The implications of heterogeneous vascular patterning on radiation therapy response

The implications of heterogeneous vascular patterning on
radiation therapy response
Jacob G. Scott
ECMTB 2014
Key Factor
Christopher McFarland
1Harvard-MIT Division of Health
Oncology, *contributed equally
Background:
Metastasis is a highly lethal and poorly understood process that
accounts for the majority cancer deaths
Patterns of metastatic spread are not explained by deterministic
explain these patterns
We develop a stochastic model at the genomic level and use
population genetics techniques to explore this phenomenon
Feature of Model O
Population size determined
by fitness of cells
La
Key F
Christopher
1Harvard-MIT Div
Oncology, *contri
Background:
Feature of
Radiation Oncology and
Integrative Mathematical
Oncology
occupied 30%, the intermediate region 20% and the
periphery 50% of the total T1Gd visible tumour
volume. These regions were incorporated in the virtual
tumour model (Figure 7). The low, intermediate and
high vascularity voxels from Part 1 were assigned
Figure 5: Patient 4 tumour and dose distribution. A:
One slice of the T1Gd MRI. The dark core and light
periphery can be clearly seen. B: One slice of the T2 MRI.
C: Part of the hand-drawn structures matrix, showing the
T1Gd and T2 tumour outlines, plus 2 cm margins around
them, giving the clinically targetted region. D: A slice of
the dose distribution. Colormap is given on the right.
F
c
t
A
d
i
p
h
g
o

4910
Figure 2. Image of the concentration (in mM) of the contrast agent in the central
the tumour.
affect radiation therapy efﬁcacy
Macroscopic hypoxia correlates with radiocontrast
uptake, and dose modulation is efﬁcacious, in silico1
Patient-Specific Radiotherapy for Glioblastoma
•Radiation dose/fraction is known to depend heavily
on local oxygen concentration as well as intrinsic
cell parameters
•Our ability to quantify these parameters in patients
is maturing, but has not translated to the clinic
Cell diffusivity and
S

We use a suite of mathematical and computational
models to bridge a range of spatial and temporal scales.
TIME/SPATIAL SCALE
CELLULAR DETAIL
Evolutionary
Game Theory
Reaction
Diffusion Models
Hybrid Cellular
Automata
Cellular Potts
Model
Immersed
Boundary Model
Hybrid Cellular
Automata
Non-spatial
continuum
Reaction
Diffusion
Network
Theory

m (2.4) that the VCP depends not only on the actualy oxygen concentration
riences, but also on the total number of cells. In the following sections we
ach of these factors individually, beginning with the number of cells within
which we will call the carrying capacity.
ygen dependence
d the changes in the radiation parameters, α and β, with oxygen we use the
oxygen enhancement ratio (OER):
αi =
αmax
OERαi
(pi)
, βi =
βmax
OERβi
(pi)2
, (2.5)
nd βmax are the values of α and β under fully oxygenated conditions and
re the values of α, β and oxygen, p, in compartment i. We can further find
function of the oxygen concentration by using the relation established by
correlation by Chapman et al. [22], and Palcic and Skarsgard [62]:
OERi =
(OERmax − OERmin)Km
pi + Km
+ OERmin (2.6)
oxygen concentration in compartment i, Km = 3.28 and OERαmin =OERβmin =
There is little information about how β changes in glioblastoma with oxy-
o for now, we will assume that the α
β
ratio for maximally sensitive cells
ant at 10 Gy−1
(this is reported in the literature ranging from 8.64 [25] to
31
Oxygen dependence
and the changes in the radiation parameters, α and β, with oxygen we use the
the oxygen enhancement ratio (OER):
αi =
αmax
OERαi
(pi)
, βi =
βmax
OERβi
(pi)2
, (2.5)
x and βmax are the values of α and β under fully oxygenated conditions and
pi are the values of α, β and oxygen, p, in compartment i. We can further find
s a function of the oxygen concentration by using the relation established by
tal correlation by Chapman et al. [22], and Palcic and Skarsgard [62]:
OERi =
(OERmax − OERmin)Km
pi + Km
+ OERmin (2.6)
the oxygen concentration in compartment i, Km = 3.28 and OERαmin =OERβmin =
6]. There is little information about how β changes in glioblastoma with oxy-
n, so for now, we will assume that the α
β
ratio for maximally sensitive cells
nstant at 10 Gy−1
(this is reported in the literature ranging from 8.64 [25] to
31
he basics of radiobiological modeling, to include tumor control
ear-quadratic model of surviving fraction (SF) [15, 31], the
ement ratio (OER) and linear energy transfer (LET).
SF = e−n(αd+βd2)
(2.1)
nd β refer to the radiobiologic parameters associated (phe-
kill secondary to ‘single hit’ events (α) and ‘double hit’ events
r fraction of radiation and n to the number of fractions.
trol probability
e that considers the total number of surviving cells in a tumour,
apter, in the total number of surviving cells in our domain. We
sure, based on the TCP, which we will call the Voxel Control
derstand the effect of radiation on a distribution of cells, we
vival probability of each discrete subpopulation of cells in the
ed by their proliferative state, and their microenvironmental
allow Nt
ij to be the number of cells of type i, where i ∈ {S, D}
30
image credit: http://www.eyephysics.com/tdf/models.htm
Radiation Biology Primer
TCP = e−SFN0

