Paper CritiqueCationic Nanoparticles Have Superior Transvascular Flux into Solid

Tumors: Insights from a Mathematical Model

T. Stylianopoulos, K. Soteriou, D. Fukumura, R. Jain

Presented by: Nils Persson

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Challenge

• Deliver nanoparticles into the tumor interstitium uniformly and selectively

• Resistances to this goal:– Flow through capillaries– Flow through capillary pores– Diffusion through interstitium– Drainage to normal tissue

3

Structure of the Model

Capillary – Hagen Poiseuille flowConvection only

PoresHindered

stokesflow

TumorPorous flow and diffusion

No drainageNormal tissueP = 0

DiffusionConvection

Grid 1

Grid 2

4

Structure of the Model

Capillary – Hagen Poiseuille flowConvection only

TumorPorous flow and diffusion

No drainageNormal tissueP = 0

DiffusionConvection

Grid 1

Grid 2

PoresHindered

stokesflow

5

Structure of the Model

Capillary – Hagen Poiseuille flowConvection only

TumorPorous flow and diffusion

No drainageNormal tissueP = 0

DiffusionConvection

Grid 1

Grid 2

Effect of charge,Ionic strength,Pore size

PoresHindered

stokesflow

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Two steps: Pressure, then Concentration

• Steady state pressure profile

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Capillary – Hagen Poiseuille flowConvection only

TumorPorous flow and diffusion

No drainageNormal tissueP = 0

DiffusionConvection

Grid 1

Grid 2

Two steps: Pressure, then Concentration

PoresHindered

stokesflow

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Two steps: Pressure, then Concentration

• RK4 time stepping with centered finite differences to solve C(r,t)– Not reported, most interesting result

• Figure of merit: Vascular permeability or transvascular flux, Peff

– Similar to a mass transfer coefficient into tumor:

Averaged concentration, C

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Representative Results

Fluid in tumor stagnatesPore diameter is too restrictive

Peff

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Critiques• Blood viscosity/compressibility• Capillary Compliance• Dilute particle assumption• Concentration BCs unspecified and unrealistic• Transvascular flux is over-fit• Deen’s theory is shaky for attractive charges• Centerline approximation invalid for attractive charges• “lacking information” on glycocalyx charge, the crux of the entire

paper• Neglected axial diffusion even in areas of zero flow• 2-dimensional percolation network was unnecessary to obtain results• Lead author did not apply results of a previous modeling study

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Critiques• Blood viscosity/compressibility• Capillary Compliance• Dilute particle assumption• Concentration BCs unspecified and unrealistic• Transvascular flux is over-fit• Deen’s theory is shaky for attractive charges• Centerline approximation invalid for attractive charges• “lacking information” on glycocalyx charge, the crux of the entire

paper• Neglected axial diffusion even in areas of zero flow• 2-dimensional percolation network was unnecessary to obtain results• Lead author did not apply results of a previous modeling study

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Blood Flow

• Modeled with Hagen-Poiseuille equation– Assumes laminar flow, smooth pipe, constant density and viscosity

(3 mmHg-s = 4 mPa-s)

• Properties of blood:– Viscoelastic: erythrocyte deformability and aggregation– Shear thinning– Viscosity varies with hematocrit level

• Plasma:– Newtonian– µ = 1.2 mPa-s (water = 0.65 mPa-s @ 37 °C)

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Dilute Particle Assumption

• Deen’s theory assumes no solute-solute interactions (continuous solvent)

• Nanoparticles are in pores with plasma, concentration of albumin (~10 nm) is about 0.5 wt%

• Deen predicts ~10% higher partitioning into pores at this concentration

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Where does Deen’s theory break down?

• “Only for attractive interactions (E<0) is there likely to be a problem.” – Deen, 1987

• Hindrance integrals blow up

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The Centerline Approximation

• Says that hindrance coefficients are those of r=0– Very good for repulsive charges– Very bad for attractive charges

- - - - - -

- - - - - -

Deen, 1987

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The glycocalyx – source of charge• Layer of carbohydrates on

inner wall of capillary– But, in pores?

