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Influence of
interstitial fluid dynamics
on growth and therapy
of angiogenic tumor
A.Kolobov, M.Kuznetsov
Working group on modeling of blood flow and vascular pathologies (INM of RAS)
RSF grant 14-31-00024 (new laboratories)
&
P.N.Lebedev Physical Institute of RAS
The 8th Workshop on Mathematical Models
and Numerical Methods in Biology and Medicine
INM RAS
October 31 – November 3, 2016
OUTLINE
• Interstitial fluid dynamics (IFD) in normal tissue
• Tumor growth, progression and angiogenesis
Necrotic core and peritumoral oedema
• Motivation
Why dynamic of interstitial liquid is important?!
• Mathematical model
• Influence of IFD and antiangiogenic therapy on tumor
growth
• Main results and further development
Formation of interstitial fluid
• As blood flows through the capillaries some
plasma passes into the tissues
• This interstitial fluid is very similar to plasma
but does not have large plasma protein
molecules in it
• This fluid bather every cell in the body
supplying them with glucose, amino acid, fatty
acids, salts and oxygen
Capillary fluid dynamics
Role of lymphatic system
Malignant tumor progression
number
of cells
lethal tumor mass (≈1 kg)
limit of detection
necrotic core angiogenic
switch
formation
time
• Bergers, G., & Benjamin, L. E. (2003). Tumorigenesis and the angiogenic switch. Nature reviews cancer, 3(6), 401-410.
• Neufeld, E., Szczerba, D., Chavannes, N., & Kuster, N. (2013). A novel medical image data-based multi-physics simulation
platform for computational life sciences. Interface focus, 3(2), 20120058.
Vascular endothelial growth factor VEGF
Dimer, molecular mass - 34-42 кDa
Tumor capillaries “inefficiency”
leads to edema
normal
tumor
tumor
maturation
factors
large
pores
fewer
pericytes
lower
permeability
disordered
integrin
expression
edema
• Hawkins-Daarud, A., Rockne, R. C., Anderson, A. R., & Swanson, K. R. (2013). Modeling tumor-associated edema in
gliomas during anti-angiogenic therapy and its impact on imageable tumor. Frontiers in oncology, 3, 66.
Angiogenic capillaries maturation
Antiangiogenic therapy
VEGF
recombinant
humanized monoclonal
antibody for VEGF
• tumor nutrient deprivation
• edema removal
• necrotic core shrinking
• Jain, R. K., Di Tomaso, E., Duda, D. G., Loeffler, J. S., Sorensen, A. G., & Batchelor, T. T. (2007). Angiogenesis in brain
tumours. Nature Reviews Neuroscience, 8(8), 610-622.
MOTIVATION
cell number
Mathematical modeling of combine
antitumor therapy without interstitial
fluid dynamics
days
Four patients radiotherapy data
Hawkins-Daarud A. et.al., Frontiers in Oncology,
2013, 66(3), 1-12
Modeling of tumor growth and therapy
Definition of the regrowth delay Dt and of the gain of lifetime due to RT DT and of
the four phases that compose the radius-versus-time curve.
Mathematical model
The dashed (respectively solid) black curve
corresponds to the cell density profile just
before (resp. after) RT. The cell density just
after RT is obtained by multiplying the cell
density just before RT by a parabola-shaped
function that crosses the horizontal axis at x =
R0. The dotted-dashed curve is the difference
between the solid and the dashed black
curves, and represents the cell density that has
been killed by RT.
M. Badoual et al. 2014
Cell Proliferation, 47, 369–380
In modeling,
convective fluxes and interstitial fluid dynamics
(IFD)
must be accounted for
But interstitial fluid is considered only in several models.
Jain, R. K., et al.
Cancer research (2007)
IFD model
Badoual, M., et al.
Cell proliferation (2014)
Glioma growth model
Hawkins-Daarud, A., et al.
