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3 types of Tumor growth equations from HW

\frac{d V}{dt}=k
1st order, non-linear, heterogeneous, separable
Solution:
V(t) = kt+V_{0}

\frac{d V}{dt}=kV
1st order, linear, homogeneous, separable
Solution:
V(t) = V_{0}e^{kt}

\frac{d V}{dt}=kV(1-\frac{V}{Vmax})
1st order, non-linear, homogeneous, separable
Solution:
V(t) = \frac{V_{max}}{1+\frac{V_{max}-V_{0}}{V_{0}}e^{-kt}}

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Diffusion-partitioning controlled arterial drug uptake and distribution

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Diffusion-partitioning controlled arterial drug uptake and distribution

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https://doi.org/10.1016/j.jconrel.2017.08.037

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Calcified plaque modification alters local drug delivery in the treatment of peripheral atherosclerosis

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Ultrasound Wave Equation

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(D: diffusion coefficient)
Math ingredients: rate, Laplacian, reaction term
Type: PDE second order, homogenous, linear

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S(t) - susceptible
I(t) - infectious
R(t) - removed (recovered)

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where alpha is the infection rate and beta is the recovery rate and
S+I+R = N (total population)

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system of equations to solve amount of products and reactions with time 

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\dot{u} = D\triangle u +f(u)

Math ingredients:

Type:

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A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors - Yin - 2019 - CPT: Pharmacometrics &amp; Systems Pharmacology - Wiley Online Library

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Momentum balance (without inertial effects)

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Momentum balance (without inertial effects)

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\boxed{\nabla\cdot\boldsymbol\sigma=0}\\
(\boldsymbol\sigma: \text{Cauchy stress tensor})

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Momentum balance (with inertial effects)

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Momentum balance (with inertial effects)

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\boxed{\nabla\cdot\boldsymbol\sigma=\rho \frac{D{\boldsymbol{v}}}{Dt}} \\ (\rho: \text{density}, \boldsymbol{v}: \text{velocity}, \\ D/Dt: \text{TOTAL time derivative})

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\dot{u} = D \triangle u
(D: diffusion coeff)
Math ingredients: rate, Laplacian
Type: PDE second order homogeneous, linear if D not a function of u, nonlinear otherwise

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Momentum balance with inertial effects and body force

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Momentum balance with inertial effects and body force

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\boxed{\nabla\cdot\boldsymbol\sigma + f=\rho \frac{D{\boldsymbol{v}}}{Dt}} \\

(f: \text{body force, e.g. gravity)})

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-MRI/NMR
-The Bloch Equations

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saturation rate= Vmax
number of molecules inside cell= n
constant determining speed of saturation (mol/cm⁴)= K
constant determining speed of saturation (mol/cm⁴)= C

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Generic Form: *See picture
Potential BME Application: Bone Models
Another application: Creep behavior of other materials

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E = electric field, B = magnetic field
Mathematical Ingredients: curl, rate
Type: homogenous linear first order PDE

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General Form: u-dot = f(u)

Math Ingredient: Reaction term

Application: Population growth (cancer cell growth)

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\triangle u = f(x,y,z,t)
Math ingredient: Laplacian, source term
Type: linear PDE second order heterogeneous

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\triangle u =0
Math ingredient: Laplacian
Type: linear PDE second order homogeneous

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Enter equation and its name if any

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another equation

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epsilon is the EM work done when a unit charge loops once
phi_B is the magnetic flux

How MRI works:
1. EM coils in machine produce strong magnetic field (MF) and force protons in the body to align with that field.
2. A radiofrequency (RF) current is then pulsed through the patient, the protons are stimulated, and spin out of equilibrium.
3. When the RF field is turned off, the MRI sensors can detect the energy released as the protons realign with the MF.
4. The time it takes for the protons to realign and the amount of energy released depends on environment and chemistry of the molecules. This is encoded by the machine as varying degrees of brightness; the faster the alignment, the brighter the image.

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Matrix representation of Bloch equations

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Directional equations describing MF-induced spin

M_x,y,z are directional components of nuclear magnetization
γ is gyromagnetic ratio (describes angular momentum of particle)
B_x,y,z are directional components of MF
T_1,2 are times of 1) return to direction of MF, and 2) return of spin back to baseline

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Visual Representation of Bloch Equations

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Visual Representation of Bloch Equations