There are currently over 2.5 million breast cancer survivors in the United States and, according to the American Cancer Society, 10 to 20 percent of these women will develop recurrent breast cancer. Early detection of recurrence can avoid unnecessary radical treatment. However, self-examination or mammography screening may not discover a recurring cancer if the number of surviving cancer cells is small, while biopsy is too invasive and cannot be frequently repeated. It is therefore important to identify non-invasive biomarkers that can detect early recurrence. The present paper develops a mathematical model of cancer recurrence. The model, based on a system of partial differential equations, focuses on tissue biomarkers that include the plasminogen system. Among them, only uPAR is known to have significant correlation to its concentration in serum and could therefore be a good candidate for serum biomarker. The model includes uPAR and other associated cytokines and cells. It is assumed that the residual cancer cells that survived primary cancer therapy are concentrated in the same location within a region with a very small diameter. Model simulations establish a quantitative relation between the diameter of the growing cancer and the total uPAR mass in the cancer. This relation is used to identify uPAR as a potential serum biomarker for breast cancer recurrence.
Citation: Hao W, Friedman A (2016) Serum uPAR as Biomarker in Breast Cancer Recurrence: A Mathematical Model. PLoS ONE 11(4): e0153508. https://doi.org/10.1371/journal.pone.0153508
Editor: Yi-Hsien Hsieh, Institute of Biochemistry and Biotechnology, TAIWAN
Received: December 10, 2015; Accepted: March 30, 2016; Published: April 14, 2016
Copyright: © 2016 Hao, Friedman. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: The authors have been supported by the Mathematical Biosciences Institute and the National Science Foundation under Grant DMS 0931642.
Competing interests: The authors have declared that no competing interests exist.
Human breast cancer is a major cause of death in the United States and worldwide . It is estimated that 230,000 women in the United States are diagnosed annually with invasive breast cancer, and more than 40,000 die from the disease . A major factor that contributes to poor prognosis is the fact that diagnosis is often delayed due to limitation in mammography screening . Poor prognosis occurs also in assessing the risk of recurrence in patients of low grade breast cancer; improving this assessment will help avoid unnecessary chemotherapy .
Risk factors associated with gene mutations such as BRCA1 and BRCA2, and with family history and aging have long been recognized . More recent work is also looking for risk assessment that can be associated with serum biomarkers [6–8]. Three tissue biomarkers have been identified: urokinase plasminogen activator (uPA), plasminogen-activator-inhibitor (PAI-1), and tissue factor (TF) [3, 4, 9, 10]. For uPA to become active it must bind to its receptor uPAR . Active uPA is extracellular matrix-degrading protease that promotes tumor progression and metastasis. It binds to plasminogen and converts it to its activated form, plasmin, a process inhibited by PAI-1 [12–16]. Plasmin mediates the activation of matrix metaloproteinase (MMP) which enables cancer cells’ migration [12, 15, 17]. TF promotes tumor by enhancing VEGF production . Harbeck et. al  reported on an extensive 6-year study to assess the risk associated with node-negative breast cancer recurrence in terms of the levels of uPA and PAI-1. Based on this report and other studies it was concluded that tissue (uPA, PAI-1) provide predictive information about early breast cancer [4, 20]. The American Society of Clinical Oncology also recommends uPA and PAI-1 as prognostic tumor markers for breast cancer .
Although uPA and PAI-1 levels are elevated in breast cancer tissue, these high levels are not detected in the blood. Indeed, as reported in Rha et al., the blood level of uPA and PAI-1 of the plasminogen activation system correlated with that of breast tissue in order of R2 = 0.35 for uPA and R2 = 0.11 for PAI-1. So uPA and PAI-1 are not reliable serum biomarkers. On the other hand, it was reported by Rha et al. , that the correlation of the level of uPAR in the blood with that of tissue is significant, with R2 = 0.61 (P = 0.001). Recently Soydine et al.  found that uPAR in serum and in urine of breast cancer patients (n = 180) was significantly higher than in healthy control (n = 60). Serum uPAR was also shown to be a prognostic biomarker in endometrial cancer .
The present paper is concerned with prognosis of breast cancer recurrence. Most commonly, recurrence occurs within 3–5 years, although the statistics of recurrence is not clear [25, 26]. When breast cancer recurs, it most often recurs in the same location as the primary cancer . In the present paper we address with a mathematical model the following question: Can uPAR be used as biomarker to recognize breast cancer recurrence? We assume that after treatment of the original cancer, some cancer cells survived in the same location, occupying a small spherical region of radius R0 and that the cancer begins to grow while maintaining a spherical shape. Simulations of our mathematical model profile the cancer radius and the expression of uPAR as functions of R0 and time t: R(R0, t) and uPAR(R0, t). If a patient’s uPAR in tissue (surrounding the primary tumor) is measured at time t, say at t = 100 days, we can then use this measurement to determine R0 and hence also the tumor radius R(R0, t) for any time t after 100 days. The articles of Rha et al.  and Soydine et al.  suggest that the uPAR level in serum is siginificantly correlated and hence proportional to the level of uPAR in the tissue, hence serum uPAR could serve as a potential biomarker. When clinical data become available to more reliably confirm this proportionality coefficient, the uPAR could then actually be used as serum biomarker for breast cancer recurrence.
The mathematical model is based on the diagram shown in Fig 1. The model includes, in addition to uPA, uPAR and PAI-1, also TF, VEGF, M-CSF, MMP and MCP-1. It also includes the cells that produce these proteins, or activated by them, namely cancer cells, fibroblasts, macrophages and endothelial cells. The variables of the model are listed in Table 1. The model is described by a system of partial differential equations (PDEs) in a radially symmetric tumor, with evolving radius R(t). The initial radius, R0 = R(0), is a parameter which is patient-dependent. We shall assume that the total density of cells at each point in the tissue is constant; since tumor cells proliferate, the radius R(t) is increasing with time and cells are moving with velocity u which is time dependent.