Frontiers in oncology (2013)
Glioma growth model
• considered IF pressure
and seepage from tumor
• bevacizumab therapy
modeled but only
phenomenologically
• no tumor dynamics
• no tissue structure
• edema considered
• no IFD
• direct
edema emergence
due to tumor cells
• considered IFD
and angiogenesis
• comparison with clinical data
of result of therapy modeled
by direct parameters changes
• no convective flows
• no metabolites
Scheme of the model
n1 – proliferating cells
n2 – migrating cells
h – host cells
m – interstitial fluid
EC – normal capillaries
FC – angiogenic capillaries
S – glucose
CRF – capillary
regression factors
V – VEGF
A – bevacizumab
Tumor and interstitial fluid description
Tumor cells:
“go or grow”
𝜕𝑛1
𝜕(𝐼𝑛1 )
= 𝐵𝑛1 − 𝑃1 (𝑆)𝑛1 + 𝑃2 (𝑆)𝑛2 −
𝜕𝑡
𝜕𝑥
low S
high S
𝜕𝑛2
𝜕 2 𝑛2
𝜕(𝐼𝑛2 )
= 𝐷𝑛
+
𝑃
(𝑆)𝑛
−
𝑃
(𝑆)𝑛
−
𝑑
(𝑆)𝑛
−
1
1
2
2
𝑛
2
𝜕𝑡
𝜕𝑥 2
𝜕𝑥
Host cells:
𝜕ℎ
𝜕(𝐼ℎ)
= −𝑑ℎ (𝑆)ℎ −
𝜕𝑡
𝜕𝑥
IF:
very
low S
cells + IF = const
(incompressible tissue)
outflow
seepage
𝜕𝑚
𝜕2𝑚
ℎ
𝜕(𝐼𝑚)
= 𝑑ℎ 𝑆 ℎ + 𝑑𝑛 𝑆 𝑛2 + [𝑄𝑚,𝐸𝐶 𝐸𝐶 + 𝑄𝑚,𝐹𝐶 𝐹𝐶](𝑚𝑖𝑐 − 𝑚) + 𝐷𝑚
−
𝑣
𝑚
−
𝑑𝑟
𝜕𝑡
𝜕𝑥 2
ℎ + ℎ∗
𝜕𝑥
inflow
Convective flow velocity field is defined by cells and IF dynamics:
𝑥
𝐼=
{𝐵𝑛1 + [𝑄𝑚,𝐸𝐶 𝐸𝐶 + 𝑄𝑚,𝐹𝐶 𝐹𝐶](𝑚𝑖𝑐 − 𝑚) − 𝑣𝑑𝑟 𝑚
0
ℎ
𝜕𝑛2
𝜕𝑚
}𝑑𝑟
+
𝐷
+
𝐷
𝑛
𝑚
ℎ + ℎ∗
𝜕𝑥
𝜕𝑥
Dynamics of capillary surface density
Preexisting:
Angiogenic:
𝜕𝐸𝐶
=
𝜕𝑡
𝜕𝐹𝐶
=
𝜕𝑡
angiogenesis
−𝜇 𝐸𝐶 + 𝐹𝐶 − 1 𝐹𝐶 ∗ 𝜃 𝐸𝐶 + 𝐹𝐶 − 1
−𝑙 𝑛1 + 𝑛2 𝐸𝐶 − 𝑘𝐶𝑅𝐹 𝐶𝑅𝐹 ∗ 𝐸𝐶
−𝑉
+𝑣𝑚𝑎𝑡 𝐹𝐶 ∗ 𝑒𝑥𝑝
𝑉𝑛𝑜𝑟𝑚
𝜕 𝑒𝑙𝑎𝑠𝑡 ∗ 𝐼 ∗ 𝐸𝐶
−
𝜕𝑥
density
maintaining
degradation
maturation
convection
random
motion
𝑅 ∗ 𝜃 𝑆 − 𝑆𝑐𝑟𝑖𝑡 𝑉
𝐸𝐶 + 𝐹𝐶
𝑉 + 𝑉∗
−𝜇 𝐸𝐶 + 𝐹𝐶 − 1 𝐹𝐶 ∗ 𝜃 𝐸𝐶 + 𝐹𝐶 − 1
−𝑙 𝑛1 + 𝑛2 𝐹𝐶 − 𝑘𝐶𝑅𝐹 𝐶𝑅𝐹 ∗ 𝐹𝐶
−𝑉
−𝑣𝑚𝑎𝑡 𝐹𝐶 ∗ 𝑒𝑥𝑝
𝑉𝑛𝑜𝑟𝑚
