Jim Brunner

Tumor Angiogenesis

A tumor in the absence of blood vessels, known as non-vascularized, grows only to about 1 mm3. The same tumor grows to 1000 times that size when vascularized and has access to blood and the oxygen and nutrients it carries. Tumors can gain access to blood vessels in 4 ways. These are vasculogenesis, angiogenesis, vascular mimicry, and growing along existing blood vessels (1). Angiogenesis is the process by which new blood vessels form and branch away from existing blood vessels. Without angiogenesis, tumors reach a steady state in which apoptosis and proliferation of tumor cells balance (1). This balance corresponds to a balance between the pro-angiogenic factors and anti-angiogenic factors that signal endothelial cells, promoting angiogenesis or suppressing it, respectively (1). When this balance is upset and pro-angiogenic factors are present in high enough concentrations, angiogenesis begins and the tumor can grow.

Angiogenesis is a normal process both in development and in healthy adults. It is a part of normal healing of a cut as well as part of the female menstrual cycle (3). In development, angiogenesis is driven mainly by a signal molecule called vascular endothelial growth factor (VEGF)(1). Another important pro-angiogenic factor is fibroblast growth factor (FGF)(1). Located on the leading front of a growing blood vessel is a specialized endothelial cell known as a tip cell, which has a variety of specialized receptors for angiogenic factors that guide the formation of the vessel (1). Endothelial cells, the cells which make up the walls of blood vessels normally undergo mitosis rarely, on average about once every three years (3). Increased levels of endothelial proliferation beyond the ~10,000 endothelial cells which divide regularly indicates angiogenesis is occurring (1).

Pro-angiogenic factors which target endothelial cells can be secreted by a tumor as well as stromal cells and immune cells in the tumors microenvironment (1). The main pro-angiogenic factors are VEGF-A, bFGF and FGF2, as well as platelet derived growth factor B (PDGF-B), cytokines, TGFβ, TGFα, MMP2 and MMP9 (1). TGFβ and TGFα are only pro-angiogenic in low doses (1). MMP2 and MMP9 degrade the extracellular matrix which would normally block the formation of blood vessels (1). The two most important of these factors are VEGF and FGF. VEGF-R, the receptor molecule for VEGF is restricted to endothelial cells (1). Further, it has been found that VEGF production and secretion is induced by hypoxia and hypoglycemia(2). The pro-angiogenic factors are regulated by stress, the oncogenes including Bcl-2, or loss of function of tumor suppressor genes(1). VEGF is important mainly in the initiation of angiogenesis.

There are also several important anti-angiogenesis factors. Interferons INFα and INFβ inhibit MMP9 and the pro-angiogenic factor IL8(1). Angiostaten binds to a variety of cell surface molecules, inhibiting endothelial cell proliferation and migration (1). Thrombospondin 1 also inhibits angiogenesis (1). Another important anti-angiogenic factor is known as Endostatin.

Endostatin inhibits the proliferation of endothelial cells, most likely by binding to VEGFR2 (a VEGF receptor) and suppressing Wnt signaling (1). Endostatin is among the anti-angiogenic factors being studied as a treatment for human cancers (2). Bevacizumab (Avastin), is a VEGF antibody that is now FDA approved in the treatment of cancers (3). VEGF antibodies block VEGF from binding to its receptor molecules (3). Release of these factors is regulated by the loss of tumor suppressor genes and the activation of oncogenes.

Tumor growth is dependent on angiogenesis, because this is the process by which the tumor is supplied with oxygen and other nutrients necessary for proliferation. Angiogenesis contrasts with the earlier hypothesis that existing blood vessels dilated to provide the blood necessary for tumor growth (3). Experimental evidence, including the results of experiments performed on rabbit corneas (a non vascularized medium) support the hypothesis that angiogenesis is essential to tumor growth beyond 1-2 mm3.