Equation for tissue factor (T)
Equation for VEGF (V)
The first term on the right-hand side accounts for production of VEGF by cancer cells [28, 30, 31], a process enhanced by tissue factor , and the second term accounts for VEGF produced by macrophages [30, 31].
Equation for plasmin (P)
Plasminogen is the inactive precursor of trypsin-like serine plasmin. When it becomes activated, it is converted to plasmin. The generation of plasmin requires the binding of uPA to plasminogen, after uPA was released from the complex uPA-uPAR and became active , and this binding is inhibited by PAI-1 [12–16]. For simplicity we take the concentration of plasminogen to be constant. Hence the concentration of plasmin concentration satisfies the equation (3)
Equation for uPAR (uPR)
Equation for uPA ( and )
Inactive uPA is produced by fibroblasts . Hence, (5) uPA is activated when inactive uPA combines with uPAR. We take the equation for to be (6)
Equation for PAI-1 (PA)
Equation for M-CSF (q)
The M-CSF concentration satisfies the equation (8) where the first term of the right-hand side account for production by cancer cells .
Equation for MCP-1 (p)
Equation for macrophages (M)
We assume that all cells are moving with common velocity u, and are subject to dispersion. The equation of macrophages density is given by (10) where DM is the dispersion coefficient. The second term of the right-hand side accounts for chemotaxis [28, 31, 39]. Monocytes from the vascular system, with density M0, are attracted to the tissue by MCP-1, and they differentiate into macrophages. Macrophages are terminally differentiated cells. On the boundary of each blood vessel within the tumor there holds a flux condition, which we take to be
As in  we can use a homogenization method to replace these fluxes by an average source of macrophages within the tissue, and this is the first term on the right-hand side of Eq (11). We assume that the macrophages are primarily tumor-associated-macrophages (TAM) of M2 phenotype.
Equation for endothelial cells (E)
Endothelial cells are chemoattracted by VEGF and their proliferation is increased by VEGF . The equation for endothelial density is given by (11) where we used the notation: X+ = X if X > 0, X+ = 0 if X ≤ 0. The second term on the right-hand side assumes a threshold V0 below which proliferation of E does not occur [41, 42].
Equation for fibroblasts (f)
There is a mutual enhancement in the interaction between cancer cells and fibroblasts. Cancer cells secrete TGF-β which increases the activation and proliferation of fibroblasts, while EGF secreted by fibroblasts increases the proliferation of cancer cells [43–50].
The growth of cancer cells occurs on a time scale of days, whereas the secretion and decay of cytokines occur on a time scale of minutes to hours . In order to understand the growth of cancer we simplify the model by using quasi-steady-state approximation for the equations of Tβ and E, so that Tβ = c1 C, G = c2 f for some constants c1 and c2. Then the fibroblasts-enhanced growth rate of fibroblast density (f) through Tβ may be replaced by λfCC, and the tumor-cell-enhanced proliferation of cancer density (C) through G may be replaced by λCf f. We use the Michaelis-Menten law to express the enhanced proliferation of fibroblasts by Tβ (i.e., by c1 C) because TGF-β activation of fibroblast and EGF enhancement of cancer cells may be limited due to the limited rate of receptors recycling associated with this process.
Equation for cancer cells (C)
There are three terms in the bracket: The first term is for tumor growth at rate which is oxygen dependent; the second term represents enhancement by EGF produced by fibroblasts; the third term accounts for proliferation of cancer cells by uPA as it binds to its receptor uPAR on cancer cells [11, 12, 15, 36], a process resisted by PAI-1 [14, 16]. The Michaelis-Menten law is used to represent the limited rate of receptor recycling associated with the enhancements by f and by uPR. We take if w ≤ wh and λC(w) = λwC if w ≥ wh.
Equation for oxygen (w)
The first term of the right-hand side accounts for infusion of oxygen through the blood, which is represented by the density of endothelial cells. The last three terms represent oxygen taken up by macrophages, tumor cells and fibroblasts.
Equations for MMP (Q) and TIMP (Qr)
Equation for ECM (ρ)
Equation for u
We also assume that all cells are approximately of the same volume and surface area, so that the dispersion coefficients of the all cells have the same coefficient, D. By adding Eqs (11)–(14) and (17), we get an equation for ∇⋅u: (19)
We assume that if a breast cancer recurs, it is because some cancer cells survived in the initial location. We also assume, to simplify the computations, that these cells are contained in a sphere of small radius R0, and that as time increases the region containing the cancer cells is a sphere with increasing radius r = R(t), and all the model variables are radially symmetric, that is, they are functions of (r, t) where 0 ≤ r ≤ R(t), t > 0. In particular, where is the radial unit vector.
The free boundary equation
We assume flux boundary conditions due to oxygen transport of the form (21) for X = w and E, where X0 is the density of oxygen or of endothelial cells within the vascular system at the boundary of the tumor. The coefficients αX on the free boundary are chosen as follows: , and . We assume no-flux boundary condition for all other variables except ρ. Since ρ satisfies the hyperbolic Eq (17) with uer as the velocity of the free boundary, we do not need to prescribe a boundary condition for ρ on the free boundary. The radial velocity is determined from Eq (19).
We choose the following initial conditions for ECM, oxygen and cells concentration in unit of g/cm3: ρ = 0.02, w = wh, M = 0.07, E = 0.05, f = 0.06, and C = 0.4. All the cytokines are taken to be initially zero. The initial conditions do not affect the simulation results after a few weeks. However, as will be shown, the very small value of R(0) plays a critical role in the tumor growth profile.
