TUMOR BOUNDARY INSTABILITY INDUCED BY NUTRIENT CONSUMPTION AND SUPPLY YU FENG MIN TANG XIAOQIAN XU AND ZHENNAN ZHOU

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TUMOR BOUNDARY INSTABILITY INDUCED BY NUTRIENT
CONSUMPTION AND SUPPLY
YU FENG, MIN TANG, XIAOQIAN XU, AND ZHENNAN ZHOU
Abstract.
We investigate the tumor boundary instability induced by nutrient
consumption and supply based on a Hele-Shaw model derived from taking
the incompressible limit of a cell density model. We analyze the boundary
stability/instability in two scenarios: 1) the front of the traveling wave; 2) the
radially symmetric boundary. In each scenario, we investigate the boundary
behaviors under two different nutrient supply regimes, in vitro and in vivo. Our
main conclusion is that for either scenario, the in vitro regime always stabilizes
the tumor’s boundary regardless of the nutrient consumption rate. However,
boundary instability may occur when the tumor cells aggressively consume
nutrients, and the nutrient supply is governed by the in vivo regime.
1. Introduction
Tumor, one of the major diseases threatening human life and health, has been
widely concerned. The mathematical study of tumors has a long history and
constantly active. We refer the reader to the textbook [10,11] and review articles
[2,7,48,62]. Previous studies and experiments indicate that the shape of tumors
is one of the critical criteria to distinguish malignant from benign. Specifically,
malignant tumors are more likely to form dendritic structures than benign ones.
Therefore, it is significant to detect and predict the formation of tumor boundary
instability through mathematical models. Before discussing the mathematical studies
of tumor morphology, we review relevant mathematical models as follows.
The first class of model was initiated by Greenspan in 1976 [32], which further
inspired a mass of mathematical studies on tumor growth (e.g., [5,8,23,65]). The
tumor is regarded as an incompressible fluid satisfying mass conversation. More
precisely, these free boundary type models have two main ingredients. One is the
nutrient concentration
σ
governed by a reaction-diffusion equation, which considers
the consumption by the cells and the supplement by vessels. The other main
component is the internal pressure
p
, which further induces the cell velocity
v
via different physical laws (e.g., Darcy’s law [5,12,26,32], Stokes law [23,28,29],
and Darcy&Stokes law [18,19,44,60,65]). Finally, the two ingredients are coupled
via the mass conservation of incompressible tumor cells, which yields the relation
∇ · v
=
λ
(
σ
), with the cell proliferation rate
λ
depending on
σ
. To close the model,
the Laplace-Young condition (
p|∂Ω
=
γκ
, where
κ
is the mean curvature, and
γ
stands for the surface tension coefficient) is imposed on the tumor-host interface.
For some variant models, people replace the Laplace-Young condition with other
curvature-dependent boundary conditions (see, e.g., [50,61,64]). More sophisticated
Date: October 5, 2022.
1
arXiv:2210.01359v1 [math.AP] 4 Oct 2022
2 FENG, TANG, XU, AND ZHOU
models were also investigated recently. In particular, we mention the studies based
on the two-phase models [50,61,64], and the works involve chemotaxis [34,38].
Most studies on the stability/instability of tumor boundary are based on the above
class of models and have been investigated from different points of view. Among
them, for different models (e.g., Darcy [17,21,22,25,27]; and Stokes [20,23,24]),
Friedman et al. proved the existence of non-radially symmetric steady states
analytically and classified the stability/instability of the boundaries from the Hopf
bifurcation point of view. Specifically, in their studies, the bifurcation parameter is
characterized by the cell proliferation rate or ratio to cell-cell adhesiveness. Then the
authors showed that the boundary stability/instability changes when the parameter
crosses a specific bifurcation point. On the other hand, Cristini et al. in [12], as
the pioneers, employ asymptotic analysis to study and predict the tumor evolution.
