Dynamical remodeling of the during tumor growth vascular network K. Bartha

Dynamical remodeling of the
vascular network
during tumor growth
K. Bartha1, H. Rieger2
D.-S. Lee2, R. Paul2, M. Welter2
1Semmelweis
University for Medicine, Budapest
2Universität des Saarlandes, Saarbrücken
Nonequilibrium statistical physics of complex systems, KIAS, Seoul, 3.-6.7.2006
Normal vascularization vs.
tumor vessels
Normal capillary network (Steiner et.al.,1992).
Dilated vessels, brush border effect in the
periphery of a melanoma (Steiner et.al.,1992)
Tumor cell proliferation confined to outer rim:
Experiment, Brú et al.
Theoretical model, Drasdo & Höhme
Biophys. J. 85, 2948 (2003)
Phys. Biol. 2, 133 (2005)
Radius of an avascular tumor in vitro grows linearly in time
Capillary network remodeling in vitro:
Migration and anastomoisis of endothelial cells (EC)
100m
Matrix:
Fibrin gel + bFGF
Microcarriers
coated with ECs
Nehls, Herrmann, Hühnken
Histochem. Cell Biol. 109, 319
(1998)
Angiogenesis / Oxygen / HIF / Cooption
Oxygen diffusion range / Hypoxia
[from Carmeliet and Jain, Nature 407, 249 (2000)]
Vessel cooption
Changes in tumor vasculature during growth. After co-opting host
vessels, tumors (gray) initially grow as well as vascularized masses
(A).As tumor growth progresses, many of the central tumor vessels
regress (B), resulting in massive TC death and necrosis (strippled
region). Surviving TCs form cuffs around the few remaining internal
vessels.
[from Holash et al., Science 284, 1994 (1999)]
Model concept
Combination of tumor growth, cooption and neovascularization
Lattice L x L, Lattice constant 10 m ( ~ size of a cell )
Vessel system = network of pipes between lattice points parallel to coordinate axis
V = { v | v vessel segment } + ideal pipe flow (blood)
Tumor cells on lattice points
T := { r | TC on lattice point r }
Oxygen concentration field c oxy(r)
produced by perfused vessels
Growth factor concentration field cgf(r)
produced by tumor cells
Lattice point/TC
Dynamical evolution: stochastic process of
insertion / removal of vessels and TCs
Vessel segment
Initial state
10 mm
(L=1000)
Blood flow computation
l
Laminar stationary flow
Through ideal pipes
Hagen-Poiseuille
Flow rate
Shear force
q(v)  ( / 8)
r4

f
P
f (v)  1 / 2rP
P1
P2
P  ( P1  P2 ) / l
P5 q5
q4
Mass conservation
P0
q3 P3
q1
P1
P4
q2
Kirchhoff‘s Law
0   qi   consti  ( Pi  P0 )
P2
Boundary conditions
Constant flow from
upper left to lower right corner
P( x, y)  1  12 ( x / L  y / L)
q
( x, y )  { Boundary sites }
System of linear equations to determin pressure on network nodes  P,f,q
q
Growth factor field computation
Diffusion of GF into ECM
0
cgf
t
 Dcgf  k0cgf  k1t (r )
Hypoxic TCs are sources of GF
1 for r  T and coxy (r )   oxy
t (r )  
0 otherwise

Consumption term -k0cgf causes exponential asymptotic decay
Solution via Greens function method
cgf (r )   Ggf ( r  r ' )  k1t (r ' )
r'
Ggf (r ) ~ e
 r / k0
/r
Approximation: Each TC generates linearly decaying distribution with
cut-off radius Rgf  m ( radius of vessel generation TC )
GGF(r)
N-1
0
r
RGF/Oxy
O2 field computation
Diffusion of O2 in ECM
0
coxy
t
 Dcoxy  coxy k (r )  J (r )
Solution
Finite difference approximation for ,
coxy computed at each lattice site
via solution of system of linear equations
Source term J(r) represents vessel network
q  (cblood  c(r )) for r  v V
J (r )  
0 otherwise

