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A stochastic Markov chain model to describe cancer metastasis

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A stochastic Markov chain model to describe cancer metastasis

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Content A STOCHASTIC MARKOV CHAIN
MODEL TO DESCRIBE CANCER
METASTASIS
by
Jeremy Mason
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
18 December 2013
Copyright 2013 Jeremy Mason
Dedication
To my grandparents.
Good. Better. Best.
ii
Acknowledgements
I would like to thank my advisor, Professor Paul Newton, for his continued support through-
out this process and for his great and joyous personality. Thank you for creating an envi-
ronment where we can both be ourselves and learn and grow at the same time. I could not
have chosen a better advisor.
I would also like to thank Professors Eva Kanso, Paul Macklin, Roger Ghanem, and
Timothy Pinkston for their support and suggestions regarding my research and for taking
the time to be a part of my dissertation committee.
Thank you to the Bill and Melinda Gates Foundation for providing me with a means of
extending my academic career to an elite level at two of the nest institutions in the nation.
Another thank you to the great people I met at USC and through USC over the past
years; the AME department, the GAPP oce, CED, my lab mates, my friends, and my
colleagues. For being my basketball buddies and my food testers. To my family spread out
across the country, thank you for being my support through it all and continually reminding
me why, and that I am not alone.
And nally, to Quentin and Candace. A very special thank you for being there for me
at every turning moment, for being my solid ground when things got tough, for showing me
who I truly am, and for letting me show it to the world through my work and my hobbies.
I could not have done it without either of you.
iii
Table of Contents
Dedication ii
Acknowledgements iii
List of Figures vi
List of Tables viii
Abstract ix
Chapter 1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Markov model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Tumor entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2 A stochastic Markov chain model to describe cancer metastasis 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Algorithm to compute the Markov transition matrix . . . . . . . . . . 14
2.2.2 Convergence of the algorithm . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Singular values and properties of the ensemble . . . . . . . . . . . . . 18
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Description of the Markov chain model . . . . . . . . . . . . . . . . . 23
2.3.2 The state vectors and denition of the steady-state . . . . . . . . . . 23
2.3.3 Structure of the lung cancer matrix and convergence to the steady-state 27
2.3.4 First- and second-order sites . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.5 Self-seeding sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.6 Mean rst-passage times . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 3 Spreaders and sponges dene metastasis in cancer 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
iv
3.2.1 Structure of the lung cancer multistep diagram . . . . . . . . . . . . 47
3.2.2 The autopsy datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Cancer metastasis as a stochastic multistep process . . . . . . . . . . 54
3.3.2 Rank-ordering the two-step metastatic pathways toward the nal state
of the disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Metastatic sites as spreaders or sponges . . . . . . . . . . . . . . . . . 56
3.3.4 The spatial pathways of lung cancer . . . . . . . . . . . . . . . . . . . 58
3.3.5 Timescales of progression: enhancing the Kaplan-Meier approach . . 61
3.3.6 Assimilating new autopsy data of adenocarcinoma lung cancer patients
undergoing complete resection . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 4 Comparisons of 8 major cancer types 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Network diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Convergence to steady-state . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Multistep Pathway diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Reduced diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5.1 Spreaders and sponges . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.5.2 Two-step pathway percentages . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Mean rst-passage times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter 5 The entropy of metastatic cancer 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.1 Brief summary of autopsy dataset used . . . . . . . . . . . . . . . . . 87
5.2.2 Denition of entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.3 Denition of relative-entropy . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3.1 Distribution of metastatic tumors . . . . . . . . . . . . . . . . . . . . 93
5.3.2 Metastatic entropy for 12 major cancer types . . . . . . . . . . . . . 94
5.3.3 Relative-entropy between each primary cancer type and the aggregate
entropy associated with all cancers . . . . . . . . . . . . . . . . . . . 95
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Chapter 6 Discussion 102
Bibliography 107
v
List of Figures
1.1 Schematic diagram of human circulatory system . . . . . . . . . . . . . . . 3
1.2 Metastatic distributions from autopsy data . . . . . . . . . . . . . . . . . . 8
2.1 Ensemble convergence graph . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Singular value distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Converged lung cancer network . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Outgoing edges from lung vs. target distribution . . . . . . . . . . . . . . . 31
2.5 Lung! LN (reg) histogram . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Lung! adrenal histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 State vector progression for k = 0 & k = 1 . . . . . . . . . . . . . . . . . . 34
2.8 State vector progression for k = 2 & k =1 . . . . . . . . . . . . . . . . . . 35
2.9 Pathways from lung to liver . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.10 Mean rst-passage time histogram . . . . . . . . . . . . . . . . . . . . . . . 44
3.1 The one-step pathways of metastatic lung cancer . . . . . . . . . . . . . . . 48
3.2 The two-step pathways through top 8 rst-order sites . . . . . . . . . . . . 50
3.3 Convergence plot for the lung cancer matrix . . . . . . . . . . . . . . . . . 52
3.4 Reduced pathway diagram showing top 30 two-step paths . . . . . . . . . . 60
3.5 Kaplan-Meier curve of lung cancer victims . . . . . . . . . . . . . . . . . . 62
3.6 Mean rst-passage times from lung to each of the metastatic sites . . . . . 63
4.1 Converged cancer networks of 8 cancer types . . . . . . . . . . . . . . . . . 69
4.2 Dynamical progression of~ v
0
of 8 cancer types (semi-log) . . . . . . . . . . 72
vi
4.3 Dynamical progression of~ v
0
of 8 cancer types (linear) . . . . . . . . . . . . 73
4.4 Multistep pathway diagrams of 8 cancer types . . . . . . . . . . . . . . . . 74
4.5 Reduced pathway diagrams of 8 cancer types showing top 30 paths . . . . . 77
4.6 Reduced pathway diagrams of 8 cancer types showing top 35% . . . . . . . 81
4.7 Mean rst-passage time histograms of 8 cancer types showing top 35% . . . 84
5.1 Distribution histogram and log-log plot of all cancer distribution . . . . . . 89
5.2 Distribution histograms and log-log plots of 12 cancer types . . . . . . . . . 90
5.3 Distributions compared with all cancer (non-site specic) . . . . . . . . . . 98
5.4 Distributions compared with all cancer (site specic) . . . . . . . . . . . . 100
vii
List of Tables
1.1 Metastatic site numbering system . . . . . . . . . . . . . . . . . . . . . 6
2.1 One- and two-step transition probabilities . . . . . . . . . . . . . . . . 37
2.2 Self-edge weightings for each site . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Mean rst-passage times from lung . . . . . . . . . . . . . . . . . . . . 42
3.1 Top two-step pathway probabilities . . . . . . . . . . . . . . . . . . . . 57
3.2 Comparative table of top two-step metastatic pathways of all types
from Lung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Entropy table for each cancer type and for all cancers grouped to-
gether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
viii
Abstract
A stochastic Markov chain model for metastatic progression is developed for primary 8 ma-
jor cancer types based on a network construction of metastatic sites with dynamics modeled
as an ensemble of random walkers on the network. We calculate a transition matrix for
each primary cancer and use it to construct a circular bi-directional network of primary and
metastatic locations based on postmortem tissue analysis of 3827 autopsies on untreated
patients documenting all primary tumor locations and metastatic sites from this popula-
tion [19]. The resulting 50 potential metastatic sites are connected by directed edges with
distributed weightings, where the site connections and weightings are obtained by calculat-
ing the entries of an ensemble of transition matrices so that the steady-state distribution
obtained from the long-time limit of the Markov chain dynamical system corresponds to
the ensemble metastatic distribution obtained from the autopsy dataset. We condition our
search for a transition matrix on an initial distribution of metastatic tumors obtained from
the dataset. Through an iterative numerical search procedure, we adjust the entries of a
sequence of approximations until a transition matrix with the correct steady-state is found
(up to a numerical threshold). Once the transition matrix for a given cancer type is com-
puted, our metastatic progression model is based Monte Carlo simulations of collections of
random walkers all leaving the primary tumor location and executing a random walk across
the directed graph from site to site. The model allows us to simulate and quantify dis-
ease progression pathways and timescales of progression from the primary tumor location
to other sites. Pathway diagrams are created that classify metastatic tumors as `spreaders'
ix
or `sponges' and quanties three types of multidirectional mechanisms of progression: (i)
self-seeding of the primary tumor, (ii) reseeding of the primary tumor from a metastatic site
(primary reseeding), and (iii) reseeding of metastatic tumors (metastasis reseeding). The
entire process is replicated for additional primary tumors in the dataset of [19] for individual
analysis and comparative purposes.
A second contribution of this work is to introduce a quantitative notion of `metastatic
entropy' for cancer and use it to compare the complexity and predictability associated with
the 12 most common cancer types worldwide. We apply these notions of entropy and pre-
dictability directly to the autopsy dataset used to create our Markov model. The raw data,
which contains the number of metastases found at all of the anatomical sites for each cadaver
(whose primary tumor location is also recorded), is normalized (yielding their empirical dis-
tribution) so that we can interpret the histograms as probability mass functions (PMFs)
representing the large scale (whole body) metastatic `signature' of each primary cancer. We
characterize the power-law distributions associated with metastatic tumor distributions for
each primary cancer type. Then we calculate the entropy associated with each and use the
Kullback-Liebler divergence (relative-entropy) to compare each cancer type with all of the
data aggregated into an `all cancer' category, whose entropy value is used as a benchmark
for comparisons.
x
Chapter 1
Introduction
1.1 Background
The identication of circulating tumor cells (CTCs) in the human circulatory system dates
back to Ashworth's 1869 paper [3] in which he identied and pointed out the potential signi-
cance of cells similar to those found in the primary tumor of a deceased cancer patient. Since
then, there has been sporadic focus on CTCs as a key diagnostic tool in the ght against can-
cer, based mostly on the so-called `seed-and-soil' hypothesis [23, 62, 77] of cancer metastasis,
in which the CTCs play the role of seeds which detach from the primary tumor, disperse
through the bloodstream, and get trapped at various distant sites (typically small blood
vessels of organ tissues), then, if conditions are favorable, extravasate, form metastases, and
subsequently colonize. The metastatic sites oer the soil for potential subsequent growth of
secondary tumors. Paget's 1889 seed-and-soil hypothesis [62] asserts that the development
of secondary tumors is not due to chance alone, but depends on detailed interactions, or
cross-talk, between select cancer cells and specic organ micro-environments. In 1929, J.
1
Ewing challenged the seed-and-soil hypothesis [22] by proposing that metastatic dissemina-
tion occurs based on purely mechanical factors resulting from the anatomical structure of
the vascular system, a proposal that is now known to be too simplistic an explanation for
the metastatic patterns that are produced over large populations. While the seed-and-soil
hypothesis remains a bedrock theory in cancer research, it has been signicantly rened over
the years to incorporate our current level of understanding on how the ability for a tumor
cell to metastasize depends on its complex interactions with the homeostatic factors that
promote tumor cell growth, cell survival, angiogenisis, invasion, and metastasis [23].
A schematic diagram associated with the metastatic process is shown in Figure 1.1. Here,
the primary tumor (from which the CTCs detach) is located in the lower part of the diagram
and the distant potential secondary locations where CTCs get trapped and form metastases
are shown. In this study, we will not be concerned with extravasation, colonization and
the formation of secondary tumors which are complex processes in their own right [77], but
rather with a probabilistic description of metastatic progression from primary neoplasm to
metastatic sites; hence, we provide a quantitative framework for charting the time-evolution
of cancer progression along with a stochastic description of the complex interactions of these
cells with the organ micro-environments. Also shown in the gure are representative scales
of a typical red blood cell (8m), capillary diameter (5 8m), CTC (20m), and human
hair diameter (100m). The total number of remote sites at which metastases are found for
any given type of primary cancer is relatively small (see the autopsy dataset described in
[19]), say on the order of 50 locations, those sites presumably being the locations at which
2
CTCs get trapped and subsequently colonize. For any individual making up the ensemble,
of course, the number of sites with metastatic tumors would be much smaller. A `ballpark'
estimate, based on the ratio of mets to primaries (from [19]) suggests a number around
9484=3827 2:5, although in the modern era, this number is probably higher. A reasonably
thorough overview of this process is described in [70].
Figure 1.1: Schematic diagram of human circulatory system
Schematic diagram of human circulatory system showing circulating tumor cells (CTCs)
detaching from primary tumor and getting trapped in capillary beds and other potential
future metastatic locations as outlined by the `seed-and-soil' framework.
3
It wasn't until recently, however, that important technological developments in the abil-
ity to identify, isolate, extract, and genetically and mechanically study CTCs from cancer
patients became available (see, for example [14, 15, 34, 45, 46, 61, 63, 72]). These new ap-
proaches, in turn, produced the need to develop quantitative models which can predict/track
CTC dispersal and transport in the circulatory and lymphatic systems of cancer patients
for potential diagnostic purposes. As a rough estimate, data (based primarily on animal
studies) shows that within 24 hours after release from the primary tumor, less than 0:1% of
CTCs are still viable, and fewer than those, perhaps only a few from the primary tumor, can
give rise to a metastasis. There are, however, potentially hundreds of thousands, millions,
or billions of these cells detaching from the primary tumor continually over time [8, 79], and
we currently do not know how to deterministically predict which of these cells are the future
seeds, or where they will take root. All of these estimates, along with our current lack of
detailed understanding of the full spectrum of the biological heterogeneity of cancer cells,
point to the utility of a statistical or probabilistic framework for charting the progression of
cancer metastasis. This is a particularly important step for any potential future comprehen-
sive computer simulation of cancer progression, something not currently feasible. Although
the dispersion of CTCs is the underlying dynamical mechanism by which the disease spreads,
the probabilistic framework obviates the need to model all of the biomechanical features of
the complex processes by which cells journey through the vascular/lymphatic system. The
mathematical/computational framework for such an approach is provided later in detail.
4
1.2 The Markov model
We develop a new Markov chain based model of metastatic progression for primary lung
cancer and for 7 other cancer types (breast, cervical, colorectal, pancreatic, prostate, ovarian,
and skin), which oers a probabilistic description of the time-history of the disease as it
unfolds through the metastatic cascade [77]. The Markov chain is a dynamical system
whose state vector is made up of all potential metastatic locations identied in the dataset
described in [19] (dened in Table 1.1), with normalized entries that can be interpreted
as the time-evolving (measured in discrete steps k) probability of a metastasis developing
at each of the sites in the network. One of the strengths of such a statistical approach is
that we need not oer specic biomechanical, genetic, or biochemical reasons for the spread
from one site to another, those reasons presumably will become available through more
research on the interactions between CTCs and their microenvironment. We account for
all such mechanisms by dening a transition probability (which is itself a random variable)
of a random walker dispersing from one site to another, thus creating a quantitative and
computational framework for the seed-and-soil hypothesis as an ensemble based rst step,
then can be further rened primarily by using larger, better, and more targeted databases
such as ones that focus on specic genotypes or phenotypes, or by more rened modeling
of the correlations between the trapping of a CTC at a specic site, and the probability of
secondary tumor growth at that location.
The Markov chain dynamical system takes place on a metastatic network based model of
the disease, which we calculate based on the available data over large populations of patients.
5
Table 1.1: Metastatic site numbering system
# Name # Name
1 Adrenal* 26 Omentum*
2 Anus 27 Ovaries
3 Appendix 28 Pancreas*
4 Bile Duct 29 Penis
5 Bladder 30 Pericardium*
6 Bone* 31 Peritoneum*
7 Brain* 32 Pharynx
8 Branchial Cyst 33 Pleura*
9 Breast 34 Prostate*
10 Cervix 35 Rectum
11 Colon 36 Retroperitoneum
12 Diaphragm* 37 Salivary
13 Duodenum 38 Skeletal Muscle*
14 Esophagus 39 Skin*
15 Eye 40 Small Intestine*
16 Gallbladder* 41 Spleen*
17 Heart* 42 Stomach*
18 Kidney* 43 Testes
19 Large Intestine* 44 Thyroid*
20 Larynx 45 Tongue
21 Lip* 46 Tonsil
22 Liver* 47 Unknown
23 Lung* 48 Uterus*
24 Lymph Nodes (reg)* 49 Vagina*
25 Lymph Nodes (dist)* 50 Vulva
Site numbering system used in transition matrix and network model. The

indicates an
entry in the target vector associated with lung cancer primary from the dataset of [19].
In particular, we use the data described in the autopsy analysis in [19] in which metastatic
distributions in a population of 3827 deceased cancer patients were analyzed. None of the
patients received chemotherapy nor radiation. The autopsies were performed between 1914
and 1943 at 5 separate aliated centers, with an ensemble distribution of 41 primary tumor
types, and 30 metastatic locations. Figure 1.2 shows histograms of the number of metastases
6
found at the various sites in the population. Figure 1.2(a) shows the metastatic distribution
in the entire population, while Figure 1.2(b) shows the distribution in the subset of the
population with primary lung cancer. We note that this data oers no particular information
on the time history of the disease for the population or for individual patients - only the
long-time metastatic distribution in a population of patients, where long-time is associated
with end of life, a timescale that varies signicantly from patient to patient (even those with
nominally the same disease). Although the initial analysis focuses on a model for primary
lung cancer, the approach would work equally well for all of the main tumor types (which is
outlined in Chapter 4).
Network based models of disease progression have been developed recently in various
contexts such as the spread of computer viruses [4], general human diseases [27], and even
cancer metastasis [10], but as far as we are aware, our Markov chain/random walk approach
to modeling the dynamics of the disease on networks constructed for each primary cancer type
from patient populations oers a new and potentially promising computational framework for
simulating disease progression. More general developments on the structure and dynamics
on networks can be found in the recent works [48, 50, 51, 52, 53, 74]. For brief introductions
to some of the mathematical ideas developed in this study, see [18, 20, 25, 67].
The classic view of metastatic progression, framed in part by the seed-and-soil hypothesis
of Paget [62], is that cancer spreads from the primary tumor site to distant metastatic
locations in a unidirectional way. The seeds responsible for the spread are CTCs ([56, 77, 79])
that detach from the primary tumor, enter the bloodstream and lymphatic system [79], and
7
(a)
0
0.05
0.1
0.15
0.2
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Anus
Appendix
Bile Duct
Branchial Cyst
Cervix
Colon
Duodenum
Esophagus
Eye
Larynx
Lip
Penis
Pharynx
Rectum
Retroperitoneum
Salivary
Tongue
Tonsil
Unknown
Vulva
Proportion
(b)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Brain
Thyroid
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Bladder
Prostate
Uterus
Vagina
Anus
Appendix
Bile Duct
Branchial Cyst
Breast
Cervix
Colon
Duodenum
Esophagus
Eye
Larynx
Lip
Ovaries
Penis
Pharynx
Rectum
Retroperitoneum
Salivary
Testes
Tongue
Tonsil
Unknown
Vulva
Proportion
Figure 1.2: Metastatic distributions from autopsy data
Metastatic distributions from autopsy dataset extracted from 3827 patients [19]. Y-axis in
each graph represents a proportion between 0 and 1. The sum of all the heights is 1. These
are the two key probability distributions used to `train' our lung cancer progression model.
(a) Overall metastatic distribution including all primaries. We call this distribution the `all'
cancer distribution as it includes all primary cancer types.; (b) Distribution of metastases
associated with primary lung cancer. We call this distribution the `target' distribution that
we label~ v
T
.
8
travel to new distant locations. If conditions are favorable, this initiates a complex [9, 23, 78]
and not well understood metastatic cascade, ultimately leading to tumor growth at distant
anatomic sites if their soil is hospitable [62]. The exclusively unidirectional nature of this
process has been challenged recently in a series of articles [2, 11, 41, 44, 60, 68], which use
mouse models to show a mechanism by which CTCs from the primary tumor can re-enter
the primary, a process called `self-seeding' [60]. These authors further comment that "it is
tempting to speculate that self-seeding might occur not only at the primary tumor site, but
also at distinct metastatic sites, ::: each site being a nesting ground." The possibility of
metastasis from metastases has also been discussed in [6, 33]. While the underlying `agent'
responsible for the spread of cancer is the CTC, the disease progression pathways in dierent
patients can be both predictable (from a statistical viewpoint), but often unpredictable and
surprisingly distinct in patients with nominally the same disease [21, 81], prompting the
question "how can metastatic pathways be predictable and unpredictable at the same time
[11]?"
Motivated in part by these questions, we develop our model for cancer progression and
use it to identify and quantify the multidirectional pathways and timescales associated with
metastatic spread for primary lung cancer.
While stochastic in nature, our model shows that a dening aspect of both pathway se-
lection and timescale determination is whether the disease spreads from the primary tumor
to a metastatic site that is either a `spreader' (adrenal gland and kidney) or a `sponge' (re-
gional lymph nodes, liver, bone). In contrast to the traditional view of cancer metastasis as
9
a unidirectional process starting at the primary site and spreading to distant sites as time
progresses, our model supports and quanties the view that there are important multidirec-
tional aspects to metastatic progression. These fall under 3 general classes: (i) self-seeding
of the primary tumor, (ii) reseeding of the primary tumor from a metastatic site (primary
reseeding), and (iii) reseeding of metastatic tumors (metastasis reseeding).
Using a discrete Markov chain [58] system of equations applied to a large autopsy dataset
of untreated patients with cancer [19], we quantify the likelihoods of the top metastatic
pathways in terms of probabilities and conduct Monte Carlo computer simulations of cancer
progression that statistically re
ect the autopsy data about a (non-Gaussian) distribution
of disease. The stochastic Markov chain dynamical system takes place on a metastatic
network-based model of disease progression that we construct based on available autopsy
data over large populations of patients. To obtain our baseline model, we use the data
described in an autopsy analysis [19] in which metastatic tumor distributions in a population
of 3827 untreated deceased cancer patients were recorded; 163 of these had primary lung
cancer of some type, distributing a total of 619 metastatic tumors across 27 dierent sites.
Information on lung cancer type in this dataset is not possible to obtain as the samples were
collected before the widespread use of immunohistochemistry (1914-1943), without which,
the subcategorization of non-small cell lung cancer is unreliable. However, it is probably
safe to assume that the distribution of lung cancer type was not signicantly dierent than
current distributions, roughly 40% adenocarcinoma, 30% squamous cell carcinoma, 9% large
cell carcinoma, and 21% small cell carcinoma.
10
1.3 Tumor entropy
Metastatic cancer is a dynamic disease of relentlessly increasing entropy. From an initial
primary tumor located at a single anatomical site, the metastatic cascade leads to a pro-
liferation of tumors at other sites on a timescale of months, or years in most cases, if left
untreated [9, 23, 77, 78]. Entropy is a quantity associated with notions of complexity and
predictability used primarily in two distinct elds: information theory [13, 37, 71] and sta-
tistical thermodynamics [40]. It is used to quantify the level of `disorder' associated with a
dynamical process that has a potentially large number of sites that it can occupy [16, 65].
Systems that can visit a large number of sites with relatively equal probability have higher
entropy (they are considered more disordered and less predictable) than systems that can
only occupy a few sites with widely separated probabilities (considered less disordered and
more predictable). Our goal in this study is to demonstrate how the notion of entropy can
be used in the context of metastatic spread to quantify and compare the complexity of the
12 most prevalent cancer types worldwide.
To x ideas further, suppose each anatomical site where a primary or metastatic tumor
could appear is indexed by `i
0
; (i = 1;:::N). Let
i
represent the probability that site `i' is
occupied (i.e. has a metastatic tumor), and let~ = (
1
;
2
;:::;
N
)2R
N
represent a prob-
ability mass distribution over a collection of potentially occupied sites, so that
P
N
i=1