• Negatively charged, ~-0.022 C/m21

-used -0.05 C/m2

• 0.5 µm thick – too thick for capillary pores

• Acts as molecular sieve and shear stress transducer

1Donath et al., 1996, image Reitsma et al., 2007

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Previous Studies

• Campbell and Jain et al., 2002– Tumor uptake not affected by charge, however,

accumulation in tumor vessels doubled for cationic particles

• Stylianopoulos, et al., 2010– Positive charge hinders

diffusion in the interstitium– Not applied to this model

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Conclusions

• Very powerful model used to study known charge phenomena– Many bad approximations, missing data

• Much greater insight is provided on the transport phenomena and pore size distributions

• Slight modifications could yield very realistic results

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Dimensionless Quantities, 400nm pores, 60nm particles

DiffusionConvection

Grid 1

Grid 2

D ~ 10-13 m2/s= 10-9 cm2/s

V:4E-5 m/sPe,r = 9Pe,L = 220Re = 4E-6∆P = 1.25 mmHg

V:2.5E-4 m/sPe,r = 53Pe,L = 1300Re = 2.5E-5∆P = 7.5 mmHg

5µm

15µm

V:2.3E-4 m/sPe,L = 2.3E6Re = 8.6E-4∆P = 20 mmHg (max)

V:2E-6 cm/sPe,L = 100Re << 1∆P = 1.25 mmHg“L” = 0.5cm

Assumes Ac is void surface area

V:2.4E-5 cm/sPe,L = 600Re << 1∆P = 7.5 mmHg“L” = 0.25cm

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Reynolds Number in Pores… <<1∆P = 0.05 = 1.25 mmHg ∆P = 0.3 = 7.5 mmHg

166.6 Pa 999.7 PaR (nm) R (m) v (m/s) Re v (m/s) Re

10 1.00E-08 1.0E-07 5.2E-10 6.2E-07 3.1E-0950 5.00E-08 2.6E-06 6.5E-08 1.6E-05 3.9E-07

100 1.00E-07 1.0E-05 5.2E-07 6.2E-05 3.1E-06150 1.50E-07 2.3E-05 1.8E-06 1.4E-04 1.1E-05200 2.00E-07 4.2E-05 4.2E-06 2.5E-04 2.5E-05250 2.50E-07 6.5E-05 8.1E-06 3.9E-04 4.9E-05300 3.00E-07 9.4E-05 1.4E-05 5.6E-04 8.4E-05350 3.50E-07 1.3E-04 2.2E-05 7.7E-04 1.3E-04400 4.00E-07 1.7E-04 3.3E-05 1.0E-03 2.0E-04450 4.50E-07 2.1E-04 4.7E-05 1.3E-03 2.8E-04500 5.00E-07 2.6E-04 6.5E-05 1.6E-03 3.9E-04

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Peclet Numbers in Pores… >>1

∆P = 1.25 mmHg ∆P = 7.5 mmHg166.61 Pa 999.67 Pa

R (nm) R (m) v (m/s) D(m^2/s) Pe, r Pe,L v (m/s) D(m^2/s) Pe, r Pe,L10 1.00E-08 1.0E-07 1.89E-11 5.5E-05 2.8E-02 6.2E-07 1.89E-11 3.3E-04 1.7E-0150 5.00E-08 2.6E-06 3.78E-12 3.4E-02 3.4E+00 1.6E-05 3.78E-12 2.1E-01 2.1E+01

100 1.00E-07 1.0E-05 1.89E-12 5.5E-01 2.8E+01 6.2E-05 1.89E-12 3.3E+00 1.7E+02150 1.50E-07 2.3E-05 1.26E-12 2.8E+00 9.3E+01 1.4E-04 1.26E-12 1.7E+01 5.6E+02200 2.00E-07 4.2E-05 9.46E-13 8.8E+00 2.2E+02 2.5E-04 9.46E-13 5.3E+01 1.3E+03250 2.50E-07 6.5E-05 7.57E-13 2.2E+01 4.3E+02 3.9E-04 7.57E-13 1.3E+02 2.6E+03300 3.00E-07 9.4E-05 6.30E-13 4.5E+01 7.4E+02 5.6E-04 6.30E-13 2.7E+02 4.5E+03350 3.50E-07 1.3E-04 5.40E-13 8.3E+01 1.2E+03 7.7E-04 5.40E-13 5.0E+02 7.1E+03400 4.00E-07 1.7E-04 4.73E-13 1.4E+02 1.8E+03 1.0E-03 4.73E-13 8.5E+02 1.1E+04450 4.50E-07 2.1E-04 4.20E-13 2.3E+02 2.5E+03 1.3E-03 4.20E-13 1.4E+03 1.5E+04500 5.00E-07 2.6E-04 3.78E-13 3.4E+02 3.4E+03 1.6E-03 3.78E-13 2.1E+03 2.1E+04

Assumes γ=0.3 (similar to paper)Assumes Stokes-Einstein diffusivityAssumes Hagen-Poiseuille velocity in pores