Comsumption rate tissue dependent
 k in ECM
k (r )   0
k1  k0 in Tumor
106 variables, still efficiently doable
Choice of parameters
cblood  1
All others such that
coxy ( zwischen Vessels )  12 cmax
Roxy ( ECM )  100m
Roxy (Tumor)  50m
Dynamics ...
Time step  = 1h
TC proliferation where O2 conc. sufficiently high
If coxy(r,)oxy at tumor-surface site r:
w{ TT{r} } = /Tt
TCs removed, if too long underoxygenated O2
If coxy(r,)<oxy ·0.1 for time longer than TuO at
Lattice sites r with TC: w{ TT\{r} } = 1/2
Vessel growth where GF Conc. sufficiently high and void of TCs
At site rvV where cGF(r)GFgrowth is possibel with
w{ VV{v} }=/Te
Generate v always between existing, parallel vessels,
if path is not blocked by TCs
And all othe vessels >1 sites away.
...
Vessel dilatation where GF Conc. sufficiently high
For each rvV wo cGF(r)GFand r(v)< rmax :
w{ R(v)  R(v)+A / 2l(v) } = /Te
(  addition of a EC with area A to total vessel surface)
(if node r between different vessels, dilate thinner vessel)
Vessels surrounded by TCs collapse if shear force too low
For rvV with at least two neighboring TCs
and f(v)<fcritf0 and r(v)<rstab: w{ VV\{v} }= pcollapse
(Involving shear force motivated by invivo data)
Vessel regression if not circulated
If vV not circultated (bi-connected w. boundary):
Remove if coxy(r)<oxy : w{ VV\{v} }=1/2
Snapshots
t=1000
t=600
t=300
t=100
t=0
1,0mm
2,0mm
3,0mm
4,0mm
5,0mm
Video
Growth factor and O2 at different times
GF/TC-State
O2
t=0h
oxy
0.1 oxy
t=100h
t=300
t=600h
t=1000h
Quantitative Analysis
Densities etc as function of the distance to tumor center
TC density
Vessel density (MVD)
T=600h
Vessel radius
Shear force
n.b.: Tumor Radius linear in time: Consequence of Eden growth
Variations ...
fcrit=0.5 pcollapse=0.01
fcrit=0.5 pcollapse=0.001
fcrit=0.2 pcollapse=0.01
...
Vary fcrit with distance to tumor center
Effect of solid stress increase inside the tumor
fc,max
fcrit(r)
f crit (r)  f c,max  (1  Rmax )
r
Rmax
0
fc,max=0.5 pcollapse=0.01
Rmax r
fc,max=0.5 pcollapse=0.001
fc,max=0.2 pcollapse=0.01
Results (summary):
Model predicts compartmentalization of the tumor into different regions:
• Highly vascularized peritumoral region (outside tumor)
• Increased MVD in tumor periphery (inside tumor)
• Low MVD in the tumor center
• Vessel radius increases with decreasing distance from the center
• Necrotic regions (void of TCs and ECs)
threaded by thick vessels
surrounded by cuffs of viable TCs
[K. Bartha, H. Rieger, J. Theor. Biol., in press (2006)]
Comparison w. experimental data obtained on melanoma vessels
From Döme et al., J. Path. 197, 355 (2002).
3d model:
Quantitatively similar to 2d version!
[D.-S. Lee, H. Rieger, K. Bartha, PRL 96, 058104 (2006)]
Fractal structure of tumor vessel networks
Experiment
500m
arteriovenous net.:
df= 1.700.03
500m
500m
Norm. capillary net.:
df= 1.980.02
carcinoma network:
df= 1.880.04
Gazit et al.,
Phys. Rev. Lett. 75, 2428 (1995)
Theory
Tumor network:
df= 1.850.05
Original network: df=2
Box-counting method
for fractal dimension
df?
Invasion percolation: df=1.81
Jain et al: Fractal structure of tumor vasculature due to invasion percolation growth process
[Nature Med. 4, 984, (1998); PRL 75, 2428 (1995)]
Conventional percolation: df=1.89
Bartha and HR: Fractal structure of tumor vasvulature due to stochastic vessel regression
[J. Theor. Biol., in press (2006); PRL 96, 058104 (2006)]
Flow correlated collapse avoids
percolation transition
Note that stochastic removal of vessels inside the tumor would lead to a
(unrealistic) percolation transition  blood flow is essential model ingredience
pcollapse < pc
pcollapse = pc
pcollapse > pc
Idealized models
Bi-connectivity percolation in a regular network with linearly growing removal region
plus delayed stabilization:
Same universality clas as conventional percolation, df=1.89
[R. Paul, H. Rieger, 2006]
Still needs fine tuning of p (removal prob.) to pc for critical percolation cluster!
Flow correlated percolation drives the tumor vasculature into critical state
Fluid flow simulation through tumor vasculature
Injection of substanbce (drug) into blood flow – restricted in time
„Drug Delivery“ through complex tumor vasculature
Delivery problem: Strongly fluctuating flow velocity
Original model: Fluid flow through porous media
(oil, gas through sand, stone etc.)
Model
Network of interconnected pipes
Calculate flow rate for stationary flow
Time step :
Propagate drug mass at each node from in-flow pipes to out-flow pipes
Snapshots
T=3.0s
T=1.5s
T=0.93s
Video
Quantitative Analysis ...
Total mass throughput:

