Problem 7 - Biocalculus Calculus for the Life Sciences

Question

7\. Cancer progression The development of many cancers, such as colorectal cancer, proceed through a series of pre- cancerous stages. Suppose there are \(n-1\) precancerous stages before developing into cancer at stage \(n .\) A simple system of differential equations modeling this is $$\begin{aligned} x_{0}^{\prime} &=-u_{0} x_{0} \\ x_{i}^{\prime} &=u_{i-1} x_{i-1}-u_{i} x_{i} \\ x_{n}^{\prime} &=u_{n-1} x_{n-1} \end{aligned}$$ where \(x_{i}\) is the fraction of the population in state \(i,\) the \(u_{i}\) 's are positive constants, and \(i=1, \ldots, n-1 .\) $$\begin{array}{l}{\text { (a) Suppose } n=2 . \text { What is the system of differential equa }} \\ {\text { tions for the three stages? }} \\ {\text { (b) Note that the variable } x_{2} \text { does not appear in the equa- }} \\\ {\text { tions for the rate of change of } x_{0} \text { or } x_{1} . \text { Consequently, }} \\ {\text { we can solve the two-dimensional system for } x_{0} \text { and }} \\ {x_{1} \text { separately. Do so, assuming that } x_{0}(0)=k \text { and }} \\ {x_{1}(0)=0 .}\end{array}$$ $$\begin{array}{l}{\text { (c) Use your solution for } x_{1}(t) \text { obtained in part (b) to write }} \\ {\text { a differential equation for } x_{2}(t) \text { . }} \\ {\text { (d) Solve the differential equation from part (c), assuming }} \\ {x_{2}(0)=0}\end{array}$$

Step-by-Step Solution

Answer

(a) \(x'_0 = -u_0 x_0, x'_1 = u_0 x_0 - u_1 x_1, x'_2 = u_1 x_1\). (b) \(x_0(t) = ke^{-u_0 t}, x_1(t) = \frac{u_0 k}{u_1 - u_0} [e^{-u_0 t} - e^{-u_1 t}]\). (c) \(x'_2 = u_1 \cdot \frac{u_0 k}{u_1 - u_0} [e^{-u_0 t} - e^{-u_1 t}]\). (d) \(x_2(t) = \frac{ku_0}{u_1 - u_0} \left(\frac{1 - e^{-u_0 t}}{u_0} + \frac{e^{-u_1 t} - 1}{u_1}\right)\).

Step 1: Formulating the Differential Equations for Three Stages

Given that \(n=2\), we have stages \(x_0, x_1,\) and \(x_2\). According to the prompt, the differential equations become: \(x'_0 = -u_0 x_0\), \(x'_1 = u_0 x_0 - u_1 x_1\), and \(x'_2 = u_1 x_1\).

Step 2: Solving for \(x_0(t)\) and \(x_1(t)\)

Given \(x_0(0) = k\) and \(x_1(0) = 0\), solve \(x'_0 = -u_0 x_0\). This is a separable differential equation with solution \(x_0(t) = ke^{-u_0 t}\). Substitute \(x_0(t)\) into the equation \(x'_1 = u_0 x_0 - u_1 x_1\), which becomes \(x'_1 = u_0 ke^{-u_0 t} - u_1 x_1\). Use integrating factors or an appropriate method to solve for \(x_1(t) = \frac{u_0 k}{u_1 - u_0} [e^{-u_0 t} - e^{-u_1 t}]\).

Step 3: Writing a Differential Equation for \(x_2(t)\)

From the equation \(x'_2 = u_1 x_1\), substitute the expression for \(x_1(t)\) obtained previously: \(x'_2 = u_1 \cdot \frac{u_0 k}{u_1 - u_0} [e^{-u_0 t} - e^{-u_1 t}]\).