Bcl-2 Orchestrated Cross Talk

VEGF induces pathways in endothelial cells that up-regulate the Bcl-2 gene (4). This gene plays a few important roles in angiogenesis and tumor growth. Bcl-2 up-regulates pro-angiogenic chemokines in endothelial cells (4). Furthermore, Bcl-2 in endothelial cells has been shown to increase the transcriptional activity of STAT3, which is a regulator for VEGF (4). This causes the secretion of VEGF by endothelial cells, which can bind to VEGFR1, a VEGF receptor, which is found on tumor cells (4). This, in turn, causes up-regulation of the Bcl-2 gene in tumor cells, indicating that Bcl-2 is integral to the cross talk between endothelial cells and tumor cells (4).

In tumor cells, Bcl-2 up-regulates the chemokines CXCL1 and CXCL8, which are potent proangiogenic factors (4). In addition, over expression of Bcl-2 in tumor cells enhances VEGF expression (4). Bcl-2 also may play a role in the resistance of some tumors to therapies such as radiation (4).

Knocking out the Bcl-2 gene in endothelial cells prevented the up-regulation of Bcl-2, CXCL1 and CXCL8 in tumor cells, and slowed down tumor growth (4). The sum of this interaction is a cross talk between endothelial and tumor cells in which Bcl-2 up-regulation in endothelial cells ultimately causes Bcl-2 up-regulation in tumor cells (4). Bcl-2 activity causes VEGF to be secreted by both cell types, which both have receptors for it (4). In addition to VEGF, Bcl-2 in tumor cells also causes the secretion of proangiogenic chemokines CXCL1 and CXCL8.

A model for HDMEC growth and the VEGF-Bcl-2-CXCL8 pathway

A mathematical model can be used to describe HDMECs and how they grow as part of the Bcl-2 mediated pathway that occurs in tumor angiogenesis (5). The model consists of equations describing microvessel density and tumor cell density, as well as sets of equations for VEGF uptake and CXCL8 uptake by endothelial cells (5). The model rests upon the assumptions that each VEGF molecule can only bind one receptor, and Micheals-Menton kinetics govern this binding. It also assumes that tumor cell growth is dependent on vessel density.

The equation for endothelial cell (HDMEC) density is:

dMdt =(μ¿¿ a ϕa+μl ϕ l−( λm−δ ϕa ))M (1− M

M 0−α1 V )−α1 ( α2 ϕa+α 3 ϕl ) M τ ¿ [1]

dVdt

=(α 2 ϕa+α3 ϕl ) M τ−α 4(λm−δ ϕa)α 1V [2]

(5).

φa and φl describe the average density of active VEGF receptors and CXCL8 receptors, respectively:

ϕa=Da

M +α 1V[1a]

ϕ l=C l

M +α 1V[1b]

(5).

In these equations the variable M is the dependant variable for free HDMEC density, V is the variable for vessel density, Da is the density of doubly bound VEGFR2, Cl is the density of bound CXCL8 receptors, and all are dependent on time (5). Mτ is the delay term for M, which must be included because experimental data shows that vessel growth does not begin until 5 days after Bcl-2 expression is up-regulated (5). This delay shows the biological necessary delay for the development and growth of cells (5). The first term in [1] represents the proliferation of HDMECs, with μa the rate that VEGF induces HDMEC proliferation and μl is the rate that CXCL8 induces the same (5). The other part of this term (λ ¿¿m−δ ϕa)¿ represents cell death; natural minus VEGF induced longer life (5). The next part of this equation is normal proliferation, which is logistic in the absence of the Bcl-2 mediated pathway (5). The last term in [1] is the loss of free HDMECs due to growth of vessels (free cells are used to create new vessels). In [2], the first term is for the growth of vessels, with α2 and α2 parameters representing the rate that VEGF and CXCL8 induce vessel growth, respectively (5). The second term is dysfunctionality in vessels due to cell death, with α1 being the average number of cells per vessel, and α4 being the rate at which a vessel becomes dysfunctional (5).

The set of equations that describe VEGF availability and uptake are:

dAdt

=−2η1a k f 1

a A Ra+η2a kr 1

a Ca−λa A+r3 N ¿ [3]

d Ra

dt=−2k f 1

a A Ra+η3a kr 1

a Ca−η5ak f 2

a Ca Ra+2 η4a k r 2

a Da+2 η4v k p

a D a [4]

d Ca

dt=2 η5

a k f 1a A Ra−kr 1

a Ca−η5a k f 2

a Ca Ra+2 η6a kr 2

a Da [5]

d D a

dt=η7

a k f 2a Ca Ra−2 kr 2

a Da−k pa D a [6]

(5).