t 0... Ende

minvessel (t ) / t 0... Ende minboundary(t )

...
Time for which conc. higher than threshold
tc c0  t  falls c(t )  c0 , 0 sonst
c0=0.01 cinit

...
Time for which conc. higher than threshold
tc c0  t  if c(t )  c0 , 0 otherwise
c0=0.1 cinit

...
Time for which conc. higher than threshold
tc c0  t  falls c(t )  c0 , 0 sonst
c0=0.5 cinit

...
Time for which conc. higher than threshold
tc c0  t  if c(t )  c0 , 0 otherwise
c0=0.9 cinit

Improved model for micro-vasculature
Vessels occupy links of a hexagonal lattice
Generate network via stochastic process
Phase1:
Add probabilistically tripoids starting from sources and sinks
until arterial and venous trees approach each other.
Phase2 ( Remodeling):
Connections between arteries and venes established via capillary segments
Compute hydrodynamic properties, O2-field, etc.,.
Delete temporarily capillary segments
Stochastically: Insertion of new branchings, vessel regression by removeing lower branches of the tree
Keep always binary tree structure, connection between trees only via capillary segments
Übergangs Wahrscheinlichkeiten aus verschiedenen Einflüssen, z.B. Wand-Scherspannung.
Transition probabilities depending on shear force.
[Gödde and Kurz, Dev.Dyn 220:387 (2001)]
Snapshot
Tumor model
Essentially as original (capillaries only) model,
but now hexagonal lattice
New: Growth of individual sprouts, ...
Video
Conclusion: Biological implications
Tumor pushes peritumoral plexus into the tissue during growth
– and remodels the densified network inside,
assisted by the redirected blood flow
Is MVD a usefull diagnostic tool?
Apparently (according to our model) not:
– metabolic demand (oxy) and GF production rate (GF) need not to be correlated
Model prediction:
Tumors with a large metabolic demand will only grow if their GF production is high –
Possibly human melanoma are in that category (indication: large peritumoral plexus).
MVD inside the tumor can be high or low – independent of growth rate
Main parameter determining the tumor growth is the TC proliferation time –
O2 will not be a crucial factor as long as RGF is sufficiently large
Outlook
• Blood flow description: inhomogeneous fluid, hematocrit, etc.
• Arteriovenous original vasculature (done)
• O2 diffusion (PDE with sources and sinks) (done)
• Solid stress ( tumor growth inhibition)
• more detailed model of ECM (extracellular matrix)
• ...
• Guided migration, origin of arteriovenous structure
Role of ephrin/EphR, VEGF, etc.
• Vasculogensis, tubulogenesis
Title
Title
Title
Sketch of a model
combining tumor growth, cooption and neovascularization
Start with a regular vessel network of given MVD (~100/mm2)
Start with a small tumor (e.g. Eden cluster of TCs)
O2 concentration field is produced by circulated vessels
GF (Growth Factor) concentration field is produced by viable TCs
