Download Multiphase Mechanics of Tumor Encapsulation and Invasion

Multiphase Mechanics of Tumor
Growth and Control Strategies
Trachette L. Jackson
University of Michigan
Department of Mathematics
Historical Relationship Between
Mathematics and Biology
Attitudes of Early Biologists Attitudes of Early Mathematicians
“Beware of
mathematicians and all
those who make empty
prophecies. The danger
already exists that the
mathematicians have made
a covenant with the devil
to darken the spirit and
confine man in the bonds
of hell.”
-Saint Augustine (354-430)
“In mathematics we find
the primitive source of
rationality; and to
mathematicians must the
biologist resort for means
to carry out their
researches.”
In: Positive Philosophy
Auguste Comte
(1798-1857)
Cancer
Arises when a cell or small group of cells have
undergone a series of mutations which render
then insensitive to mitotic and apoptotic cues
from the surrounding tissue.
These cells and their progeny eventually form a
tumor.
Avascular Tumor Growth
Avascular = before blood
vessels
Nutrient obtained via
diffusion
Cells in the center starve
Cells on the periphery thrive
Cells in the interior are
quiescent
Growth limited to a few mm
in diameter
Angiogenesis
Formation of new blood
vessels from the existing
vasculature
Hypoxia induces a
chemical cascade which
stimulates endothelial
cells of near by vessels to
aggregate, proliferate, and
migrate towards the tumor
Vascular Tumor Growth
50% Cells
10% Blood vessels
40% Extra cellular
matrix (ECM)
Vessel structure is
disorganized
May be benign and
encapsulated or
milignant
Progression of Spatio-Temporal Tumor Modeling
First Generation: Avascular Growth
Burton, 1966; Greenspan, 1972; Glass, 1973; Adam, 1986-87;
Maggelakis, 1990; Byrne and Chaplain, 1993-94
Reaction-diffusion equation coupled with an integrodifferential equation describing tumor radius response to
externally supplied nutrient
Radial symmetry
Must track two free boundaries, Rn(t) defined implicitly
Progression of Spatio-Temporal Tumor Modeling
First Generation: Avascular Growth
Nutrient Equation: ct = Dc Dc – k f(c)
Diffusion
Consumption
Tumor Growth Equation
R
R
d/dt (4pR3/3) = 4p ∫ (f(c) – d1) r2 dr - ∫ d2 r2dr
R
n
0
Net Growth
Necrosis
Dc = diffusion coefficient of nutrient
k = nutrient consumption parameter
d1 = natural death rate; d2 = decomposition rate
Parameter Values
Parameter
Value
Reference
fo
0.07 h-1
Ward et al., 1997
co
1.5 x 10-8 g/mm3
d1
0.01 h-1
Casciari et al,
1992
Ward et al., 1997
Dc
7.2 mm2/h
Hatky et al., 1985
vc
1.77 x 10-6 mm3
k
5.8 x 10-5 g/mm3
Casciari et al.,
1992
Casciari et al.,
1992
First Generation Model Predictions
Time of necrotic core formation:
tn=
1
f0 –d1 ln
(
6Dcc0
kfoRn2
)~
3- 7 days
Size of tumor when core develops:
6Dcco
Rn =
~ 0.4mm
2
kfoRn
Steady state radius of tumor and of necrotic core:
Rs
Rns
~ 1.87mm
~ 1.64mm
Progression of Spatio-Temporal Tumor Modeling
First Generation: Avascular Growth
Advantages
Simplicity
Steady state results
agree with
experimentally grown
tumors
Reproduces layered
structure
Analytical predictions
possible
Disadvantages
Cells not explicitly
mentioned
Difficult to generalize
to other geometries
Progression of Spatio-Temporal Tumor Modeling
Second Generation: Asymmetry
Byrne and Chaplain, 1996; Jackson, 2002
Cell proliferation and death produce a velocity field,
u = f( c) - d1
D
The outer surface of the tumor is a free boundary (B(r,t)
= r – R(q,f,t) = 0) moving with the speed of the tumor
velocity,
n r = n u(R(t),t)
Same nutrient equation as before
Velocity is often related to pressure via Darcy’s law
u=-m p
D
Curvature effects incorporated via boundary conditions
Progression of Spatio-Temporal Tumor Modeling
Second Generation: Asymmetry
Goal: Determine under what conditions radially
symmetric solutions are unstable to asymmetric
perturbations. For cases when instability is predicted,
the shape of the tumor may alter radically from the
symmetric configuration.