i
= 1,
with 0
i
1. The level of disorder associated with the distribution ~ is captured by a
scalar quantity H
N
, called the `entropy' of the state ~ . As explained later, it is a quantity
that is a function both of N, and the way the probabilities are distributed among the N sites.
11
The lowest entropy state, corresponding to the one of least disorder, would be represented
by a distribution such as~ = (0; 0; 1; 0; 0;:::; 0), in which caseH
N
= 0. In this distribution,
statei = 3 is occupied with probability 1, making it predictively certain. Typically, this site
would be the anatomical location of the primary tumor in a patient whose disease has not
yet progressed to other sites. The highest entropy state, corresponding to the one of most
disorder, would be represented by the `uniform' distribution ~ = (
1
N
;
1
N
;
1
N
;
1
N
;
1
N
;:::;
1
N
).
For this uniform distribution, each site is occupied with equal probability. This distribution,
which constitutes an upper bound on the entropy, represents a state of maximal disorder,
corresponding to the least predictable state. The point we want to emphasize is that associ-
ated with any specic probabilistic distribution of occupied sites (typically falling between
the above two extremes), is a quantitative notion of `disorder', which in turn is related to
the system's predictability and complexity [16, 65]. Since each cancer type has a dierent
empirical metastatic tumor distribution, each will have a dierent metastatic entropy value
and these entropy values can be thought of as convenient `surrogates' representing metastatic
complexity and disorder.
12
Chapter 2
A stochastic Markov chain model to describe
cancer metastasis
2.1 Introduction
This section focuses on building a Markov chain model for primary lung cancer and high-
lighting some preliminary results. An initial guess of the model is constructed from the data
provided in [19] and then iterated on until a nal model is reached. We then use the entries
of the transition matrix and it's dynamics to classify the metastatic sites of the model and
to translate the results into easily understood and useful information.
2.2 Methods
Because we are computing the entries of a 50 50 matrix using only the 50 entries of
our target steady-state, the solution to this problem is not unique, a problem which is
addressed in the works of [17, 30, 31] for example. In those papers, the solution to this
13
constrained linear inverse problem is obtained by identifying the transition matrix that
satises a certain maximum entropy condition, and also one obtained by satisfying a least-
squares condition. More relevant to our problem is a criterion which targets a family of
solutions by pre-conditioning the search on an approximate transition matrix informed by
the data, followed by an iteration process which then adjusts the entries until a transition
matrix with the correct steady-state is obtained. We show that this process converges, and
we use the algorithm to create an ensemble of transition matrices whose entries are best
interpreted as (approximately) normally distributed random variables. We then characterize
the ensemble of stochastic transition matrices using the means and variances of the singular
value distributions [28] associated with the ensemble.
2.2.1 Algorithm to compute the Markov transition matrix
The three key steps in computing the transition matrix are:
(i) Step 1 - The choice of initial matrix A
0
: First, an approximate transition matrix, A
0
, is
obtained based on information we extract directly from the data set [19]. For the `lung row'
of A
0
, we use the lung target distribution shown in Figure 1.2(b), which is the metastatic
distribution in a population of people with lung cancer primary tumors. This is our rst
approximation to how the outgoing edges from the lung are weighted. On all of the other
49 rows, we use the all cancer distribution shown in Figure 1.2(a). Since we do not know, a
priori, how any of the other metastatic sites communicate with any of the others, we use this
`agnostic' distribution for all of these non-lung rows. Two key properties of A
0
constructed
14
this way are that it has Rank = 2 (i.e. only two linearly independent rows), and it does not
have our target distribution shown in Figure 1.2(b) as a steady-state, hence we know A
0
is
not the correct transition matrix for lung cancer. Therefore, we perform an iteration process
in Step 2 which adjusts the entries of A
0
to arrive at a nal transition matrix A
f
that has
higher rank (typically the same rank as the number of entries in the target vector), and has
the target distribution (Figure 1.2(b)) as a steady-state.
(ii) Step 2 - The iteration process to A
f
: A
0
, is then used to start an iteration process where
the entries are adjusted iteratively, using randomized adjustments, until its steady-state
distribution converges to the target distribution. The converged matrix obtained after this
process is what we call the `trained' lung cancer matrix, A
f
. We will discuss this key step
further below.
(iii) Step 3 - Creating an ensemble of A
f
's: Because the iterative procedure is based on ran-
dom adjustments of the matrix entries, and because we adjust the entries only up to some
pre-determined numerical value dened as our convergence threshold (typically chosen to
be O(10
5
)), the transition matrices produced from Step 2 should be thought of as having
entries that have some inherent probability distribution associated with them, with a sam-
ple mean and variance obtained by collecting an ensemble of these matrices. We will show
two of the key edge probability distributions (lung to regional lymph nodes, and lung to
adrenal) and also discuss the statistical spread of the ensemble of transition matrices using
their singular value distributions as a diagnostic tool.
15
2.2.2 Convergence of the algorithm
We now describe Step 2 of our algorithm in more detail, the iterative training stage which
takes us from our initial matrix A
0
, to our nal matrix A
f
. Dene the transition matrix
after stepj in the iteration process to beA
j
, with corresponding steady-state~ v
(j)
1
dened as
~ v
(j)
1
(A
j
I) = 0: (2.1)
Our goal is to nd the entries of A
j
so that
~ v
T
(A
j
I) = 0: (2.2)
i.e. so thatk~ v
(j)
1
~ v
T
k
2
= 0. We do this iteratively as follows. Since ~ v
T
6=~ v
(j)
1
, we can
dene a `residual' at step j:
~ v
T
(A
j
I) =~ r
j
(~ v
T
~ v
(j)
1
)(A
j
I); (2.3)
wherek~ r
j
k
2
6= 0. Our goal is to nd the entries of A
j
so thatk~ r
j
k
2
<< 1, where
is dened as our numerical convergence threshold. In practice, we do this by calculating
k~ v
T
~ v
(j)
1
k
2
directly and iterate the entries of A
j
untilk~ v
T
~ v
(j)
1
k
2
<, where typically we
16
take =O(10
5
). Stated more generally, our goal is to solve the following linear constrained
optimization problem. Given a target vector ~ v
T
, nd the entries a
ij
of the matrix A to
minimize the Euclidean norm of the residual vector~ r, where:
~ v
T
(A I) =~ r: (2.4)
The constraints are 0 a
ij
1, and
P
50
j=1
a
ij
= 1. Most importantly, we have pre-
conditioned the iterative process in Step 1 on our particular initial matrix A
0
. The general
framing of this problem as a constrained optimization problem is discussed in [17, 30, 31].
To do this, we iteratively adjust the entries of A
j
at each step (so as to maintain the
constraint that all rows sum to one) according to the following algorithm:
1. Calculate the residual~ r
j
at step j, starting with A
0
; (j = 0);
2. Pick the column of A
j
corresponding to the maximum entry of ~ r
j
;
3. Pick the column of A
j
corresponding to the minimum entry of ~ r
j
;
4. Pick a row of A
j
at random;
5. Decrease the entry of A
j
selected in step (ii) by , increase the entry of A
j
selected in
step (iii) by , where is scaled with the size ofk~ r
j
k
2
. This new matrix is A
j+1
;
6. Calculate the newk~ r
j+1
k
2
and stop ifk~ r
j+1
k
2
<. Otherwise go to step (ii) and repeat
the process.
17
Because of the randomized nature of the algorithm, and because of the nite threshold
of convergence, the converged nal matrix A
f
will be slightly dierent each time the iter-
ative process is carried out, even when all the trained matrices start with the same initial
A
0
. Thus, we carry out the iteration and convergence process, producing an ensemble of
1000 nal transition matrices A
f
, and we show the convergence (down to O(10
5
)) of the
ensemble in Figure 2.1 (plotted on a semi-log plot). The solid curve is the average conver-
gence rate computed from the 1000 training sessions, while the error bars show the standard
deviations associated with the ensemble, showing the spread of the convergence rates, which
are relatively tight.
2.2.3 Singular values and properties of the ensemble
A very useful diagnostic tool to characterize the structure and understand the statistical
spread associated with the matrices in the ensemble are the singular values,
n
(
1
>
2
>
:::>
2
7> 0), associated with the collection ofA
f
's. These are shown in Figure 2.2, plotted
from largest to smallest. Values shown (as open circles) are the sample means associated
with the singular values of the ensemble of 1000 converged matrices A
f
, all trained using
the same initial matrix A
0
. The error bars show the sample standard deviations, which are
small. The 27 non-zero singular values re
ect the fact that there are 27 entries in the steady-
state distribution for primary lung cancer. An equivalent way to say this is that the rank
of A
f
is 27, while the nullspace dimension is (approximately) 23. The standard deviations
show the statistical spread associated with two sources of uncertainty, one is the random
18
0 1 2 3 4 5 6
x 10
4
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
j = # of steps
Residual
Figure 2.1: Ensemble convergence graph
Ensemble convergence to A
f
, starting from A
0
. y-axis isk~ r
j
k
2
, x-axis is step j. We use
an ensemble of 1000 trained matrices A
f
, each conditioned on the same initial matrix A
0
.
The average convergence curve is shown, along with standard deviations marked along each
decade showing the spread associated with the convergence rates.
search algorithm we use to obtain convergence, and the other is the convergence threshold,
which we typically take to be O(10
5
). The small standard deviations indicate that the
algorithm is converging to the same nal A
f
, within a relatively small range of statistical
spread. On this graph, we also show the least squares curve t to singular values
4
through
19

24
, which follow a slope 0:1389, indicating that the singular values roughly decrease
like
n
exp(
n
)( 0:1901). The two diamond shaped data points on the graph
correspond to the two singular values of A
0
re
ecting the linear independence of the two
distributions from Figure 1.2 that we use in A
0
. We point out that the A
f
's should not be
viewed as small perturbations ofA
0
- our convergence algorithm starts with a rank 2 matrix
and generates an ensemble of (approximately) rank 27 matrices all within a relatively tight
statistical spread.
We also show one other set of singular values on the graph with the asterix data points.
To test the robustness of the ensemble with respect to perturbations of the initial matrixA
0
,
we start the search with an initial matrix of the formA
0
+A
1
. Here, the perturbation matrix
A
1
is a 50 50 rank 2 matrix obtained by giving each entry in the lung row a uniformly
distributed random number in the interval [-1,1], and each entry in all the other rows another
uniformly distributed random number in the interval [-1,1]. This creates a random rank 2
matrix. The perturbation parameter is chosen so that the perturbation size is (roughly) 5%
as compared with the average row value of A
0
. The asterix data points, which correspond
to a converged A
f
below a threshold of O(10
10
), all fall within the one standard deviation
bars of the unperturbed values, again showing that the nal converged matrix is relatively
robust to small changes in the initial matrixA
0
. For deniteness, when we make conclusions
associated with Monte Carlo simulations, we use the ensemble averaged set ofA
f
's obtained
over a set of 1000 converged matrices, each converged to within O(10
5
). Because of this,
20
0 5 10 15 20 25 30
10
−3
10
−2
10
−1
10
0
10
1
Index
Singular Value