𝜕 𝑒𝑙𝑎𝑠𝑡 ∗ 𝐼 ∗ 𝐹𝐶
−
𝜕𝑥
𝜕 2 𝐹𝐶
+𝐷𝐹𝐶
𝜕𝑥 2
Balance of substances
Glucose:
𝜕𝑆
𝑆
𝜕2𝑆
= − 𝑞𝑛1 𝑛1 + 𝑞𝑛2 𝑛2 + 𝑞ℎ ℎ
+ 𝑄𝑆,𝐸𝐶 𝐸𝐶 + 𝑄𝑆,𝐹𝐶 𝐹𝐶 𝑆𝑏𝑙𝑜𝑜𝑑 − 𝑆 + 𝐷𝑆 2
𝜕𝑡
𝑆 + 𝑆∗
𝜕𝑥
Capillary regression factors:
𝜕𝐶𝑅𝐹
𝜕 2 𝐶𝑅𝐹
= 𝑝𝐶𝑅𝐹 𝑛1 + 𝑛2 − 𝜔𝐶𝑅𝐹 𝐶𝑅𝐹(𝐸𝐶 + 𝐹𝐶) − 𝑑𝐶𝑅𝐹 𝐶𝑅𝐹 + 𝐷𝐶𝑅𝐹
𝜕𝑡
𝜕𝑥 2
VEGF:
𝜕𝑉
𝜕2𝑉
= 𝑝𝑉 𝑛2 − 𝜔𝑉 𝑉(𝐸𝐶 + 𝐹𝐶) − (𝑘𝐴 𝐴𝑛 )𝐴𝑉 − 𝑑𝑉 𝑉 + 𝐷𝑉 2
𝜕𝑡
𝜕𝑥
binding
Bevacizumab:
𝜕𝐴
𝜕2𝐴
= [𝑄𝐴,𝐸𝐶 𝐸𝐶 + 𝑄𝐴,𝐹𝐶 𝐹𝐶](𝐴𝑏𝑙𝑜𝑜𝑑 − 𝐴) − (𝑘𝐴 𝑉𝑛 )𝐴𝑉 + 𝐷𝐴 2
𝜕𝑡
𝜕𝑥
Antiangiogenic therapy:
𝜕𝐴𝑏𝑙𝑜𝑜𝑑
= 𝐹𝐴,𝑖𝑣 − 𝑑𝐴 𝐴𝑏𝑙𝑜𝑜𝑑
𝜕𝑡
Influence of IFD on tumor growth rate
Antiangiogenic therapy effect
shrinking
20%
slowdown
Antiangiogenic therapy effect
Model run:
moderate therapy effect
free
right side
“bone”
on left side
microvasculature
glucose
tumor
interstitial fluid
Model run:
moderate therapy effect
Model run:
tumor shrinking due to therapy
Model run:
static tumor (no therapy)
Main results
Model makes it possible to adequately reproduce
clinically observed phenomena:
• formation of peritumoral edema
and it disappearance due to bevacizumab therapy
via account of physiological characteristics of angiogenic capillaries
• significant slowing of tumor growth
as a result of therapeutic intervention
via account of formation of interstitial fluid from tumor necrosis
and its transport in the peritumoral region,
as well as considering convective flows
Future work
• incorporation of cell debris dynamics
in necrotic core
•Improvement of IFD description
(including of hydrodynamic issues);
• modeling cytotoxic
and other types of antitumor therapy,
as well as combined therapy;
• account for influence of IF concentration
on dynamics of substances.
Thank you
for your attention!