Their work is of great significance to the dynamic simulation of tumors and nurtured
more related works in this direction [49,51,52,61,64]. All the research demonstrated
that many factors could induce the tumor’s boundary instability, including but
not limited to vascularization [12,49,50,61], proliferation [12,20,24,25,27,49],
apoptosis [12,20,24,25,27,49,49,61,64], cell-cell adhesion [12,20,24,25,27,61],
bending rigidity [50,64], microenvironment [51,52,61,64], chemotaxis [49,51].
In recent decades, tumor modeling from different perspectives has emerged and
developed. In particular, one could consider the density model proposed by Byrne
and Drasdo in [6], in which the tumor cell density
ρ
is governed by a porous medium
type equation, and the internal pressure
p
is induced by the power rule
p
=
ρm
with the parameter
m >
1. The power rule enables
p
naturally vanish on the
tumor boundary. Moreover, the boundary velocity
v
is governed by Darcy’s law
v
=
−∇p|∂Ω
. Previous research indicates that the porous media type equations
have an asymptote concerning the parameter m tending to infinity [3,30,35,41,42].
Motivated by this, Perthame et al. derived the second kind of free boundary model
in [57] by taking the incompressible limit (sending
m
to infinity), or equivalently
mesa-limit of the density models. An asymptotic preserving numerical scheme
was designed by J.Liu et al. in [45], the scheme naturally connects the numerical
solutions to the density models to that of the free boundary models.
In the mesa-limit free boundary models proposed in [57], the limit density
ρ∞
can
only take value in [0
,
1], and the corresponding limit pressure
p∞
is characterized
by a monotone Hele-Shaw graph. More specifically,
p∞
vanishes on the unsaturated
region where
ρ∞<
1(see equation
(2.7)
). The Hele-Shaw graph representation of
pressure brings the following advantages. Firstly, in the Hele-Shaw type model, the
formation of a necrotic core can be described by an obstacle problem [33], which
leads
ρ∞
to decay exponentially in the necrotic core. Due to the Hele-Shaw graph,
the pressure
p∞
naturally vanishes there instead of taking negative values. Secondly,
a transparent regime called "patch solutions" exists, in which
ρ∞
remains in the
form of
χD(t)
, i.e., the indicator function of the tumor region. Again, to satisfy
the corresponding Hele-Shaw graph,
p∞
has to vanish on the tumor’s interface
(where
ρ∞
drops from 1to 0), which is significantly different from the first kind
of free boundary models developed from [32], in which the internal pressure relies
on the boundary curvature
κ
as mentioned previously. Moreover, in the mesa-
limit free boundary models, the boundary velocity is still induced by Darcy’s law
v∞
=
−∇p∞|∂Ω
. For completeness, the derivation of the mesa-limit model is
summarized in Section 2.1. Albeit various successful explorations based on such
NUTRIENT INDUCED TUMOR BOUNDARY INSTABILITY 3
mesa-limit free boundary models [13
–
16,33,36,38
–
41,43,47,54,58,59], the study on
its boundary stability/instability is yet thoroughly open.
The primary purpose of this paper is to investigate whether boundary instability
will arise in the mesa-limit free boundary models, which should shed light on the
boundary stability of the cell density models when
m
is sufficiently large. To simplify
the discussion, we consider tumors in the avascular stage with saturated cell density
so that the density function
ρ∞
is a patch solution, and the tumor has a sharp
interface. As the first attempt in this regard, we explore the instability caused
by nutrient consumption and supply. A similar mechanism can induce boundary
instability in other biological systems, see [4] for nutrient induce boundary instability
in bacterial colony growth models. The role of nutrition in tumor models has been
widely studied, and we refer the reader to the latest article in this direction [36].
Inspired by [58], we divide the nutrient models into two kinds, in vitro and in
vivo, according to the nutrient supply regime. In either regime, the nutrient is
consumed linearly in the tumor region with a rate
λ >
0. However, in the in vitro
model, we assume that a liquid surrounds the tumor with nutrient concentration
cB
.
Mathematically, the nutrient concentration remains
cB
at the tumor-host interface.
For the in vivo model, the nutrient is transported by vessels outside the tumor and
reaches
cB
at the far field. Correspondingly, we assume the exchange rate outside
the tumor is determined by the concentration difference from the background, i.e.,
cB−c. The two nutrient models will be specified more clearly in Section 2.1.2.