1000 Ensemble
A
0
SVD’s
1000 Perturbed Ensemble
Figure 2.2: Singular value distribution
Average distribution of the 27 non-zero singular values associated with the ensemble of 1000
matricesA
f
all obtained using the same A
0
. x-axis is the indexn,y-axis is
n
. Data points
(open circles) indicate the sample average, with error bars showing the sample standard
deviations. Line is a least squares curve t through
4
through
24
, showing linear decrease
with exponent =0:1389. The 27 non-zero singular values re
ect the fact that there are
27 entries in the steady-state target distribution for primary lung cancer. The two diamond
shaped data points are the two singular values associated with the initial matrixA
0
. The 27
`asterix' data points are those obtained from a trained matrix using a perturbed A
0
, with
Rank 2 perturbation. See text for details.
we view the transition probabilities of the Markov chain, i.e. the edge values in our network,
as themselves being random variables, with a standard deviation that we can characterize.
21
2.3 Results
In this section we describe three main results from the model. First, the model separates
the 27 non-zero sites from Figure 1.2(b) into what we call `rst-order' sites (20 of these),
and `second-order' sites (7 of these). Second, the model quanties the ability of each site to
self-seed by ranking the average edge weight of each site back to itself (see [60]). Of these,
the strongest self-seeders are the lymph nodes, bone, kidney, and lung. Third, the model
allows us to calculate a time-ordering (model based) associated with metastatic progression.
This is achieved by performing Monte Carlo simulations of the mean rst-passage times
from the lung site to each of the other sites in the network. The mean rst-passage time
is the average number of edges a random walker must traverse in order to hit a given site,
hence the number is not restricted to take on discrete integer values. We think of these
mean rst-passage times as the proxy timescale for progression. In principle, they can be
calculated analytically using the fundamental matrix (see [29]), but in practice, since this
involves inverting the 5050 transition matrix, it is far more convenient to obtain the results
numerically via Monte Carlo simulations. The results will be described in terms of a `random
walker' leaving the lung site and traversing the network, moving from site to site along one
of the outgoing edges available to it at the site it is leaving, choosing a given edge with the
probability corresponding to its weighting.
22
2.3.1 Description of the Markov chain model
With the stochastic transition matrix A
f
, we brie
y describe the basic features and inter-
pretations of a Markov dynamical system model which we write as:
~ v
k+1
= ~ v
k
A
f
; (k = 0; 1; 2;::: ) (2.5)
The matrixA
f
is our transition matrix which is applied to a state-vector~ v
k
at each discrete
time-step k to advance to step k + 1. Thus, it is easy to see that:
~ v
k
=~ v
0
A
k
f
(2.6)
where ~ v
0
is the initial-state vector. The underlying dynamics associated with disease pro-
gression is interpreted as a random walk on the weighted directed network dened by the
entries of the transition matrix.
2.3.2 The state vectors and denition of the steady-state
To interpret the meaning of the initial-state vector and the transition matrix, one should
think of the patient's initial tumor distribution in terms of probabilities, or `uncertainties'.
Thus, an initial-state vector with a 1 in the 23rd entry:
~ v
0
= (0; 0; 0; 0; 0; 0; 0;:::; 1;::: )
23
in our 50 node model indicates, with absolute certainty, that the patient has a primary tumor
located in the `lung' (position 23). At the other extreme, we may have an initial-state vector:
~ v
0
= (1=50; 1=50; 1=50; 1=50; 1=50; 1=50;::: )
which indicates that all locations of the initial tumor distribution are equally likely. One
interpretation of this is that we have no information at all about where the primary tumor
is located. A third possibility is that we have some limited information about the initial
tumor distribution, but not completely certain information, thus an initial-state vector:
~ v
0
= (1=2; 0; 0; 0; 0; 0; 1=2; 0; 0; 0; 0; 0; 0; 0; 0; 0;::: )
would indicate that we think it likely that there is a primary tumor in the `adrenal' (position
1) or `brain' (position 7), but we are not sure which.
Then, we can ask how this initial information propagates forward in time as the disease
progresses. To advance one-step forward in time, we apply the transition matrix once to the
initial-state vector, thus:
~ v
1
=~ v
0
A
f
:
24
This gives us our new state-vector~ v
1
after step one. For the next step, we apply the transition
matrix again, this time to~ v
1
:
~ v
2
=~ v
1
A
f
=~ v
0
A
2
f
:
The dynamical system proceeds according to Equation (2.6) in a manner consistent with
the schematic diagram from Figure 1.1. As described in the introduction, it is best to think
of the entries of the state-vector as probabilities for metastases developing at each of the
discrete sites in our model (and in the data set), thus for the seed to take root in the soil. The
entries of the state-vector ~ v
k
continually get redistributed in time, as measured in discrete
steps k, until they reach the target steady-state distribution. A dierent interpretation of
the entries of the state-vector at each discrete step is that they re
ect the ensemble statistical
distribution of a collection of agents executing a random walk across the network. We should
point out, however, that for the ensemble of random-walkers all leaving from the lung site,
the best way to measure the passage of time is via mean rst-passage times to each of the
sites, which we compute using Monte Carlo simulations. It is important to keep in mind
that since the transition matrix is constructed based on an autopsy data set, there is no
direct information available on time-histories of progression, only tumor distribution at time
of death. A big advantage of using this data set is that we are able to build a model based
on the `natural' progression of the disease (i.e. untreated patients), whereas clinical data
on time-histories of progression for untreated patients do not exist, as far as we are aware.
Therefore, our challenge is to extract as much information as we can using the autopsy data
25
set [19], keeping in mind that time should be interpreted only as the model timescale of
progression.
Now comes a natural and important question. After long-times (k large), is there some
steady-state distribution that is achieved by the model? Correspondingly, given a particular
primary tumor, what are long-term probabilistic distributions of possible metastases? We
call this distribution vector~ v
(0)
1
, and dene it as:
~ v
(0)
1
= lim
k!1
~ v
0
A
k
f
: (2.7)
Notice that if a steady-state distribution is achieved, then for suciently largek,~ v
(0)
k+1
~ v
(0)
k
,
and since
~ v
(0)
k+1
=~ v
(0)
k
A
f
; (2.8)
this implies that
~ v
(0)
1
=~ v
(0)
1
A
f
: (2.9)
Thus
~ v
(0)
1
(A
f
I) = 0; (2.10)
26
which means that~ v
(0)
1
is a left-eigenvector of A
f
corresponding to eigenvalue = 1. This is
a crucial and practical observation that allows us to calculate the steady-state distribution
~ v
(0)
1
directly from the transition matrix. Since the rows of A
f
add to one, it always has at
least one eigenvalue that is 1, hence there is always at least one steady-state distribution,
but there may be more than one | this depends in detail on the matrix structure, something
the eigenvalue distribution [28] can reveal.
The target distribution for lung cancer shown in Figure 1.2(b) and labeled ~ v
T
is not a
steady-state for the matrix A
0
, i.e.
~ v
T
(A
0
I) = (~ v
T
~ v
(0)
1
)(A
0
I)6= 0; (2.11)
sincek~ v
T
~ v
(0)
1
k
2
6= 0.
2.3.3 Structure of the lung cancer matrix and convergence to the
steady-state
Figure 2.3 shows the network diagram associated with the ensemble averaged converged
matrix - this is the lung cancer network conditioned on our initial guess A
0
averaged over
1000 training sessions. Each of the sites has incoming and outgoing edges (denoted with
arrow heads) which connect it to other sites in the target distribution where the cancer can
spread, and each of the edges have a probabilistic weighting (not shown), with the constraint
that the weightings associated with all outgoing edges at each site must sum to 1. The disease
27
spreads across the network from an initial site following a random walk. To minimize the
number of edges depicted in the gure, we have combined incoming and outgoing edges
whenever possible, and placed arrow heads on both ends of an edge, instead of plotting the
two edges separately.
In Figure 2.4 we plot the (mean) edge weightings of the outgoing edges from the lung, as
compared with the values of the target distribution shown in Figure 1.2(b). The dierences
show that the values in the lung row of A
f
have adjusted from their initial values in A
0
.
Figure 2.5 and Figure 2.6 highlight our interpretation of the transition probabilities, or edge
values of the network, as random variables. We show in these gures the distributions
associated with the ensemble of lung to regional lymph node (Figure 2.5) edge values, and
those associated with the lung to adrenal (Figure 2.6) edge values. In each case, we histogram
the edge values from the 1000 converged matrices, and use the sample means and variances
to overlay a corresponding normal distribution. The vertical dashed lines in Figures 2.5 and
2.6 show the initial value of the transition probability from lung to regional lymph nodes
(Figure 2.5) and lung to adrenal (Figure 2.6). These initial values used in the matrix A
0
are
obtained using the entire data set of DiSibio and French [19], i.e. over all primary cancer
types. The converged Gaussian distributions shown in these gures, however, are specic
to lung cancer only. The fact that the mean is clearly shifted to the left of the vertical line
in Figure 2.5 indicates that the lung to regional lymph node connection for lung cancer is
less signicant, statistically, than for other cancer types. A possible anatomical explanation
for this left shift could be the fact that regional lymph nodes, for lung cancer, are located
28
23
24
25
26
28
30
31
33
34
38
39
40
41
42
44
48
49
27
29
32
35
36
37
43
45
46
47
50
1
5
6
7
12
16
17
18
19
22
2
3
4
8
9
10
11
13
14
15
20
21
Figure 2.3: Converged lung cancer network
The converged lung cancer network shown as a circular, bi-directional, weighted graph.
We use sample mean values for all edges connecting sites in the target distribution.
The disease progresses from site 23 (lung) as a `random walker' on this network. Arrow
heads placed on the end or ends of the edges denote the direction of the connections.
Edge weightings are not shown. There are 50 sites (dened in Table 1.1) obtained
from the full data set of [19], with `Lung' corresponding to site 23 placed on top. The
27 sites that are connected by edges are those from the target vector for lung cancer
dened in Table 1.1.
29
very close to the lung itself, compared with their typical distance away from other primary
tumor locations. Because of this unusually close proximity, regional lymph nodes could
easily have been mistakingly considered as part of the lung in some of the autopsies in the
series, eectively reducing the signicance of the lung to regional lymph node connection. By
contrast, the right shift of the mean, shown in Figure 2.6 for the lung to adrenal connection,
would indicate that the lung to adrenal connection is statistically more important for lung
cancer than for other primary cancer types. This could be due to the documented anatomic
connection between lung and adrenal that is known, but has not, to date, been a particular
focus of lung cancer metastasis studies.
The dynamical system dened by the Markov process:
~ v
k+1
= ~ v
k
A
f
; (k = 0; 1; 2;;;; ) (2.12)
can be thought of as governing the statistical distribution associated with random walkers
traversing the network. Figures 2.7 and 2.8 show the dynamical progression of the initial
state-vector, starting with an initial state-vector corresponding to a lung tumor, i.e. 1 in
position 23, with 0's elsewhere. In the sequence, the target vector~ v
T
is depicted with lled
bars, while the vector~ v
k
(for k = 0; 1; 2;1) is depicted with unlled bars. Convergence to
the target is exponential. By k = 2, convergence to the steady-state is essentially complete.
30
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Adrenal
Anus
Appendix
Bile Duct
Bladder
Bone
Brain
Branchial Cyst
Breast
Cervix
Colon
Diaphragm
Duodenum
Esophagus
Eye
Gallbladder
Heart
Kidney
Lg Intestine
Larynx
Lip
Liver
Lung
LN (reg)
LN (dist)
Omentum
Ovaries
Pancreas
Penis
Pericardium
Peritoneum
Pharynx
Pleura
Prostate
Rectum
Retroperitoneum
Salivary
Skeletal Musc
Skin
Sm Intestine
Spleen
Stomach
Testes
Thyroid
Tongue
Tonsil
Unknown
Uterus
Vagina
Vulva
Proportion
Figure 2.4: Outgoing edges from lung vs. target distribution
Weight of outgoing edges from the lung (using sample mean values from ensemble) as com-
pared with the `target' distribution.
2.3.4 First- and second-order sites
The 27 metastatic sites associated with lung cancer shown in the distribution of Figure
1.2(b) can be separated into two distinct groups in light of the ensemble averaged transition
probabilities listed in decreasing order in Table 2.1. The middle column of this table shows
the transition probability going directly from the lung to each of the 27 sites of the target
vector (ensemble averaged standard deviations). The right column of the table shows
31
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
0
5
10
15
20
25
Lung to LN (reg) Transition Probability
Bin Height
Figure 2.5: Lung! LN (reg) histogram
Histogram of edge values from lung to regional lymph nodes for 1000 trained A
f
's, showing
that edge values (transition probabilities) are best thought of as random variables which
are (approximately) normally distributed. Dashed vertical line shows initial edge value
associated withA
0
. Normal distribution with sample mean (0:15115) and variance (0:01821)
is shown as overlay.
the most likely two-step path from lung to each of the sites listed on the left, via the most
probable intermediate site. Thus it shows the product of the direct transition probability
from lung to an intermediate site (in parentheses on right), times the transition probability
from that intermediate site to the site listed on the left. When one compares these values
32
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
0
5
10
15
20
25
Lung to Adrenal Transition Probability
Bin Height
Figure 2.6: Lung! adrenal histogram
Histogram of edge values from lung to adrenal for 1000 trainedA
f
's showing that edge values
(transition probabilities) are best thought of as random variables which are (approximately)
normally distributed. Dashed vertical line shows initial edge value associated with A
0
. Nor-
mal distribution with sample mean (0:13165) and variance (0:01953) is shown as overlay.
(all are ensemble averaged) it is clear that the top 20 sites (listed above the cut-o line)
have direct transition values higher than their most probable two-step transition, hence we
call these `rst-order' sites. If the disease reaches one of these sites, the most likely path is
directly from the lung after one-step. A random walker, leaving the lung site, after it chooses
33
(a)
0
0.05
0.1
0.15
0.2
0.25
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Brain
Thyroid
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Bladder
Prostate
Uterus
Vagina
Anus
Appendix
Bile Duct
Branchial Cyst
Breast
Cervix
Colon
Duodenum
Esophagus
Eye
Larynx
Lip
Ovaries
Penis
Pharynx
Rectum
Retroperitoneum
Salivary
Testes
Tongue
Tonsil
Unknown
Vulva
Proportion

v
target
v
0
(b)
0
0.05
0.1
0.15
0.2
0.25
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Brain
Thyroid
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Bladder
Prostate
Uterus
Vagina
Anus
Appendix
Bile Duct
Branchial Cyst
Breast
Cervix
Colon
Duodenum
Esophagus
Eye
Larynx
Lip
Ovaries
Penis
Pharynx
Rectum
Retroperitoneum
Salivary
Testes
Tongue
Tonsil
Unknown
Vulva
Proportion

v
target
v
1
Figure 2.7: State vector progression for k = 0 & k = 1
Panel showing progression of state vector ~ v
k
for lung cancer primary using the ensemble
averaged lung cancer matrix. Filled rectangles show the long-time metastatic distribution
from the autopsy data in Figure 1.2(b), unlled rectangles show the distribution at step k
using the Markov chain model. (a) k = 0; (b) k = 1.
34
(a)
0
0.05
0.1
0.15
0.2
0.25
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Brain
Thyroid
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Bladder
Prostate
Uterus
Vagina
Anus
Appendix
Bile Duct
Branchial Cyst
Breast
Cervix
Colon
Duodenum
Esophagus
Eye
Larynx
Lip
Ovaries
Penis
Pharynx
Rectum
Retroperitoneum
Salivary
Testes
Tongue
Tonsil
Unknown
Vulva
Proportion

v
target
v
2
(b)
0
0.05
0.1
0.15
0.2
0.25
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Brain
Thyroid
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Bladder
Prostate
Uterus
Vagina
Anus
Appendix
Bile Duct
Branchial Cyst
Breast
Cervix
Colon
Duodenum
Esophagus
Eye
Larynx
Lip
Ovaries
Penis
Pharynx
Rectum
Retroperitoneum
Salivary
Testes
Tongue
Tonsil
Unknown
Vulva
Proportion