∂c(x, t)
∂t
= Dc∇2
c(x, t) − fc(x, t), (1.1)
entration of oxygen at a given time t and position x, Dc is the
ygen, which we assume to be constant (providing linear, isotropic
governed by Michaelis-Menten kinetics and is defined as:

µir(c, t) if there is a cell of type i at x at time t,
0 otherwise,
(1.2)
nd the labels H, S, P and T are used to refer to healthy, TIC, TAC
y. Here µi is defined as the cell type-specific oxygen consumption
T ), which modulates r(c, t), the oxygen dependent consumption
r(c, t) = rc

c(x, t)
c(x, t) + Km

e the maximal uptate rate and effective Michaelis-Menten con-
upplement equation (1.1) with the following initial and boundary
ith the oxygen in the domain set to c(x, 0) = c0 and all lattice
mal cells. In the case of a cancer simulation, we replace the center
e TIC. Vessels are placed throughout the domain at a prescribed
5
nt of oxygen equal to that carried in the arterial blood. This oxygen is then allowed
ffuse into the surrounding tissue.
he spatiotemporal evolution of the oxygen field is described by the reaction-diffusion
al differential equation (PDE)
∂c(x, t)
∂t
= Dc∇2
c(x, t) − fc(x, t), (1.1)
e c(x, t) is the concentration of oxygen at a given time t and position x, Dc is the
ion coefficient of oxygen, which we assume to be constant (providing linear, isotropic
ion) and fc(x, t) is governed by Michaelis-Menten kinetics and is defined as:
fc(x, t) =

0 otherwise,
(1.2)
e i ∈ {H, S, P, T} and the labels H, S, P and T are used to refer to healthy, TIC, TAC
TD cells respectively. Here µi is defined as the cell type-specific oxygen consumption
ant (µH, µS, µP , µT ), which modulates r(c, t), the oxygen dependent consumption
defined as
r(c, t) = rc

c(x, t)
c(x, t) + Km

e rc and Km denote the maximal uptate rate and effective Michaelis-Menten con-
respectively. We supplement equation (1.1) with the following initial and boundary
tions. We begin with the oxygen in the domain set to c(x, 0) = c0 and all lattice
s occupied by normal cells. In the case of a cancer simulation, we replace the center
e point with a single TIC. Vessels are placed throughout the domain at a prescribed
5
source of oxygen occupying one lattice point, are placed randomly
ce at the start of a given simulation, with a specified spatial density
we neglect vascular remodelling. Each vessel is assumed to carry an
ual to that carried in the arterial blood. This oxygen is then allowed
rrounding tissue.
ral evolution of the oxygen field is described by the reaction-diffusion
quation (PDE)
∂c(x, t)
∂t
= Dc∇2
c(x, t) − fc(x, t), (1.1)
concentration of oxygen at a given time t and position x, Dc is the
f oxygen, which we assume to be constant (providing linear, isotropic
) is governed by Michaelis-Menten kinetics and is defined as:
=