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Data

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Ionic Strength

• I (mol/kg) = ½ Σmizi2

– Seawater: 0.72 mol/kg– In paper: varied from 0 to 0.15M

• Debye length κ-1 ~ I-1/2

– 1 nm• ln γ ~ κ ~ I1/2

Increased concentration = decreased Debye length = lower potential

Debye LengthsI K-1 (m) e0 8.85E-12

0.005 4.38E-09 eR 780.01 3.10E-09 R 8.3140.06 1.26E-09 T 3100.15 7.99E-10 dS 1000

Na 6.02E+23e 1.60E-19

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H and W for negative particles

Lambda = 0.3I = 0.15

3

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Flux of negatively charged particles

Q = -.05

Lambda = 0.3

Both 400nm/60nm

4

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More negative charge fluxes

positive

NegativeI = .15

5

400nm/60nm

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Varying pore size

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Fitting the Vascular Permeability

• C(t) linear – why?

• They fit an exponential0

100200

300400

500600

700800

9001000

0.00

0.05

0.10

0.15

0.20

0.25

f(x) = 0.0002199202 x + 0.00040044141R² = 0.999948648166217

Time (sec)

Dim

ensi

onle

ss C

on-

cent

ratio

n

Source: Data Thief

Large Kd yields vastly simpler eqn.Kd ~ 80,000

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No flowpredicted

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No flow to the tumor

Wu et al. 2008

- Also predicted by a paper they cite

- Nanoparticles reach tumor’s edge, then spill out

- This paper also treats viscosity as a function of hematocrit

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Transport Properties of a Tumor

Chauhan et al. 2011

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Starling’s Law

• Jv = Kf([Pc-Pi]-σ[πc-πi])• σ = reflection coef.– More likely varies with pore size– Human serum albumin: ~10nm in diameter

• Assumes Hagen-Poiseuille flow in pore• R = 200 nm, µ = 4 Pa-s, ∆P varies, L = 5 µm• ∆P is normalized to 25 mmHg

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Vascular Equations

Hagen - Poiseuille

Mass balance, missing transvascular loss

- ϕ

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Interstitial Equations

+ ϕTumor mass balane:

Pe ~ 800 in the interstitium

Transvascular“generation” term

Darcy’s Law, Ac = void surface area Ac = pi*d/Sv

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Transvascular Equations

S = inner surface area of vessel

L = thickness of capillary wall

For steady state

R (nm) Lp (m^3/s/Pa/m^2) and in cm/mmHg v (m/s)10 1.88E-10 2.50E-06 6.25E-0750 4.69E-09 6.25E-05 1.56E-05

100 1.88E-08 2.50E-04 6.25E-05150 4.22E-08 5.62E-04 1.41E-04200 7.50E-08 1.00E-03 2.50E-04250 1.17E-07 1.56E-03 3.91E-04300 1.69E-07 2.25E-03 5.63E-04350 2.30E-07 3.06E-03 7.66E-04400 3.00E-07 4.00E-03 1.00E-03450 3.80E-07 5.06E-03 1.27E-03500 4.69E-07 6.25E-03 1.56E-03

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Transvascular EquationsQt

FluxMass/time*area

Flowmass/time

W = convective hindranceH = diffusive hindrance

1-σ=WDerived from…

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Hindrance Factors

diffusive

convective pore

Centerline:Pull out K-1 and GAs constants

τ=κ

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Derivation of Partition Coefficients

Drag forceDiffusive force

• N = U*C• Poiseuille velocity profile• Average flux over pore area• Assume C(z,β) separable• Neglect Taylor dispersion• Radial Pe > 2• Axial Pe >29• Both are satisfied (See table)

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Runge-Kutta 4th Order

Industry standard, O(h5) error

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Transport through pores

• Assumptions:– low Re <1, Per>>1 (samples all radii)– dilute in fluid continuum (?)– significant radius relative to pore– no Taylor dispersion (Per>2, PeL>29)– fully developed flow

• Diffusion and hydrodynamic drag forces

Smith & Deen, 1982

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Length Scales

• RBC: D = 8 µm• Human serum albumin: 10 nm• Nanomedicines: 10 – 100 nm– In paper: 6 – 200 nm, usually 0.3*Dp

• Capillaries: 10 µm (15 µm in paper)• Vascular pores: 1-1000 nm– In paper: 400 nm average, distribution

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Vascular Network

• 200x200 two-layer grid, each point is defined by the neighbors to which it connects

• Volumetric flow conserved through each point• Vasculature only exists at pre-defined points• Interstitium occupies all grid points inside radius

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Overview of Tumor Modeling

Wu et al. 2007