In Figs 2 and 3 we take R(0) = 10−2 cm = 100μm; at this initial size the tumor may contain a few thousand cells, including cancer cells. We define the average density/concentration (Ave) and total mass (TM) as follows
Fig 2 shows the profiles of the average density of cells and concentration of proteins for the first 600 days, as the tumor radius R(t) continues to grow, while Fig 3 shows the total mass of cells and proteins. Although the total mass of all the variables is growing continuously, the averages are not all monotone increasing. Cancer density dips for the first 80 days probably due to hypoxic conditions as the density of endothelial cells is still small (VEGF is still small). The average of oxygen is decreasing as cancer cells are increasing and consuming more oxygen, while the consumption of oxygen by macrophages, fibroblasts, and epithelial cells, combined, is also slightly increasing. uPA is growing after 80 days when fibroblast density begins to increase; PAI-1 mimics the profile of active uPA, while the concentrations of uPAR is relatively stable.
In estimating some of the parameters we assumed steady state of averages of densities of cells and concentrations of proteins, except for p and w. Fig 2 shows agreement with these assumed steady state averages. Thus the model simulations in Fig 2 are consistent with the assumptions made in the parameters estimation.
Harbeck et al.  measured the concentrations of uPA and PAI-1 in breast cancer survivors by the protein nanogram of protein antigen per milligram of tissue protein of breast tissue. They determined that when the risk of recurrence is high, PAI-1≈5×uPA. This proportion between PAI-1 and active uPA is seen also in Fig 2.
For the purpose of developing serum biomarkers we are interested in the total mass of uPAR in the growing tumor, rather than in the average density. Fig 3 shows that the total mass of cells and cytokines are increasing in time, but at different rates.
So far we have taken R(0) fixed at R0 = 10−2 cm. We next want to use the model for diagnostic purposes. Our goal is to determine from one measurement of the total mass of uPAR at time t0 after the primary breast cancer treatment the radius of the tumor at the same time t0 and at any subsequent time t1, t1 > t0. To do this we take R0 ∈ (10−2, 5 × 10−2) cm as a potential initial radius. For each R0, we compute the tumor radius at time t, R = R(R0, t) and the total mass of uPAR at time t, uPAR(R0, t).
Fig 4 shows R and Fig 5 shows the total mass of uPAR, at any (R0, t) in the range R0 ∈ (10−2, 5 × 10−2) cm and 0 < t < 1000 days. From these two figures we can generate a mapping from total uPAR(t) to R(t) which is independent of R0, as follows: From a value of uPAR at some time t after primary breast cancer treatment, we estimate, by using Fig 4, the corresponding parameter R0. We can then use Fig 5 to predict the value of R corresponding to this R0 and the time t. The mapping is uPAR(t)→R(t) is shown in Fig 6, where the color bar determines the value of R(t) for a given pair of uPAR and t.
For any time t, 0 < t < 1000 days, measurement uPAR in gm/cm3 (on the horizontal axis) determines the size of the radius of the cancer in cm, using the column color. All the parameters are as in Tables 2 and 3.
The color map in Fig 6 is a prognostic map for recurrent breast cancer: when a patient’s uPAR is measured t days after the primary breast cancer treatment, the color bar in Fig 6 predicts the size of the recurrent tumor.
There are several mathematical models of breast cancer focusing on different aspects of the disease [28, 38, 56–58]. Our model is the first one to focus on biomarkers associated with the risk of breast cancer recurrence.
Cancer recurrence occurs in 10 to 20 percent of all breast cancer survivors. In order to avoid unnecessary radical treatment, it is important to diagnose a recurrent cancer as early as possible. Although several tissue biomarkers have been identified, biopsy cannot be frequently repeated. This motivated the present study of focusing on potential serum biomarkers which are non-invasive. Of all the plasmiogen system tissue biomarkers only uPAR concentration significantly correlates with uPAR concentration in blood [22, 23]. For this reason we developed in this paper a mathematical model that quantifies the relation between tissue uPAR and the size of a recurrent cancer (Fig 6).
The mathematical model is represented by a system of partial differential equations in a growing tumor with radius R(t). We assume a very small initial radius R0 ∈ (10−2, 5 × 10−2) cm, corresponding to cancer cells that survived after primary breast cancer treatment. The radius R0 may vary from one patient to another. Nevertheless, Fig 6 shows that a patient’s uPAR measured at any time within 1000 days after primary breast cancer treatment, can be used to estimate the cancer size (i.e., its radius) by the color bar. Furthermore, the initial R0 can then be determined from Fig 4, and hence, by Fig 5 also the radius R(t) for all subsequent times.
While the measurement of serum biomarkers for patient survival after primary breast cancer treatment is still being debated , we propose here uPAR as a potential serum biomarker. When clinical data become available to enable us to estimate the precise proportion of uPAR tissue concentration to plasma concentration, uPAR could then be used as plasma biomarker which informs the size of a recurrent tumor. The simulation results of Figs 4–6 can be extended to include smaller values of R0, e.g. 0.005 ≤ R0 ≤ 0.01. For such values, it takes a significantly longer time for the tumor radius to reach a size that can be detected by self examination.
The simulation results rely heavily on parameters taken from the literature, sometimes in a different context, and on parameters estimated in this paper, sometimes rather crudely. For this reason we performed sensitivity analysis to determine how the radius of the tumor will vary when some of the parameters are increased or decreased.