TCs proliferate if local O2 conc. is sufficient (TC = Tumor Cell)
ECs proliferate when GF conc. is sufficient (EC = Endothelial Cell)
Outside the tumor: EC form new vessels
Inside the tumor: ECs increase vessel radius
Vessels surrounded by TCs collapse if wall shear force is low
Vessels regress if not circulated
TCs die if too long under-oxygenated
Modelvorhersagen / Morphometrie echter Tumore
Einteilung des Tumor in verschiedene Regionen:
• Hoch vaskularisierte Peritumorale-Region (außerhalb des Tumor)
• Erhöhte MVD in Tumor-Periphery (innerhalb des Tumor)
• Niedrige MVD in the Tumor Zentrum
• Gefäßradius wächst mit sinkendem Abstand zum Zentrum
• Necrotische Regionen ( ohne TC und EC )
durchzogen von dilatierten Gefäßen
umgeben von Manschetten lebender TC
Results (summary):
Model predicts compartmentalization of the tumor into different regions:
• Highly vascularized peritumoral region (outside tumor)
• Increased MVD in tumor periphery (inside tumor)
• Low MVD in the tumor center
• Vessel radius increases with decreasing distance from the center
• Necrotic regions (void of TCs and ECs)
threaded by thick vessels
surrounded by cuffs of viable TCs
Comparison with data for melanoma:
[Döme et al. J. Pathol. `02]
cont.
From Döme et al., J. Path. 197, 355 (2002).
Vergleich mit Experimentellen Daten von Melanoma
Döme et al., J. Path. 197, 355 (2002).
Propagations Prozedur...
für jedes Vessel v, das Drug-Masse mv>0 beinhaltet:
Drug Volumen
x0  x0  v
V  A( x1  x0 )
r
A
m
x0
0
q, v
x1  x1  v
x0
x1
l
x1
l
falls x0>l: Transferiere gesamte Masse in Knotenpunkt, setze x0=x1=0
falls x1>l: Transferiere überschüssige Masse in Knotenpunkt
m
m  m
x0
x1
Gefäße
m
m  A( x1  l )
V
m  m  m
mn  mn  m
x1  l
mn
...
falls Drug-Injektion: für jeden Randknoten durch den Blut ins System reinfließt:
q1
qin   qk
mn  mn   qin  concin
q2
mn
für jeden Knoten mit mn>0:
Berechne Ausfluß-Rate
q1
Qout,total  qout,sys   qk
qout,sys
mn
( >0 dort wo Blut das
System verläßt)
q2
für jedes Gefäß vk durch das Blut aus Knoten rausfließt: Verteile Drug nach Flussrate
m
q1
qout,sys
mn
q2
m 
qk
Qout,total
 mn
mk  mk  m
mk
m
x0
x1
...
„Verschmiere“ Drug über Gefäß:
falls schon Drug in Gefäß vorhanden ( mk>0 ):
x0  0
mk
x0
mk
x0
x1
x1
sonst:
x1  v
mk
x0
x1
mk
x0
Setze Knotenmasse zurück auf 0
mn  0
Wahl des Zeitschritt  :
  min Vv / qv 
vVessels
Zeit um gesamtes Vesselvolumen Vv
einmal auszutauschen
x1
Fractal dimension of the tumor vessel network
Box-Counting
Method.
df=1.850.05
df of percolation cluster:
1.891 [random removal of segments with prob. pc]
df of vasculature in carcinoma: 1.890.04 [Jain et al. (1995)]
Fractal structure is exclusively determined by collapse events,
Correlation of collapse with blood flow drives the structure into the critical state
N.b.: The fractal structure appears inside the tumor due to remodelling,
not due to vessel growth (c.f. Jain et al.).