Our study of boundary stability/instability consists of two scenarios. We begin
with a relatively simple case, the front of traveling waves, in which quantitative
properties can be studied more explicitly. In this case, the unperturbed tumor region
corresponds to a half plane with the boundary being a vertical line propagating
with a constant normal velocity. Then we test the boundary stability/instability by
adding a perturbation with frequency
l∈R+
and amplitude
δ
. Our analysis shows
that in the in vitro regime,
δ
always decreases to zero as time propagates. In other
words, the boundary is stable for any frequency perturbation. In contrast, in vivo
regime, there exists a threshold value
L
such that the perturbation with a frequency
smaller than
L
becomes unstable when the nutrient consumption rate,
λ
, is larger
than one.
The above case corresponds to the boundary stability/instability while the tumor
is infinitely large. In order to further explore the influence of the finite size effect on
the boundary stability/instability, we consider the perturbation of radially symmetric
boundary with different wave numbers
l∈N
and radius
R
. Our analysis shows
that the in vitro regime still suppresses the increase of perturbation amplitude
and stabilizes the boundary regardless of the consumption rate, perturbation wave
number, and tumor size. For the in vivo regime, when the consumption rate
λ
is less
than or equal to one, the boundary behaves identically the same as the in vitro case.
However, when
λ
is greater than one, the continuous growth of tumor radius will
enable perturbation wave number to become unstable in turn (from low to high).
Further more, as
R
is approaching infinity, the results in the radial case connect to
the counterparts in the traveling wave case.
The main contribution of this work is to show that tumor boundary instability
can be induced by nutrient consumption and supply. As a by-product, our results
indicate that the cell apoptosis and curvature-dependent boundary conditions present
4 FENG, TANG, XU, AND ZHOU
abundantly in previous studies (e.g., [12,23]) are unnecessary for tumor boundary
instability formation.
The paper is organized as follows. In Section 2, we first derive our free boundary
models by taking the incompressible limit of density models characterized by porous
medium type equations in Section 2.1. Besides that, we also introduce the in vitro
and in vivo nutrient regimes in this subsection. Furthermore, the corresponding
analytic solutions are derived in Section 2.2. Section 3is devoted to introducing the
linear perturbation technique in a general framework. Then, by using the technique
in Section 3, we study the boundary stability of the traveling wave and the radially
symmetric boundary under the two nutrient regimes, respectively, in Section 4and
Section 5(with main results in Section 4.1 and Section 5.1). Finally, we summarize
our results and discuss future research plans in Section 6.
2. Preliminary
2.1. model introduction.
2.1.1. The cell density model and its Hele-Shaw limit. To study the tumor growth
under nutrient supply, let
ρ
(
x, t
)denote the cell population density and
c
(
x, t
)be
the nutrient concentration. We assume the production rate of tumor cells is given
by the growth function
G
(
c
), which only depends on the nutrient concentration. On
the other hand, we introduce
(2.1) D(t) = {ρ(x, t)>0}
to denote the support of
ρ
. Physically, it presents the tumoral region at time
t
. We
assume the tumoral region expands with a finite speed governed by the Darcy law
v
=
−∇p
via the pressure
p
(
ρ
) =
ρm
. Thus, the cell density
ρ
satisfies the equation:
(2.2) ∂
∂tρ−∇·(ρ∇p(ρ)) = ρG(c), x ∈R2, t >0.
For the growth function G(c), we assume
(2.3) G(c) = G0c, with G0>0,
note that in contrast to the nutrient models in [57,63], we eliminate the possibility
of the formation of a necrotic core by assuming that
G
(
·
)is always positive and
linear (for simplicity), since this project aims to study the boundary instability
induced by the nutrient distribution itself.
Many researches, e.g. [14,15,33,39,43,57], indicate that there is a limit as
m→ ∞
which turns out to be a solution to a free boundary problem of Hele-Shaw type. To
see what happens, we multiply equation (2.2) by mρm−1on both sides to get
(2.4) ∂
∂tp(ρ) = |∇p(ρ)|2+mp(ρ)∆p(ρ) + mG0p(ρ)c.