In developing the mathematical model we made several simplifying assumptions: (i) the tumor is radially symmetric; (ii) the total density of cells at each point of the tissue is constant; and (iii) the cells are approximately of the same volume and surface area making the dispersion coefficient of cells equal. Furthermore, the mathematical model is minimal in the sense that includes just the plasminogen system (of uPA, uPAR, PAI-1), plasmin and MMP, tissue factor, VEGF, MCP-1, M-CSF, and the cells that produce these proteins or are activated by them, namely, macrophages, endothelial cells, fibroblasts and cancer cells. For these reasons, this work should be viewed as providing just a first step upon which a more elaborate and more comprehensive model could be developed in the future. When new data of uPAR expression level in patients of breast cancer recurrence become available, some of the parameters of the model will accordingly be adjusted to make the model simulations agree with patients data. Additional cytokines may then also be included, and the assumption of a spherical tumor may be changed to better reflects tumor histology.
Materials and Methods
The diffusion coefficients of proteins (Y) are proportional to the molecular surface area , which is proportional to , where MY is the molecular weight . Accordingly, we can have the following relation: where MV and DV are the molecular weight and diffusion coefficient of VEGF. Since DV = 8.64 × 10−2 cm2 day−1, MV = 24 kDa  and MP = 92 kDa , MuP = 38 kDa , MT = 35 kDa  and MPA = 43 kDa , we get DP = 0.212 cm2day−1, DuP = 0.117 cm2day−1, DT = 0.111 cm2day−1 and DPA = 0.127 cm2day−1.
- dT: The tissue factor half life is 9 hours , hence dT = 1.85 day−1.
- AT: The concentration of tissue factor in cancer is 3.5 × 10−5 g/ml . The ratio of tissue factor in healthy to disease in plasma is 1:2 . We assume larger ratio in breast tissue, so that in the healthy case T = 2 × 10−4 g/ml and take KT = 10−4 g/ml. From Eq (1), in steady state for the healthy case, we get AT = dT T = 3.23 × 10−5 g/ml/day.
- λTC: We assume that most of the tumor is populated with cancer cells, and take C = 0.5 g/ml. As in the deviation of Eq (12), to simplify the model and to estimate some of the parameters, we assume that cytokine equations are in steady-state. From the steady-state of Eq (1) in disease state we have AT + λC C = dT × 3.5 × 10−5, so that λTC = 5.7 × 10−5 day−1.
According to (31, 39) λVC varies in the range of 10−21 to 10−20 in units of g/s/cell, and the production of VEGF by tumor associated macrophages is far larger than the production of VEGF by cancer cells. Accordingly we take λVC = 2 × 10−8/day and λVM = 2 × 10−6/day. We also assume that T enhances cancer-cells production by less than 200% and take λVT = 2.
- dP: The half life of plasmin is 0.5 day , hence dP = 1.39/day.
- λPu: uPAR in disease is 3/2 uPAR in healthy case . Plasminogen binding to cancer cells was increased by 3/2 in disease compared to the healthy . Hence we assume that where PA = 8.39 × 10−6, and g/ml [70, 71]. By taking , we get λPu = 10.5.
- λP: The molecular weight of plasminogen is 92 kDa  while molecular weight of uPA is 38 kDa . The concentration of uPA in breast cancer is 1.8 × 10−6 g/ml [70, 71]. We assume that the number of uPA proteins in plasma is the same as the number of plasmin proteins, so that
Therefore, the concentration of plasmin in breast cancer is approximately 4.4 × 10−6 g/ml. By Eq (3), we have which implies λP = 2.42 × 10−6 g/ml/day.
- duPR: The half life of uPAR is 12 hours , therefore duPR = 1.38/day.
- λuPR M and λuPR C: The concentration of uPAR in breast cancer is 1.8 × 10−6 g/ml [70, 71]. The concentration of uPAR in normal healthy breast is significantly smaller ; we take it to be approximately 10 times smaller. In healthy case, M = 0.04 g/ml , and from
we get, λuPR M = 6.21 × 10−6 day−1.
In disease case, M = 0.3 g/ml . Taking C = 0.5 g/ml in the steady state equation we get λuPR C = 1.242 × 10−6 day−1.
- and : The half life of activated uPA is 5 hours [76, 77], therefore /day. We assume that inactivated uPA degrades slower, and take /day.
- λuf: The concentration of uPA in normal healthy tissue is estimated to g/ml , and fibroblast density is 0.07 g/ml . From we get λuf = 2.057 × 10−4 day−1.
- λu: We use the steady state equation where we take KuPR = uPR = 1.8 × 10−6 g/ml, and g/ml [70, 71]. Since = 3.3, we get λu = 1.92 day−1.
The PAI-1 concentration in breast cancer is 12 ng/mg  (and ), which implies that PA = 12 × 10−9/1.43/10−3 = 8.39 × 10−6 g/ml. Since dPA = 8.32 day−1, we have dPA PA = 7 × 10−5 g/ml/day. In the steady state, we have
However, since the tumor is growing, the left-hand side should be larger than 7 × 10−5. We take the left-hand side to be 4 times larger than 7 × 10−5. We also assume that the first term λPC C contributes -th, and that, the remaining two terms contribute each -th of the total 7 × 10−5. From λPC C = 4 × 10−5 g/ml/day, where C = 0.5 g/ml, we get λPC = 8 × 10−5/day. Similarly, from λPf f = λPM M = 1.2 × 10−4 g/ml/day and f = 0.14 g/ml, M = 0.3 g/ml, we get λPf = 8.4 × 10−4/day and λPM = 4 × 10−4/day.
From steady state of Eq (9) with M = 0.3 g/cm3, λp(w) ∼ λp, dp = 1.73/day and q/(q0 + q) ∼ 1/2, we take an approximate steady state of p to be 2 × 10−7 g/cm3, and Kp = 2 × 10−7 g/cm3. The parameter β is unknown; in  it was chosen to be 0.2/day; here we take it to be 0.3/day.
In Eq (12) production term is due to cytokines secreted by cancer cells. We assume that this term is only a fraction of the death rate df f of fibroblasts, where df = 16.6 × 10−3/day, and take λfC = 5 × 10−3/day.