Here the dependant variables are A – free VEGF concentration, Ra – VEGFR2 density, Ca – single bounded VEGFR2 complex density, and Da – doubly bounded VEGFR2 complexes. [3] consists of terms for the coupling of VEGF to VEGFR, uncoupling of complexes (last uncoupling, from singly bounded to free), natural decay of VEGF, and hypoxia induced production of VEGF by tumor cells, written in that order (5). [4] has terms for coupling of the first VEGF to a VEGFR2, uncoupling, the second coupling, uncoupling of a second VEGF, and recycling of the receptor after activation, written in that order (5). The equation for singly coupled VEGFR2 complexes, [5], has terms for the coupling of the first VEGF to an uncoupled VEGFR, subtraction of uncoupling, subtraction of coupling of a second VEGF, and addition of the uncoupling of the second VEGF (5). The last equation, [6], has a term for the coupling of a second VEGF to a VEGFR2, subtracted from this are a term for the uncoupling of the second VEGF and a term for the recycling of activated complexes, respectively (5). The k constants are rate constants of the chemical reactions that take place in the biological steps theses terms represent, while the η constants represent ratios of masses between different molecules involved (5). The parameter Vchar is the threshold vessel density above which hypoxia does not induce VEGF production (5). The multiplicative factor of 2 that is used in many terms represents the 2 chemical mechanisms many of the reactions involved can follow (5).

The set of equations for CXCL8 uptake are similar:

dLdt

=−η1l k f

l L Rl+η2l kr

l C l−λl L+β l M +βa ϕa M [7]

d R l

dt=−k

f

l

L Rl+η3l kr

l C l+η3l k p

l C l [8]

d C l

dt=η4

l k fl L R l−kr

l C l−k pl C l [9]

(5).

The dependant variables are L – the density of free CXCL8, Rl – the density of unbound receptors for CXCL8, and Cl – the density of CXCL8 active complexes (5). The first equation, [7], has terms for the rate of coupling of CXCL8 to its receptors, the rate of uncoupling, natural decay of the chemical, natural production by endothelial cells, and induced production by endothelial cells, in that order (5). The second, [8], has terms for the coupling, uncoupling, and recycling of active receptors (5). The last equation in this group, [9], also has terms for the rate of coupling, uncoupling, and recycling (5). The η constants again are ratios of weights, and ks are rates of reactions (5).

Lastly, the model includes an equation for tumor growth based on vessel density:

dNdt

=r1C 2

C12+C2−r2(1−

σ C2

C22+C2 ) [10]

with

C (V )=Cm (V 0+V )k+V 0+V

[10a]

(5).

Here C represents oxygen concentration, which is dependent upon vessel density.

This model is useful in predicting the effectiveness of various treatments for tumors. Using this model, treatments that inhibit CXCL8 production and treatments that block Bcl-2 gene expression can be evaluated (5). To determine the usefulness of a anti CXCL8 treatment, an extra parameter is added to [7], multiplying the production terms by an inhibition factor, which ranges from 1 (no treatment) to 0 (full blocking of production) (5). [7] becomes:

dLdt

=−η1l k f

l L Rl+η2l kr

l C l−λl L+ϵ l(β l M +βa ϕa M )

(5)

The new parameter, ϵl, allows the model to be manipulated (5). Experimental treatments give this value a realist range of about 0.8-0 (5). Analysis of solutions to the model showed that varying this parameter increased the time it took for the tumor to reach maximum density and be fully vascularized, but did not reduce the maximum tumor size (5).