Advantages
Disadvantages
--Only non-necrotic solutions apply
--More general geometries
--Can recover 1st generation models --Cell density not explicitly treated
Progression of Spatio-Temporal Tumor Modeling
Third Generation: Polyclonality
Thompson, 1999; Jackson and Byrne, 2000; Jackson, 2002
Assume tumor contains cell populations with different
phenotypes:
nt + (uni)x = S(ni)
Assume there are no voids within the tumor:
S ni = 1
ux = S S(ni)
Advantages
--More realistic tumor composition
--Can predict the effect of cellular
heterogeneity
Disadvantages
--Analysis of asymmetric effects
--Cell-Tissue interactions not
included
Progression of Spatio-Temporal Tumor Modeling
Fourth (Present) Generation:
Tissue Mechanics
Chaplain and Sleeman, 1993; Jones et al., 2000; Chen et al., 2001;
Breward et al., 2002; Jackson and Byrne, 2002; Lubkin and Jackson, 2002
Conservation of Mass---Transport equations for tumor
cells, normal cells, extracellular matrix, interstitial fluid
Conservation of Momentum---Force balance equations
prescribing the internal stress and inter phase forces
Today’s Topics: Tumor Encapsulation and
Novel Chemotherapeutic Strategies
Tumor Encapsulation
Definition: The cascade
of events that result in the
formation of a multilayered sheath of
epithelium surrounding a
tumor.
A multilobular tumor is
one in which lobes of
different sizes are
separated by strands of
connective tissue.
Enhanced chest CT showing an encapsulated and lobulated
(arrow) thymoma (T).
Capsule Composition and
Importance
Capsules are composed of
ECM (connective tissue)
Characteristic of many
benign tumors
Key feature in determining
clinical outcome
A physical barrier to invasion
Often limits symptoms
Capsule Formation Theories
Expansive Growth Hypothesis
Passive response
Tumor expansion compresses the
surrounding ECM
Foreign Body Hypothesis
Active response
Host over-expresses collagen in
surrounding tissue in direct
response to the tumor's growth
Experimental Observations
Wakasa et. al., 2000.
Tumors form capsules more often when
their diameter is greater than 2 cm.
Nagata et al., 2000.
There is no correlation between tumor size
and capsule incidence or thickness.
Clear Contradictions.
Experimental findings are difficult to
evaluate.
This is the perfect time for
MATHEMATICAL MODELING!
Mathematical Goals
Develop a novel modeling framework which accounts
the mechanical interactions of tumors with surrounding
ECM and accurately describes the process of tumor
encapsulation.
Use the model to.
Compare the active vs. passive hypotheses for capsule
formation.
Assist in the understanding of the role of the tumor cell
and ECM interactions in capsule formation.
Investigate possible mechanism for the transition between
benign and invasive growth.
Suggest an explanation for the transformation of a simple
nonlobular tumor to the more complex multilobular form.
Model Development
View tissue as a mixture of three interacting
continua: cells, extracellular matrix, and
interstitial fluid
Derive mass and momentum balance equations
for each phase
Close the system with suitable constitutive
relationships
Assume a Cartesian geometry with the tumor
expanding in the the direction parallel to the xaxis
Mass Balance Equations
Tumor Cells (n):
nt + (vnn)x = Sn
Extracellular Matrix (m):
Water (w):
mt + (vmm)x = Sm
wt + (vww)x = Sw
vi = the velocity of each phase
Si = the rates of production of each phase
Assumptions
A1: The system is closed in that volume is
simply transferred from one state to another,
therefore:
Sw = - (Sn + Sm)
A2: There are no voids in the tissues so that
n+m+w=1
Together these assumptions reduce the number of
unknowns from six to four.