λC(w) accounts for the proliferation rate minus the death rate by necrosis, while dC is the death rate by apoptosis. In transgenic mice λC(w) is large [31, 38], and cancer develops within a few days. Since breast tumor in human develops much slower, on the time scale of years, we take day−1 if w < wh and λC(w) = 0.6 day−1 if w > wh, while dC = 0.5 day−1. We assume that the enhanced proliferation rate by fibroblast and by binding to uPR are small, and take λCf = 0.06 day−1 and λCup = 0.05 day−1.
In steady healthy state, we have by Eq (12), Af = df f where Af = 10−3 g/cm3/day, df = 1.66 × 10−2/day. So f = 6.02 × 10−2 g/cm3. Accordingly, we take the half-saturation Kf = 0.1 g/cm3
We assume that fibroblasts consume oxygen at the same rate as macrophages, so that dwf = dwM = 80 cm3/g/day by .
From the steady state of Eq (3), with no activated uPA, we get . Since dP = 1.39/day  and λP was estimated (in Eq (3)) by 2.42 × 10−6 g/cm3/day, we get P = 1.74 × 10−6 g/cm3. With active uPA in Eq (3), this value of P should be increased by the factor , so accordingly we take the half-saturation KP to be 4.4 × 10−6 g/cm3.
Since most VEGF is produced by tumor associated macrophages [31, 39]. the steady state of Eq (2) yields where ε < 1, dV = 12.6/day  and λVM = 2 × 10−6/day. Taking M = 0.3 g/ml and ε = 1/2, we get the approximate value V = 7 × 10−8 g/ml. We accordingly take KV = 7 × 10−8. We also take and to be of order 1, and for simplicity choose ; however, other choices affect the model simulations only very little (not shown here).
We performed sensitivity analysis on the all production parameters of the system (1)-(17). Following the method in , we performed Latin hypercube sampling and generated 1000 samples to calculate the partial rank correlation (PRCC) and the p-values with respect to the radius of the tumor at day 600. The results are shown in Fig 7 (The p-value was <0.01).
The most positively correlated production parameters are λVM (the production of VEGF by macrophages), λu (the production uPA activator), λCf (λCf and λfC represent the cross-talk between cancer cells and fibroblasts, which increases the number of cancer cells). The most negatively correlated production parameters are λPf and λPC (which increase, together with λPM, the production of PAI-1, thus increasing the blockage on uPA and the consequently proliferation of cancer cells), and λP (which increases plasmin, and hence also PAI-1).
The remaining parameters are mildly correlated to tumor growth, and their correlation (+ or -) is agreement with the model description in Fig 1.
In order to illustrate our numerical method, we consider the following convection-diffusion equation: (22) where FX accounts for all the ‘active’ terms. Since the model we consider is a free boundary problem, we employ the moving mesh method to compute it. We write Eq (33) can be written in the total derivative form
Let , denote numerical approximations of i-th grid point and , respectively, when t = nτ, where τ is the time stepsize. The discretization is derived by the explicit Euler finite difference scheme, i.e., where , , and , . Then the mesh is moving by , where is solved by the velocity equation. In order to make the Euler method stable, we take .
The authors have been supported by the Mathematical Biosciences Institute and the National Science Foundation under Grant DMS 0931642. The authors thank J.F. Rabajante for this many comments that helped improve the presentation of the paper.
Analyzed the data: WH AF. Contributed reagents/materials/analysis tools: WH AF. Wrote the paper: WH AF.
- 1. Foundation BCR. Breast cancer statistics & resources. 2016.
- 2. Foundation USBCR. U.S. Breast cancer statistics. 2016.
- 3. Dos Anjos Pultz B, da Luz FA, de Faria PR, Oliveira AP, de Araujo RA, Silva MJ. Far beyond the usual biomarkers in breast cancer: a review. J Cancer. 2014;5(7):559–571. pmid:25057307
- 4. Kantelhardt EJ, Vetter M, Schmidt M, Veyret C, Augustin D, et al. Prospective evaluation of prognostic factors uPA/PAI-1 in node-negative breast cancer: phase III NNBC3-Europe trial (AGO, GBG, EORTC-PBG) comparing FEC versus FECDocetaxel. BMC Cancer. 2011;11:140. pmid:21496284
- 5. Rubinstein WS, O’Neill SM, Peters JA, Rittmeyer LJ, Stadler MP. Mathematical modeling for breast cancer risk assessment. State of the art and role in medicine. Oncology (Williston Park, NY). 2002;16(8):1082–1094.
- 6. Chung L, Moore K, Phillips L, Boyle FM, Marsh DJ, Baxter RC. Novel serum protein biomarker panel revealed by mass spectrometry and its prognostic value in breast cancer. Breast Cancer Res. 2014;16(3):R63. pmid:24935269
- 7. Jesneck JL, Mukherjee S, Yurkovetsky Z, Clyde M, Marks JR, Lokshin AE, et al. Do serum biomarkers really measure breast cancer? BMC Cancer. 2009;9:164. pmid:19476629
- 8. Mirabelli P, Incoronato M. Usefulness of traditional serum biomarkers for management of breast cancer patients. Biomed Res Int. 2013;2013:685641. pmid:24350285
- 9. Dublin E, Hanby A, Patel NK, Liebman R, Barnes D. Immunohistochemical expression of uPA, uPAR, and PAI-1 in breast carcinoma. Fibroblastic expression has strong associations with tumor pathology. Am J Pathol. 2000;157(4):1219–1227. pmid:11021826
- 10. Mego M, Karaba M, Minarik G, Benca J, Sedlackova T, et al. Relationship between circulating tumor cells, blood coagulation, and urokinase-plasminogen-activator system in early breast cancer patients. Breast J. 2015;21(2):155–160. pmid:25623304
- 11. Nielsen BS, Sehested M, Duun S, Rank F, Timshel S. Urokinase plasminogen activator is localized in stromal cells in ductal breast cancer. Lab Invest. 2001;81(11):1485–1501. pmid:11706057
- 12. Didiasova M, Wujak L, Wygrecka M, Zakrzewicz D. From plasminogen to plasmin: role of plasminogen receptors in human cancer. Int J Mol Sci. 2014;15(11):21229–21252. pmid:25407528
- 13. Duffy M, McGowan P, Harbeck N, Thomssen C, Schmitt M. uPA and PAI-1 in breast cancer: validated for clinical use in level-of-evidence-1 studies. Breast Cancer. 2014;16:428.