Similarly, a parameter can be added to the terms related to Bcl-2 expression to analyze the usefulness of anti Bcl-2 treatments. Doing this, [7], [1], and [2] change to:

dLdt

=−η1l k f

l L Rl+η2l kr

l C l−λl L+β l M +ϵ a βa ϕa M

dMdt =(μ¿¿ aϕa+μl ϕl−( λm−ϵ a δ ϕa )) M (1− M

M 0−α 1V )−α 1 ( α2 ϕa+α 3 ϕl ) M τ ¿

dVdt

=(α 2 ϕa+α3 ϕl ) M τ−α 4(λm−ϵ a δ ϕa)α1 V

(5)

As with ϵl, ϵa can be changed from 1 (no treatment) to 0 (total blockage) (5). Analysis of the solutions of the model shows that not only does anti Bcl-2 treatment slow the development of the tumor, but also it lowers the maximum tumor size and vessel density significantly (5). In fact, the model shows that anti Bcl-2 treatment used on fully formed tumors is affective in lower the tumor size (5).

The drawback to this model is that it does not incorporate the crosstalk between endothelial cells and tumor cells that is described in the previous section. The model does not take into account the impact that VEGF and chemokines secreted by HDMECs have on the growth of tumor cells, and Bcl-2 expression in tumor cells. The model also does not describe the action of the Bcl-2 pathway within the cell.

An improvement to the above model

A further model has been developed that describes the intracellular Bcl-2 pathway, which is relevant to the above model (6). This model describes the apoptotic action of the Bcl-2 protein and the anti apoptotic Bad (6). The model also includes terms for the inhibition of Bcl-2 by the small molecule inhibitor BL193 (6). The biological mechanism that is studied is the ability of Bcl-2 to bind to Bad, thus preventing Bad’s apoptotic action (6). BL193 binds to Bcl-2 to prevent it from blocking Bad action (6). The model is:

dBdt

=−k fb BX +kr

b Cbx−k fi BI+kr

i Cbi+βa

d φa

dt [11]

dXdt

=−k fb BX+ K r

bCbx [12]

d Cbx

dt=−k f

b BX −krb Cbx [13]

dIdt

=−k fi BI+kr

i Cbi+Di(I 0−I ) [14]

d Cbi

dt=kf

i BI−kri Cbi [15]

with dependant variables free Bcl-2 concentration (B), free Bad concentration (X), Bcl-2-Bad bound complex concentration (Cbx), BL193 concentration (I), and BL193-Bcl-2 bound complex (Cbi) (6). The equation for B, [11], includes terms for the binding and unbinding of Bcl-2 with Bad, unbinding and binding terms for Bcl-2 and BL193, and finally a term for the induced activation of the Bcl-2 gene by VEGF, with ϕa being the density of VEGFR2 receptors on the HDMEC cells (6). Equations [12] and [13] deal with Bad and its bound complexes with Bcl-2 (6). The first term in each represents coupling of Bcl-2 and Bad to form the complex, and the second term represents uncoupling to give free molecules (6). [14] and [15] are very similar, with the added difference of the third term in [14], which represents the diffusion of BL193 into the cell (6).

The model also presented an equation for CXCL8 production by HDMECs and an equation for the death rate of HDMECs:

CXCL8 production: β l (B )=βm+ap(1−e−bp B) [16]

Death of HDMECs : λm ( X )=aa ebd X [17]

(6).

This model is useful for analyzing the effectiveness of Bcl-2 inhibitors as cancer therapies (6). In fact, analysis of the model’s solutions lead to the conclusion that the most effective way to improve the treatment is to increase the efficiency with which it inhibits Bcl-2, rather than increasing its ability to diffuse across cell membranes (6).

This second model clarifies the terms in [1] that correspond to VEGF induced cell proliferation, and the VEGF affect on cell life, because VEGF up-regulates Bcl-2 activity in HDMECs.

The remaining piece of the puzzle is a similar model for the internal pathways of tumor cell up-regulation of Bcl-2, as well as a description of the crosstalk by which endothelial cell secretion of VEGF causes up-regulation of Bcl-2 in tumor cells. Adding these parts will be a step toward a full model of the complex interactions between HDMECs and tumor cells, and can be useful in predicting the effectiveness of Bcl-2 targeted treatments.