Step 4: Solving the Differential Equation for \(x_2(t)\)

Since \(x_2(0) = 0\), integrate \(x'_2 = u_1 \cdot \frac{u_0 k}{u_1 - u_0} [e^{-u_0 t} - e^{-u_1 t}]\) with respect to time to find \(x_2(t)\). Integrate each part separately: \[x_2(t) = \frac{u_0 k}{u_1 - u_0} \left[\frac{e^{-u_0 t}}{-u_0} + \frac{e^{-u_1 t}}{u_1}\right]\]. Evaluate the definite integral from 0 to \(t\) to obtain \[x_2(t) = \frac{ku_0}{u_1 - u_0} \left(\frac{1 - e^{-u_0 t}}{u_0} + \frac{e^{-u_1 t} - 1}{u_1}\right)\].

Key Concepts

Precancerous Stages Modeling

Cancer often develops in stages, starting from a healthy cell to a precancerous state and ultimately turning into cancerous cells. Precancerous stages modeling helps in understanding how these stages progress before a full transformation to cancer. In this model, a population fraction represents each stage's progression. This involves assessing how cells transition from one precancerous stage to another until they reach the malignant stage.
Using mathematics, we model these stages through differential equations to describe the transitions: cells in stage 0 precancerous, stage 1 precancerous, and finally stage 2 which is cancerous. Different levels of change rates, denoted by positive constants like \( u_i \), are applied to track each transition.
The equations tell us how the population of cells in each stage changes over time based on these constants. This is crucial for understanding how swiftly normal cells might convert into cancer cells, helping in forecasting and potentially mitigating the progress of conditions like colorectal cancer.

Separable Differential Equations

Separable differential equations are a handy tool for solving complex biological models like our precancerous stage model in cancer progression. These equations allow us to simplify the differential equation by separating variables to solve them separately, making complex equations more manageable.
In the given system, the rate of change of cells in stage 0, represented as \(x'_0 = -u_0 x_0\), is an example of a separable equation. We can rearrange this into two parts, one containing all terms involving the variable \(x_0\) and another with the variable \(t\). This rearranges to \( \frac{dx_0}{x_0} = -u_0 \, dt \). Solving each side separately, we can easily find the general solution for \(x_0(t)\).
This method simplifies solving the progress of cells through each stage, well reflecting the exponential growth or decay found in biological systems. Such equations are not only central to biological research but also to physical sciences and engineering, providing useful insights into rates of reaction and change over time.

Integrating Factors

An integrating factor aids in solving linear differential equations, especially when they can't be directly separated. It’s a method used to transform a non-exact differential equation into an exact one, making integration feasible.
For the differential equation \(x'_1 = u_0 ke^{-u_0 t} - u_1 x_1\), solving directly isn't straightforward. We multiply through by an integrating factor to simplify; this involves finding a function \( \mu(t) \) that when multiplied by every term, converts the equation into one that can be integrated directly.
Typically, the integrating factor is given by \( e^{\int u_1 dt} = e^{u_1 t} \). Multiply the entire differential equation by this factor, making it more manageable: \( e^{u_1 t} x'_1 + u_1 e^{u_1 t} x_1 = u_0 k e^{(u_1-u_0) t} \).
Integrating both sides allows us to solve for \(x_1(t)\), offering insight into the behavior of the cancer progression stages in our model. Integrating factors, thus, are essential in mathematical methods to solve real-world problems efficiently.

Three-Stage Cancer Model

The three-stage cancer model provides a structured approach to understanding how cancer progresses through a simple system of differential equations. In this model, we consider three stages: beginning from a normal cell state, transitioning through a precancerous intermediate state, followed by the malignant cancerous state.
This model simplifies the complex processes that occur in cancer development, aiding researchers in mapping trajectories and predicting outcomes of anti-cancer strategies. With the equations \(x'_0 = -u_0 x_0\), \(x'_1 = u_0 x_0 - u_1 x_1\), and \(x'_2 = u_1 x_1\), each \(x_i\) represents the fraction of population in a particular stage, and the constants \(u_i\) describe the transition rates between stages.
Analyzing these rates separately and together allows one to forecast potential cancer progression, crucial for timely intervention. Solving such a model offers a simplified yet powerful lens to see the possible impacts of growth-inhibiting drugs and lifestyle changes, providing a predictive framework to combat cancer more effectively.