Momentum Conservation
Newton’s Second Law (F = ma): The time rate of
change of momentum of a particle is equal to the sum of
the forces acting on it.
In highly damped mechanical systems (like many
biological tissues) both the convective momentum
transport and inertia (indisposition to motion) can be
ignored
In this case, momentum conservation reduces force
balance equations.
0 = Fnn + Fnm + Fnw
0 = Fmm - Fnm + Fmw
0 = Fww - Fnw - Fmw
Fii = intraphase force
Fij = interphase force
Defining the Forces
Denote the stress (force per unit area) in each phase by
si for i = n, m, w
The intraphase force Fii can be expressed in terms of the
average stress weighted by the volume fraction of each
phase e.g.
Fnn = (n sn)x
The interphase force Fij is often expressed as the sum of
two components: the static pressure exerted on material
of phase I by material of phase j and the frictional drag
due to the relative motion of the phases , e.g.
Fnm = pnx + k(vm – vn)nm
Prescribing Stresses
The cell and ECM phases are characterized by
additional, isotropic pressures which distinguish
them from water.
sn = - (p +Sn)
sm = - (p +Sm)
sw = - p
Prescribing Stresses
Sn = snn
A3: Pressure in cell phase
increases as phase density
increases
Sm = smm(1 + qn)
A4: Pressure in ECM increases as
phase density increases and tumor
causes additional pressure on the ECM
Force Balance Equations
0 = (n sn)x + k(vm – vn)nm + k(vw – vn)nw + pnx
0 = (msm)x - k(vm – vn)nm + k(vw – vm)mw + pmx
0 = (w sw)x - k(vw – vn)nw - k(vw – vm)mw + pwx
Solving for Pressure and Velocities
Add the force balance equations to get an
expression for the pressure.
px = -(nSn + mSm)x
Substitute into cell and ECM balances and
solve for the velocities
vn = v – v(n)nx and vm = v – v(m)mx
v = common velocity = - 1 px
k
Final Model Equations
nt = S n +
1
k
[(1-n)(nSn)x – n(mSm)x]x
mt = S m +
1
k
[(1-m)(mSm)x – m(nSn)x]x
Parabolic System of Equations with Cross Diffusion
Tumor Growth
Sn = ann(1-n-m) - dn
ECM Production
Sm = amnm(1-n-m)
Initial and Boundary Conditions
Tumor is initially localized to a region near x = 0 where
it has replaced most of the ECM.
ECM is at normal, steady state levels except in the
vicinity of the tumor where levels have dropped nearly to
zero.
No flux boundary conditions are imposed at x = 0 and at
x = L. The former is due to symmetry across the origin
and the latter results from assuming that the tissue is
isolated.
Foreign Body Hypothesis
Tumor
ECM
Model simulations when only the active response is operative. As
the strength of the active response, am, is varied from low (top) to
high (bottom), ECM accumulates in the center of the tumor.
Summary of Active Response
Growth suppression without capsule formation
Substantial ECM accumulation
Consistent with biopsies of fibrous tumors (Dvorok,
1986; Mckinnell et al., 1999)
Consistent with experimental observation that ECM
accumulation is associated with reduced tumor cell
density (Dvorok, 1986; Lunevicius, 2001)
A sufficiently strong active response could eliminate the
tumor and is therefore a potentially important mechanism
for control, but it not necessary for capsule formation.
The Expansive Growth Hypothesis
Tumor
ECM
Model simulations when only the passive response is operative.
As pressure that tumor cells generate within the ECM, q, varies
from high (top) to low (bottom).
Changes in Capsule Structure
Capsule Width
Capsule Volume
Summary of Passive Response
Passive alone is sufficient to form a capsule
Minimal accumulation of ECM
As pressure the tumor generates in the ECM increases,
more ECM is compressed a thin, dense capsule forms.
The less sensitive tumor cells are to changes in their own
density, the thicker the capsule will be.