Preliminary Tumor Model

To develop a model for the Bcl-2 mediated crosstalk, we start with a basic model for tumor and endothelial cell growth, adapted from the previously presented models. The basic starting point for a tumor model is:

dNdt

=(μN−δN (b )) N (1− NN 0

) [18]

dBdt =μBN N−λB B+μN N (1−

NN0 )b¿−δN (b )(1−

NN 0 )B [19]

With N Tumor cell number per well, B total Bcl-2 in nM per well, b Bcl-2 expression in nM per cell (b=B/N), and b* constitutive Bcl-2 expression in nM per cell. The first equation in this model, [18], shows logistic growth for tumor cells. The equation deviates from the general form for logistic growth in the term(μN−δ N (b )), which replaces a simple constant (7). Here μN is the rate of proliferation of tumor cells, and δN(b) is the Bcl-2 mediated death function of b. The second equation, [19], gives the rate of change the amount of Bcl-2 in the tumor wells. The first term gives the production of Bcl-2 by tumor cells; the second term is the natural decay of Bcl-2, with λB the rate of decay constant. The third term introduces the constitutive Bcl-2 that is expressed when a cell proliferates. The last term is the loss of Bcl-2 due to cell death.

Different functions for δN(b) were tried, including a constant, linear and inverse proportional functions, and a more complicated function using a logistic increase in Bcl-2 mediated death. The same stable steady state was found for any function used. This steady state was:

N=N0=11490.000

B=μBN0/λB=59.89670296

The stability of this steady state was analyzed for the different functions used, and it was found analytically to be stable for any realistic set of parameter values. The stability condition of this steady state, based on analysis of partial derivatives was:

μN+k μB

λB− j>0

In the model, μN is the natural proliferation of tumor cells, and j is the death constant in the absence of Bcl-2 expression. Therefore, j should be less than μN for normal cell growth.

For each choice of δN(b), there was a second, unstable steady state that depended on δN(b). Because the stable steady state was the same for any choice of δN(b), the linear function δN (b)= j−kb was used, making the model:

dNdt

=( μN−( j−k8BN

))N (1− NN0

) [20]

dBdt =μBN N−λB B+μN N (1−

NN0 )b¿−( j−kb)(1−

NN0 )B [21]

Figure A

The second, unstable steady state is

N=N 0 ( μB k−λB j− λB μN+μN k−μN )

μN (k+1 )

B=N 0( j−μN)( μB k−λB j−λB μN+μN k−μN )

μN k (k+1 )

Preliminary HDMEC Model

We also needed a basic model for HDMEC growth. The starting point for this is:

dMdt

=( μM ( L )−δ M (b ) ) M (1− MM 0

) [22]

dBdt =μ

BMM−λB B+μM ( L ) M (1−

MM 0 )b¿−δ M (b )(1− M

M 0 )B [23]

dLdt

=μL (b ) M−λL L [24]

Here the dependant variables are: M, HDMEC number per well; B, total Bcl-2 in nM per well; and L, CXCL1/8 in nM per well. The equation for HDMEC growth, [22], follows the logistic model with proportionality that is not constant, but instead dependant on l=L/M and b=B/M. The term μM (l ) is CXCL1/8 dependant proliferation of HDMEC cells as a function of l, and δ M (b ) is Bcl-2 mediated death of HDMEC cells. The equation which gives the change in Bcl-2 concentration, [23], is similar in form to the equation in the preliminary tumor model for the same thing, [19]. The first term gives production of Bcl-2 by HDMECs, the second the natural decay of Bcl-2. The third term is constitutive Bcl-2 expressed when a cell proliferates. This term includes μM ( l ) because CXCL1/8 affect the proliferation of HDMECs. The last term is loss of Bcl-2 due to the death of cells. The third equation, [24], shows the change in CXCL1/8 concentration is dependent on production of CXCL1/8 by HDMECs and natural decay of CXCL1/8. The production of CXCL1/8 by HDMECs is in turn dependant on Bcl-2.