0 otherwise,
(1.2)
} and the labels H, S, P and T are used to refer to healthy, TIC, TAC
vely. Here µi is defined as the cell type-specific oxygen consumption
P , µT ), which modulates r(c, t), the oxygen dependent consumption
r(c, t) = rc

c(x, t)
c(x, t) + Km

note the maximal uptate rate and effective Michaelis-Menten con-
We supplement equation (1.1) with the following initial and boundary
n with the oxygen in the domain set to c(x, 0) = c0 and all lattice
ormal cells. In the case of a cancer simulation, we replace the center
ingle TIC. Vessels are placed throughout the domain at a prescribed
ll diameters [33] and the information from the literature concerning the ratio of
r to normal oxygen consumption (see Section 1.2.2.1).
ntroducing the non-dimensional variables ˜x = x/L, ˜t = t/τ and ˜c = c/c0, we define
ew non-dimensional parameters
˜Dc =
Dcτ
L2
, ˜rc =
τn0rc
c0
. (1.3)
otational convenience, we henceforth drop the tildes and refer to the non-dimensional
meters only as DC and rc. See Table 1.1 for a full list of parameter estimates and
ndix ?? for our procedure for esimating the cancer cell oxygen consumption rate.
.3 Numerical solution
der to solve equation (1.1) numerically, we discretize space and time by considering
k∆t, xi = i∆x and yj = j∆x and approximate the concentration of oxygen at
tep k and position (i∆x, j∆x) by ck
i,j ≈ c(xi, yj, tk). We use a central difference
oximation for the Laplacian and thus approximate equation (1.1) by
ck+1
i,j − ck
i,j
∆t
=
DC
∆x2

ck
i+1,j + ck
i−1,j + ck
i,j+1 + ck
i,j−1 − 4ck
i,j

−

fc
k
i,j
, (1.4)
e

fc
k
i,j
is the cell-specific oxygen consumption µcellrc at time k given a cell at
on (i∆x, j∆x) as discussed in equation 1.2. We then rearrange equation (1.4) to
n a solution for ck+1
i,j , yielding
rc
Maximal oxygen
consumption rate
2.3 × 10−16
mol cells−1
s −1
[32]
c0
Background oxygen
concentration
1.7 × 10−8
mol cm −2
[6]
∆x
Average cell
diameter
50µm [25]
τ
Average cell
doubling time
16h [19]
cap
Hypoxic
threshold
0.1 [20]
rp
Proliferative oxygen
consumption
5 × rc [32]
Km
Effective Michaelis-
Menten constant
0.8mmHg [54]
n0
Cancer cell
density
1.6 × 105
cells cm −2
[21]
s
TIC symmetric
division probability
0 ≤ s ≤ 1 Model-specific
a
TAC proliferative
capacity
0 − 10 Model-specific
µcancer/µH
Cancer metabolic
ratio
2 [12]
ck+1
i,j = ck
i,j

1 − 4
DC∆t
∆x2

+
DC∆t
∆x2

ck
i+1,j + ck
i−1,j + ck
i,j+1 + ck
i,j−1

− ∆t(fc)k
i,j. (1.5)
During each update, then, the oxygen tension in a given lattice point is updated
Diffusion and uptake, exact
Diffusion and uptake, numeric approximation

autophagy (directly translated as ‘self-eating’), a state in which they become resistant
to nutrient starvation [91], and cells are known to die on different time scales and by
different mechanisms (apoptosis vs. necrosis) depending on the magnitude and duration
of the hypoxic insult. While these differences have been shown to affect tumour growth
[19], as this is not the main aim of this model, we will simplify this scenario by assigning
a rate, pd, for cell death at each cellular automaton update defined in Section 2.2.3, when
under extreme hypoxia (i.e. c cap).
2.2.2.4 Quiescence
When cells sense that there is not enough oxygen to divide, or experience contact inhi-
bition, they undergo a state of quiescence during which there is no division. We model
this as an oxygen threshold (c cp) below which cellular division is not possible and by
the spatial constraint which requires the cell to be quiescent if there is not at least one
neighbouring lattice point (Moore neighbourhood, see Figure 2.7(a)) empty or inhabited
by a normal cell. When a cell senses that sufficient oxygen and a neighbouring site have
become available, the state of quiescence is reversed.
Figure 2.3: A summary of oxygen based cell fate threshholds. At each cellular au-
tomaton update, each cell in the domain undergoes a series of fate decisions based on the local
oxygen concentration. When c c , cells die at rate p , when c c c cells are quiescent

DomainSize
Healthy tissue with regular vascular patterning

1 2 3
4 5
Cancer invading into healthy tissue with regular vascular
patterning
1
2
3
4
5

