Differential Equations (2/2) by Alex https://padlet.com/bmes698/27yxvax7x5ki8xs5 Summary type of equation vs BME application en-us 2023-09-19 21:38:20 UTC 2023-10-27 14:26:43 UTC hello@padlet.com Insert potential BME application alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2711591440 Enter equation and its name if any]]> 2023-09-19 21:40:33 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2711591440 Another application alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2711591709 another equation]]> 2023-09-19 21:40:57 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2711591709 Generic form alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2711592980 \triangle u =0
Math ingredient: Laplacian
Type: linear PDE second order homogeneous
]]> 2023-09-19 21:42:26 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2711592980 Generic form alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712936373 \triangle u = f(x,y,z,t)
Math ingredient: Laplacian, source term
Type: linear PDE second order heterogeneous
]]> 2023-09-20 14:33:27 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712936373 Reaction sgphilli https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712942080 General Form: u-dot = f(u)

Math Ingredient: Reaction term

Application: Population growth (cancer cell growth)]]> 2023-09-20 14:36:42 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712942080 Maxwell–Faraday equation sgorlews https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712949709 E = electric field, B = magnetic field
Mathematical Ingredients: curl, rate
Type: homogenous linear first order PDE]]> 2023-09-20 14:40:35 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712949709 Standard Linear Solid (SLS) Model https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712950133 Generic Form: *See picture 
Potential BME Application: Bone Models 
Another application: Creep behavior of other materials ]]> 2023-09-20 14:40:49 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712950133 Diffusion through cell membrane https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712953305 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]]> 2023-09-20 14:42:30 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712953305 alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712954945 2023-09-20 14:43:28 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712954945 Applications carledge3 https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712958643 -MRI/NMR
-The Bloch Equations]]> 2023-09-20 14:45:31 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2712958643 Generic form alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2714701749 \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
]]> 2023-09-21 13:48:09 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2714701749 Momentum balance (without inertial effects) alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715145977 \boxed{\nabla\cdot\boldsymbol\sigma=0}\\ (\boldsymbol\sigma: \text{Cauchy stress tensor})]]> 2023-09-21 18:55:28 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715145977 Momentum balance (with inertial effects) alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715146561 \boxed{\nabla\cdot\boldsymbol\sigma=\rho \frac{D{\boldsymbol{v}}}{Dt}} \\ (\rho: \text{density}, \boldsymbol{v}: \text{velocity}, \\ D/Dt: \text{TOTAL time derivative})

]]> 2023-09-21 18:55:55 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715146561 Momentum balance with inertial effects and body force alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715162474 \boxed{\nabla\cdot\boldsymbol\sigma + f=\rho \frac{D{\boldsymbol{v}}}{Dt}} \\ (f: \text{body force, e.g. gravity)})]]> 2023-09-21 19:09:06 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715162474 Solution alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715208765 ???]]> 2023-09-21 19:54:01 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715208765 Solution alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715209478 ???]]> 2023-09-21 19:54:45 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2715209478 Solution alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2717530466 ???]]> 2023-09-23 23:08:04 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2717530466 Resources about tumor growth alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2721343590 A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors - Yin - 2019 - CPT: Pharmacometrics & Systems Pharmacology - Wiley Online Library]]> 2023-09-26 14:14:38 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2721343590 wtwilson https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732188113 2023-10-04 14:27:13 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732188113 Navier-Stokes Equation wtwilson https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732191536 2023-10-04 14:29:04 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732191536 System for reaction-diffusion https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732196513 2023-10-04 14:31:43 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732196513 General form? alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732199745 \dot{u} = D\triangle u +f(u)

Math ingredients:

Type:

]]> 2023-10-04 14:33:28 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732199745 How to solve it? alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732200988 2023-10-04 14:34:09 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732200988 BME applications? alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732201383 2023-10-04 14:34:24 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732201383 Equation sgphilli https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732202409 2023-10-04 14:35:00 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732202409 enzyme reaction kinetics https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732203110 2023-10-04 14:35:22 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732203110 Wave Equation sgphilli https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732204515 2023-10-04 14:36:05 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732204515 COVID model? alexguevel https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732205603 2023-10-04 14:36:41 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732205603 Generic Form carledge3 https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732207404 (D: diffusion coefficient)
Math ingredients: rate, Laplacian, reaction term
Type: PDE second order, homogenous, linear]]> 2023-10-04 14:37:37 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732207404 Application (Wave Equation) https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732210300 Ultrasound Wave Equation]]> 2023-10-04 14:39:05 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732210300 Possible Windkessel Equation sgphilli https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732210654 2023-10-04 14:39:17 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732210654 Diffusion-partitioning controlled arterial drug uptake and distribution https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732210823 https://doi.org/10.1016/j.jconrel.2017.08.037]]> 2023-10-04 14:39:24 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732210823 Brownian Motion https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732211638 2023-10-04 14:39:51 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732211638 Faraday's Law of Induction sgorlews https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732214742
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.]]> 2023-10-04 14:41:35 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732214742 Bloch Equations sgorlews https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732255775 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

]]> 2023-10-04 15:04:15 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732255775 Visual Representation of Bloch Equations sgorlews https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732256647 2023-10-04 15:04:45 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2732256647 Tumor Growth jordisblackburn https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2759200936 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}}

]]> 2023-10-23 14:22:10 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2759200936 SIR model example jordisblackburn https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2759224712 S(t) - susceptible
I(t) - infectious 
R(t) - removed (recovered)]]> 2023-10-23 14:34:18 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2759224712 SIR Model equations jordisblackburn https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2759228209 where alpha is the infection rate and beta is the recovery rate and 
S+I+R = N (total population)]]> 2023-10-23 14:36:16 UTC https://padlet.com/bmes698/27yxvax7x5ki8xs5/wish/2759228209