- 14. Gandhari M, Arens N, Majety M, Dorn-Beineke A, Hildenbrand R. Urokinase-type plasminogen activator induces proliferation in breast cancer cells. Int J Oncol. 2006;28(6):1463–1470. pmid:16685447
- 15. H KC, B M. The role of plasminogen-plasmin system in cancer. Coagulation in Cancer. 2009;4:43–65.
- 16. Hildenbrand R, Gandhari M, Stroebel P, Marx A, Allgayer H, Arens N. The urokinase-system—role of cell proliferation and apoptosis. Histol Histopathol. 2008;23(2):227–236. pmid:17999379
- 17. Li WY, Chong SS, Huang EY, Tuan TL. Plasminogen activator/plasmin system: a major player in wound healing? Wound Repair Regen. 2003;11(4):239–247. pmid:12846910
- 18. Ruf W, Yokota N, Schaffner F. Tissue factor in cancer progression and angiogenesis. Thromb Res. 2010;125 Suppl 2:S36–38. pmid:20434002
- 19. Harbeck N, Dettmar P, Thomssen C, Berger U, Ulm K, Kates R, et al. Risk-group discrimination in node-negative breast cancer using invasion and proliferation markers: 6-year median follow-up. Br J Cancer. 1999;80(3-4):419–426. pmid:10408848
- 20. Harbeck N, Schmitt M, Vetter M, Krol J, Paepke D, et al. Prospective Biomarker Trials Chemo N0 and NNBC-3 Europe Validate the Clinical Utility of Invasion Markers uPA and PAI-1 in Node-Negative Breast Cancer. Breast Care (Basel). 2008;3(s2):11–15.
- 21. Harris L, Fritsche H, Mennel R, Norton L, Ravdin P, et al. American Society of Clinical Oncology 2007 update of recommendations for the use of tumor markers in breast cancer. J Clin Oncol. 2007;25(33):5287–5312. pmid:17954709
- 22. Rha SY, Yang WI, Gong SJ, Kim JJ, Yoo NC, Roh JK, et al. Correlation of tissue and blood plasminogen activation system in breast cancer. Cancer Lett. 2000;150(2):137–145. pmid:10704735
- 23. Guney N, Soydine HO, Derin D, Tas F, Camlica H, Duranyildiz D, et al. Serum and urine survivin levels in breast cancer. Med Oncol. 2006;23(3):335–339. pmid:17018890
- 24. Memarzadeh S, Kozak KR, Chang L, Natarajan S, Shintaku P, Reddy ST, et al. Urokinase plasminogen activator receptor: Prognostic biomarker for endometrial cancer. Proc Natl Acad Sci USA. 2002;99(16):10647–10652. pmid:12130664
- 25. Delozier T, Switsers O, Genot JY, Ollivier JM, Hery M. Delayed adjuvant tamoxifen: ten-year results of a collaborative randomized controlled trial in early breast cancer (TAM-02 trial). Ann Oncol. 2000;11(5):515–519. pmid:10907942
- 26. Romero L, Klein L, Ye W, Holmes D, Soni R, Silberman H, et al. Outcome after invasive recurrence in patients with ductal carcinoma in situ of the breast. Am J Surg. 2004;188(4):371–376. pmid:15474428
- 27. Freedman GM, Fowble BL. Local recurrence after mastectomy or breast-conserving surgery and radiation. Oncology (Williston Park, NY). 2000;14(11):1561–1581.