v
target
v
∞
Figure 2.8: State vector progression for k = 2 & k =1
Panel showing progression of state vector ~ v
k
for lung cancer primary using the ensemble
averaged lung cancer matrix. Filled rectangles show the long-time metastatic distribution
from the autopsy data in Figure 1.2(b), unlled rectangles show the distribution at step k
using the Markov chain model. (a) k = 2; (b) k =1.
35
one of the available outgoing edges with probability corresponding to the edge weighting,
will rst visit one of these rst-order sites. The most heavily weighted edges, hence the most
likely rst site visits, will be regional lymph nodes and adrenal, accounting for roughly 28%
of the rst site visits. The next two most heavily weighted sites are distant lymph nodes
and liver. These four sites account for roughly 50% of the rst site visits of an ensemble of
random walkers.
The remaining 7 sites (below the cut-o, starting from skin) have two-step transition
path probabilities that are equal to or more probable than their direct one-step path from
lung (taking into account standard deviations). We call these the `second-order' sites. The
interpretation of these sites is if there is a metastatic tumor at one of these sites, it is equally
probable, or more probable that there is also a metastatic tumor at an intermediate site, most
probably the regional lymph nodes or adrenal gland. Skin is the most signicant second-
order site, suggesting a possible pathway from a primary tumor in the lung to a metastatic
tumor on the skin via the regional lymph nodes or adrenal gland (not shown, but almost as
probable).
The classication of sites allows us to quantify possible disease progression paths (de-
scribed in terms of `random walkers') from lung to a given metastatic location. This is shown
in Figure 2.9 where we focus on the multiple pathways by which cancer can spread from a
primary lung tumor to the liver. We show in the gure the outgoing connection from lung
to liver (with weight 0:08028 0:00946), since liver is a rst-order site. Roughly 92% of the
random walkers, however, do not transition to liver on the rst step, but go instead to a
36
Table 2.1: One- and two-step transition probabilities
Target Sites One-step transition prob (Avg) Two-step transition probs
LN (reg) 0.15115 0.01821 0.02819 (LN (reg))
Adrenal 0.13165 0.01953 0.01397 (LN (reg))
LN (dist) 0.11928 0.00279 0.01860 (LN (reg))
Liver 0.08028 0.00946 0.01440 (LN (reg))
Kidney 0.06677 0.01231 0.00709 (LN (reg))
Bone 0.05914 0.00196 0.00931 (LN (reg))
Lung 0.05223 0.01504 0.01214 (LN (reg))
Pleura 0.04735 0.00338 0.00657 (LN (reg))
Pancreas 0.04660 0.00785 0.00549 (LN (reg))
Heart 0.03639 0.00739 0.00407 (LN (reg))
Spleen 0.03415 0.00454 0.00432 (LN (reg))
Brain 0.03274 0.00728 0.00360 (LN (reg))
Thyroid 0.03180 0.00628 0.00356 (LN (reg))
Pericardium 0.02733 0.00557 0.00306 (LN (reg))
Diaphragm 0.02169 0.00216 0.00289 (LN (reg))
Lg Intestine 0.01724 0.00266 0.00219 (LN (reg))
Gallbladder 0.01015 0.00048 0.00145 (LN (reg))
Stomach 0.00949 0.00139 0.00119 (LN (reg))
Sm Intestine 0.00786 0.00158 0.00149 (LN (reg))
Skeletal Musc 0.00413 0.00093 0.00047 (LN (reg))
Skin 0.00439 0.00443 0.00203 (LN (reg))
Peritoneum 0.00384 0.00567 0.00308 (LN (reg))
Omentum 0.00305 0.00223 0.00103 (LN (reg))
Prostate 0.00064 0.00060 0.00025 (LN (reg))
Vagina 0.00052 0.00059 0.00025 (LN (reg))
Bladder 0.00009 0.00029 0.00023 (Adrenal)
Uterus 0.00007 0.00025 0.00022 (Adrenal)
The 27 target sites listed in decreasing order of their edge weights (ensemble average values)
from lung site. The 20 sites above the `cut-o' are called `rst-order' sites. Their direct
connections from the lung are strong enough so that they represent the most likely route to
that site. The 7 sites listed below are called `second-order' sites. Their connections from
the lung are suciently weak that it is equally or more likely (taking into account standard
deviations) to get to the site via some other rst-order site (shown in parentheses).
37
dierent rst-order site. Some of these will pass to the liver on the second step, as shown by
the directed (solid) arrows. Still others pass to a second-order site, and then to the liver, as
shown by the directed (dashed) arrows. In this way, all possible pathways to the liver from
lung can be compared probabilistically and one can make quantitative predictions on which
other sites might have metastases if a lung cancer patient develops a metastatic liver tumor.
2.3.5 Self-seeding sites
A recent focus in the literature has been on the possibility that tumors can `self-seed' (see
[39, 60]) since that process would help explain the exceptionally rapid (`Gompetzian' [59])
growth of certain primary tumors. In addition, these papers discuss the possibility, not yet
proven experimentally, that self-seeding could potentially occur from a metastatic site back
to itself, i.e. `metastasis reseeding'. The focus on self-seeding of the primary tumor (CTCs
that colonize their tumors of origin) demonstrated convincingly in mouse models [39] has led
to the general concept that cancer progression, and hence progression pathways, may not be
a strictly unidirectional process of progression from primary tumor to sequentially distant
metastatic sites. It may well involve aspects that are more multidirectional in nature, such as
tumor self-seeding, reseeding of the primary tumor from a metastatic tumor, or reseeding of
a metastatic site from the metastatic tumor. Experimental evidence and the development of
theoretical models that support this, is currently an active area of research. In our model, a
site that is self-seeding is one in which a random walker leaving that site can return directly.
The simplest way (but not the only way) to do this would be after one step, if the site
38
Adrenal
Lung
Liver
Bladder
Bone
Brain
Diaphragm
Gallbladder
Heart
Kidney
Large intestine
Lymph nodes (reg)
Lymph nodes (dist)
Pancreas
Pericardium
Skeletal muscle
Small intestine
Spleen
Stomach
Thyroid
Vagina
Omentum
Peritonium
Prostate
Skin
Uterus
Figure 2.9: Pathways from lung to liver
Probabilistic decomposition of pathways from lung to liver. First transition probability is
directly from lung to liver (0:08028 0:00946). Paths from the rst-order sites to liver are
shown as solid arrows. Paths from second-order sites to liver are shown as dashed arrows.
has an edge connecting back to itself. This would correspond to a non-zero probability in
the diagonal entry of the transition matrix. We list in Table 2.2 the sites that have this
property, along with the edge weighting, listed from strongest to weakest. For primary
lung cancer, the most strongly weighted self-connecting edges are the lymph nodes (regional
39
and distant), liver, adrenal, bone, and lung. A more thorough analysis of this potentially
important multidirectional mechanism of progression for each given type of primary cancer,
along with the average time it takes to self-seed will be discussed later.
Table 2.2: Self-edge weightings for each site
Target Sites Self-edge weight (avg)
LN (reg) 0.1865 0.0152
LN (dist) 0.1231 0.0028
Liver 0.0945 0.0094
Adrenal 0.0929 0.0212
Bone 0.0616 0.0019
Lung 0.0522 0.0150
Kidney 0.0470 0.0143
Pleura 0.0434 0.0049
Pancreas 0.0360 0.0097
Spleen 0.0286 0.0057
Heart 0.0262 0.0088
Thyroid 0.0233 0.0076
Brain 0.0230 0.0092
Peritoneum 0.0211 0.0122
Pericardium 0.0203 0.0071
Diaphragm 0.0192 0.0031
Lg Intestine 0.0141 0.0033
Skin 0.0140 0.0071
Sm Intestine 0.0098 0.0019
Gallbladder 0.0097 0.0007
Stomach 0.0081 0.0019
Omentum 0.0068 0.0030
Skeletal Musc 0.0032 0.0013
Bladder 0.0020 0.0025
Uterus 0.0020 0.0025
Vagina 0.0017 0.0012
Prostate 0.0017 0.0009
27 target sites and their self edge weights (ensemble average) listed in decreasing order.
40
2.3.6 Mean rst-passage times
An important quantity associated with our model is called `mean rst-passage time' to each
of the sites - how many steps, on average, does it take for a random walker to pass from the
lung site to each of the other sites. This gives us a model based timescale (not limited to
take on discrete values) associated with disease progression, something a static autopsy data
set cannot give us directly. It is important to keep in mind that these values are model based
only, they do not arise from comparisons of disease time histories, something that could be
done with a dierent data set that contains time progression information. To calculate these
times, we follow a random walker starting at the lung position, progressing from site to site
until all of the sites have been visited at least one time. Using this method for roughly 10,000
of these random walkers, we collect statistical information on the mean rst-passage time to
each of the sites, i.e. the average number of steps it takes to rst arrive at each site. We
show below in Table 2.3 the mean rst-passage times from the lung site, which we obtain
by Monte Carlo simulations using an ensemble of 10,000 realizations, where each realization
is run long enough in time so that all sites identied by the lung cancer target vector are
visited at least once. We emphasize that the mean rst-passage times are distributed over
a range of positive values quite distinct from the discrete values required in the underlying
Markov process.
Despite the fact that these mean rst-passage times are model-based (i.e. time passage
information is not directly in the data set) they are interesting from several points of view.
The normalized values, shown in the right column of the table, are obtained by dividing each
41
Table 2.3: Mean rst-passage times from lung
Target Sites MFPT (un-normalized) MFPT (normalized)
LN (reg) 5.6414 0.4919 1.0000 0.0872
LN (dist) 8.3541 0.8096 1.4809 0.1435
Adrenal 10.0349 1.0068 1.7788 0.1785
Liver 10.6139 1.0226 1.8814 0.1813
Lung 13.0284 1.1497 2.3094 0.2038
Bone 16.0277 1.4508 2.8411 0.2572
Kidney 20.3944 1.9664 3.6151 0.3486
Pleura 22.9329 2.4375 4.0651 0.4321
Pancreas 26.4350 2.6438 4.6859 0.4686
Spleen 33.7009 3.4925 5.9739 0.6191
Heart 36.5513 3.6359 6.4791 0.6445
Brain 40.5540 4.3179 7.1886 0.7654
Thyroid 41.3240 4.0700 7.3251 0.7215
Pericardium 46.8599 4.1645 8.3064 0.7382
Diaphragm 51.3372 5.6196 9.1001 0.9961
Peritoneum 51.9555 5.4518 9.2097 0.9664
Lg Intestine 69.0501 7.3192 12.2399 1.2963
Skin 79.2006 8.4505 14.0392 1.4979
Gallbladder 104.9654 10.0373 18.6063 1.7792
Sm Intestine 105.8723 9.9567 18.7670 1.7649
Stomach 122.4070 12.7034 21.6980 2.2518
Omentum 155.6364 15.8049 27.5883 2.8016
Skeletal Musc 313.7172 30.6400 55.6098 5.4313
Bladder 620.7585 63.7243 110.0362 11.2958
Prostate 630.6260 68.4618 111.7854 12.1356
Vagina 630.8929 64.6222 111.8327 11.4550
Uterus 633.1578 63.9966 112.2342 11.3441
Mean rst-passage times (un-normailzed and normalized) from lung to each target site,
obtained by Monte Carlo simulation. Histogram plot is shown in Figure 2.10.
entry of the un-normalized column by the regional lymph node passage time time of 5.6414.
This way, everything is measured with respect to the time associated with the progression
from lung to regional lymph nodes, providing a relative predictive timescale for average
progression. If a patient with a primary lung tumor progresses to a metastatic tumor in
42
the regional lymph nodes after one year, one might expect it to take roughly another 6
months to progress to the distant lymph nodes, or roughly 9 months to the adrenal gland.
The interpretation is not that the disease will spread from lung to lymph nodes to liver
to adrenal, etc. all in one individual patient (since the model is based on an ensemble
data set), but that one, or perhaps several of these secondary sites will eventually produce
metastatic tumors, and we have a predictive handle on the progression timescales. The mean
rst-passage time histogram is plotted in Figure 2.10 and gives a visual representation of the
relative timescales to each of the sites. The sites seem to be grouped into approximately three
clusters. In the rst group, consisting of sites LN (reg) - Bone, there is an approximate linear
increase in the mean rst-passage times. The second grouping (Kidney - Peritoneum) also
increases linearly, but on a slightly shifted line. The third grouping (Lg intestine - Uterus)
increases (roughly) exponentially. Sites in this group, with very large mean rst-passage
times, like prostate or bladder, would be ones in which, if a metastatic tumor does appear,
would indicate poor prognosis as other areas would have had a lot of time and `probabilistic'
opportunities to develop tumors as well.
Not shown in the table and gure are mean rst-passage times from sites other than lung.
But it is worth pointing out that we have calculated these times starting at all 50 sites, and
the shortest mean rst-passage time occurs from pleura to adrenal, with a un-normalized time
of 1.02, or normalized value of 0.1811. This exceptionally short passage time indicates that if
the lung tumor does progress to the pleura, one might expect a short time later for progression
to occur to the adrenal gland. As mentioned earlier, this is another possible indication of
43
0
20
40
60
80
100
120
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Thyroid
Brain
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Prostate
Vagina
Uterus
Bladder
Passage Times
Figure 2.10: Mean rst-passage time histogram
Mean rst-passage time histogram for Monte Carlo computed random walks all starting from
lung. Error bars show one standard deviation. Values are normalized so that LN (reg) has
value 1, and all others are in these relative time units.
the potential importance of adrenal gland involvement in lung cancer progression. We are
currently comparing our model based mean rst-passage times with other data sets that
contain the time-history of the disease in individual patients and ensembles.
44
2.4 Discussion
This chapter has set up the framework for the Markov model that will be used for detailed
analysis, hypothesis testing, and comparative studies. After building a successful model,
the network is analyzed from a mathematical standpoint and translated in such a way that
gives clinicians useful information. Labeling and ordering the sites relate to the more and
less important metastatic sites for the progression of the primary cancer, and timescales of
progression illustrate timeframes to a metastasis. Even though the calculations were focused
only on building a lung cancer model, it is important to know that these same techniques
can be used for any primary cancer.
45
Chapter 3
Spreaders and sponges dene metastasis in
cancer
3.1 Introduction
This section focuses mainly on the key role that `spreaders' and `sponges' play in primary lung
cancer. By rank-ordering the top two-step pathways emanating from the primary cancer, we
can observe the main sites that dictate the spread of the disease and then classify these sites
using their respective pathway probabilities. We then look at the timescales of progression
given by the model (a re
ection of the time taken for a primary tumor to metastasize) and
compare it against the industry standard Kaplan-Meier curves. By assimilating new data
into the model from [73], we further compare these timescales as well as the top two-step
paths used to classify the `spreaders' and `sponges'.
46
3.2 Methods
3.2.1 Structure of the lung cancer multistep diagram
The 27 metastatic sites in the diagram shown in Figure 3.1 are organized in ring formation,
with 20 sites surrounding lung on the inner ring and the remaining 7 sites organized on the
outmost ring, each connected to a site from the inner ring. The sites listed on the inner
ring are called `rst-order' sitesthey have direct edge connections from the lung, with edge
probabilities decreasing from 12:00 clockwise around the ring. The most heavily weighted
edge, hence the most likely rst step of metastatic disease, is the transition from lung to
regional lymph nodes [LN (reg)]. The least heavily weighted, hence least likely rst step,
is the transition from lung to skeletal muscle shown just to its left. The 7 sites making
up the outermost ring are called `second-order' sites, also organized with edge probabilities
decreasing in clockwise order. These sites are classied as `second-order' due to the fact that
they have two-step probabilities via a rst-order site that are equal or higher in probability
than any direct one-step probability from the lung. In short, for disease to spread to a
second-order site from lung, it most probably passes via a rst-order site.
The general structure of the concentric diagram, with lung placed at the center, highlights
the underlying classical uni-directional view of disease progression. However, the diagram
also highlights the 3 key mechanisms of multidirectional progression: (i) self-seeding of the
primary lung tumor shown in the diagram as a self-loop in the seventh position, with an
edge weight of 5:2% and (ii) reseeding of the primary tumor from a rst-order site, shown
47
Lung
LN (reg)
0.151
0.052
0.052
Adrenal
0.132
0.079
0.093
LN (dist)
0.119
0.080
0.187
Liver
0.080
0.079
0.014
Kidney
0.067
0.079
0.026
Bone
0.059
0.080
0.002
0.094
Pleura
0.047
0.079
0.036
Pancreas
0.047
0.080
0.123
Heart
0.036
0.079
0.010
Spleen
0.034
0.079
0.043
Brain
0.033
0.079
0.062
Thyroid
0.032
0.080
0.003
Pericardium
0.027
0.079
0.007
Diaphragm
0.022
0.080
0.023
Lg Intestine
0.017
0.080
0.047
Gallbladder
0.010
0.079
0.019
Stomach
0.009
0.080
0.002
Sm Intestine
0.008
0.079
0.021
Skeletal Musc
0.004
0.079
0.020
Peritoneum
0.020
Skin
0.013
Omentum
0.007
Prostate
0.002
Vagina
0.002
Bladder
0.001
Uterus
0.001
Peritoneum
0.020
Skin
0.014
Omentum
0.007
Bladder
0.002
Vagina
0.002
Uterus
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.013
Omentum
0.007
Uterus
0.002
Prostate
0.002
Vagina
0.002
Bladder
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Vagina
0.002
Prostate
0.002
Uterus
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Uterus
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Prostate
0.002
Vagina
0.002
Peritoneum
0.020
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Uterus
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Uterus
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Uterus
0.002
Prostate
0.002
Vagina
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Prostate
0.002
Vagina
0.002
Peritoneum
0.022
Skin
0.014
Omentum
0.007
Bladder
0.002
Vagina
0.002
Uterus
0.002
Prostate
0.002
Peritoneum
0.020
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Uterus
0.002
Vagina
0.002
Prostate
0.002
Peritoneum
0.021
Skin
0.013
Omentum
0.007
Uterus
0.002
Bladder
0.002
Prostate
0.002
Vagina
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Bladder
0.002
Uterus
0.002
Prostate
0.002
Vagina
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Prostate
0.002
Vagina
0.002
Peritoneum
0.021
Skin
0.014
Omentum
0.007
Uterus
0.002
Bladder
0.002
Vagina
0.002
Prostate
0.002
Figure 3.1: The one-step pathways of metastatic lung cancer
Ensemble averaged one-step pathway diagram. Primary lung tumor is at the center, next ring
out are the 20 rst-order sites showing their direct connection from the lung, with transition
probabilities getting weaker in clockwise direction. Next ring out are the 7 second-order sites
and their connections from the rst-order sites. The 3 elements of multidirectional spread are
highlighted in this diagram: (i) self-seeding of the primary tumor (self-loop back to center),
(ii) reseeding of the primary tumor from a rst-order site (arrows back to center), and (iii)
reseeding of rst-order sites (self-loops back to rst-order site). Not shown in the diagram
are the one-step paths from rst-order site to another rst-order site.
48
as arrows directed back to the center. Because we are using an ensemble average of 1,000
trained lung cancer matrices to produce this diagram, the reseeding edges are all roughly
comparable in weight (8%), (iii) metastasis reseeding of rst-order sites shown as a self-loop
back to each metastatic site. The strongest metastasis re-seeders are lymph nodes (regional
and distant), followed by liver, adrenal, bone, and kidney.
From this diagram, we can obtain the two-step pathway probabilities from the lung, by
direct multiplication of the 2 edges making up any of the two-step paths starting from lung.
The 729 distinct two-step paths from the lung, the top ones of which are shown in Figure 3.2,
produce the statistical distribution ~ v
2
produced by the Markov chain model. We calculate
all of these and rank them in decreasing order in the next subsections. By comparing the
probability distributions~ v
2
and~ v
1
(shown in Figure 2.8a), we can see that after 2 steps, the
distribution has nearly converged to the steady-state, so we expect our rankings of two-step
pathways not to change much if we compare them to the top three-step and higher step
pathways.
Figure 3.3 shows a (ensemble) convergence and non-convergence plot associated with our
search algorithm to calculate the Markov transition matrix based on the baseline dataset [19].
What is signicant is the non-convergence of our algorithm when we constrain our searches
to not allow for any multidirectional edges. In other words, when we forced our algorithm
to not allow edges directly back to a site (no self-metastases nor primary reseeding), either
separately or together, the algorithm would not converge to a solution. In contrast, the
49
(a) (b)
Lung
Peritoneum
Skin
Omentum
Prostate
Vagina
Bladder
Uterus
Adrenal
LN (dist)
Liver
Kidney
Bone
Pleura
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
0.052
0.024
0.024
0.020
0.019
0.015
0.010
0.029
0.027
0.036
0.043
0.062
0.047
0.008
0.095
0.151 0.080
0.010
0.123
0.003
0.092
0.020
0.013
0.007
0.002
0.002
0.187
0.001
0.001
Lung
Peritoneum
Skin
Omentum
Prostate
Vagina
Bladder
Uterus
Adrenal
LN (dist)
Liver
Kidney
Bone
Pleura
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
0.052
0.024
0.020
0.019
0.014
0.010
0.008
0.023
0.029
0.027
0.037
0.044
0.062
0.010
0.047
0.132 0.079
0.003
0.095
0.187
0.123
0.020
0.014
0.007
0.002
0.002
0.093
0.002
0.002
(c) (d)
Lung
Peritoneum
Skin
Omentum
Prostate
Vagina
Bladder
Uterus
Liver
Kidney
Bone
Pleura
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
Adrenal
LN (dist)
0.052
0.021
0.019
0.014
0.010
0.008
0.010
0.024
0.024
0.029
0.027
0.036
0.043
0.003
0.062
0.119 0.078
0.187
0.047
0.092
0.095
0.021
0.013
0.007
0.002
0.002
0.123
0.002
0.002
Lung
Peritoneum
Skin
Omentum
Prostate
Vagina
Bladder
Uterus
Kidney
Bone
Pleura
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach Sm Intestine Skeletal Musc
LN (reg)
Adrenal
LN (dist)
Liver
0.052
0.019
0.014
0.010
0.008
0.010
0.003
0.021
0.023
0.024
0.029
0.027
0.037
0.187
0.043
0.080 0.080
0.092
0.062
0.123
0.047
0.021
0.014
0.007
0.002
0.002
0.095
0.002
0.002
Figure 3.2: The two-step pathways through top 8 rst-order sites
Diagram of all 28 two-step pathways from lung to a tertiary site. a. lung through regional
lymph nodes. b. lung through adrenal gland. c. lung through distant lymph nodes. d. lung
through liver. (Continued on the following page.)
50
(e) (f)
Lung
Peritoneum
Skin
Omentum
Prostate
Vagina
Bladder
Uterus
Bone
Pleura
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
Adrenal
LN (dist)
Liver
Kidney
0.052
0.014
0.010
0.008
0.010
0.003
0.187
0.019
0.021
0.023
0.024
0.029
0.027
0.092
0.036
0.067 0.080
0.123
0.043
0.094
0.062
0.021
0.014
0.007
0.002
0.002
0.047
0.002
0.002
Lung
Peritoneum
Skin
Omentum
Uterus
Bladder
Prostate
Vagina
Pleura
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
Adrenal
LN (dist)
Liver
Kidney
Bone
0.052
0.010
0.008
0.010
0.003
0.186
0.093
0.015
0.019
0.021
0.024
0.023
0.028
0.123
0.027
0.059 0.079
0.095
0.036
0.047
0.043
0.021
0.014
0.007
0.002
0.002
0.062
0.002
0.002
(g) (h)
Lung Peritoneum
Skin
Omentum
Uterus
Bladder
Vagina
Prostate
Pancreas
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
Adrenal
LN (dist)
Liver
Kidney
Bone
Pleura
0.052
0.008
0.010
0.003
0.187
0.092
0.123
0.010
0.014
0.019
0.020
0.024
0.023
0.095
0.029
0.047 0.079
0.047
0.027
0.062
0.036
0.020
0.014
0.007
0.002
0.002
0.043
0.002
0.002
Lung
Peritoneum
Skin
Omentum
Bladder
Uterus
Vagina
Prostate
Heart
Spleen
Brain
Thyroid
Pericardium
Diaphragm
Lg Intestine
Gallbladder
Stomach
Sm Intestine Skeletal Musc
LN (reg)
Adrenal
LN (dist)
Liver
Kidney
Bone
Pleura
Pancreas
0.052
0.010
0.003
0.187
0.092
0.123
0.095
0.008
0.010
0.014
0.019
0.021
0.023
0.047
0.024
0.047 0.079
0.062
0.029
0.043
0.026
0.021
0.014
0.007
0.002
0.002
0.036
0.002
0.002
Figure 3.2 (Continued.)
e. lung through kidney. f. lung through bone. g. lung through pleura. h. lung through
pancreas.
algorithm, in general, converged quickly to a solution when all connections were allowed and
produces a transition matrix with many multidirectional connections from site to site.
51
0 1 2 3 4 5 6
x 10
4
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
j = # of steps
Residual