Cancer invading into normal tissue with irregular
patterning

Heterogeneity in vessel pattern induces change in skewness
and cell number

Simplest scenario, two vessels

500 simulations of small domain at dynamic equilibrium
with N randomly seeded vessels
N/(domain size)
Celldensity

nter and Research Institute, Tampa, FL
itute, University of Oxford, Oxford, UK
n efficacy
e based CA
ur
density
Oxygen concentration (mmHG)
VesselDensity
TCP
TCP
Tumour control probability depends not only on vessel
density, but also on vessel organization
TCP ~62% TCP ~75%
TCP ~99% TCP ~88%
our in
t6

Skewness and
TCP
Towards patient-specific biology-driven heterogeneous radiation
planning: using a computational model of tumor growth to identify
novel radiation sensitivity signatures.
Jacob G Scott1,2, David Basanta1, Alex G Fletcher2, Philip K Maini2, Alexander RA Anderson1
1Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL
2Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
Adapting radiotherapy to hypoxic tumours 4909
Figure 1. Pre- and post-contrast T1-weighted MR images taken in the coronal plane through the
head of the dog with a spontaneous sarcoma. The gross tumour volume (GTV) is enclosed by the
white contour, while the tongue (T) and mandible (M) are indicated.
This is reflected in figure 2, showing a corresponding image of the tracer concentration in
the tumour. With respect to blood (and thus oxygen) supply, the tumour periphery may
qualitatively be characterized as normoxic, while the core is probably hypoxic or necrotic.
The tentative pO2 distribution (in frequency form) in the canine tumour, as obtained from
the MR scaling procedure, is given in figure 3. In the same figure, the oxygen distribution
obtained from Eppendorf histograph measurements (Brurberg et al 2005) is shown. The two
4910 E Malinen et al
Figure 2. Image of the concentration (in mM) of the contrast agent in the central coronal plane of
the tumour.
Figure 3. Frequency histograms of the tumour oxygen tension in the canine patient, as determined
by the Eppendorf histograph (Brurberg et al 2005) and the MR analysis.
plots appear similar and rather log-normally distributed, but both have a high frequency of
readings at the lowest oxygen level. The measured median and mean pO2 levels obtained
from the histograph were 8.5 and 13.9 mm Hg, respectively, against 13.6 and 16.6 mm Hg,
respectively, estimated from the tentative MR analysis. The correlation coefficient between
the histograms was 0.88, and a rank sum test and a Kolmogorov–Smirnov test showed that
the histograms were not significantly different (p values of 0.20 and 0.14, respectively). The
‘hypoxic fraction’, i.e. the fraction of pO2 readings smaller than 5 mm Hg were 0.42 and
0.28 for the histograph and MR analysis, respectively. For the current case, it is tentatively
assumed that the MR analysis provides pO2-related images that are biologically relevant.
The compartmental volumes and corresponding mean pO2 levels are given in table 1.
In figure 4, coronal images displaying the tumour compartments are shown. It appears that
the compartmental volumes vary considerably with the coronal plane position although the
Biology and microenvironment
In this work, we use a proliferation-invasion-radiotherapy
Figure 1. Parameter generation for the patient-specific biomathematical model. 1. Determine radial measurements from serial T1Gd and
T2/FLAIR magnetic resonance imaging. 2. Compute the invisibility index (D/r) from intra-study T1Gd and T2/FLAIR radial measurements. 3. Compute
the radial velocity (2
ffiffiffiffiffiffiffi
Dr
p
) from serial T1Gd or T2/FLAIR radial measurements.
doi:10.1371/journal.pone.0079115.g001
cell parameters
PNASPLUS
XRT dose
modulation using
putative stem
distribution3 and
dynamics4 shown
effective in silico and
in vivo4
replication can be
inferred from MRI
imaging, allowing for
understanding of growth
prediction and dose
shape modification2
Non-invasive PET
imaging reported with
Several layers of heterogeneity effect radiation efficacy
N
Quantitative Histology
yields vessel
organization
Microenvironmental feedback Lattice based CAStem hierarchy
Non stem-driven tumour
high vessel density
Stem-driven tumour low vessel density
VesselDensity
TCP
TCP
Enabling translation - information from several scales
Optimized plan
TCP ~62% TCP ~75%
TCP ~99% TCP ~88%
Mathematical model of a stem driven tumour in
a heterogeneous vascular environment6