- 28. Bluff JE, Brown NJ, Reed MW, Staton CA. Tissue factor, angiogenesis and tumour progression. Breast Cancer Res. 2008;10(2):204. pmid:18373885
- 29. Zhang X, Yu H, Lou JR, Zheng J, Zhu H. MicroRNA-19 (miR-19) regulates tissue factor expression in breast cancer cells. J Biol Chem. 2011;286(2):1429–1435. pmid:21059650
- 30. Eubank TD, Roberts RD, Khan M, Curry JM, Nuovo GJ. Granulocyte macrophage colony-stimulating factor inhibits breast cancer growth and metastasis by invoking an anti-angiogenic program in tumor-educated macrophages. Cancer Res. 2009;69(5):2133–2140. pmid:19223554
- 31. Szomolay B, Eubank T, Roberts RD, Marsh CB, Friedman A. Modeling the inhibition of breast cancer growth by GM-CSF. J Theor Biol. 2012;303:141–151. pmid:22763136
- 32. Mantovani A, Schioppa T, Porta C, Allavena P, Sica A. Role of tumor-associated macrophages in tumor progression and invasion. Cancer Metastasis Rev. 2006;25(3):315–322. pmid:16967326
- 33. Hildenbrand R, Wolf G, Bohme B, Bleyl U, Steinborn A. Urokinase plasminogen activator receptor (CD87) expression of tumor-associated macrophages in ductal carcinoma in situ, breast cancer, and resident macrophages of normal breast tissue. J Leukoc Biol. 1999;66(1):40–49. pmid:10410988
- 34. LeBeau AM, Duriseti S, Murphy ST, Pepin F, Hann B. Targeting uPAR with antagonistic recombinant human antibodies in aggressive breast cancer. Cancer Res. 2013;73(7):2070–2081. pmid:23400595
- 35. Sieuwerts AM, Klijn JG, Henzen-Logmand SC, Bouwman I, Van Roozendaal KE, Peters HA, et al. Urokinase-type-plasminogen-activator (uPA) production by human breast (myo) fibroblasts in vitro: influence of transforming growth factor-beta(1) (TGF beta(1)) compared with factor(s) released by human epithelial-carcinoma cells. Int J Cancer. 1998;76(6):829–835. pmid:9626349
- 36. Ceruti P, Principe M, Capello M, Cappello P, Novelli F. Three are better than one: plasminogen receptors as cancer theranostic targets. Exp Hematol Oncol. 2013;2(1):12. pmid:23594883
- 37. Illemann M, Laerum OD, Hasselby JP, Thurison T. Urokinase-type plasminogen activator receptor (uPAR) on tumor-associated macrophages is a marker of poor prognosis in colorectal cancer. Cancer Med. 2014;3(4):855–864. pmid:24889870
- 38. Chen D, Roda JM, Marsh CB, Eubank TD, Friedman A. Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model. Bull Math Biol. 2012;74(11):2752–2777. pmid:23073704
- 39. Fujimoto H, Sangai T, Ishii G, Ikehara A, Nagashima T. Stromal MCP-1 in mammary tumors induces tumor-associated macrophage infiltration and contributes to tumor progression. Int J Cancer. 2009;125(6):1276–1284. pmid:19479998
- 40. Hao W, Rovin BH, Friedman A. Mathematical model of renal interstitial fibrosis. Proc Natl Acad Sci USA. 2014;111(39):14193–14198. pmid:25225370
- 41. Acker T, Beck H, Plate KH. Cell type specific expression of vascular endothelial growth factor and angiopoietin-1 and -2 suggests an important role of astrocytes in cerebellar vascularization. Mech Dev. 2001;108(1-2):45–57. pmid:11578860
- 42. Seghezzi G, Patel S, Ren CJ, Gualandris A, Pintucci G. Fibroblast growth factor-2 (FGF-2) induces vascular endothelial growth factor (VEGF) expression in the endothelial cells of forming capillaries: an autocrine mechanism contributing to angiogenesis. J Cell Biol. 1998;141(7):1659–1673. pmid:9647657
- 43. Bhowmick NA, Neilson EG, Moses HL. Stromal fibroblasts in cancer initiation and progression. Nature. 2004;432(7015):332–337. pmid:15549095
- 44. Hofland LJ, van der Burg B, van Eijck CH, Sprij DM, van Koetsveld PM. Role of tumor-derived fibroblasts in the growth of primary cultures of human breast-cancer cells: effects of epidermal growth factor and the somatostatin analogue octreotide. Int J Cancer. 1995;60(1):93–99. pmid:7529213
- 45. Kim Y, J W, F L, M O, Friedman A. Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: a mathematical model and experiments. J Math Biol. 2010;61:401–421. pmid:19902212
- 46. Mueller MM, Fusenig NE. Friends or foes—bipolar effects of the tumour stroma in cancer. Nat Rev Cancer. 2004;4(11):839–849. pmid:15516957
- 47. Ronnov-Jessen L, Petersen OW, Koteliansky VE, Bissell MJ. The origin of the myofibroblasts in breast cancer. Recapitulation of tumor environment in culture unravels diversity and implicates converted fibroblasts and recruited smooth muscle cells. J Clin Invest. 1995;95(2):859–873. pmid:7532191
- 48. Samoszuk M, Tan J, Chorn G. Clonogenic growth of human breast cancer cells co-cultured in direct contact with serum-activated fibroblasts. Breast Cancer Res. 2005;7(3):R274–283. pmid:15987422
- 49. Soon PS, Kim E, Pon CK, Gill AJ, Moore K. Breast cancer-associated fibroblasts induce epithelial-to-mesenchymal transition in breast cancer cells. Endocr Relat Cancer. 2013;20(1):1–12. pmid:23111755
- 50. Sadlonova A, Novak Z, Johnson MR, Bowe DB, Gault SR. Breast fibroblasts modulate epithelial cell proliferation in three-dimensional in vitro co-culture. Breast Cancer Res. 2005;7(1):46–59.
- 51. Louzoun Y, Xue C, Lesinski GB, Friedman A. A mathematical model for pancreatic cancer growth and treatments. J Theor Biol. 2014;351:74–82. pmid:24594371
- 52. Mallmann MR, Schmidt SV, Schultze JL. Macrophages in human cancer: Current and future aspects. PLoS ONE. 2012;7(9):e45466.