No Constraints
No Self−Metastases nor Primary Reseeds
No Self−Metastases Reseeds
No Primary Reseeds
Figure 3.3: Convergence plot for the lung cancer matrix
Ensemble averaged (1000 trained matrices) convergence plot associated with algorithm to
compute the lung cancer transition matrix. Curve marked with squares shows the conver-
gence (ensemble averaged convergence plots) with no constraints to the lung cancer transition
matrix used in this study. Curve marked with circles shows non-convergence when the search
is constrained so that metastasis re-seeding and primary re-seeding are not allowed. Curve
marked with X's shows non-convergence when the search is constrained so that only metas-
tasis re-seeding is not allowed. Curve marked with diamonds shows non-convergence when
the search is constrained so that only primary re-seeding is not allowed.
3.2.2 The autopsy datasets
The data in [19] compiles the metastatic tumor distributions in a population of 3,827 deceased
cancer patients, none of whom received chemotherapy or radiation, hence the model can be
52
said to be based on the natural progression of the disease, although mastectomy for many
breast cancer primaries was most likely conducted at that time. In addition, brain metastases
are likely underrepresented by this dataset as brain autopsies probably were not universally
conducted at that time. The autopsies were conducted between 1914-1943 in 5 separate
aliated centers, with an ensemble distribution of 41 primary tumor types and 30 distinct
metastatic locations. The total number of distinct primary and metastatic tumor locations
is 50, which sets the size of our square Markov transition matrix (50 50) as well as the
number of entries in the Markov state vector ~ v
k
. The data oer no direct information on
the time history of the disease, either for individual patients comprising the ensemble or in
ensemble format. The data we use, therefore, only contain information on the `long-time'
distribution of metastatic tumors, where long-time is associated with end of life, a timescale
that varies signicantly from patient to patient. The model does, however, allow us to infer
time histories from autopsy data based on the logic that if more metastatic tumors show up
in a population of patients at a specic site, then on average, they would develop earlier in
the progression history. Although this association is not perfect, if does allow us to extract
meaningful temporal inferences from our Markov chain model. Details of how we infer the
correct ensemble Markov transition matrix are described in reference [54].
We use the dataset in 2 distinct ways to construct our model. First, we associate the dis-
tribution of metastatic tumors (after appropriate normalization) for primary patients with
lung cancer with the steady-state (long-time) probability distribution of our Markov chain
53
[58]. From this, we compute the `transition matrix' for our Markov chain (ensemble aver-
aged) that produces this steady-state. As the problem is mathematically underdetermined,
the calculation procedure requires an initial `candidate' transition matrix obtained from the
autopsy data and discussed in [54], which is then systematically iterated until a numerical
convergence criterion is satised. Interestingly, we also show that when our search algorithm
is con- strained so as to not allow any multidirectional edges in the directed graph associ-
ated with the transition matrix, no self- consistent model can be produced (i.e., the search
algorithm does not converge). Then, we update our baseline model with the more targeted
dataset described [73] of 137 patients with adenocarcinoma of the lung (stage I and II), all
treated with complete lung resection, and show how the baseline model is able to adapt to
this assimilated dataset.
3.3 Results
3.3.1 Cancer metastasis as a stochastic multistep process
The ensemble averaged lung cancer transition matrix associated with the Markov chain
model (see Figure 3.1) depicts the complete metastatic pathway diagram [54]. Each of the
2,500 entries, a
ij
, of the 50 50 transition matrix determines the probability of the disease
(modeled as a random walker over the network) spreading from site `i' to site `j' in an
eectively multistep process before the statistical tumor distribution of the autopsy dataset
is lled out. The diagram rank orders (in decreasing clockwise order) all of the possible
54
pathways emanating from the lung. One-step paths are dened by the edges leading directly
out from the lungthe sum of these outgoing edges must be one. The single most likely one-
step path of disease progression from the lung is to the regional lymph nodes, shown at the
top of the diagram, with a probability of 15:1%, followed by the lung to adrenal gland path,
with probability of 13:2%. On the diagram ordering the rst steps out of the lung, we also
show the `self-seeding' step directly back to the lung, represented by the edge from lung
looping back to itself, with edge probability 5:2%.
Two-step paths are made up of an edge from the lung to another site (or back to itself),
followed by the edge from that site to a second site. There are 729 two-step paths emanating
from the lung. The probability of taking a particular two-step path from the lung is obtained
by multiplying the weights of the 2 edges making up the path. The sum of all of these two-
step path probabilities must be 1, and so on for three-step paths, four-step paths, etc. We
focus on quantifying all of the two-step paths in this article, because as shown in Figure
2.8a, after 2 iterations of the Markov chain (k = 2), the state vector has nearly converged
to the steady-state target vector for metastatic tumors making metastatic progression for
lung cancer eectively a two-step process. In Figure 3.2, we show all of the two-step paths
emanating from the lung passing through each of the 8 most probable metastatic sites. To
obtain the probability of cancer progression on 1 of these two-step paths, one multiplies the
products of the 2 edges making up the two-step path.
55
3.3.2 Rank-ordering the two-step metastatic pathways toward the
nal state of the disease
We list the top multidirectional two-step pathways obtained from our model in Table 3.1.
The rst entries of Table 3.1 list the top 10 reseeding pathways back to the lung from a
rst-order site, along with the running cumulative values. We highlight from this list several
points. First, lymph nodes, adrenal gland, and liver are the most important intermediate
sites that reseed back to the lung. Their cumulative probability value (3:8%) accounts for
more than half of the total cumulative value from the entire list (6:2%). This total cumulative
value is slightly greater than, but roughly comparable in size to the lung to lung reseeding
path value of 5:2%, indicating that cells that reseed to the lung land therewith roughly equal
probabilities of having arrived via an intermediate site (see Table 3.1) versus directly from
the lung. The second half of Table 3.1 lists the top 10 two-step reseeding pathways back to a
metastatic site, a mechanism we call `metastasis reseeding.' From this table, we can see that
for lung cancer, lymph nodes and adrenal gland are the most active metastasis re-seeders,
followed by liver, bone, and kidney.
3.3.3 Metastatic sites as spreaders or sponges
A careful analysis of the top 30 two-step pathways allows us to compute the key probabilistic
quantity of interest associated with each two-step path which characterizes each site as a
sponge or a spreader. The quantity is the ratio of probability out (P
out
) over probability in
(P
in
) to each of the sites. If P
out
> P
in
, the site is a spreader, whereas if P
in
> P
out
, we
56
Table 3.1: Top two-step pathway probabilities
Top reseeding pathways back to lung Transition probability Cumulative values
Lung ! LN (reg) ! Lung 0.01214
Lung ! Adrenal ! Lung 0.01042 0.02256
Lung ! LN (dist) ! Lung 0.00952 0.03208
Lung ! Liver ! Lung 0.00645 0.03853
Lung ! Kidney ! Lung 0.00533 0.04386
Lung ! Bone ! Lung 0.00467 0.04853
Lung ! Pleura ! Lung 0.00375 0.05228
Lung ! Pancreas ! Lung 0.00367 0.05595
Lung ! Heart ! Lung 0.00288 0.05883
Lung ! Lung ! Lung 0.00273 0.06156
Top metastasis reseeders Transition probability Cumulative values
Lung ! LN (reg) ! LN (reg) 0.02819
Lung ! LN (dist) ! LN (dist) 0.01468 0.04287
Lung ! Adrenal ! Adrenal 0.01223 0.05510
Lung ! Liver ! Liver 0.00758 0.06268
Lung ! Bone ! Bone 0.00364 0.06632
Lung ! Kidney ! Kidney 0.00314 0.06946
Lung ! Pleura ! Pleura 0.00206 0.07152
Lung ! Pancreas ! Pancreas 0.00168 0.07320
Lung ! Spleen ! Spleen 0.00098 0.07418
Lung ! Heart ! Heart 0.00095 0.07513
Top two-step reseeding pathways back to lung: Primary! First-order site! Primary.
Top reseeding pathways back to metastatic site: Primary! First-order site! Back to
rst-order site. Cumulative values (obtained by adding the previous transition
probabilities) are listed in third column.
characterize it as a sponge. The ratio (P
out
=P
in
) of their exiting and incoming probabilities,
in the case of a spreader, gives us what we call the amplication factor, as it is larger than
one, whereas in the case of a sponge, we call the ratio the absorption factor, as it is less
than 1. Using these quantities, the top 2 spreaders are the adrenal gland and kidney, with
amplication factors of 1.91 (adrenal gland) and 2.86 (kidney). The total number of two-step
pathways into and out of the adrenal gland is 10, whereas the total into and out of kidney is
57
only 3. For these reasons, we identify the adrenal gland as the key distant anatomic spreader
of primary lung cancer.
The sponges associated with primary lung cancer are the regional lymph nodes, liver,
and bone. Their respective absorption factors are 0.74 (regional lymph nodes), 0.87 (liver),
and 0.75 (bone). The total number of two-step pathways into and out of the regional lymph
nodes is 16, compared with 8 into and out of the liver, and 5 into and out of bone. For these
reasons, we identify the regional lymph nodes as the key anatomical sponge associated with
primary lung cancer, followed by both bone and liver.
3.3.4 The spatial pathways of lung cancer
To compare the relative importance of two-step unidirectional pathways versus two-step
multidirectional pathways, we list the top 30 two-step pathways in decreasing order in Table
3.2. The top metastatic pathway (of any type) is the lung! LN (reg)! LN (reg) metastasis
reseeding pathway, whereas the top unidirectional pathway is the lung! adrenal! LN (reg)
path. Looking at all of the multidirectional pathways, it is clear that the lymph nodes and
adrenal gland are the key metastatic sites responsible for multidirectional spread, whereas
lymph nodes, adrenal gland, and liver are important sites responsible for unidirectional
spread. In general terms, lymph nodes, adrenal gland, and liver feature very prominently as
intermediate metastatic sites in many of the two-step pathways.
The information can then be combined into a reduced 2-step diagram for progression,
shown in Figure 3.4. The diagram shows the centrality of lymph nodes and adrenal gland
58
Table 3.2: Comparative table of top two-step metastatic pathways of all types
from Lung
Top 30 Untreated/Baseline Top 30 Stage I Top 30 Stage II Untreated Stage I Stage II
1 *LN (reg) ! LN (reg) *LN (reg) ! LN (reg) *LN (reg) ! LN (reg) 0.02819 0.02766 0.02764
2 Adrenal ! LN (reg) Adrenal ! LN (reg) Adrenal ! LN (reg) 0.02461 0.02320 0.02408
3 LN (dist) ! LN (reg) LN (dist) ! LN (reg) LN (dist) ! LN (reg) 0.02234 0.01937 0.02057
4 LN (reg) ! LN (dist) LN (reg) ! LN (dist) LN (reg) ! LN (dist) 0.01860 0.01613 0.01712
5 Adrenal ! LN (dist) Liver ! LN (reg) Liver ! LN (reg) 0.01620 0.01550 0.01566
6 Liver ! LN (reg) LN (reg) ! Liver Adrenal ! LN (dist) 0.01501 0.01476 0.01491
7 *LN (dist) ! LN (dist) *LN (reg) ! Lung LN (reg) ! Liver 0.01468 0.01461 0.01488
8 LN (reg) ! Liver Adrenal ! LN (dist) *LN (reg) ! Lung 0.01440 0.01349 0.01385
9 LN (reg) ! Adrenal LN (reg) ! Adrenal LN (reg) ! Adrenal 0.01397 0.01283 0.01355
10 Adrenal ! Liver Kidney ! LN (reg) Adrenal ! Liver 0.01253 0.01282 0.01292
11 Kidney ! LN (reg) Adrenal ! Liver *LN (dist) ! LN (dist) 0.01245 0.01232 0.01271
12 *Adrenal ! Adrenal *Adrenal ! Lung Kidney ! LN (reg) 0.01223 0.01209 0.01241
13 *LN (reg) ! Lung *LN (dist) ! LN (dist) *Adrenal ! Adrenal 0.01214 0.01125 0.01185
14 LN (dist) ! Liver Bone ! LN (reg) *Adrenal ! Lung 0.01130 0.01115 0.01184
15 LN (dist) ! Adrenal *Adrenal ! Adrenal Bone ! LN (reg) 0.01101 0.01083 0.01109
16 Bone ! LN (reg) *Lung ! LN (reg) LN (dist) ! Liver 0.01100 0.01042 0.01097
17 *Adrenal ! Lung LN (dist) ! Liver *LN (dist) ! Lung 0.01042 0.01023 0.01019
18 Liver ! LN (dist) *LN (dist) ! Lung LN (dist) ! Adrenal 0.00988 0.01013 0.01003
19 *LN (dist) ! Lung LN (reg) ! Bone Liver ! LN (dist) 0.00952 0.00940 0.00970
20 LN (reg) ! Bone Brain ! LN (reg) *Lung ! LN (reg) 0.00931 0.00924 0.00965
21 Pleura ! LN (reg) Liver ! LN (dist) Pleura ! LN (reg) 0.00886 0.00904 0.00944
22 Pancreas ! LN (reg) LN (dist) ! Adrenal LN (reg) ! Bone 0.00873 0.00894 0.00937
23 Kidney ! LN (dist) Pleura ! LN (reg) *Lung ! Adrenal 0.00820 0.00886 0.00839
24 Adrenal ! Bone *Lung ! Adrenal *Liver ! Liver 0.00811 0.00872 0.00835
25 *Lung ! LN (reg) *Liver ! Liver Adrenal ! Bone 0.00789 0.00819 0.00815
26 *Liver ! Liver *Liver ! Lung Pancreas ! LN (reg) 0.00758 0.00817 0.00814
27 Liver ! Adrenal Adrenal ! Bone *Liver ! Lung 0.00735 0.00787 0.00781
28 LN (dist) ! Bone Pancreas ! LN (reg) Kidney ! LN (dist) 0.00734 0.00775 0.00769
29 Bone ! LN (dist) Kidney ! LN (dist) Liver ! Adrenal 0.00728 0.00748 0.00759
30 LN (reg) ! Kidney LN (reg) ! Kidney *Lung ! LN (dist) 0.00709 0.00740 0.00715
Comparative table of top two-step metastatic pathways of all types from Lung. * paths are multidirectional. See Figure 3.1 for corresponding
diagram. Since each path starts from the lung, we show only the 2nd and 3rd site in the two-step pathway. First column lists the two steps out
from lung according to the baseline untreated dataset [19]. Columns 2 and 3 list the two steps out from lung according to the assimilated model
which incorporates dataset [73]. Columns 4-6 list the two-step probabilities of the corresponding pathways.
as key rst met- astatic sites, with many incoming and outgoing edges. The gure also
captures all of the information about the spreader or sponge character of each site, with
red indicating the color of the key spreaders (adrenal gland, kidney) and blue indicating the
color of sponges (lung, regional lymph nodes, liver, bone). Amplication and absorption
factors are shown in each of the ovals.
59
LN(reg)
Kidney
LN(dist)
Pleura
Pancreas
Liver
Adrenal
Bone
Lung
0.028
0.012
0.010
0.010
0.008
0.015
0.008
0.025
0.014
0.022
0.011
0.009
0.009
0.012
0.008
0.007
0.007
0.014
0.012
0.007
0.016
0.019
0.007
0.011
0.011
0.009
0.015
0.008
0.010
0.013
0.746
1.887
0.869
2.912
0.738
36.83%
Figure 3.4: Reduced pathway diagram showing top 30 two-step paths
Top 30 two-step pathways emanating from lung (representing 36:83% of the total pathway
probabilities), obtained by multiplying the edges of the one-step edges comprising each two-
step path. Edges without numbers are one-step paths emanating from lung. All other
numbered edges mark the second edge in a two-step path, with numbers indicating the two-
step probabilities. Colors indicate classication of each node as a `spreader' (red) or `sponge'
(blue). Spreader amplication factor and sponge absorption factor are listed in each oval.
Edge colors indicate primary self-seeding (red), primary reseeding (green), and metastasis
reseeding (yellow). See text for more detailed descriptions.
60
3.3.5 Timescales of progression: enhancing the Kaplan-Meier ap-
proach
Our model gives a useful measure of metastatic progression timescale, called rst-passage
time from lung to any given site, dened as the number of edges a `random walker' leaving
the lung must traverse to rst arrive at that site. Monte Carlo simulations of random
walk paths from the lung are conducted computationally to obtain mean rst-passage times
(averages over 10,000 runs) to every other site in the model. The mean rst-passage times
(MFPTs) act as a proxy timescale (model-based) for metastatic progression. It is a model-
based relative measure of the time that it takes for a primary tumor to metastasize to a
secondary site, or, roughly speaking, a model-based measure of the timescale associated
with successful extravasation and colonization [23]. Timescales associated with metastatic
disease are typically quantied by so-called Kaplan-Meier survival curves [43, 57], which
follow a cohort of patients from presentation until death, plotting the survival percentage
associated with the cohort (example shown in Figure 3.5). Alternative methods have been
proposed, but by and large, tracking survival of a cohort of patients remains the industry-
standard way of tracking progression. There is very little in the literature that tracks the
timescale of progression from metastatic site to metastatic site [6, 21, 32, 36, 81].
Mean rst-passage times from lung to each of the other sites are shown in Figure 3.6.
The sites are ordered from shortest to longest mean rst-passage time from lung. In dark, we
show the baseline (untreated patients) model using the dataset [19]. The dashed-dot line is a
linear curve t to the rst 9 sites, showing a clear linear increasing regimen (roughly the top
61
Figure 3.5: Kaplan-Meier curve of lung cancer victims
Example of an industry standard Kaplan-Meier survival curve. Curve shows the survival
rate of lung cancer patients over a period of 2000 days. Solid blue curve corresponds to
patients that were considered non-heavy smokers. Dashed red curve corresponds to patients
who were considered heavy smokers.
16 sites), followed by a group of sites where mean rst-passage times increase nonlinearly.
The rst 9 sites used in the reduced model set the basic linear timescales of progression for
the high probability metastatic locations. Times increase following the general linear formula
MFPT = at +b, where a = 2:56;b = 2:07 for the baseline (untreated) model, where `a'
is the slope and `b' is the y-intercept. In this formula, larger slopes indicate longer overall
mean rst-passage times from lung to metastatic sites. Spread to regional lymph nodes is
62
fastest (with a normalized value of 1), followed by normalized times to distant lymph nodes
(1.47), adrenal (1.72), and liver (1.75). One should interpret these timescales to indicate
that it takes roughly 75% longer for cancer to metastasize to adrenal gland and liver than
to regional lymph nodes. Self-seeding back to lung has a normalized mean rst-passage time
of 2.30, which is faster than to most of the rst-order sites, but over twice the time as the
lung to regional lymph node timescale.
0
20
40
60
80
100
120
140
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Thyroid
Brain
Pericardium
Peritoneum
Diaphragm
Lg Intestine
Skin
Sm Intestine
Gallbladder
Passage Time

Untreated
Lung Resection Stage 1
Lung Resection Stage 2
Lung Resection Stage 1 & 2
Figure 3.6: Mean rst-passage times from lung to each of the metastatic sites
Blue shows the baseline (untreated population) model, red shows the baseline model with
assimilated stage I resections, and green shows baseline model with assimilated stage II
resections. Lines are linear curve ts to rst 9 entries. Error bars show 1 SD from the mean.
See text for details.
63
3.3.6 Assimilating new autopsy data of adenocarcinoma lung can-
cer patients undergoing complete resection
Figure 3.6 (more details are shown in Table 3.2) also shows metastatic pathways and mean
rst-passage times using the model with assimilated data from [73], an autopsy dataset track-
ing a cohort of patients with adenocarcinoma of the lung (ACL) who underwent complete
lung resection. Of these, 35 survived more than 30 days after resection, 22 were classied
as stage I, and 13 as stage II. We assimilated their metastatic tumor distribution from an
autopsy study into our baseline (untreated population) model, recalculated the Markov tran-
sition matrix and all mean rst-passage times. The results are shown in Figure 3.6 (and the
middle and right columns of Table 3.2). Stage I are shown in medium dark, stage II in light
gray.
Comparing the columns of Table 3.2, the main change in the spatial pathways shows
up in the fth entry down, where the Lung! Adrenal! LN (dist) pathway drops in
probability on the list of the stage I treated patients but not as much as for the stage II
treated patients. Lung resection seems to alter this important pathway, particularly for stage
I patients, making it less likely to occur, perhaps by disruption of lymphatic connections
between the primary tumor and ipsilateral adrenal gland. The overall probabilities of each
of the pathways in the treated population also decrease from the untreated population.
The eect of treatment on the overall mean rst-passage times is shown in Figure 3.6.
The corresponding curve t to the rst 9 sites follow the same general linear trend as in
the untreated population, MFPT =at +b, but with a = 2:68;b = 1:55 (stage I, medium
64
dark);a = 2:54;b = 1:91 (stage II, light gray). The conclusions we can draw are clear: mean
rst-passage times increase overall with the stage I treated cohort, shown by the increase
in slope over the untreated slope, but not with the stage II treated cohort. Interestingly,
the MFPT back to lung in the treated cohort actually decreases with treatment. As lung is
classied as a sponge in our model, this does not seem to have a negative overall eect on the
general trend of increasing passage times with treatment. In contrast, the MFPT back to
adrenal gland (the key spreader) with the treated cohort increases. This enhances the overall
increase in MFPTs for the treated cohort. The mean rst-passage times increase most in
the subgroup of stage I patients, indicating that complete lung resection is more eective in
this group compared with the stage II subgroup. To summarize, our model shows that lung
resection for patients with ACL seems to generally increase overall MFPTs of metastases for
stage I patients, and it does this by (i) altering a key pathway from lung to adrenal gland to
distant lymph nodes, (ii) increasing mean rst-passage times to the adrenal gland (spreader),
(iii) decreasing mean rst-passage times back to the lung (sponge), and (iv) reducing the
overall top pathway probabilities. Lung resection seems to have very little impact on stage
II patients. The failure of resection to improve metastasis-free survival in stage II patients
with lung cancer may occur because the regional lymph nodes act as a sponge (Figure 3.4),
potentially suppressing early metastasis when not removed. However, because the risk of
local disease is high in lung cancer, surgery remains the preferred treatment in stage II
disease.
65
3.4 Discussion
This chapter focused on the importance of the primary and the role it's metastatic sites
played in the big picture. As a result, we looked at the top ranked two-step paths and their
role in cancer being a multidirectional spreader and classied sites as spreaders and sponges
to analyze how a metastasis eects disease progression. The data assimilation section shows
how new data can be incorporated into the existing model without having to start at the
beginning of the process and also how widely cancer can dier depending on the stage that
it is in. This type of information and classication can tell a great amount about the model
that is not easily seen without a bit of analysis.
66
Chapter 4
Comparisons of 8 major cancer types
4.1 Introduction
As mentioned earlier, the entire process of creating a Markov model and analyzing it's
transition probabilities and dynamics can be repeated for any primary cancer type that
we have the proper data for. This section focuses on analyzing and comparing the models
for 12 major cancer types (lung, breast, prostate, colorectal, pancreatic, ovarian, cervical,
skin). These cancers were chosen because they are some of the most common cancers found
today. We focus on the models' convergence to their steady-state, their network, multistep
pathway, and reduced order diagrams, and their mean rst-passage times. Included in this
analysis is classication of rst- and second-order sites, spreaders and sponges, and a visual
`predictability' associated with the model.
67
4.2 Network diagrams
Just as with primary lung cancer, each metastatic model is built from the dataset listed in
[19] by iterating upon an initial guess matrix that is constructed from the overall cancer
distribution and it's respective distribution. An ensemble of matrices is calculated for each
and used for comparison for the rest of this chapter. The rst, and most obvious thing
to look at is the structure of each network, which is shown in Figure 4.1. From a global
standpoint, these look practically identical.
Closer inspection and an analysis of the strengths of the connections (not shown in
gures) shows more subtle dierences. The skin network is composed of the most nodes
(30) while the prostate network has the fewest (21). Despite this, with roughly half the
number of connections, the prostate network's strongest connection is more than twice that
of the strongest connection in skin, and 5:77 times stronger than the average weighting of
it's network (compared to 3:93 for skin). The ovarian network boasts the largest maximum
weight connection along with the largest maximum-average ratio right at 9:05 The other
networks, excluding skin and prostate, range from 26-28 nodes and 676-784 connections
between them. This is a perfect indication of how similar cancer appears from a global
standpoint, but a more detailed look can reveal how dierent each can be.
68
(a) (b)
23 24
25
26
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(c) (d)
34
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6 12
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Figure 4.1: Converged cancer networks of 8 cancer types
Converged cancer networks shown as circular, bi-directional, weighted graphs. Arrow heads
placed on the end or ends of the edges denote the direction of the connections. Edge weight-
ings are not shown. Primary cancer placed on top. a. Primary lung cancer. b. Primary
breast cancer. c. Primary prostate cancer. d. Primary colorectal cancer. (Continued on the
following page.)
69
(e) (f)
28
30
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33
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48
1 5
6
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Figure 4.1 (continued.)
Converged cancer networks shown as circular, bi-directional, weighted graphs. Arrow heads
placed on the end or ends of the edges denote the direction of the connections. Edge weight-
ings are not shown. Primary cancer placed on top. e. Primary pancreatic cancer. f. Primary
ovarian cancer. g. Primary cervical cancer. h. Primary skin cancer.
4.3 Convergence to steady-state
As with every model that is created, verication that it is performing properly is essential.
The most logical way to do this with these metastatic cancer models is to ensure that each
70
progresses to it's steady-state. Figure 4.2 shows the dynamical progression of each initial-
state vectors plotted on a semi-log plot as the Euclidian norm between the current state
vector and the steady-state (k~ v
k
~ v
1
k). A line of best t indicates exponential progression
in the form of
k~ v
k
~ v
1
k exp(k) (4.1)
As shown in the gure, the values are roughly similar ranging from3:5 to5:4. In
fact, three pairs, colorectal and ovaries, breast and prostate, and lung and skin, have nearly
identical decay rates diering by at most 0:05. This would indicate that, from a global view,
each cancer model behaves more or less the same in terms of overall disease progression. The
same dynamical progression is shown again in Figure 4.3 on a linear plot. This more clearly
shows the exponential decay, but more importantly, it shows the convergence of each model
to each steady-state in approximately 2 steps. This indicates that in each model, it is indeed
the rst two steps in the progression that are the most relevant.
4.4 Multistep Pathway diagrams
With the multistep pathway diagrams, shown in Figure 4.4, its much easier to see dierences
between the networks. The ovarian and skin networks are the only two models that do
not contain second-order nodes. This indicates that every connection emanating from the
primary is equally large or greater than if it went through an intermediate site. On the other
71
0 1 2 3 4 5 6 7
10
−14
10
−12
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
2
k
Convergence

Pancreas (−5.3565)
Colorectal (−4.9959)
Ovaries (−4.9493)
Cervix (−4.5435)
Breast (−4.0126)
Prostate (−3.9772)
Lung (−3.5395)
Skin (−3.4933)
Figure 4.2: Dynamical progression of~ v
0
of 8 cancer types (semi-log)
Dynamical progression of initial-state vectors plotted on a semi-log plot. Values plotted are
Euclidian norm (k~ v
k
~ v
1
k) between state vector and steady-state. Line of best t follows
exponential decay in the form exp(k). values for each primary listed in legend.
hand, the models for prostate, colorectal, pancreatic and cervical cancers do not illustrate
primary reseeding nor primary self-seeding. Evidence of this is found in the original dataset
in the lack of metastases formed at other locations in the primary organ.
In most of the networks, the regional and distal lymph nodes have some of the strongest
connections to the primary. The exceptions to this are the skin and ovarian network. Even
72
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
1
2
3
4
5
6
7
8
9
k
Convergence