In order to complete the preliminary model, we need to defineμM ( l ), δM (b ), and μL (b ). Different sets of functions were tried, but a very similar steady state was reached for each. The steady state which appeared stable was the same for M and B no matter what was used for μM ( l ) or δM (b ), but the steady state of L depended on what was used for μL (b ). Luckily, δ M (b ) and μL (b ) could be found or adapted from (6). The functions used were:

δ M (b )=k2 e−k3 b=k2e−k3 B

M

μM (L)=k 1 L

( L+k5 )

μL(b)=βM+ap (1−e−k4 b )=βM +ap(1−e

−k4 BM )

This makes the model:

dMdt

=( k1 L( L+k5 )

−k2 e−k3 B

M )M (1− MM 0

) [25]

dBdt

=μBM

M−λB B+k 1 L

( L+k5 )M (1− M

M 0 )b¿−k2 e−k3 B

M (1− MM 0 )B [26]

dLdt

=(β M+ap(1−e

−k4 BM )) M−λL L [27]

Figure B

This model resulted had the following steady state:

M=M 0 =17000

B=μB M 0

λB =88.62001308

L=( βM+ap

(1−e−k4 B

M )) M 0

λL

=0.8908210499

This steady state was analyzed for the given parameter values, and found to be stable.

Combining the Preliminary Models

In order to model the interaction and chemical cross talk between HDMECs and tumor cells, the preliminary models needed to be combined. Some terms also needed to be modified to include the effects of both CXCL1/8 and VEGF. The full model will give the growth in amount of tumor cells and endothelial cells over time taking into account the chemical cross talk between the two cell types mediated by Bcl-2. This crosstalk pathway includes VEGF and CXCL1/8.

The Effect of CXCL1/8

First the models were combined and only the effects of CXCL1/8 were investigated. The normal proliferation term in the tumor cell equation, μN, was changed to reflect the dependence on CXCL1/8 of tumor cell proliferation. To this end, the term was changed to

μN ( L )=k6 L

L+k8, which mirrors the corresponding term in the HDMEC growth equation. The

model then became:

dMdt

=(μM ( L )−δ M (b1 )) M (1− MM 0

)

dNdt

=(μN ( L )−δN (b2 )) N (1− NN 0

)

d BM

dt=μ

BMM −λB BM +μM ( L ) M (1− M

M 0 )b¿−δ M ( b1 )(1− MM 0 )BM

d BN

dt=μBN N−λB BN+μN (L)N (1− N

N0 )b¿−δN (b )(1− NN 0 )BN

dLdt

=μL( b1 ) M−λL L

with

μM (L)=k 1 L

( L+k5 )

μN(L)=k6 L

L+k7

δ M (b1)=k2 e−k3 b1=k2e−k3 B

M

δ N ( b2 )= j−k 6b2= j−k6 BN

μL(b1)=βM+ap (1−e−k4 b1 )=β M+ap(1−e

−k4 BM )

The stable steady state solution for this equation was the same as the steady states for the preliminary equations,

M=M 0 =17000

N=N 0 =11490.000

BM=μBM M 0

λB =88.62001308

BN=μBN N 0

λB =59.89670296

L=( βM+ap

(1−e−k4 μ BM

λB ))M 0

λL

=0.8908210501

This steady state was analyzed and found to be stable.

The major difference from the preliminary model to the combined model was the time it took for the solution to reach the steady state. Figure A showed that N reached its steady state in about 10 days in the preliminary model, whereas in the combined model N reached the same steady state in about 35 days. However, the increase of N was steeper in the combined model.

This difference is caused by the change in μN. With μN as a constant, proliferation of tumor cells was steady and the model reached steady state quickly. With the adjustment for CXCL1/8 dependant proliferation, initial proliferation was lower than before. For comparison, the constant μN was 1.2924 per day, and at t=1, μN ( L )=0.000428149333008118216. However,

Figure C

as M, the number of HDMEC cells increased, μN ( L ) increased above the constant μN, causing the increased proliferation and steepness of tumor cell increase. At t=35.6, μN(L)= 12.60234495.

Because CXCL1/8 is produced and secreted by HDMECs, this is a clear indication of the cross talk that occurs between HDMECs and tumor cells. It shows that endothelial cells, in addition to delivering the oxygen and nutrients tumor cells need to survive through formation of blood vessels, also directly stimulate increased proliferation in tumor cells, and thus faster growth as the number of HDMECs increase.