For certain parameter values, the model predicts that the
capsule width will remain constant as the tumor grows.
Both Active and Passive Responses
Tumor
ECM
Model simulations with both mechanisms operative as the strength
of the active response, am, is increased.
Key Results
Active hypothesis
Tumor suppression without capsule formation
Passive hypothesis
Tumor encapsulation without growth suppression
Active + Passive hypotheses
Fibrous tumor moving with constant speed is
surrounded by a thin, dense capsule
Transition from Benign to Invasive
Growth
As a result of genetic mutations, some tumor cells
may acquire the skills necessary to breach the
capsule.
Tumor cells begin producing proteolytic enzymes
or proteases to remove the ECM barrier
Pressure
Protease
Production
ECM
Degradation
ECM Degradation Model
Assumptions
A5: Tumor cells produce the protease at a
constant rate only when the pressure they
experience exceeds a threshold value.
Pn – Pc
1
rt = Drrxx + mn[1 + tanh(
)
]
- lr
2
DP
Production
Diffusion
Decay
l = natural decay rate,
Drr = diffusion coefficient
m = maximum production rate
DP = sensitivity of tumor cells to changes in pressure
Pc = critical pressure for induces protease production
Pn = pressure experienced by tumor cells =
p +Sn
= snn(1-n) – smm2(1 + qn)
ECM Degradation Model
Modification
Assumptions
A6: ECM degradation is proportional to the product
of the protease density and ECM volume fraction.
Sm = amnm(1-n-m) - grm
A7: Protease enhances motility of cells by weakening
bonds that attach cells to each other and that maintain
the integrity of the ECM. Therefore drag coefficient
is assumed to be decreasing function of protease
density.
k
k(r) = 1 + br
Breaking Down the Capsule
Tumor
ECM
Protease production leads to destruction of the capsule and tumor
invasion. As sensitivity of the drag coefficient to the protease
increases (from top to bottom), so does the rate transcapsular
spread.
Continued Investigations
Modify model to include pressure-sensitive
tumor growth
This should lead to experimentally observable
steady state encapsulated tumors
Extend model to address issue of multiple
lobe formation
Polyclonality
Variable motility
Viscous and poroelastic effects
Novel Chemotherapeutic Strategies:
The Promise and Potential of MAbs
Monoclonal Antibodies
(MAbs)
Bind to tumor antigens
thus marking cancer
cells for death
Direct Drug Targeting
Magic Bullets
Drugs attached to Mabs
drug
The Failure of Mabs Against Solid
Tumors
Advantages of MAb Therapy
Stubborn Solid Tumors
High Specificity
Lower Toxicity
Works well for leukaemia and
lymphomas
Main Reasons for Failure Against
Solid Tumors
Antigen Heterogeneity
Vascular Permeability
Interstitial Pressure
New Approach: Two-step Drug
Targeting
Antibody Enzyme Conjugates for the
Activation of anticancer prodrugs
Two-phase Approach for Studying
Chemotherapeutic Strategies:
The Tumor Model
Tumor Cells (n):
nt + (vnn)x = Sn
Water (w):
wt + (vww)x = Sw
No voids: n + w = 1
wvw = -nvn
Momentum Balance
nt = Sn +
1
k
[n(1-n)2(nSn)x ]x
Two-phase Approach for Studying
Chemotherapeutic Strategies:
The Drug Transport Model
Extracellular Drug
qt + (vwq)x = Dq(q)xx- l12q + l21p
Intracellular Drug
pt + (vnp)x = l21q - l21p - l23p
Sequestered Drug
st + (vws)x = l23p - bgs
Goals of Modeling Approach
Accurately predict tumor response to traditional
chemotherapy
Modify model to investigate direct-drug targeting
and two-step targeting
Isolate the role of convection vs. diffusion of
large molecules
Acknowledgements
Collaborators
Helen Byrne, Center for Mathematical Medicine,
University of Nottingham
Peter Senter, Seattle Genetics
Funding
NSF Grant #
Alfred P. Sloan Foundation