Ripley’s K(t)mulations (from Figure 3.2) that yield a cell density greater than
ape.
i and j, X is the variable of interest, in this case oxygen
global mean of X. The matrix w is a matrix that contains
the value of 1 if the elements i and j are adjacent, and 0
seful in a number of ecological contexts, for our purposes it is
dre et al. CITE, the residual spatial autocorrelation between
hort relative length scale on which oxygen varies as compared
makes all landscapes appear to be well correlated, that is,
hat approaches zero.
asure is Ripley’s K, and its variance stabilized cousin, Ripley’s
re functions of distance, describe, instead of adjacent elements,
hin a given distance. For Ripley’s K, we have
ˆK(t) = λ−1

i=j
I(dij t)
n
, (3.8)
Figure 3.3: Proportion of simulations (from Figure 3.2) that yield a cell density greater than
90%, exhibiting a sigmoid shape.
of spatial units indexed by i and j, X is the variable of interest, in this case oxygen
concentration and ¯X is the global mean of X. The matrix w is a matrix that contains
spatial weights which take the value of 1 if the elements i and j are adjacent, and 0
otherwise.
While this measure is useful in a number of ecological contexts, for our purposes it is
not. As suggested by Legendre et al. CITE, the residual spatial autocorrelation between
cells and oxygen, and the short relative length scale on which oxygen varies as compared
to vessel presence/absence, makes all landscapes appear to be well correlated, that is,
have a Moran’s I measure that approaches zero.
3.5.2.2 Ripley’s K
A more appropriate measure is Ripley’s K, and its variance stabilized cousin, Ripley’s
L. These measures, which are functions of distance, describe, instead of adjacent elements,
the number of elements within a given distance. For Ripley’s K, we have
ˆK(t) = λ−1

i=j
I(dij t)
n
, (3.8)
where λ is the average density of points in the domain, I is the indicator function which
yields
I(dij t) =

1 if the Euclidian distance fromi → j t,
0 otherwise.
(3.9)
We will utilize the variance stabilized version of this measure, ˆL(t) which is given by
Figure 3.4: Our assumptions about cellular oxygen di
ferences in tumour control probability. We plot two
uniform (Left) and Poisson (center) and compare thes
the CA (right). Each distribution is mapped on to the
the TCP is calculated as per equation (3.6). The Poisson
using a lambda value eqivalent to the mean of the mea
CA. The CA distribution was created by averaging 20
the CA after it reached dynamic equilibrium. We ﬁnd
99% and 91%, respectively.
ˆL(t) =
ˆK(t)
π
1/2
,
which has an expected value of ˆL(t) = t for homogeneou
eﬀects, we implement the correction suggested by Ripley

Ripley’s L and TCP
Towards patient-specific biology-driven heterogeneous radiation
planning: using a computational model of tumor growth to identify
novel radiation sensitivity signatures.
Jacob G Scott1,2, David Basanta1, Alex G Fletcher2, Philip K Maini2, Alexander RA Anderson1
1Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL
2Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
Adapting radiotherapy to hypoxic tumours 4909
Figure 1. Pre- and post-contrast T1-weighted MR images taken in the coronal plane through the
head of the dog with a spontaneous sarcoma. The gross tumour volume (GTV) is enclosed by the
white contour, while the tongue (T) and mandible (M) are indicated.
This is reflected in figure 2, showing a corresponding image of the tracer concentration in
the tumour. With respect to blood (and thus oxygen) supply, the tumour periphery may
qualitatively be characterized as normoxic, while the core is probably hypoxic or necrotic.
The tentative pO2 distribution (in frequency form) in the canine tumour, as obtained from
the MR scaling procedure, is given in figure 3. In the same figure, the oxygen distribution
obtained from Eppendorf histograph measurements (Brurberg et al 2005) is shown. The two
4910 E Malinen et al
Figure 2. Image of the concentration (in mM) of the contrast agent in the central coronal plane of
the tumour.
Figure 3. Frequency histograms of the tumour oxygen tension in the canine patient, as determined
by the Eppendorf histograph (Brurberg et al 2005) and the MR analysis.
plots appear similar and rather log-normally distributed, but both have a high frequency of
readings at the lowest oxygen level. The measured median and mean pO2 levels obtained
from the histograph were 8.5 and 13.9 mm Hg, respectively, against 13.6 and 16.6 mm Hg,
respectively, estimated from the tentative MR analysis. The correlation coefficient between
the histograms was 0.88, and a rank sum test and a Kolmogorov–Smirnov test showed that
the histograms were not significantly different (p values of 0.20 and 0.14, respectively). The
‘hypoxic fraction’, i.e. the fraction of pO2 readings smaller than 5 mm Hg were 0.42 and
0.28 for the histograph and MR analysis, respectively. For the current case, it is tentatively
Biology and microenvironment
Figure 1. Parameter generation for the patient-specific biomathematical model. 1. Determine radial measurements from serial T1Gd and
cell parameters
PNASPLUS
XRT dose
modulation using
putative stem
distribution3 and
dynamics4 shown
effective in silico and
in vivo4
replication can be
inferred from MRI
imaging, allowing for
understanding of growth
prediction and dose
shape modification2
Several layers of heterogeneity effect radiation efficacy
yields vessel
Microenvironmental feedback Lattice based CAStem hierarchy
Non stem-driven tumour
high vessel density
Stem-driven tumour low vessel density
VesselDensity
TCP
TCP
TCP ~62% TCP ~75%
TCP ~99% TCP ~88%
Mathematical model of a stem driven tumour in
a heterogeneous vascular environment6