- 53. Leek RD, Harris AL. Tumor-associated macrophages in breast cancer. J Mammary Gland Biol Neoplasia. 2002;7(2):177–189. pmid:12463738
- 54. Hao W, Friedman A. The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE. 2014;9(3):e90497. pmid:24621857
- 55. Olson MW, Gervasi DC, Mobashery S, Fridman R. Kinetic analysis of the binding of human matrix metalloproteinase-2 and -9 to tissue inhibitor of metalloproteinase (TIMP)-1 and TIMP-2. J Biol Chem. 1997;272(47):29975–29983. pmid:9368077
- 56. Connolly BM, Choi EY, Gardsvoll H, Bey AL, Currie BM. Selective abrogation of the uPA-uPAR interaction in vivo reveals a novel role in suppression of fibrin-associated inflammation. Blood. 2010;116(9):1593–1603. pmid:20466854
- 57. Chen D, Bobko AA, Gross AC, Evans R, Marsh CB. Involvement of tumor macrophage HIFs in chemotherapy effectiveness: mathematical modeling of oxygen, pH, and glutathione. PLoS ONE. 2014;9(10):e107511. pmid:25295611
- 58. Enderling H, Chaplain MA, Anderson AR, Vaidya JS. A mathematical model of breast cancer development, local treatment and recurrence. J Theor Biol. 2007;246(2):245–259. pmid:17289081
- 59. Liao KL, Bai XF, Friedman A. Mathematical modeling of Interleukin-35 promoting tumor growth and angiogenesis. PLoS ONE. 2014;9(10):e110126. pmid:25356878
- 60. Shui YB, Wang X, Hu JS, Wang SP, Garcia CM. Vascular endothelial growth factor expression and signaling in the lens. Invest Ophthalmol Vis Sci. 2003;44(9):3911–3919. pmid:12939309
- 61. Lijnen HR, Van Hoef B, Ugwu F, Collen D, Roelants I. Specific proteolysis of human plasminogen by a 24 kDa endopeptidase from a novel Chryseobacterium Sp. Biochemistry. 2000;39(2):479–488. pmid:10631010
- 62. Paborsky LR, Tate KM, Harris RJ, Yansura DG, Band L. Purification of recombinant human tissue factor. Biochemistry. 1989;28(20):8072–8077. pmid:2690932
- 63. Curino A, Mitola DJ, Aaronson H, McMahon GA, Raja K. Plasminogen promotes sarcoma growth and suppresses the accumulation of tumor-infiltrating macrophages. Oncogene. 2002;21(57):8830–8842. pmid:12483535
- 64. Lindhout T, Blezer R, Schoen P, Nordfang O, Reutelingsperger C. Activation of factor X and its regulation by tissue factor pathway inhibitor in small-diameter capillaries lined with human endothelial cells. Blood. 1992;79(11):2909–2916. pmid:1586738
- 65. Lwaleed BA, Francis JL, Chisholm M. Monocyte tissue factor levels in cancer patients. Saudi Med J. 2000;21(8):722–729. pmid:11423883
- 66. Ueno T, Toi M, Koike M, Nakamura S, Tominaga T. Tissue factor expression in breast cancer tissues: its correlation with prognosis and plasma concentration. Br J Cancer. 2000;83(2):164–170. pmid:10901365
- 67. Collen D, Wiman B. Turnover of antiplasmin, the fast-acting plasmin inhibitor of plasma. Blood. 1979;53(2):313–324. pmid:153773
- 68. Thielemann A, Baszczuk A, Kopczynski P, Kopczynski Z. High concentration of urokinase-type plasminogen activator receptor in the serum of women with primary breast cancer. Contemp Oncol (Pozn). 2013;17(5):440–445.
- 69. Stillfried GE, Saunders DN, Ranson M. Plasminogen binding and activation at the breast cancer cell surface: the integral role of urokinase activity. Breast Cancer Res. 2007;9(1):R14. pmid:17257442
- 70. Hildenbrand R, Schaaf A, Dorn-Beineke A, Allgayer H, Sutterlin M. Tumor stroma is the predominant uPA-, uPAR-, PAI-1-expressing tissue in human breast cancer: prognostic impact. Histol Histopathol. 2009;24(7):869–877. pmid:19475533
- 71. Benz CC. Impact of aging on the biology of breast cancer. Crit Rev Oncol Hematol. 2008;66(1):65–74. pmid:17949989
- 72. Kariko K, Kuo A, Barnathan E. Overexpression of urokinase receptor in mammalian cells following administration of the in vitro transcribed encoding mRNA. Gene Ther. 1999;6(6):1092–1100. pmid:10455412
- 73. Hu Q, Lu YY, Noh H, Hong S, Dong Z. Interleukin enhancer-binding factor 3 promotes breast tumor progression by regulating sustained urokinase-type plasminogen activator expression. Oncogene. 2013;32(34):3933–3943. pmid:22986534
- 74. Degnim AC, Brahmbhatt RD, Radisky DC, Hoskin TL, Stallings-Mann M. Immune cell quantitation in normal breast tissue lobules with and without lobulitis. Breast Cancer Res Treat. 2014;144(3):539–549. pmid:24596048
- 75. Kwon GY, Lee SD, Park ES. Mast cell and macrophage counts and microvessel density in invasive breast carcinoma-comparison analysis with clinicopathological parameters. Cancer Res Treat. 2005;37(2):103–108. pmid:19956488
- 76. Blasi F, Vassalli JD, Dan K. Urokinase-type plasminogen activator: proenzyme, receptor, and inhibitors. J Cell Biol. 1987;104(4):801–804. pmid:3031083
- 77. Estreicher A, Muhlhauser J, Carpentier JL, Orci L, Vassalli JD. The receptor for urokinase type plasminogen activator polarizes expression of the protease to the leading edge of migrating monocytes and promotes degradation of enzyme inhibitor complexes. J Cell Biol. 1990;111(2):783–792. pmid:2166055
- 78. Strojan P, Budihna M, Smid L, Vrhovec I, Skrk J. Urokinase-type plasminogen activator (uPA) and plasminogen activator inhibitor type 1 (PAI-1) in tissue and serum of head and neck squamous cell carcinoma patients. Eur J Cancer. 1998;34(8):1193–1197. pmid:9849478
- 79. Quillin ML, Matthews BW. Accurate calculation of the density of proteins. Acta Crystallogr D Biol Crystallogr. 2000;56(Pt 7):791–794. pmid:10930825
- 80. Chorostowska-Wynimko J, Swiercz R, Skrzypczak-Jankun E, Wojtowicz A, Selman SH. A novel form of the plasminogen activator inhibitor created by cysteine mutations extends its half-life: relevance to cancer and angiogenesis. Mol Cancer Ther. 2003;2(1):19–28. pmid:12533669
- 81. Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008;254(1):178–196. pmid:18572196