BMOLE 452-689 – Transport Chapter 15. Drug Transport in Solid Tumors BMOLE 425-625 Biomolecular Engineering Engineering in the Life Sciences Era BMOLE 452-689 – Transport Chapter 15. Drug Transport in Solid Tumors Text Book: Transport Phenomena in Biological Systems Authors: Truskey, Yuan, Katz Focus on what is presented in class and problems… Dr. Corey J. Bishop Assistant Professor of Biomedical Engineering Principal Investigator of the Pharmacoengineering Laboratory: pharmacoengineering.com Dwight Look College of Engineering Texas A&M University Emerging Technologies Building Room 5016 College Station, TX 77843 cbishop@tamu.edu Benjamin Franklin and curbs in the street? © 2008 Prof. Anthony Guiseppi-Elie; guiseppi@clemson.edu; T: 001 864 656 1712 F: 001 864 656 1713

Ancient Egyption Medical Text Reports Cancer Edwin Smith Papyrus (1600 BC – likely copies from 2500 BC) Reported cancer has no treatment Reported removing tumors from breast via cauterization Compliments of the New York Academy of Medicine

Enhanced Permeability and Retention Effect Size of particles? How does the cargo more likely arrive? How does the cargo more likely reside? Issues of EPR effect Heterogeneity of tumor EPR effect is enhanced by ectopic tumor formation (lots of research is ectopic-based; interpretation may be biased in some cases) Article: Engineered Nanocrystal Technology: In-vivo fate, targeting and applications in drug delivery.

ICG – NIR Fluorescence (biological tissue attenuates NIR wavelengths less)

Drug transport upon injection Discussion: what would happen if you deliver 1 mL of a drug formulation via a syringe pump over a long period of time versus a fast bolus? Advantages/disadvantages?

Vascular Endothelial Growth Factor Bevacizumab (Avastin) versus ranibizumab (Lucentis) 48 kDa Diffusive properties? Cost? 149 kDa

Vascular Endothelial Growth Factor Bevacizumab (Avastin) versus ranibizumab (Lucentis) 48 kDa Diffusive properties? Cost? 149 kDa How do proteins localize within the cell?

Vascular Endothelial Growth Factor Bevacizumab (Avastin) versus ranibizumab (Lucentis) 48 kDa Diffusive properties? Cost? 149 kDa How do proteins localize within the cell? MLS (core hydrophobic sequence of 5-16 aa at N-terminus), NLS (PKKKRKV, etc.)

Vascular Endothelial Growth Factor Bevacizumab (Avastin) versus ranibizumab (Lucentis) 48 kDa Diffusive properties? Cost? 149 kDa How do proteins localize within the cell? MLS (core hydrophobic sequence of 5-16 aa at N-terminus), NLS (PKKKRKV, etc.), Indirect association possible?

Balamurali Krishna Ambati: Nature. 2006 Oct 26;443(7114):993-7. Epub 2006 Oct 18. Corneal avascularity is due to soluble VEGF receptor-1. sFlt

Programmed Death (PD): PD-L1 and PD-1 Immune checkpoint inhibitors

Anti-vascular and immunotherapies against tumors: do they conflict Anti-vascular and immunotherapies against tumors: do they conflict? Discussion

Quantitative analysis of interstitial fluid transport Interstitial (space between objects)

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning?

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning? Resists shear flow and strain linearly with stress

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning? Resists shear flow and strain linearly with stress σ= nonlinear f

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning? Resists shear flow and strain linearly with stress σ= nonlinear f σ =f(E) (constitutive equation of tissues) Where E = strain tensor in tissue

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning? Resists shear flow and strain linearly with stress σ= nonlinear f σ =f(E) (constitutive equation of tissues) Where E = strain tensor in tissue K=4.6E-13[GAG]-1.202*(μ37C/ μ)

Quantitative analysis of interstitial fluid transport Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning? Resists shear flow and strain linearly with stress σ= nonlinear f σ =f(E) (constitutive equation of tissues) Where E = strain tensor in tissue φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) K=4.6E-13[GAG]-1.202*(μ37C/ μ)

ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝐾𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝐾= 𝑘μ ρ𝑔

ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝐾𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝐾= 𝑘μ ρ𝑔 Steady state: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 = φ 𝐵 =𝛻 ϵ 𝑉 𝑓

ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝐾𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝐾= 𝑘μ ρ𝑔 Steady state: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 = φ 𝐵 =𝛻 ϵ 𝑉 𝑓 ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝐾𝛻 𝑝 𝑖 ; V f =− 𝐾𝛻 𝑝 𝑖 ϵ −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center.

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. 29

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor.

