11.6 Case Study: Tumor Growth

Case Study: Tumor Growth

  • We have developed the theory of diffusive mass transport in Ch9.8.
  • We now show how this mechanism can be used in medical research.
  • We examine growth of tumors through oxygen availability.

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Case Study: Tumor Growth

  • The case study is based on an early model of tumor growth by Greenspan (1972).
  • The purpose of the study was to explain why some tumors do not grow beyond a certain size.

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Background

  • Cancer cells proliferate (divide) at faster rate than normal.
  • This growth often results in spherical geometry called tumors.
  • Earliest stages of tumor development are regulated by availability of nutrients.
  • Tumor growth requires oxygen from surrounding tissues.

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Background

  • Some tumors grow to a fixed size of just a few millimeters across, while others grow very large.
  • What causes some to be limited in their growth is a decline in available nutrients.

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Tumour Types

  • Nutrients such as oxygen diffuse into tumor from surface with declining oxygen availability towards center.

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Tumour Types

  • Other tumors secrete chemicals that cause surrounding blood vessels to grow into them and supply oxygen equally to all parts of the tumor.
  • These tumors can grow much larger.

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Case Study: Tumour Growth

  • We focus on tumors that only grow to a fixed size.
  • This type of tumor is called nonvascular, or avascular.
  • We investigate mechanisms that limit growth.
  • When tumor is small oxygen can reach all parts easily.
  • As it increases in size, live cells close to surface absorb most of oxygen with less oxygen available towards center.

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Case Study: Tumour Growth

  • If oxygen concentration falls below a critical level, then cells stop dividing and a central quiescent core develops.
  • The growth rate of tumor will then decline markedly (Greenspan, 1972).

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3 Phases of Growth for Avascular Tumors

  • Proliferating cells cause tumor mass to grow quickly.
  • A core of quiescent cells form with a surrounding proliferating layer.
  • A necrotic core of dead and decomposing cells forms
  • This core is surrounded by inner layer of quiescent cells and an outer layer of proliferating cells.

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3 Phases of Growth for Avascular Tumors

  • Focus: Calculate oxygen concentration inside tumor during first phase of growth.

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Modeling the First Growth Phase

Assumptions

  • Assume spherical tumor.
  • Assume oxygen diffuses into tumor from surface towards center.

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Additional Variables & Assumptions

  • Assume oxygen concentration quickly reaches equilibrium (faster than tumor grows in size).
  • \( C(r)= \) equilibrium oxygen concentration
  • \( r= \) radial distance from center of tumor
  • \( C(r) \) is at equilibrium with respect to time
  • \( R(t)= \) tumor radius at time \( t \)

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First Growth Phase


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  • Oxygen concentration \( C(r) \) decreases as \( r \) decreases
  • Spherical Diffusion (9.39):

\[ D \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dC}{dr} \right) + M(r) = 0 \]

  • \( M(r) < 0 \)
    • cells absorb oxygen
  • \( D \) = diffusion constant

First Growth Phase (Cont.)

  • Assume constant oxygen absorption rate, \( M(r) = -A_0 \)
  • Assume boundary conditions given for \( 0 < r < R(t) \)
  • Can solve for \( C(r) \) (see Exercise 11.17):

\[ C(r) = c_1 - \frac{A_0}{6D} \left( R^2 - r^2 \right) \]

  • \( c_1 \) = initial concentration of oxygen on surface (\( r= R \))

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First Growth Phase (Cont.)

  • From previous slide,

\[ C(r) = c_1 - \frac{A_0}{6D} \left( R^2 - r^2 \right) \]

  • This solution is only valid while \( C(r) > c_q \)
  • \( c_q \) = oxygen concentration at which cells become quiescent.

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First Growth Phase (Cont.)

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  • What is \( r_q \) at which tumor cells become quiescent?

\[ C(r) = c_1 - \frac{A_0}{6D} \left( R^2 - r^2 \right) \]

  • Note \( C(0) = c_q \) at \( r = 0 \)
  • Solve for \( R = r_q \):

\[ r_q = \sqrt{ \left( c_1 - c_q \right) \frac{6D}{A_0} } \]

First Growth Phase (Cont.)

  • Radius \( r_q \) decreases as \( c_1 - c_q \) decreases
  • Radius \( r_q \) increases as absorption rate \( A_0 \) decreases.

\[ r_q = \sqrt{ \left( c_1 - c_q \right) \frac{6D}{A_0} }, \text{ where } c_1 > c_q \]

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Core Expanding

  • As cells in proliferating layer starve and become more inactive, quiescent core actually expands.

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Finding the Equations

  • Let \( R_q(t) \) be time-dependent radius of inactive core.
  • Let \( R(t) \) be time-dependent outer radius of tumor.

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Finding the Equations

  • Once core has formed, there are two different regions with distinct dynamics, therefore we will need two different diffusion equations.

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Finding the Equations (cont.)

  • Let \( C_p(r) \) be equilibrium oxygen concentration in proliferating layer.
    • In proliferating layer, cells absorb oxygen.
  • Let \( C_q(r) \) be the equilibrium oxygen concentration in the inactive core.
    • In quiescent core, cells cannot absorb oxygen.

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Finding the Equations (cont.)

  • Oxygen diffuses into region of inactive layer with \( C_q(r) < c_q \) but no oxygen is consumed there.
  • Mass production rate \( M(r) = 0 \) for \( 0 < r< R_q(t) \)
  • Also, \( M(r) = - A_0 \) for \( R_q(t) < r < R(t) \).

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Diffusion Equations for Layers

  • Recall spherical diffusion equation:

\[ \frac{D}{r^2} \frac{d}{dr} \left( r^2 \frac{dC}{dr} \right) + M(r) = 0 \]

  • For the quiescent layer, \( M(r) = 0 \):

\[ \begin{aligned} \frac{D}{r^2} \frac{d}{dr} \left( r^2 \frac{dC_q}{dr} \right) = 0, \,\, 0 < r < R_q(t) \end{aligned} \]

  • For the proliferating layer, \( M(r) = - A_0 \):

\[ \frac{D}{r^2} \frac{d}{dr} \left( r^2 \frac{dC_p}{dr} \right) - A_0 = 0, \,\, R_q(t) < r < R(t) \]

Boundaries

  • There is a boundary at outer surface.
  • There is a shared boundary between quiescent and proliferating layers.
  • The center of core is also considered a boundary.

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Boundary Conditions

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  • At outer surface, concentrations must match:

\[ C_{p}(R) = c_1 \]

  • At shared boundary, concentrations match:

\[ C_p(R_q) = C_q(R_q) = c_q \]

  • At shared boundary, mass fluxes must match:

\[ J_q(R_q) = J_p(R_q) \]

  • At center, flux is zero & use Fourier's Law:

\[ J_{q}(0) = 0 \Rightarrow C'_{q}(0) = 0 \]

Avascular Tumor Growth

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  • The avascular tumors discussed in this case study are small.
  • These can be 1-3 mm across, and cause few health complications.
  • Usually they cease expansion altogether.

Avascular Tumor Growth

  • They become vascular when they modify blood vessels to form new capillaries.
  • These capillaries extend into the tumor and increase availability of oxygen to the tumor.
  • This can lead to explosive growth and spread the cells into other parts of the body through direct access to blood supply.

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Conclusion

  • It is important to understand process of vascularization and the conditions under which it occurs.
  • The model we have examined is an early model, since then more research has been done.

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