Pancreas (−5.3565)
Colorectal (−4.9959)
Ovaries (−4.9493)
Cervix (−4.5435)
Breast (−4.0126)
Prostate (−3.9772)
Lung (−3.5395)
Skin (−3.4933)
Figure 4.3: Dynamical progression of~ v
0
of 8 cancer types (linear)
Dynamical progression of initial-state vectors plotted on a linear plot. Values plotted are
Euclidian norm (k~ v
k
~ v
1
k) between state vector and steady-state. Line of best t follows
exponential decay in the form exp(k). values for each primary listed in legend.
Convergence is nearly complete after 2 steps.
though the distal lymph nodes forms the second strongest connection with skin, the regional
lymph nodes falls down to position 12. Another important observation is the location of
skin's self-seeding loop at position 3. This is unusually high as compared to the other
73
(a) (b)
Lung
LN (reg)
0.151
0.052
0.052
Adrenal
0.132
0.079
0.093
LN (dist)
0.119
0.080
0.187
Liver
0.080
0.079
0.014
Kidney
0.067
0.079
0.026
Bone
0.059
0.080
0.002
0.094
Pleura
0.047
0.079
0.036
Pancreas
0.047
0.080
0.123
Heart
0.036
0.079
0.010
Spleen
0.034
0.079
0.043
Brain
0.033
0.079
0.062
Thyroid
0.032
0.080
0.003
Pericardium
0.027
0.079
0.007
Diaphragm
0.022
0.080
0.023
Lg Intestine
0.017
0.080
0.047
Gallbladder
0.010
0.079
0.019
Stomach
0.009
0.080
0.002
Sm Intestine
0.008
0.079
0.021
Skeletal Musc
0.004
0.079
0.020
Peritoneum 0.020
Skin 0.013
Omentum 0.007
Prostate 0.002
Vagina 0.002
Bladder 0.001
Uterus 0.001
Peritoneum 0.020
Skin 0.014
Omentum 0.007
Bladder 0.002
Vagina 0.002
Uterus 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.013
Omentum 0.007
Uterus 0.002
Prostate 0.002
Vagina 0.002
Bladder 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Vagina 0.002
Prostate 0.002
Uterus 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Uterus 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Prostate 0.002
Vagina 0.002
Peritoneum 0.020
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Uterus 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Uterus 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Uterus 0.002
Prostate 0.002
Vagina 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Prostate 0.002
Vagina 0.002
Peritoneum 0.022
Skin 0.014
Omentum 0.007
Bladder 0.002
Vagina 0.002
Uterus 0.002
Prostate 0.002
Peritoneum 0.020
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Uterus 0.002
Vagina 0.002
Prostate 0.002
Peritoneum 0.021
Skin 0.013
Omentum 0.007
Uterus 0.002
Bladder 0.002
Prostate 0.002
Vagina 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Bladder 0.002
Uterus 0.002
Prostate 0.002
Vagina 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Prostate 0.002
Vagina 0.002
Peritoneum 0.021
Skin 0.014
Omentum 0.007
Uterus 0.002
Bladder 0.002
Vagina 0.002
Prostate 0.002
Breast
LN (reg)
0.119
0.022
0.102
Bone
0.111
0.022
0.087
LN (dist)
0.103
0.022
0.171
Lung
0.099
0.022
0.091
Pleura
0.093
0.022
0.021
Skin
0.076
0.022
0.026
Liver
0.072
0.022
0.005
Adrenal
0.071
0.023
0.061
0.036
Pericardium
0.030
0.022
0.022
Ovaries
0.026
0.022
0.115
Brain
0.023
0.022
0.017
Spleen
0.022
0.022
0.065
Pancreas
0.019
0.022
0.003
Uterus
0.018
0.022
0.022
Thyroid
0.016
0.022
0.005
Diaphragm
0.014
0.022
0.015
Kidney
0.010
0.022
0.017
Peritoneum
0.009
0.022
0.020
Stomach
0.008
0.022
0.052
Gallbladder
0.007
0.022
0.008
Bladder
0.006
0.022
0.007
Heart
0.006
0.022
0.009
Vagina
0.003
0.022
0.007
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.004
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.003
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
Sm Intestine 0.005
Lg Intestine 0.005
Omentum 0.004
(c) (d)
Prostate
LN (reg)
0.275
0.018
Bone
0.248
0.012
LN (dist)
0.202
0.003
Lung
0.135
0.003
Liver
0.048
0.009
Spleen
0.014
0.078
Bladder
0.012
0.035
Kidney
0.012
0.003
Testes
0.011
0.128
Gallbladder
0.008
0.199
Adrenal 0.034
Pleura 0.017
Pancreas 0.013
Thyroid 0.004
Skin 0.004
Peritoneum 0.003
Stomach 0.002
Heart 0.002
Sm Intestine 0.002
Lg Intestine 0.002
Diaphragm 0.002
Adrenal 0.035
Pleura 0.017
Pancreas 0.012
Thyroid 0.004
Peritoneum 0.004
Skin 0.004
Sm Intestine 0.002
Lg Intestine 0.002
Stomach 0.002
Heart 0.002
Diaphragm 0.002
Adrenal 0.034
Pleura 0.017
Pancreas 0.013
Thyroid 0.004
Skin 0.004
Peritoneum 0.004
Stomach 0.002
Lg Intestine 0.002
Sm Intestine 0.002
Diaphragm 0.002
Heart 0.002
Adrenal 0.036
Pleura 0.018
Pancreas 0.013
Peritoneum 0.005
Thyroid 0.005
Skin 0.005
Diaphragm 0.003
Lg Intestine 0.002
Heart 0.002
Stomach 0.002
Sm Intestine 0.002
Adrenal 0.035
Pleura 0.018
Pancreas 0.013
Peritoneum 0.005
Skin 0.005
Thyroid 0.005
Heart 0.003
Diaphragm 0.003
Lg Intestine 0.002
Sm Intestine 0.002
Stomach 0.002
Adrenal 0.035
Pleura 0.018
Pancreas 0.013
Peritoneum 0.006
Skin 0.006
Thyroid 0.005
Diaphragm 0.003
Sm Intestine 0.003
Heart 0.003
Lg Intestine 0.003
Stomach 0.002
Adrenal 0.035
Pleura 0.019
Pancreas 0.013
Peritoneum 0.007
Skin 0.006
Thyroid 0.005
Diaphragm 0.003
Lg Intestine 0.003
Sm Intestine 0.003
Heart 0.003
Stomach 0.002
Adrenal 0.035
Pleura 0.018
Pancreas 0.013
Peritoneum 0.006
Skin 0.006
Thyroid 0.005
Diaphragm 0.003
Heart 0.003
Lg Intestine 0.003
Sm Intestine 0.003
Stomach 0.002
Adrenal 0.035
Pleura 0.018
Pancreas 0.014
Peritoneum 0.007
Skin 0.005
Thyroid 0.005
Diaphragm 0.003
Sm Intestine 0.003
Lg Intestine 0.003
Heart 0.003
Stomach 0.002
Adrenal 0.034
Pleura 0.019
Pancreas 0.014
Peritoneum 0.007
Skin 0.006
Thyroid 0.005
Diaphragm 0.003
Heart 0.003
Sm Intestine 0.003
Lg Intestine 0.003
Stomach 0.003
Colorectal
LN (reg)
0.255
0.182
Liver
0.230
0.023
LN (dist)
0.127
0.117
Lung
0.124
0.009
Peritoneum
0.057
0.015
Adrenal
0.048
0.049
Kidney
0.023
0.008
Uterus
0.017
0.002
Ovaries
0.014
0.127
Bladder
0.014
0.012
Thyroid
0.013
0.009
Prostate
0.012
0.015
Omentum
0.010
0.244
Bone
0.009
0.034
Lg Intestine
0.009
0.005
Pancreas
0.009
0.010
Gallbladder
0.007
0.008
Vagina
0.004
0.011
Spleen
0.003
0.015
Sm Intestine
0.003
0.052
Diaphragm
0.003
0.004
Skeletal Musc
0.003
0.003
Pleura 0.013
Skin 0.010
Heart 0.004
Brain 0.003
Pericardium 0.002
Stomach 0.001
Pleura 0.014
Skin 0.010
Heart 0.005
Brain 0.003
Pericardium 0.002
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.002
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.002
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.014
Skin 0.011
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Pleura 0.015
Skin 0.010
Heart 0.005
Brain 0.004
Pericardium 0.003
Stomach 0.001
Figure 4.4: Multistep pathway diagrams of 8 cancer types
Ensemble averaged one-step pathway diagram. Primary tumor is at the center, next ring
out are the rst-order sites showing their direct connection from the lung, with transition
probabilities getting weaker in clockwise direction. Next ring out are the second-order sites
and their connections from the rst-order sites. a. Primary lung cancer. b. Primary breast
cancer. c. Primary prostate cancer. d. Primary colorectal cancer. (Continued on the
following page.)
74
(e) (f)
Pancreas
Liver
0.242
0.004
LN (reg)
0.201
0.013
Peritoneum
0.098
0.108
LN (dist)
0.091
0.195
Lung
0.074
0.025
Diaphragm
0.055
0.023
Pleura
0.031
0.022
Omentum
0.028
0.090
Kidney
0.026
0.040
Adrenal
0.025
0.037
Sm Intestine
0.022
0.006
Spleen
0.022
0.077
Stomach
0.016
0.034
Gallbladder
0.015
0.003
Lg Intestine
0.014
0.012
Ovaries
0.009
0.210
Thyroid
0.006
0.010
Bladder
0.005
0.006
Uterus
0.004
0.019
Testes
0.004
0.003
Prostate
0.003
0.012
Bone 0.021
Skin 0.009
Pericardium 0.006
Breast 0.003
Heart 0.003
Bone 0.020
Skin 0.009
Pericardium 0.006
Breast 0.003
Heart 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Breast 0.003
Heart 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Breast 0.003
Heart 0.003
Bone 0.022
Skin 0.009
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.023
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.023
Skin 0.010
Pericardium 0.006
Heart 0.003
Breast 0.003
Bone 0.023
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.023
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.023
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.023
Skin 0.009
Pericardium 0.006
Heart 0.003
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.024
Skin 0.009
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.003
Breast 0.003
Bone 0.023
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.003
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.003
Breast 0.003
Bone 0.022
Skin 0.010
Pericardium 0.006
Heart 0.004
Breast 0.003
Ovaries
Peritoneum
0.349
0.020
0.088
Pleura
0.064
0.020
0.039
Diaphragm
0.062
0.020
0.007
Omentum
0.061
0.020
0.084
Uterus
0.057
0.020
0.013
Liver
0.057
0.020
0.027
LN (dist)
0.048
0.020
0.022
Lg Intestine
0.032
0.020
0.007
Vagina
0.031
0.020
0.020
Spleen
0.030
0.020
0.007
Kidney
0.028
0.020
0.007
Bladder
0.028
0.020
0.019
Sm Intestine
0.025
0.020
0.007
0.062
Stomach
0.018
0.020
0.208
Adrenal
0.013
0.020
0.030
Skin
0.005
0.022
0.022
Gallbladder
0.004
0.020
0.042
Brain
0.003
0.020
0.021
Pericardium
0.003
0.020
0.122
Thyroid
0.002
0.020
0.052
LN (reg) 0.119
Lung 0.061
Bone 0.018
Heart 0.006
Pancreas 0.006
LN (reg) 0.123
Lung 0.062
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.123
Lung 0.060
Bone 0.021
Heart 0.007
Pancreas 0.007
LN (reg) 0.119
Lung 0.061
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.122
Lung 0.060
Bone 0.022
Heart 0.007
Pancreas 0.007
LN (reg) 0.120
Lung 0.060
Bone 0.020
Pancreas 0.007
Heart 0.007
LN (reg) 0.122
Lung 0.059
Bone 0.020
Pancreas 0.007
Heart 0.006
LN (reg) 0.120
Lung 0.061
Bone 0.020
Pancreas 0.008
Heart 0.007
LN (reg) 0.121
Lung 0.060
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.118
Lung 0.061
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.121
Lung 0.061
Bone 0.021
Heart 0.007
Pancreas 0.007
LN (reg) 0.121
Lung 0.061
Bone 0.022
Pancreas 0.007
Heart 0.007
LN (reg) 0.121
Lung 0.060
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.121
Lung 0.060
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.124
Lung 0.062
Bone 0.022
Pancreas 0.007
Heart 0.007
LN (reg) 0.123
Lung 0.061
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.122
Lung 0.059
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.121
Lung 0.060
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.121
Lung 0.060
Bone 0.021
Pancreas 0.007
Heart 0.007
LN (reg) 0.122
Lung 0.059
Bone 0.021
Pancreas 0.007
Heart 0.007
(g) (h)
Cervix
LN (reg)
0.328
0.120
LN (dist)
0.183
0.117
Liver
0.125
0.024
Lung
0.122
0.014
Peritoneum
0.047
0.010
Ovaries
0.036
0.166
Bone
0.026
0.045
Kidney
0.024
0.004
Sm Intestine
0.016
0.029
Lg Intestine
0.015
0.009
Bladder
0.014
0.012
Spleen
0.012
0.008
Vagina
0.011
0.003
Adrenal
0.011
0.029
Diaphragm
0.009
0.002
Omentum
0.009
0.290
Heart
0.005
0.012
Pleura 0.012
Pancreas 0.007
Skin 0.005
Gallbladder 0.003
Stomach 0.002
Thyroid 0.002
Pericardium 0.002
Uterus 0.001
Brain 0.001
Pleura 0.013
Pancreas 0.007
Skin 0.006
Gallbladder 0.004
Thyroid 0.003
Stomach 0.002
Pericardium 0.002
Uterus 0.001
Brain 0.001
Pleura 0.014
Pancreas 0.007
Skin 0.006
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.002
Brain 0.001
Uterus 0.001
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.001
Uterus 0.001
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.015
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.015
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.015
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Uterus 0.002
Brain 0.002
Pleura 0.014
Pancreas 0.008
Skin 0.008
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.015
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.015
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Pericardium 0.003
Thyroid 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.014
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Pleura 0.015
Pancreas 0.008
Skin 0.007
Gallbladder 0.004
Thyroid 0.003
Pericardium 0.003
Stomach 0.003
Brain 0.002
Uterus 0.002
Skin
Lung
0.118
0.050
0.111
LN (dist)
0.089
0.051
0.108
0.079
Adrenal
0.068
0.052
0.059
Kidney
0.060
0.050
0.042
Brain
0.059
0.051
0.034
Bone
0.055
0.051
0.060
Thyroid
0.054
0.051
0.033
Heart
0.053
0.050
0.033
Pancreas
0.043
0.051
0.033
Sm Intestine
0.039
0.051
0.025
LN (reg)
0.036
0.052
0.128
Spleen
0.036
0.051
0.028
Liver
0.028
0.051
0.070
Pericardium
0.027
0.051
0.018
Gallbladder
0.024
0.051
0.016
Diaphragm
0.022
0.051
0.019
Pleura
0.021
0.051
0.029
Stomach
0.018
0.051
0.012
Skeletal Musc
0.012
0.051
0.007
Peritoneum
0.011
0.051
0.027
Prostate
0.011
0.051
0.007
Uterus
0.011
0.051
0.010
Breast
0.008
0.051
0.007
Testes
0.008
0.050
0.005
Lg Intestine
0.005
0.051
0.007
Ovaries
0.004
0.051
0.010
Bladder
0.002
0.051
0.005
Vagina
0.002
0.051
0.002
Omentum
0.001
0.052
0.005
Figure 4.4 (continued.)
Ensemble averaged one-step pathway diagram. Primary tumor is at the center, next ring
out are the rst-order sites showing their direct connection from the lung, with transition
probabilities getting weaker in clockwise direction. Next ring out are the second-order sites
and their connections from the rst-order sites. e. Primary pancreatic cancer. f. Primary
ovarian cancer. g. Primary cervical cancer. h. Primary skin cancer.
75
models that fall at 7, 8, and 9. This may indicate that skin more easily and readily spreads
throughout it's primary organ unlike other cancers.
4.5 Reduced diagrams
The natural progression of the model analysis leads to the creation of the reduced pathway
diagrams and the classication of spreaders and sponges. Figure 4.5 shows the reduced
diagrams for the 8 cancer models with their highlighted spreaders and sponges. Once again,
these diagrams are constructed from the top 30 two-step pathways emanating from the
primary cancer. These values are then used to calculate the probability in and out of each
node to classify the spreaders and sponges. The percentage located below the primary
indicates the total percentage that the 30 paths correspond to out of all the two-step paths.
This percentage will be addressed later in the section.
Looking at the reduced diagrams, the most obvious dierence between the models is the
number of nodes involved. The lung, breast and pancreas models only use 8 of their nodes,
while prostate and colorectal use 7, ovarian and cervical use 9, and skin uses 10. Regardless
of the number of nodes, the most `active' node is the rst one located at 12:00. Seeing as this
is contains the strongest connection to the primary, this makes sense in terms of it having
a good amount of two-step pathways in the top 30. The next most obvious thing is the
multidirectional pathways associated with each model. While all of them show metastasis
reseeding (yellow loops attached to metastatic nodes) in their top 30 paths, only lung,
76
(a) (b)
LN(reg)
Kidney
LN(dist)
Pleura
Pancreas
Liver
Adrenal
Bone
Lung
0.028
0.012
0.010
0.010
0.008
0.015
0.008
0.025
0.014
0.022
0.011
0.009
0.009
0.012
0.008
0.007
0.007
0.014
0.012
0.007
0.016
0.019
0.007
0.011
0.011
0.009
0.015
0.008
0.010
0.013
0.746
1.887
0.869
2.912
0.738
36.83%
Breast
LN(reg)
Pleura
LN(dist)
Liver
Adrenal
Lung
Bone
Skin
0.020
0.010
0.010
0.012
0.012
0.012
0.013
0.016
0.019
0.010
0.008
0.012
0.011 0.008
0.013
0.009
0.017
0.018
0.011
0.011
0.009
0.010
0.014
0.011
0.009
0.011
0.010
0.009
0.009
0.009
1.363
0.432
0.528
0.753
35.23%
(c) (d)
Prostate
LN(reg)
Bone
LN(dist)
Lung
Liver
Kidney
Adrenal
80.84%
0.072
0.065
0.056
0.053
0.051
0.049
0.046
0.040
0.037
0.035
0.035
0.032
0.026
0.026
0.025
0.021
0.019
0.017
0.016
0.012
0.011
0.009
0.009
0.009
0.009
0.007
0.006
0.006
0.005
0.005
0.187
0.542
1.222
Colorectal
LN(reg)
Liver
LN(dist)
Lung
Peritoneum
Adrenal
Bone
62.98%
0.062
0.056
0.046
0.031
0.042
0.029
0.023
0.016
0.030
0.030
0.027
0.022
0.015
0.013
0.012
0.015
0.007
0.014
0.006
0.009
0.010
0.007
0.008
0.009
0.007
0.012
0.012
0.011
1.216
0.032
0.016
0.852
Figure 4.5: Reduced pathway diagrams of 8 cancer types showing top 30 paths
Top 30 two-step pathways emanating from primary tumors (total pathway probability listed
in center node), obtained by multiplying the edges of the one-step edges comprising each
two-step path. Edges without numbers are one-step paths. All other numbered edges mark
the second edge in a two-step path, with numbers indicating the two-step probabilities. a.
Primary lung cancer. b. Primary breast cancer. c. Primary prostate cancer. d. Primary
colorectal cancer. (Continued on the following page.)
77
(e) (f)
Pancreas
Liver
Lung
Peritoneum
Pleura
Adrenal
LN (dist)
LN (reg)
Diaphragm
0.047
0.010
0.042
0.008
0.011
0.014
0.040
0.051
0.016
0.026
0.011
0.016
0.019
0.018
0.019
0.007
0.022
0.008
0.019
0.008
0.021
1.230
0.674
1.290
1.353
53.42%
0.008
0.022
0.008
0.009
0.007
0.009
0.018
0.010
0.012
0.808
Ovaries
Periton Cav
LN (reg)
Liver
Kidney
Lung
LN (dist)
Pleura
Spleen
Stomach
51.92%
0.015
0.037
0.015
0.028
0.034
0.025
0.016
0.011
0.659
0.023
0.010
0.014
0.012
0.014
0.012
0.018
0.019
0.018
0.024
0.015
0.009
0.010
0.018
0.011
0.010
0.011
4.485
1.408
0.243 0.020
0.017
0.010
0.010
0.034
(g) (h)
Cervix
LN (reg)
Peritoneum
Liver
Bone
Kidney
Lung
LN (dist)
Ovaries
Adrenal
65.53%
0.095
0.053
0.055
0.035
0.036
0.038
0.021
0.015
0.015
0.008
0.010
1.108
0.729
0.009
0.030
0.015
0.014
0.008
0.008
0.006
0.015
0.015
0.007
0.014
0.008
0.010
0.020
0.008
0.006
0.040
0.022
0.021
1.615
0.331
Lung
Skin
LN (dist)
Adrenal
Kidney
Brain
Bone
Thyroid
Heart
Pancreas
Sm Intestine
LN (reg)
Spleen
Liver
0.009
23.81%
0.013
0.010
0.007
0.006
0.006
0.007
0.009
0.007
0.007
0.006
0.006
0.006
0.010
0.013
0.008
0.007
0.007
0.006
0.006
0.006
0.008
0.006
0.011
0.015
0.007
0.007
0.007
0.007
0.008
Figure 4.5 (continued.)
Top 30 two-step pathways emanating from primary tumors (total pathway probability listed
in center node), obtained by multiplying the edges of the one-step edges comprising each
two-step path. Edges without numbers are one-step paths. All other numbered edges mark
the second edge in a two-step path, with numbers indicating the two-step probabilities. e.
Primary pancreatic cancer. f. Primary ovarian cancer. g. Primary cervical cancer. h.
Primary skin cancer.
78
ovarian, and skin models show primary self-seeding (red loops attached to the primaries),
and only lung and skin show primary reseeding (green arrows pointing back to the primary).
4.5.1 Spreaders and sponges
One of the most important pieces of information to gather from these reduced diagrams is the
classication of the spreaders and sponges. Since these tell us what spreads and what absorbs
the CTCs, they are very important to analysis of the models. The rst key observation is
that the number of each diers from primary to primary. While lung and pancreas have ve
stand-out sights each, the colorectal model has only two. Most of the models have a fairly
equal number of spreaders and sponges (diering at1), yet breast has three sponges to it's
singular bone spreader.
Another noteworthy dierence between the models is how a node acts like a sponge in one
model and a spreader in another. A perfect example of this is bone in breast, prostate, and
skin cancers, which acts like a spreader, yet acts like a sponge in lung and cervical cancers.
Another major example is the liver acting like a sponge in lung, breast, prostate, and ovarian
cancer, and a spreader in colorectal and pancreatic. This shows how dierent cancer can
act on a smaller scale. The same sites are major players in the disease progression, but for
dierent reasons depending on which primary the metastasis came from.
79
4.5.2 Two-step pathway percentages
Looking back at the percentage of two-step paths that these 30 paths take up, we begin to
see a dierent picture. These values range from 23:81% to 80:84% and give an indication
of how complex and unpredictable each cancer really is. On one end, prostate's 30 paths
representing 80% of all it's paths, says that this diagram is a good representation of how
prostate cancer metastasizes throughout the body and that this process is fairly predictable.
On the other end of the spectrum, skin's 30 paths representing only 24%, indicates that the
exact opposite, that this is not a good representation of skin cancer metastasizing and that
the process can be very unpredictable.
Another way of looking at these diagrams is to normalize them so all of them are showing
approximately the same percentage instead of the same number of paths. Figure 4.6 shows
this new approach and how easy it is to see the predictability in the models. The percentage
for the gures is xed at 35%, and the appropriate connections for each is adjusted. While
breast remains the same and lung only loses 2 of it's connections, the other models change
much more drastically. Ovarian falls to 18 paths, pancreatic to 12, colorectal to 9, cervical
to 7, and prostate to 6. The biggest change, is that skin has to nearly double it's two-step
pathways (54) in order to reach the 35% value.
This shows perfectly how predictable each cancer model is. The prostate, colorectal, and
cervical models contain most of their metastatic progression information in only a handful
of connections between a few sites. conversely, the skin model is very unpredictable in the
sense that the same amount of information is spread across many more pathways. While
80
(a) (b)
LN(reg)
Kidney
LN(dist)
Pleura
Pancreas
Liver
Adrenal
Bone
Lung
0.028
0.012
0.010
0.010
0.008
0.015
0.008
0.025
0.014
0.022
0.011
0.009
0.009
0.012
0.008
0.007
0.014
0.012
0.007
0.016
0.019
0.011
0.011
0.009
0.015
0.008
0.010
0.013
0.746
1.887
0.869
2.912
0.738
35.39%
Breast
LN(reg)
Pleura
LN(dist)
Liver
Adrenal
Lung
Bone
Skin
0.020
0.010
0.010
0.012
0.012
0.012
0.013
0.016
0.019
0.010
0.008
0.012
0.011 0.008
0.013
0.009
0.017
0.018
0.011
0.011
0.009
0.010
0.014
0.011
0.009
0.011
0.010
0.009
0.009
0.009
1.363
0.432
0.528
0.753
35.23%
(c) (d)
Prostate
LN(reg)
Bone
LN(dist)
Lung
Liver
Kidney
Adrenal
34.60%
0.072
0.065
0.056
0.053
0.051
0.049
0.187
0.542
1.222
Colorectal
LN(reg)
Liver
LN(dist)
Lung
Peritoneum
Adrenal
Bone
35.92%
0.062
0.056
0.046
0.031
0.042
0.029
0.030
0.030
1.216
0.032
0.852
Figure 4.6: Reduced pathway diagrams of 8 cancer types showing top 35%
Top two-step pathways totaling 35% emanating from primary tumors (actual pathway
probability listed in center node), obtained by multiplying the edges of the one-step edges
comprising each two-step path. Edges without numbers are one-step paths. All other num-
bered edges mark the second edge in a two-step path, with numbers indicating the two-step
probabilities. a. Primary lung cancer. b. Primary breast cancer. c. Primary prostate
cancer. d. Primary colorectal cancer. (Continued on the following page.)
81
(e) (f)
Pancreas
Liver
Lung
Peritoneum
Pleura
Adrenal
LN (dist)
LN (reg)
Diaphragm
0.047
0.042
0.040
0.051
0.026
0.019
0.019
0.022
0.019
0.021
1.230
0.674
1.290
1.353
34.59%
0.022
0.018
0.808
Ovaries
Periton Cav
LN (reg)
Liver
Kidney
Lung
LN (dist)
Pleura
Spleen
Stomach
35.93%
0.037
0.015
0.028
0.034
0.025
0.016
0.659
0.023
0.019
0.018
0.024
0.015
0.018
0.010
4.485
1.408
0.243 0.020
0.017
0.034
(g) (h)
Cervix
LN (reg)
Peritoneum
Liver
Bone
Kidney
Lung
LN (dist)
Ovaries
Adrenal
35.18%
0.095
0.053
0.055
0.035
0.036
0.038
1.108
0.729
0.040
1.615
0.331
Lung
Skin
LN (dist)
Adrenal
Kidney
Brain
Bone
Thyroid
Heart
Pancreas
Sm Intestine
LN (reg)
Spleen
Liver
0.009
35.19%
0.013
0.010
0.007
0.006 0.004
0.006
0.005
0.005
0.005
0.004
0.004 0.004
0.005
0.007
0.005
0.004
0.005
0.009
0.004
0.005
0.007
0.005
0.007
0.006
0.006
0.006
0.006
0.006
0.004
0.005
0.010
0.013
0.008
0.007
0.007
0.006
0.006
0.006
0.005
0.004
0.008
0.006
0.005
0.004
0.005
0.011
0.015
0.007
0.007
0.007
0.007
0.006
0.008
Figure 4.6 (continued.)
Top two-step pathways totaling 35% emanating from primary tumors (actual pathway
probability listed in center node), obtained by multiplying the edges of the one-step edges
comprising each two-step path. Edges without numbers are one-step paths. All other num-
bered edges mark the second edge in a two-step path, with numbers indicating the two-step
probabilities. e. Primary pancreatic cancer. f. Primary ovarian cancer. g. Primary cervical
cancer. h. Primary skin cancer.
82
the other models fall somewhere in between these two extremes, skin and prostate serve as
benchmarks in terms of how predictable each model is as compared to one another.
4.6 Mean rst-passage times
Upon analyzing the mean rst-passage times of each model, more similarities and dierences
begin to emerge. Figure 4.7 shows the graphical representation of this along with standard
deviation bars and markers (listed as `o's) for the analytical values. By looking at the shape
of the graphs, we can group them into two dierent categories: i.) a gradual progression of
passage times and ii.) distinct groups of progression times. The prostate, pancreatic, and
ovarian models fall in the second category while the rest follow the gradual progression. This
is indicating that prostate, pancreatic, and ovarian cancers may metastasize to some sites,
lay dormant for a while, and then continue metastasize to additional sites. The other models
show a more consistent metastatic pattern that stays active throughout the disease.
The next easily identiable dierence is the range of progression times. Since the rst
site is metastasized to a value of 1, it is easy to compare the dierence between the rst and
last sites' metastasis times. While the ovarian network can fully metastasize in as little as
40 time steps, it takes the cervical model nearly 7 times the amount of time steps. This says
that all of the sites in the ovarian model metastasize at a more uniform rate as compared
to the cervical model. In order to achieve this same uniformity, the cervical model would
have to be truncated after the skin node (the 20
th
largest passage time). Although this is
no indication of how long the cancer will actually take to metastasize, because these are
83
(a) (b)
0
20
40
60
80
100
120
LN (reg)
LN (dist)
Adrenal
Liver
Lung
Bone
Kidney
Pleura
Pancreas
Spleen
Heart
Thyroid
Brain
Pericardium
Diaphragm
Peritoneum
Lg Intestine
Skin
Gallbladder
Sm Intestine
Stomach
Omentum
Skeletal Musc
Prostate
Vagina
Uterus
Bladder
Passage Times
0
10
20
30
40
50
60
LN (reg)
LN (dist)
Lung
Bone
Liver
Pleura
Adrenal
Skin
Peritoneum
Ovaries
Breast
Spleen
Pericardium
Pancreas
Brain
Kidney
Diaphragm
Thyroid
Uterus
Heart
Gallbladder
Stomach
Bladder
Sm Intestine
Lg Intestine
Omentum
Vagina
Passage Times
(c) (d)
0
20
40
60
80
100
120
LN (reg)
Bone
LN (dist)
Lung
Liver
Adrenal
Pleura
Spleen
Kidney
Pancreas
Bladder
Gallbladder
Testes
Thyroid
Skin
Peritoneum
Stomach
Heart
Diaphragm
Lg Intestine
Sm Intestine
Passage Times
0
50
100
150
200
LN (reg)
Liver
LN (dist)
Lung
Peritoneum
Adrenal
Bone
Kidney
Ovaries
Pancreas
Pleura
Uterus
Thyroid
Bladder
Omentum
Skin
Spleen
Prostate
Lg Intestine
Diaphragm
Gallbladder
Sm Intestine
Heart
Vagina
Brain
Pericardium
Skeletal Musc
Stomach
Passage Times
Figure 4.7: Mean rst-passage time histograms of 8 cancer types showing top 35%
Mean rst-passage time histogram for Monte Carlo computed random walks all starting from
primary tumor. Error bars show one standard deviation. Values are normalized so that rst
passage time has value 1, and all others are in these relative time units. a. Primary lung
cancer. b. Primary breast cancer. c. Primary prostate cancer. d. Primary colorectal cancer.
(Continued on the following page.)
arbitrary, model based times, it is good to know how spread out the metastatic cascade can
occur on.
A more detailed inspection of these graphs will reveal a dierent picture. While all of the
models indicate that the regional and distal lymph nodes, liver and lung are metastasized
84
(e) (f)
0
10
20
30
40
50
60
70
LN (reg)
Liver
LN (dist)
Lung
Peritoneum
Diaphragm
Adrenal
Pleura
Kidney
Spleen
Omentum
Bone
Sm Intestine
Lg Intestine
Gallbladder
Stomach
Ovaries
Thyroid
Skin
Pericardium
Uterus
Bladder
Heart
Prostate
Breast
Testes
Passage Times
0
5
10
15
20
25
30
35
40
Peritoneum
LN (reg)
LN (dist)
Liver
Lung
Pleura
Diaphragm
Omentum
Uterus
Adrenal
Kidney
Spleen
Lg Intestine
Vagina
Bladder
Sm Intestine
Ovaries
Bone
Stomach
Skin
Thyroid
Pericardium
Pancreas
Heart
Gallbladder
Brain
Passage Times
(g) (h)
0
50
100
150
200
250
LN (reg)
LN (dist)
Liver
Lung
Peritoneum
Bone
Ovaries
Adrenal
Kidney
Spleen
Sm Intestine
Lg Intestine
Pleura
Bladder
Diaphragm
Omentum
Vagina
Heart
Pancreas
Skin
Gallbladder
Pericardium
Thyroid
Stomach
Uterus
Brain
Passage Times
0
5
10
15
20
25
30
35
40
45
50
LN (reg)
Lung
LN (dist)
Liver
Adrenal
Bone
Skin
Kidney
Brain
Heart
Thyroid
Pancreas
Spleen
Pleura
Sm Intestine
Peritoneum
Pericardium
Diaphragm
Gallbladder
Stomach
Uterus
Ovaries
Prostate
Skeletal Musc
Lg Intestine
Breast
Bladder
Testes
Omentum
Vagina
Passage Times
Figure 4.7 (continued.)
Mean rst-passage time histogram for Monte Carlo computed random walks all starting
from primary tumor. Error bars show one standard deviation. Values are normalized so
that rst passage time has value 1, and all others are in these relative time units. e. Primary
pancreatic cancer. f. Primary ovarian cancer. g. Primary cervical cancer. h. Primary skin
cancer.
to very soon in the disease progression, there are some nodes that are unique to each cancer
that are equally as fast. The lung model boasts a very important adrenal gland located at
position #3. This site is uniquely important to lung cancer, but not a major player in the
other models. In fact, the highest passage time location that is reached by the adrenal gland
85
outside of the lung model is #5 in skin cancer. Likewise for ovarian cancer with peritoneum
and prostate cancer with bone. These important sites indicate, once again, that even though
each cancer is globally the same, there lies subtle dierences on a smaller, more personal
scale.
4.7 Discussion
This chapter shows how easy it is to analyze and compare dierent cancer models in a very
robust and sensible way. Not only do we analyze the networks and their properties, but
also the analytical progression of the disease and the classication of the sites. This shows
us how predictable some cancers are and also how uniform the metastatic progression. By
performing such analyses on mathematical models, it is much easier to compare similarities
and dierences between them than in living people. In the long run, and with patient specic
models, these models have the potential to save money, time, and lives.
86
Chapter 5
The entropy of metastatic cancer
5.1 Introduction
This chapter focuses on the entropy associated with the Markov models. The entropy is
based on the distributions gathered from the dataset that the models are built from, which
is also the steady-state of the model itself. After analyzing these values individually, we
calculate the relative entropy between the distributions to see how they compare to general
cancer. This will allow for further comparisons of cancers and also more predictions of how
they will behave throughout the lifetime of the disease.
5.2 Methods
5.2.1 Brief summary of autopsy dataset used
We re-analyze the DiSibio and French [19] dataset of metastatic tumor distributions based on
autopsy studies collected for 3827 untreated cadavers from 5 dierent cancer facilities in New
87
England between 1914-1943. The data re
ect 9484 distinct metastatic tumors distributed
over 30 anatomical sites for all of the major tissue cancers. The data represent `natural' dis-
ease progression, which is useful, but we caution that brain metastases are under-represented
in the data since examination of the intracranial contents at that time was not routinely per-
formed. The data has been used in [54] to develop a Markov chain model for lung cancer
progression, where the autopsy data is used as the Markov chain steady-state, from which
transition probabilities are calculated. In this paper, we directly characterize the data, shown
in Figure 5.1 (all cancers) and Figure 5.2 (12 dierent primary cancers) in terms of their
empirical distributions, which predominantly follow power-law form [49]. Other related work
focusing on the development of dynamical models based on metastatic progression patterns
includes references [10, 32, 36, 55]. While notions of entropy have been used previously in
the context of gene expression proles and epidemiology [47, 66, 69, 75], we know of no pre-
vious work that uses these notions to characterize the complexity of large-scale progression
patterns.
5.2.2 Denition of entropy
The notion of entropy we use is borrowed from the eld of information theory and statistical
mechanics [13, 37, 40, 71]. Given a probabilistic distribution of states ~ = (
1
;
2
;:::;
N
)
spread over N sites, the entropy associated with the distribution is given by the quantity
H
N
=
P
N
i=1

