There are two different strategies for tumor treatment. These strategies can be effectively modeled using differential equations to predict tumor behavior.

 

1. Inducing cell death in increasing cancer cells to reduce tumor volume. This is commonly done by chemotherapy or immunotherapy.

 

2. Reducing the carrying capacity which decreases the tumor support.

 

 

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• C: Tumor cell population

• K: Carrying capacity

• λ: Tumor growth rate

• log⁡(c/K) : Represents the logistic growth limitation, preventing unbounded growth.

• ​ξc: The anti-tumor treatment term, modeling how treatment reduces tumor cells.

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The first term in the equation shows that as the tumor cell population or density approaches the carrying capacity, the growth of the tumor cells slows down, and the second term represents how anti-tumor treatment drugs actively kill tumor cells proportional to the tumor cell population​

 

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• ​​​​​K: The carrying capacity, influenced by tumor and treatment.

• ϕc: The positive effect of the tumor on increasing K (e.g., through angiogenesis).

• φKc^2/3: The natural resource limitation effect.

• υKg(t): The anti-angiogenic treatment term, reducing tumor-supporting resources.

• g(t):concentration of the drug.​

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The tumor itself stimulates angiogenesis (blood vessel formation), increasing K.

The second term represents the natural limitations on tumor growth. The third term models how anti-angiogenic therapy (which blocks blood vessel formation) reduces the tumor’s resource supply. If the treatment is effective, υKg(t) lowers K, restricting the tumor's ability to sustain itself.​

 

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• a(t’) : rate at which the inhibitor concentration is administered.

• clr(t-t’) : clearance rate, determining how quickly it is removed from the system.

• (t−t′) : The delay term, representing how long ago the drug was administered

• exp(−clr(t−t′)) : The clearance function, which accounts for how the drug concentration decays over time due to metabolism and elimination.​

 

The drug concentration at time, t depends on all past administrations (t′) but decays over time due to clearance. The factor exp⁡(−clr(t−t′) ensures that older drug doses contribute less to the current concentration than more recent doses. The equation integrates over all previous drug administrations, meaning it tracks the cumulative effect of the drug, rather than just the most recent dose.​​​​​