alinen et al. Phys Med Biol 2006 2. Corwin et al. PLoS ONE 2013 3. Alfonso et al., PLoS ONE 2014 4. Leder et al, Cell 2014 5. Gaedicke et al., PNAS 2014. 6. Scott et al. PLoS Comp Bi
ACKNOWLEDGEMENT: This work sponsored in part by the Moffitt Cancer Center PSOC, NIH/NCI U54CA143970
ue. (A) Representative immunohistochemistry (IHC) staining of pimonidazole in orthotopic
ap water or water containing sodium bicarbonate to drink. (B) Computational segmentation
ues. (C) Positive pixel analysis of segmented viable tissue showing intensity of pimonidazole
orange indicates moderate staining, and red indicates strong staining.
N
yields vessel
organization
Radiomics
yields vessel density
Oxygen concentration (mmHG) Oxygen concentration (mmHG)
Optimized plan
Image credit: Unkelbach et al.
doi: 10.1088/0031-9155/59/3/771

Key Factors in the Metastatic Pr
from Population Gene
Christopher McFarland1*, Jacob Scott2,3,*, David Basanta3, Alexand
1Harvard-MIT Division of Health Science Technology, 2H. Lee Moffitt Cancer and Resea
Oncology, *contributed equally to this work
Fig 1. Prevalence and significance of micrometastases are poorly
understood. Most micrometastases never progress to macroscopic
size. In this case, small colonies of breast endothelial cells are
found in the lung of a non-metastatic breast cancer patient
Background:
explain these patterns
We develop a stochastic model at the genomic level and use
population genetics techniques to explore this phenomenon
Modeling Metastasis:
A tumor was grown in silico by creating a population of single
cells that stochastically undergo mitosis and cell death. Cells
can gain passenger and driver mutations during division which
are passed to their offspring
Results: Effect of p
Success of metastases inc
Fig 5.
Driver Mutation
Passenger Mutation
B(d,p)*
An aliquot of 103 cells was taken from a primary tumor when it reached
5x105 and 106 cells and were then allowed to deposit into a foreign stroma
and observed
A stromal penalty s was applied only to driver mutations acquired in the
Fig 3.
Results: Effect of stro
Cells derived from the larger prima
Metastatic success is highly depen
Three regimes were observed in th
metastasis was impossible (green),
which only cells from certain primary
We further define a value, scritical, ab
Fig 4.
Feature of Model Observed Phenomenon
Population size determined
by fitness of cells
Larger Tumors more likely to
metastasize
Cells can acquire passenger
mutations that are slightly
deleterious
Many micrometastases never
grown to macroscopic size
Cells with more mutations are
less likely to metastasize
Stromal environment reduces
efficacy of driver mutations
Certain stromal conditions
prohibit metastasis
Metastasis continues to
mutate and evolve
Metastases with same
founding cell can have
different fates
Cells divide and acquire
mutations on individual basis
Large heterogeneity in
probability of metastasis
within primary tumor
No pre-defined growth rate
Late primary tumors less
likely to metastasize than
early tumors of equivalent
size
Pmet
We derived a way to calculate Pmet
for every cell in the 100 primary
Integrative Mathematical
Oncology
NIH 1 U54 CA143970-01S