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 )

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 p i = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑠

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 p i = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑠

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 p i = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑠 −𝐾 𝛻 2 φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑆 = φ 𝐵

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 p i = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑠 −𝐾 𝛻 2 φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑆 = φ 𝐵

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑃𝑉=𝑛𝑅𝑇 𝑠𝑜 𝑃= 𝑛 𝑉 𝑅𝑇=𝐶𝑅𝑇 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 p i = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑠 −𝐾 𝛻 2 φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑆 = φ 𝐵

Application of −𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 Assume tumor is spherical and that the transport of the fluid is spherically symmetrical around the center. The model has only one feeding artery and one vein. Assume the interstitial fluid pressure is approximately zero at the surface of the tumor. φ 𝐵 = 𝐽 𝑣 𝑉 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝐵 − 𝑝 𝑖 − σ 𝑠 ( 𝜋 𝐵 − 𝜋 𝑖 ) 𝐶𝑜𝑛𝑑𝑒𝑛𝑠𝑒: 𝑝 𝑒 = 𝑝 𝐵 − σ 𝑠 𝜋 𝐵 − 𝜋 𝑖 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝑃𝑉=𝑛𝑅𝑇 𝑠𝑜 𝑃= 𝑛 𝑉 𝑅𝑇=𝐶𝑅𝑇 𝑝 𝑖 𝐿 𝑝 𝑆 𝑉 = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 p i = φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑠 −𝐾 𝛻 2 φ 𝐵 − 𝐿 𝑃 𝑆 𝑃 𝑒 𝑉 𝑉 𝐿 𝑝 𝑆 = φ 𝐵 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 ≈ 𝜕 𝑝 𝑖 𝑑𝑡 = 𝜑 𝐵 =−𝐾 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 ≈ 𝜕 𝑝 𝑖 𝑑𝑡 = 𝜑 𝐵 =−𝐾 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 ≈ 𝜕 𝑝 𝑖 𝑑𝑡 = 𝜑 𝐵 =−𝐾 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒

1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = α 2 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝑤ℎ𝑒𝑟𝑒 α= 𝐿 𝑝 𝑆 𝐾𝑉 Eqn: 15.3.11 𝑑𝐶 𝑑𝑡 =𝐷Δ𝐶=𝐷 𝛻 2 𝐶 Fick’s 2nd law: Reminder: 𝑑 𝑝 𝑖 𝑑𝑡 =−𝐾 𝛻 2 𝑝 𝑖 = φ 𝐵 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = L p Pg . 448 (section 9.3.2) ≈ 𝜕 𝑝 𝑖 𝑑𝑡 = 𝜑 𝐵 =−𝐾 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝑉 𝑝 𝑒 − 𝑝 𝑖 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒

𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = 𝐿 𝑝 Pg . 448 (section 9.3.2) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 LHS units? RHS units?

𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = 𝐿 𝑝 Pg . 448 (section 9.3.2) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 = 𝐾 𝑚 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐾 𝑚 3 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2

In the book it states that: 𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = 𝐿 𝑝 Pg . 448 (section 9.3.2) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 = 𝐾 𝑚 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐾 𝑚 3 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 In the book it states that: 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 𝑤ℎ𝑒𝑟𝑒 𝛼= 𝐿 𝑝 𝑆 𝐾𝑉 𝑎𝑛𝑑 𝑠𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒 𝑜𝑓𝑓 𝑏𝑢𝑡 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑢𝑠𝑖𝑛𝑔 𝐿 𝑝 𝑚 =𝐾 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑠𝑜 𝑖𝑡 𝑖𝑠 𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑.

In the book it states that: 𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = 𝐿 𝑝 Pg . 448 (section 9.3.2) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 = 𝐾 𝑚 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐾 𝑚 3 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 In the book it states that: 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 𝑤ℎ𝑒𝑟𝑒 𝛼= 𝐿 𝑝 𝑆 𝐾𝑉 𝑎𝑛𝑑 𝑠𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒 𝑜𝑓𝑓 𝑏𝑢𝑡 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑢𝑠𝑖𝑛𝑔 𝐿 𝑝 𝑚 =𝐾 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑠𝑜 𝑖𝑡 𝑖𝑠 𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑. 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 = 𝐿 𝑝 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐿 𝑝 𝑚 𝑚 3 𝑚 2 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2

In the book it states that: 𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = 𝐿 𝑝 Pg . 448 (section 9.3.2) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 = 𝐾 𝑚 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐾 𝑚 3 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 In the book it states that: 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 𝑤ℎ𝑒𝑟𝑒 𝛼= 𝐿 𝑝 𝑆 𝐾𝑉 𝑎𝑛𝑑 𝑠𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒 𝑜𝑓𝑓 𝑏𝑢𝑡 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑢𝑠𝑖𝑛𝑔 𝐿 𝑝 𝑚 =𝐾 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑠𝑜 𝑖𝑡 𝑖𝑠 𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑. The book should have been more clear. 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 = 𝐿 𝑝 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐿 𝑝 𝑚 𝑚 3 𝑚 2 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2