i
ln (
i
) where 0H
N
lnN. There are two factors that lead to increased
entropy: (i) the larger the number N of sites over which the disease is distributed, the larger
88
0 0.05 0.1 0.15 0.2 0.25
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
P(Met Location | Met Occurs)
10
0
10
1
10
−3
10
−2
10
−1
10
0
Anatomical Sites of Metastases (Ordered by most −least freqeunt)
Proportion of Total Mets

Empirical Data: All Cancer
Maximum Likelihood Estimate, p(x)~x
−1.46
Figure 5.1: Distribution histogram and log-log plot of all cancer distribution
Histogram (left) of distribution of metastatic tumors over all cancer types from 3827 patients,
9484 metastatic tumors distributed over 30 anatomical sites. Data is plotted on log-log plot
(right) showing power law formp(x)x
1:46
obtained using a maximum likelihood estimator
and goodness-of-t criteria to obtain the best range of the power law distribution [49].
the entropy; (ii) the more even the probabilities are distributed among those sites, the larger
the entropy. Thus, the lowest entropy state, given byH
N
= 0, corresponds to the distribution

k
= 1;
i
= 0(i6=k). Since the probability of site `k' being occupied is 1 and the probability
of sites i6= k being occupied is 0, this state is associated with predictive certainty. In the
language of statistical thermodynamics [40], this would be called a completely `ordered' state.
By contrast, the highest entropy state corresponds to the uniform distribution in which each
site is equally probable, hence
i
=
1
N
;i = 1;:::;N. This uniform distribution gives rise to
89
(a) (b)
(c) (d)
(e) (f)
Figure 5.2: Distribution histograms and log-log plots of 12 cancer types
Histograms of distribution of metastatic tumors for primary cancers. Data is plotted on a
log-log plot for each, showing power law form. (a) Lung cancer: 163 patients, 619 metastases,
27 anatomical sites, p(x)x
1:25
; (b) Breast: 432 patients, 2235 metastases, 28 anatomical
sites, p(x)x
1:51
; (c) Prostate: 193 patients, 462 metastases, 21 anatomical sites, p(x)
x
2:31
; (d) Colorectal: 560 patients, 859 metastases, 28 anatomical sites, p(x) x
1:46
; (e)
Pancreatic: 109 patients, 323 metastases, 26 anatomical sites,p(x)x
1:35
; and (f) Ovarian
cancers: 86 patients, 302 metastases, 26 anatomical sites, p(x)x
1:05
. (Continued on the
following page.)
90
(g) (h)
(i) (j)
(k) (l)
Figure 5.2 (continued.)
Histograms of distribution of metastatic tumors for primary cancers. Data is plotted on
a log-log plot for each, showing power law form. (g) Bladder cancer: 183 patients, 256
metastases, 22 anatomical sites, p(x) x
1:73
; (h) Cervical: 418 patients, 806 metastases,
26 anatomical sites,p(x)x
1:39
; (i) Skin: 161 patients, 420 metastases, 30 anatomical sites,
p(x)x
1:00
; (j) Stomach: 348 patients, 928 metastases, 28 anatomical sites, p(x)x
1:35
;
(k) Uterine: 120 patients, 289 metastases, 24 anatomical sites,p(x)x
1:32
; and (l) Kidney
cancers: 62 patients, 212 metastases, 26 anatomical sites, p(x)x
1:76
.
91
a maximal entropy value of H = lnN. For this distribution, since each site is equally likely
to occur with probability 1=N, the predictive certainty associated with this distribution is
minimal, yielding the highest possible entropy value. We note that the entropy value is
independent of the ordering of the sites. Thus, higher values of entropy are intimately tied
to notions of disorder and complexity and have been used productively across a wide range
of disciplines.
5.2.3 Denition of relative-entropy
The concept of relative entropy, or Kullback-Liebler distance, is used to measure the `dis-
tance' between two distributions of random variables. One way to think of the relative
entropy D(PkQ) between two random variables P and Q is to view D(PkQ) as a measure
of ineciency associated with assuming that the distribution is Q, when in fact the true
distribution is P [13, 66]. It is dened as
D(PkQ) =
N
X
i=1
p
i
ln
p
i
q
i
(5.1)
In our comparisons, we use the symbol Q to represent the all cancer empirical distribution,
whereas P will represent a specic primary cancer type. Thus, the notion of relative entropy
quanties the relative ineciency of using the all cancer distribution instead of the more
targeted and informative primary cancer type.
92
5.3 Results
5.3.1 Distribution of metastatic tumors
Figure 5.1 shows the tumor distribution for all cancers collected from 3827 patients, with
a total of 9484 metastatic tumors distributed over 30 distinct anatomical sites. On the left
we show the histograms, normalized so that the total area under the bars is one, hence the
distribution represents the probability mass function associated with all cancers. On the
right we show the same data plotted on a log-log plot to more clearly bring out the fact
that there is a power-law region, where the distribution follows the form p(x) x