∂c
∂t
= D
∂2
c
∂x2
(1.6)
[0, L], with boundary conditions c(0, t) = c(0, L) = 1 and and initial
0.
this equation using separation of variables and solving for ˜c = c + 1
) and the boundary conditions ˜c(0, t) = ˜c(0, L) = 0 and initial condition
find that the solution (see Appendix A) is given by
) =
∞
n=1

2
nπ

cos nπ − 1

sin

nπx
L

exp

−
n2
π2
DCt
L2

. (1.7)
we plot the oxygen concentation profile through the centre of a 100×100
evoid of cells at one-second intervals. The analytical solution (equation
d to the numerical solution of equation (1.5) at each lattice point for
, 500 and 1, 000 seconds. We find excellent agreement between the two
cales and updates
time scales that govern the diffusion of nutrients and that at which
anaged by updating the continuous part of the model many times per
This can become computationally expensive in this explicit scheme,
seek to minimize this number. However, for stability, we require that
∂c
∂t
= D
∂2
c
∂x2
(1.6)
for t ≥ 0 and x ∈ [0, L], with boundary conditions c(0, t) = c(0, L) = 1 and and initial
condition c(x, t) = 0.
We can solve this equation using separation of variables and solving for ˜c = c + 1
which satisfies (1.6) and the boundary conditions ˜c(0, t) = ˜c(0, L) = 0 and initial condition
˜c(x, t) = −1. We find that the solution (see Appendix A) is given by
c(x, t) =
∞
n=1

2
nπ

cos nπ − 1

sin

nπx
L

exp

−
n2
π2
DCt
L2

. (1.7)
In Figure 1.8 we plot the oxygen concentation profile through the centre of a 100×100
domain which is devoid of cells at one-second intervals. The analytical solution (equation
(1.7)) is compared to the numerical solution of equation (1.5) at each lattice point for
time t = 1, 10, 100, 500 and 1, 000 seconds. We find excellent agreement between the two
solutions.
1.2.3.4 Time scales and updates
The difference in time scales that govern the diffusion of nutrients and that at which
cells operate is managed by updating the continuous part of the model many times per
cellular time step. This can become computationally expensive in this explicit scheme,
and therefore, we seek to minimize this number. However, for stability, we require that
the ∆tDc/∆x2
term from equation (1.5) is less than 0.25 [56]. We therefore choose
∆tDc/∆x2
= 0.1, which equates to updating oxygen every 0.25 seconds, or approximately
230, 400 times per cell cycle based on the parameters chosen (see Table 1.1). While we
assume the average cell cycle time to be τ = 16 hours, it is well known that cells in tissues
are not synchronized, and also that cell fate decisions such as apoptosis are made on
shorter time scales. To model this heterogeneity in division time and to more accurately
match the finer time scale associated with cell death due to microenvironmental cues
∂c
∂t
= D
∂2
c
∂x2
(1.6)
c(x, t) =
∞
n=1

2
nπ

cos nπ − 1

sin

nπx
L

exp

−
n2
π2
DCt
L2

. (1.7)
solutions.
the ∆tDc/∆x2
∆tDc/∆x2
∂c
∂t
=
for t ≥ 0 and x ∈ [0, L], with boundary con
We can solve this equation using separa
which satisfies (1.6) and the boundary condit
˜c(x, t) = −1. We find that the solution (see
c(x, t) =
∞
n=1

2
nπ

cos nπ − 1

In Figure 1.8 we plot the oxygen concenta
domain which is devoid of cells at one-second
(1.7)) is compared to the numerical solutio
time t = 1, 10, 100, 500 and 1, 000 seconds. W
solutions.
The difference in time scales that govern t
cells operate is managed by updating the co
cellular time step. This can become compu
and therefore, we seek to minimize this num
the ∆tDc/∆x2
term from equation (1.5) i
∆tDc/∆x2
= 0.1, which equates to updating
230, 400 times per cell cycle based on the p
assume the average cell cycle time to be τ =
are not synchronized, and also that cell fa
shorter time scales. To model this heterogen
match the finer time scale associated with
[68], we choose to update the HCA 100 t
∂c
∂t
= D
∂ c
∂x2
(1.6)
c(x, t) =
∞
n=1

2
nπ

cos nπ − 1

sin

nπx
L

exp

−
n2
π2
DCt
L2

. (1.7)
solutions.
the ∆tDc/∆x2
∆tDc/∆x2
Check of numerics

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