Most parameter values were adapted from Jain et al (2008) and Jain et al (2009). The parameter k5 was estimated based on the information found in Kaneko et al that the steady state should be reached in about 30 days (4). It was found that adjusting this parameter adjusted the time to reach the steady state when the other parameters were held at their values adapted from Jain et al. The estimate of this parameter given gives the model a steady state reached in about 35 days, which is consistent with experimental data.

Further Study

Further improvements to the model could be made by incorporating the effects of VEGF on proliferation and up regulation of Bcl-2. The terms in the model ϕ Mand ϕ N correspond to VEGF enhancement of proliferation, and could be functions of VEGF concentration. In this version of the model, they are left at 0. The constants μBMand μBNalso could be adjusted to depend of concentration of VEGF. These adjustments could be made and studied for different constant levels of VEGF, as in an in vitro experiment.

For a simulation of an in vivo experiment, a term that is a function of HDMEC amount would need to be added to the tumor cell equation to reflect the tumor cell’s dependence on endothelial cells for oxygen and other nutrients. In addition, a sixth equation would be needed to give the rate of production and depletion of VEGF. Terms similar to these adjustments can be found in Jain et al (2008).

Complete Model

dMdt

=(μM ( L )+ϕM−δ M (b1 )) M (1− MM 0

) [28]

dNdt

=(μN ( L )+ϕN−δN (b2 )) N (1− NN 0

) [29]

d BM

dt=μ

BMM −λB BM +μM ( L ) M (1− M

M 0 )b¿−δ M ( b1 )(1− MM 0 )BM [30]

d BN

dt=μBN N−λB BN+μN (L)N (1− N

N0 )b¿−δN (b )(1− NN 0 )BN [31]

dLdt

=μL( b1 ) M−λL L [32]

With:

μM (L)=k 1 L

( L+k5 ) [33]

δ M (b1)=k2 e−k3 b1=k2e−k3 B

M [34]

μL(b1)=βM+ap (1−e−k4 b1 )=β M+ap(1−e

−k4 BM ) [35]

μN(L)=k6 L

L+k7 [36]

δ N ( b2 )= j−k 6b2= j−k6 BN

[37]

Parameter Value Units Source

μN 1.2924 per day Jain et al (2008)

j 0.01 per day Jain et al (2008)

k 8 1 tumor cell number per fg Bcl-2 per day

N 0 1.149*104 tumor cells Jain et al (2008)

μBN 0.0813 fg Bcl-2 per tumor cell per day

Jain et al (2008)

λB 15.5958 per day Jain et al (2008)

b¿ 0.54 fg Bcl-2 per cell Jain et al (2009)

M 0 17000 HDMEC Cells Jain et al (2008)

μBM 0.0813 fg Bcl-2 per (HDMEC Jain et al (2008)

cells*days)

λL 15.5958 per day Jain et al (2008)

k1 24 per day Jain et al (2008)

k 2 3.8780*10-6 per day Jain et al (2009)

k3 256.9 per fg Bcl-2 Jain et al (2009)

k 4 1.6185 per fg Bcl-2 Jain et al (2009)

k5 0.35 pg CXCL1/8 per HDMEC

βM 8.1139*10-4 pg of CXCL8 per

(HDMEC*days)

Jain et al (2009)

a p 6.9620*10-4 pg of CXCL8 per

(HDMEC*days)

Jain et al (2009)

k 6 24 per day Jain et al (2008)

k7 0.35 pg CXCL1/8 per tumor

cell

Refrences:

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Cancer Therapy. Berlin: Springer Berlin Heidelberg, 2008. 67-88. Web. 18 May 2010.

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Oxford University Press, 1997. 1-2. Web. 18 May 2010.

3. National Cancer Institute. Understanding Cancer Series: Angiogenesis. US National

Institutes of Health, 28 Jan. 2005. Web. 18 May 2010.

4. Tomoatsu Kaneko, Zhaocheng Zhang, Maria G. Mantellini, Elisabeta Karl, and Benjamin

Zeitlin. "Bcl-2 Orchestrates a Cross-talk between Endothelial and Tumor Cells that

Promotes Tumor Growth." Cancer Res 67.20 (2007): 9685-93. Print.

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Pathway in Intratumoral Agiogenesis." Bulletin of Mathematical Biology 70 (2008): 89-

117. Print.

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