynamic profile of the virus and tumor system is vital, and it directly reflects the selection of experimental indicators.

By addressing these two key aspects, a comprehensive modeling framework can be developed to capture the intricate dynamics of oncolytic virus infection and its impact on tumor systems. This approach will facilitate a better understanding of the underlying mechanisms and provide insights for optimizing therapeutic strategies.

Based on these two points, we can make some initial assumptions. Starting with the first point, in experiments, we can enhance the targeting and infectivity by modifying relevant biological components (such as increasing expression, extending stalk regions, etc.). Mathematically, we can abstract these modifications as a new parameter, referred to as the targeting factor, denoted as T. Taking into account the specificity and saturation of protein interactions, the infection process can be considered as a large molecular elementary reaction

Here, the variables represent the following:

  • V: Concentration of free viruses
  • T: Concentration of targeting factor
  • VT: Concentration of virus and target complex
  • Y: Concentration of successfully infected cells

Dynamic Equations

Based on the assumptions mentioned earlier, we can express the dynamic equations for the infection process of oncolytic viruses as follows:

Due to the typically decisive nature of the second step, an equilibrium can be reached in the first step itself.

At the same time, the effective concentration of the targeting factor remains constant.

Whereas represents the total titer of free viruses [V] in the system at time t, incorporating the simplifications (k2<<k1、k1-) yields the following expression:

T0 is defined by human intervention, while k2 and K are adjustable parameters that can be optimized.

Based on the second point, the chosen basis for our cancer-virus system model is the interaction between the infected and uninfected cell populations. This interaction consists of two components: the cancer growth term and the viral infection term. These components can be expressed using a predator-prey type system, which captures the dynamics between two interacting populations.

To consider the dynamics of the cancer-virus system based on the quantities of infected (Y) and uninfected (X) cells,the function F(X, Y) represents the dynamics of cancer growth and death processes,the function G(X, Y) represents the infection rate. The term αy represents the constant death rate of infected cells.

Indeed, combining the modeling approach for the infection part, we can replace the second term of the differential equation for X and the first term of the differential equation for Y with established quasi-Michaelis-Menten-like equations.

Having addressed the viral infection component, the remaining task is to provide the cancer growth term. After preliminary exploration, I have ultimately chosen the Gompertz growth model to describe it. The general expression of this model is as follows:

To express the Gompertz growth model in a form that is compatible with the original differential equations, it can be written as follows:

Incorporating the adjustable parameters W and η into the system of differential equations, we obtain the following:

For the overall viral titer of the system, which consists of both free virions (C1) and intracellular virions (C2), an additional equation is required to describe the intracellular viral titer. To address this, we can employ the widely applicable logistic growth model to simulate the intracellular viral titer. The mathematical formulation of the logistic growth model is as follows:

γrepresents the growth rate, and K denotes the generalized environmental capacity. By rearranging and transforming the equation, we can obtain:

Integrating the equation mentioned above yields:

We can obtain the expression for N(t), and for all infected cells y(t), under the assumption of the fast spread model, it can be approximated that they have an average intracellular titer of N .Specifically, this can be expressed as:

By substituting N instead of the integral, we can model the expression for C2 as follows:

The final expression for the overall viral titer of the system, denoted as C, can be obtained by combining the free virions (C1) and the intracellular virions (C2):

Our theoretical framework has been fully established.


To validate the rationality of the model, we proceed with the verification work using the data obtained from RT-QPCR experiments.

We have obtained the geometric shape of C(t)

In order to obtain the possible mathematical forms of C(t), we conducted a new round of research and carried out enhanced sampling and parameter fitting work using Python. As a result, we have determined two specific mathematical forms that are considered the most reasonable.

One of the determined mathematical forms belongs to the Gompertz curve family.

The parameter optimization for the Gompertz curve is as follows:

The second mathematical form belongs to the logistic curve family.

The parameter optimization for the logistic curve is as follows:

The mathematical forms reflected by the two fitting results align with the findings from our literature research. This indicates that our modeling approach is consistent with the actual tumor growth dynamics, viral infectivity, and other objective biological principles. Therefore, we can consider our model to be comprehensive.


In summary, from the aforementioned mathematical modeling process, it can be observed that the infection process of oncolytic viruses on target cells is a mathematically open process. Furthermore, due to the unknown and complex nature of biological systems, there is fundamentally no perfect modeling approach. Even if one exists, it cannot simultaneously achieve simplicity and accuracy. Pursuing a completely comprehensive model will inevitably lead to intricate and challenging optimization. Therefore, adopting an open modeling approach, supported by experimental data and computational assistance, it is sufficient to determine plausible expression forms and a reasonable parameter space for the dynamic equations.


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