, with
= 1:46, obtained using maximum likelihood estimators, and a goodness-of-t criterion for
the optimal range over which the power-law holds [49]. We note that power-law distributions
arise in other contexts, most relevant might be the distribution of edges from nodes on the
World-Wide-Web [5]. The analogy of web-surng from site-to-site and modeling cancer
progression as a random walk process from site-to-site has been used fruitfully in [54, 55].
We caution, however, that the amount of data available from the worldwide web is orders
of magnitude larger than that available from our autopsy study, large as it is. The panels
shown in Figure 5.2a-l break the data of Figure 5.1 into 12 groupings associated with 12
major primary cancer types (a. Lung; b. Breast; c. Prostate; d. Colorectal; e. Pancreatic;
f. Ovarian; g. Bladder; h. Cervical; i. Skin; j. Stomach; k. Uterine; l. Kidney) and the
ensemble metastatic distributions associated with each. Each of the empirical distributions
93
shows a clear power-law range (details are described in Figure caption), each with a distinct
power-law exponent and approximate range of validity.
5.3.2 Metastatic entropy for 12 major cancer types
Because of well known diculties inherent with pinning down precise values for power-law
exponents, we do not use their value for comparative purposes. For the purposes of quanti-
fying the complexity associated with each primary cancer type, we calculate the `metastatic
entropy' associated with each, given by the formula H
N
=
P
N
i=1

i
ln (
i
), where
i
repre-
sents the proportion of metastatic tumors found at anatomical site `i', for a given primary
cancer type. The constraints are given by 0
i
1; (i = 1;:::;N);
P
N
i=1

i
= 1. It should
be intuitively clear that an increase in complexity is associated with two distinct features as-
sociated with each of the distributions: (i) the total number of sites, N, at which metastatic
tumors are found, and (ii) relatively `
at' distributions, meaning that the probabilities of
spreading to each site are more equally probable than what a `steep' distribution would show.
Both of these factors play an important role in the entropy values. Table 5.1 shows the value
of the metastatic entropy for each of the 12 cancer types, as well as the all cancer aggregated
data. The rst column lists the primary cancer type, the second column lists the number of
sites, N, over which the metastatic tumors are distributed, while the third column lists the
metastatic entropy associated with the empirical distributions shown in Figures 5.1 and 5.2.
We list the sites according to the descending values of the entropy shown in column 3, thus
Skin (2.9945), Breast (2.7798), Kidney (2.7554), and Lung (2.7453) all have entropy values
94
higher than the value for all cancers combined (2.7136), which we use as a benchmark for
comparisons. The cancer type with the lowest entropy value is Prostate (2.0960), consistent
with the relatively small number of sites to which it distributes (N = 21), and the relatively
sharp drop in the empirical distribution shown in Figure 5.2c. It is useful to compare this
distribution with Skin, shown in Figure 5.2i, which has more sites to which it distributes
itself (N = 30), and has a distinctly
atter distribution to those sites. For ovarian cancer,
whose entropy is relatively low (2.3275), we have grouped large intestine, small intestine,
diaphragm, ovary, omentum, and peritoneum all as one site which we call `peritoneal cav-
ity', due to the fact that metastases in each of these regions likely represent random spread
of disease within an anatomically connected region, as opposed to hematogenously seeded
metastases.
5.3.3 Relative-entropy between each primary cancer type and the
aggregate entropy associated with all cancers
Columns 4 and 5 in Table 5.1 show the Kullback-Liebler divergence between each cancer
type and the all cancer category, as detailed in section 5.2.3. We use `Q' as the all cancer
distribution, while `P ' is the distribution associated with each specic cancer type. While the
value of entropy shown in column 3 is independent of the ordering in which the sites are listed,
the K-L divergence is not. In column 5 we calculate this quantity using the P distribution
and the Q distribution arranged in decreasing order in each case, as shown in Figure 5.3.
This way of comparing the distributions focuses on the `shape' of the distribution, i.e. the
95
Table 5.1: Entropy table for each cancer type and for all cancers grouped together
Primary N Entropy K-L K-L Divergence
Cancer Type Divergence (Site Specic)
Skin 30 2.9945 0.0758 0.1373
Breast 27 2.7798 0.0329 0.0759
Kidney 27 2.7554 0.0549 0.1352
Lung 27 2.7453 0.0360 0.1097
All 30 2.7136 0.0000 0.0000
Stomach 28 2.6099 0.0213 0.1191
Uterine 24 2.5709 0.0339 0.1459
Pancreatic 26 2.5540 0.0375 0.1392
Colorectal 28 2.4686 0.0351 0.0821
Cervical 26 2.3696 0.0546 0.0979
Ovarian 21 2.3275 0.0684 0.3416
Bladder 22 2.2301 0.0957 0.1477
Prostate 21 2.0960 0.1620 0.2750
Entropy table for each cancer type and for all cancers grouped together as one. First
column lists the number of metastatic sites for that cancer type; second column lists the
computed entropy value; third column lists the Kullback-Liebler divergence between that
cancer type and the all cancer group, as compared in descending order for each; fourth
column lists the K-L divergence between that cancer type and the all cancer group
compared on a site specic basis. See text for more details.
rate at which it drops to zero, rather than the actual sites to which the disease spreads. As
Table 5.1 column 4 indicates, the K-L divergence between Prostate and `All' is the highest
(0.1620), indicating that its shape is most dierent from the all cancer category. By contrast,
Stomach cancer has the smallest K-L divergence from the all cancer group (0.0213), making
it in this sense, the most similar to the aggregate. Column 5 in Table 5.1 shows the K-L
computations between each of the cancer types and `All' on a site specic basis, as shown in
Figure 5.4. Here, we list the sites in decreasing order according to the all cancer category,
meaning that the comparative histogram heights for each of the specic primary cancers
96
generally are not arranged in strictly decreasing order. Thus, on this site-specic way of
computing the K-L divergence, Ovarian cancer (0.3416) and Prostate cancer (0.2750) have
the largest values, making them the most distinct from the all cancer aggregate on a site-
by-site comparison. By contrast, Breast cancer (0.0759) and Colorectal cancer (0.0821) have
the smallest values of site specic K-L divergence, meaning these are the most similar to the
all cancer aggregate.
5.4 Discussion
The approach taken in this chapter is dierent than in the previous chapters. The analysis
was done not directly on the Markov model, but instead on the distribution that the model
was built from. The entropy values that were computed allowed us to rank order the cancer
types in terms of their predictability and also in terms of their similarity to general cancer.
In addition to this, we also calculated the relative entropy for the distributions by matching
up both in decreasing order and not by their site listings. This allowed us to analyze how
similar the shape of the graphs were as opposed to the values contained within them.
97
(a) (b)
0
0.05
0.1
0.15
0.2
0.25
0.3
1
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3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Proportion

All Cancer
Lung Cancer
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0.05
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0.25
0.3
1
2
3
4
5
6
7
8
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14
15
16
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18
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All Cancer
Breast Cancer
(c) (d)
0
0.05
0.1
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8
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13
14
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All Cancer
Prostate Cancer
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1
2
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5
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7
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All Cancer
Colorectal Cancer
(e) (f)
0
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0.1
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1
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All Cancer
Pancreatic Cancer
0
0.05
0.1
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22
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24
25
26
27
28
29
30
Proportion

All Cancer
Ovarian Cancer
Figure 5.3: Distributions compared with all cancer (non-site specic)
Histograms of distribution of metastatic tumors for primary cancers compared with distri-
bution of all cancer. Data is plotted in descending order for each distribution, hence is not
site specic. (a) Lung cancer; (b) Breast cancer; (c) Prostate cancer; (d) Colorectal cancer;
(e) Pancreatic cancer; and (f) Ovarian cancer. (Continued on the following page.)
98
(g) (h)
0
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All Cancer
Bladder Cancer
0
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0.1
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0.2
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0.3
1
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Proportion

All Cancer
Cervical Cancer
(i) (j)
0
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3
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5
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8
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12
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14
15
16
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24
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All Cancer
Skin Cancer
0
0.05
0.1
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0.2
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0.3
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2
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Proportion

All Cancer
Stomach Cancer
(k) (l)
0
0.05
0.1
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0.3
1
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3
4
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Proportion

All Cancer
Uterine Cancer
0
0.05
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2
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5
6
7
8
9
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11
12
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14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Proportion

All Cancer
Kidney Cancer
Figure 5.3 (continued.)
Histograms of distribution of metastatic tumors for primary cancers compared with distri-
bution of all cancer. Data is plotted in descending order for each distribution, hence is not
site specic. (g) Bladder cancer; (h) Cervical cancer; (i) Skin cancer; (j) Stomach cancer;
(k) Uterine cancer; and (l) Kidney cancer.
99
(a) (b)
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Lung Cancer
0
0.05
0.1
0.15
0.2
0.25
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LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Breast Cancer
(c) (d)
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Prostate Cancer
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Colorectal Cancer
(e) (f)
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Pancreatic Cancer
0
0.05
0.1
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0.2
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LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Ovarian Cancer
Figure 5.4: Distributions compared with all cancer (site specic)
Site specic histograms of distribution of metastatic tumors for primary cancers compared
with distribution of all cancer. Data is plotted according to sites in descending order cor-
responding to the all cancer distribution. (a) Lung cancer; (b) Breast cancer; (c) Prostate
cancer; (d) Colorectal cancer; (e) Pancreatic cancer; and (f) Ovarian cancer. (Continued on
the following page.)
100
(g) (h)
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Bladder Cancer
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Cervical Cancer
(i) (j)
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Skin Cancer
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Stomach Cancer
(k) (l)
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Uterine Cancer
0
0.05
0.1
0.15
0.2
0.25
0.3
LN (reg)
LN (dist)
Liver
Lung
Bone
Adrenal
Peritoneum
Pleura
Skin
Kidney
Pancreas
Spleen
Ovaries
Diaphragm
Heart
Thyroid
Sm Intestine
Omentum
Pericardium
Brain
Lg Intestine
Gallbladder
Uterus
Bladder
Breast
Stomach
Vagina
Prostate
Skeletal Musc
Testes
Proportion

All Cancer
Kidney Cancer
Figure 5.4 (continued.)
Site specic histograms of distribution of metastatic tumors for primary cancers compared
with distribution of all cancer compared on a site-specic basis. Data is plotted according
to sites in descending order corresponding to the all cancer distribution. (g) Bladder cancer;
(h) Cervical cancer; (i) Skin cancer; (j) Stomach cancer; (k) Uterine cancer; and (l) Kidney
cancer.
101
Chapter 6
Discussion
The computational model we develop and discuss in this Ph.D. thesis is an ensemble based
Markov chain/random walk model of disease progression in which we use a stochastic tran-
sition matrix with entries that are (approximately) normally distributed. The model can
help us quantify pathways of progression for lung cancer, and can be used as a baseline
model in which to compare more targeted models which use correlations among sites making
up the ensemble (i.e. the individual patients making up the ensemble), and use timescale
information on disease progression. The model underscores the importance of the complex
and heterogeneous nature of the connections among the many potential metastatic locations
and bolsters the case for a fairly complex view of the importance of a whole host of subtle
connections among sites that may or may not produce clinically detectable tumors, but that
seem crucial in the eventual detailed understanding of cancer progression. We believe this
autopsy based ensemble study gives important baseline quantitative insight into the struc-
ture of cancer progression networks that will be useful for future comparisons. Three key
ndings based on the model are:
102
(i) Metastatic sites can be classied into `rst-order' and `second-order' sites based on
the comparative values of the one-step vs. two-step transition probabilities. This allows us
to lay out the layers of progression from lung to a given site, such as liver, shown in Figure
2.9 which lays the groundwork for a complete probabilistic classication of all pathways from
primary tumor sites to metastatic locations;
(ii) The classication and quantication of `self-seeding' transition values gives us a net-
work based interpretation of some recent biological insights [39] that will be the focus of a
separate study on probabilistic mechanisms of multidirectionality;
(iii) Model based mean rst-passage times give us relative time information (based on
average passage time to regional lymph nodes) about progression that can be used for future
comparisons with datasets that contain time progression histories.
Our model depicts cancer progression as eectively a multistep (two-step), multidirec-
tional, stochastic process, spreading probabilistically from site to site in individual patients,
but lling out a well-dened and predictable metastatic tumor distribution for large ensem-
bles of patients. This stable, robust, and predictable ensemble tumor distribution available
over large autopsy datasets is exploited to build a Markov transition matrix for lung cancer
progression. We identify the top unidirectional and multidirectional metastatic pathways
of primary lung cancer by means of a probabilistic comparison of all two-step paths ema-
nating from the lung. The results support the view that multidirectional pathways play an
important role in cancer progression. We identify 3 main mechanisms of multidirectionality
needed to obtain consistency with ensemble autopsy data: (i) primary tumor self-seeding,
103
(ii) reseeding of the primary tumor from a metastatic tumor, and (iii) metastasis reseeding.
Of these, the most important are metastasis reseeding of the lymph nodes (both regional
and distant) and adrenal gland and primary lung reseeding via the regional lymph nodes.
Also signicant is metastasis reseeding of the liver and primary self-seeding of the lung, but
neither seem to be as signicant as passage of the disease through the regional lymph nodes.
While very few patients die from their rst metastasis, the characterization of the rst
metastatic site as a spreader or sponge yields important insights into metastatic pathway
selection and the determination of progression timescales for patients. The model may have
implications for decisions surrounding surgical resection of oligometastatic disease [76] as
one might predict dierent outcomes for patients whose solitary site of disease is a sponge or
spreader. Historically, resection of isolated adrenal metastasis has entered clinical practice
in lung cancer, and removal of this spreader site has beneted patients [7]. Conversely, there
has never been an established role for resection of isolated liver metastasis, a sponge site,
despite there being a track record of success doing this in colon cancer [1, 24, 26, 35, 64].
A careful inspection of the top two-step pathways supports the dominance of unidirec-
tional metastatic spread over multidirectional processes, which perhaps explains why the
prevailing historical view is one of unidirectional spread [78]. However, we should emphasize
that our search algorithm for a Markov transition matrix could not converge to any solu-
tion when we constrained it so that multidirectional edges were ruled out but did converge
consistently to an ensemble of transition matrices when unconstrained so that all possible
paths were allowed. In other words, we were not able to nd a Markov transition matrix
104
that produced a steady-state consistent with the autopsy data unless multidirectional edge
connections were allowed. Therefore, we stress the viewpoint that multidirectional processes
play a key role in pathway selection and timescale determination for metastatic lung cancer.
Quantication of cancer entropy provides a framework for understanding the relative
metastatic risks particular to a given tumor type. We anticipate that it will be possible
to expand the application of these data to make clinical predictions and guide cancer care.
An understanding of cancer entropy has important clinical implications for establishing the
standards for clinical diagnosis of cancer, in guiding predictions from the growing eld of
radiomics, in selecting appropriate use of local therapies for oligometastatic disease, and in
guiding drug development. We will explore each of these potential uses of the data below.
While the model described above is quite consistent with clinical experience, there are a
number of limitations to the way in which cancer entropy is presented in this model. The
rst is that the model applies to a cancer primary in ensemble populations as opposed to
individuals [19]. Within patients, one may have low entropy tumors, with lung as the sole
metastatic site, while another may have 7 or 8 dierent metastatic sites. Thus the best uses
of the model will be in creating generalized approaches to specic diseases. Future derivatives
of this model may however be applicable to individuals whose burden of metastatic disease
may be thought of as having low or high entropy with alternate approaches for each situation.
The second limitation of the model is that it groups metastasis according to organ of
involvement. A patient with metastasis to the skull, pelvis, rib and femur is described as
having 1 site of metastasis (bone) and therefore is scored lower for entropy that a patient
105
with 2 metastasis, one to the liver and the other to a portal lymph node. The information
presented is thus ideally suited for understanding the implications of organ specic interven-
tions bone directed radiopharmaceuticals, hepatic artery directed chemotherapy, etc. Any
future model that uses these data to inform other forms of local therapies such as stereotactic
radiation will need to consider information about the total number of metastatic sites.
Despite this work focusing mainly on the metastatic progression of a primary cancer type,
the usefulness of it does not stop there. Future eorts can be made to tie this model together
with multiple other models to present a more complete view of cancer in the body as opposed
to one aspect. For example, one model can simulate the growth of a tumor, the next can
simulate the mutation of the cancer cells, our model can predict the metastatic progression,
and a nal one can simulate the formation of individual metastatic sites. More models can be
incorporated or omitted to model cancer on a smaller, more intricate scale, such as protein
markers, signaling pathways, and drug reception/retention. One model alone cannot solve
the problem of cancer, but many models working together provides a much better framework
for making useful and relevant predictions to help better the medical community.
106
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112

Abstract (if available)

Abstract A stochastic Markov chain model for metastatic progression is developed for primary 8 major cancer types based on a network construction of metastatic sites with dynamics modeled as an ensemble of random walkers on the network. We calculate a transition matrix for each primary cancer and use it to construct a circular bi-directional network of primary and metastatic locations based on postmortem tissue analysis of 3827 autopsies on untreated patients documenting all primary tumor locations and metastatic sites from this population. The resulting 50 potential metastatic sites are connected by directed edges with distributed weightings, where the site connections and weightings are obtained by calculating the entries of an ensemble of transition matrices so that the steady-state distribution obtained from the long-time limit of the Markov chain dynamical system corresponds to the ensemble metastatic distribution obtained from the autopsy dataset. We condition our search for a transition matrix on an initial distribution of metastatic tumors obtained from the dataset. Through an iterative numerical search procedure, we adjust the entries of a sequence of approximations until a transition matrix with the correct steady-state is found (up to a numerical threshold). Once the transition matrix for a given cancer type is computed, our metastatic progression model is based Monte Carlo simulations of collections of random walkers all leaving the primary tumor location and executing a random walk across the directed graph from site to site. The model allows us to simulate and quantify disease progression pathways and timescales of progression from the primary tumor location to other sites. Pathway diagrams are created that classify metastatic tumors as 'spreaders' or 'sponges' and quantif es three types of multidirectional mechanisms of progression: (i)self-seeding of the primary tumor, (ii) reseeding of the primary tumor from a metastatic site (primary reseeding), and (iii) reseeding of metastatic tumors (metastasis reseeding). The entire process is replicated for additional primary tumors in the dataset of for individual analysis and comparative purposes. ❧ A second contribution of this work is to introduce a quantitative notion of 'metastatic entropy' for cancer and use it to compare the complexity and predictability associated with the 12 most common cancer types worldwide. We apply these notions of entropy and predictability directly to the autopsy dataset used to create our Markov model. The raw data, which contains the number of metastases found at all of the anatomical sites for each cadaver (whose primary tumor location is also recorded), is normalized (yielding their empirical distribution) so that we can interpret the histograms as probability mass functions (PMFs) representing the large scale (whole body) metastatic 'signature' of each primary cancer. We characterize the power-law distributions associated with metastatic tumor distributions for each primary cancer type. Then we calculate the entropy associated with each and use the Kullback-Liebler divergence (relative-entropy) to compare each cancer type with all of the data aggregated into an 'all cancer' category, whose entropy value is used as a benchmark for comparisons.

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Asset Metadata

Creator Mason, Jeremy (author)

Core Title A stochastic Markov chain model to describe cancer metastasis

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 10/01/2013

Defense Date 08/26/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cancer,Markov chain,Metastasis,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Newton, Paul K. (committee chair), Ghanem, Roger G. (committee member), Kanso, Eva (committee member), Macklin, Paul (committee member), Pinkston, Timothy M. (committee member)

Creator Email jmason823@gmail.com,masonj@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-331809

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