In the book it states that: 𝐿 𝑃 = 𝐽 𝑣 𝑆∆𝑝 (section 9.4) 𝐾 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ = 𝐿 𝑝 Pg . 448 (section 9.3.2) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 −𝐾𝑉 𝑝 𝑒 − 𝑝 𝑖 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑝 𝑖 − 𝑝 𝑒 = 𝐾 𝑚 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐾 𝑚 3 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 In the book it states that: 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 𝑤ℎ𝑒𝑟𝑒 𝛼= 𝐿 𝑝 𝑆 𝐾𝑉 𝑎𝑛𝑑 𝑠𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒 𝑜𝑓𝑓 𝑏𝑢𝑡 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑢𝑠𝑖𝑛𝑔 𝐿 𝑝 𝑚 =𝐾 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑠𝑜 𝑖𝑡 𝑖𝑠 𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑. The book should have been more clear. 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 = 𝐿 𝑝 𝑚 2 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐿 𝑝 𝑚 𝑚 3 𝑚 2 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑚 2 Solve for pi(r)

Apply B.C.s => f(sinh()) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 15.3.11 Solve for pi and Apply B.C.s => f(sinh()) On board. How do we get the velocity profile after we know pi?

Apply B.C.s => f(sinh()) 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝑝 𝑖 𝜕𝑟 = 𝐿 𝑝 𝑆 𝐾𝑉 𝑅 2 𝑝 𝑖 − 𝑝 𝑒 ;𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 15.3.11 Solve for pi and Apply B.C.s => f(sinh()) On board. How do we get the velocity profile after we know pi? Darcy’s Law: 𝑉 𝑓 =𝐾𝛻 𝑝 𝑖 ⇒𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 15.3.16

Quantitative analysis of interstitial fluid transport (Repeat) Mass conservation law and momentum balance equations: 𝛻 1−ϵ 𝜕𝑢 𝜕𝑡 +ϵ 𝑉 𝑓 = φ 𝐵 − φ 𝐿 =∆φ ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor ϵ 𝑉 𝑓 − 𝜕𝑢 𝜕𝑡 =−𝑘𝛻 𝑝 𝑖 ;𝑘 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑣μΔ𝑥 Δ𝑝 𝑎𝑛𝑑 𝑘= 𝐾μ ρ𝑔 𝛻σ=0 Biological tissues are viscoelastic – meaning? Resists shear flow and strain linearly with stress σ= nonlinear f σ =f(E) (constitutive equation of tissues) Where E = strain tensor in tissue

Based on what we know, can this simplify?

Table 15.1 First observed that IFP in tumors is higher than normal tissues in 1950ish.

Table 15.1 First observed that IFP in tumors is higher If α2 is >> 1 or <1 then what does that mean? What is the gradient of the pressure near the center of the tissue? First observed that IFP in tumors is higher than normal tissues in 1950ish. What is the approx. magnitude of α2 for a murine mammary adenocarcinoma? What does this mean? What is the rate-limiting factor for transport within the tumor?

Table 15.1 First observed that IFP in tumors is higher than normal tissues. Drugs get into the very center of tumors via what?

Unsteady-state fluid transport No general constitutive equation for biological tissues: Tissue deformation is coupled with flow

Unsteady-state fluid transport No general constitutive equation for biological tissues: Tissue deformation is coupled with flow   ΦB/L = rate of fluid extravasation from Blood vessels / unit tissue vol. and the rate of lymphatic drainage / unit tissue vol. Vf = average fluid velocity in the fluid space… u is the average displacement in the solid phase ϵ =fractional volume of interstitial fluid K = hydraulic conductivity of tissues σ = effective stress tensor      

What is e (volume dilatation) physically? Momentum governing equation in unsteady-state fluid transport…       Lamé Constants: generally:    

What type of nostalgic feelings do you have about this equation? Momentum governing equation in unsteady-state fluid transport… What type of nostalgic feelings do you have about this equation?

What type of nostalgic feelings do you have about this equation? Momentum governing equation in unsteady-state fluid transport… What type of nostalgic feelings do you have about this equation? What is this analogous to?

What type of nostalgic feelings do you have about this equation? Momentum governing equation in unsteady-state fluid transport… What type of nostalgic feelings do you have about this equation? What is this analogous to? Compliments of http://www.hdwallpaper.nu/trippy-wallpapers/

Quantitative analysis of interstitial transport of solutes

Quantitative analysis of interstitial transport of solutes

Quantitative analysis of interstitial transport of solutes

Quantitative analysis of interstitial transport of solutes

Quantitative analysis of interstitial transport of solutes KK equation

Quantitative analysis of interstitial transport of solutes KK equation Patlak equation

KK equation

KK equation

KK equation Vascular surface area Per unit tissue volume KAV=available solute fraction in the tissue Fluid flow rate per Unit tissue volume

KK equation Vascular surface area Per unit tissue volume KAV=available solute fraction in the tissue Fluid flow rate per Unit tissue volume Patlak equation

KK equation Vascular surface area Per unit tissue volume KAV=available solute fraction in the tissue Fluid flow rate per Unit tissue volume Patlak equation