Hence, if we send
m→ ∞
, we formally obtain the so called complementarity
condition (see [14,57] for a slight different model):
(2.5) p∞(∆p∞+G0c)=0.
On the other hand, the cell density
ρ
(
x, t
)converges to the weak solution (see [57])
of
(2.6) ∂
∂tρ∞−∇·(ρ∞∇p∞) = ρ∞G(c),
NUTRIENT INDUCED TUMOR BOUNDARY INSTABILITY 5
and
p∞
compels the limit density
ρ∞
only take value in the range of
[0,1]
for
any initial date
ρ0∈[0,1]
(see Theorem 4.1 in [57] for a slightly different model).
Moreover, the limit pressure p∞belongs to the Hele-Shaw monotone graph:
(2.7) p∞(ρ∞) = 0,06ρ∞<1,
[0,∞), ρ∞= 1.
The incompressible limit and the complementarity condition of a fluid mechanical
related model have been rigorously justified in [14,57]. And the incompressible limit
of
(2.2)
(coupled with nutrient models that will be introduced in the next section)
was verified numerically in [46].
We define the support of p∞to be
(2.8) D∞(t) = {p∞(x, t)>0},
then (2.5) and (2.7) together yield
−∆p∞=G0cfor x∈D∞(t),(2.9a)
p∞= 0,for x∈R2\D∞(t),(2.9b)
and
ρ∞
= 1 in
D∞
. Therefore, once the nutrient concentration
c
(
x, t
)is known one
can recover p∞from the elliptic equation above.
Now we justify the relationship between
D
(
t
)and
D∞
(
t
). Observe that when
m
is finite,
ρ
and
p
(
ρ
)have the same support
D
(
t
), whereas as
m
tends to infinity,
ρ∞
may have larger support than
p∞
. However, a large class of initial data, see
e.g. [56], enable the free boundary problem (correspond to
(2.6)
and
(2.9)
) possess
patch solutions, i.e., ρ∞=χD∞, where χAstands for the indicator function of the
set
A
. In this case, the support of
p∞
coincides with that of
ρ∞
. Moreover, the
boundary velocity
v
is governed by Darcy law
v
=
−∇p∞
. Further, the boundary
moving speed along the normal direction at the boundary point
x
, denote by
σ
(
x
),
is given by:
(2.10) σ(x) = −∇p∞·ˆn(x),
where
ˆn
(
x
)is the outer unit normal vector at
x∈∂D∞
(
t
). The boundary speed for
more general initial data was studied in [39].
As the end of this subsection, we emphasize that in our free boundary model, as the
limit of the density models, the pressure
p∞
always vanishes on
∂D∞
. However, as
mentioned previously, in the first kind free boundary models, the internal pressure
˜p
is assumed to satisfy the so-called Laplace-Young condition (or some other curvature
dependent boundary condition). Mathematically, the boundary condition
(2.9b)
is
replaced by
(2.11) ˜p(x) = γκ(x),
where
γ >
0is a constant coefficient, and
κ
(
x
)denotes the curvature at the boundary
point
x
. In the related studies, the curvature condition
(2.11)
plays an essential role
(e.g., [12,23]).
2.1.2. Two nutrient models. Regarding the nutrient, it diffuses freely over the
two dimensional plane. However, inside the tumoral region, the cells consume
the nutrient. While outside the tumor, the nutrient exchanges with the far field
concentration
cB
provided by the surrounding environment or vasculature. It follows
摘要：

TUMORBOUNDARYINSTABILITYINDUCEDBYNUTRIENTCONSUMPTIONANDSUPPLYYUFENG,MINTANG,XIAOQIANXU,ANDZHENNANZHOUAbstract.WeinvestigatethetumorboundaryinstabilityinducedbynutrientconsumptionandsupplybasedonaHele-Shawmodelderivedfromtakingtheincompressiblelimitofacelldensitymodel.Weanalyzetheboundarystability/in...

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