Breast Cancer Exosomal microRNAs Facilitate Pre-Metastatic Niche Formation in the Bone: A Mathematical Model

Abstract

Pre-metastatic niche is a location where cancer cells, separating from a primary tumor, find “fertile soil” for growth and proliferation, ensuring successful metastasis. Exosomal miRNAs of breast cancer are known to enter the bone and degrade it, which facilitates cancer cells invasion into the bone interior and ensures its successful colonization. In this paper, we use a mathematical model to first describe, in health, the continuous remodeling of the bone by bone-forming osteoblasts, bone-resorbing osteoclasts and the RANKL-OPG-RANK signaling system, which keeps the balance between bone formation and bone resorption. We next demonstrate how breast cancer exosomal miRNAs disrupt this balance, either by increasing or by decreasing the ratio of osteoclasts/osteoblasts, which results in abnormal high bone resorption or abnormal high bone forming, respectively, and in bone weakening in both cases. Finally we consider the case of abnormally high resorption and evaluate the effect of drugs, which may increase bone density to normal level, thus protecting the bone from invasion by cancer cells.

Appendix

1.1 Parameter Estimates With No miRNAs

1.1.1 Half-Saturation

We denote by $Z^0$ the average density/concentration of species Z. In an expression of the form $Y\dfrac{X}{K_X+X}$ where Y is activated by X and the parameter $K_X$ is the half-saturation of X, we assume that

$$\begin{aligned}\dfrac{X^0}{K_X+X^0}\end{aligned}$$

to be not too close to 0 or to 1 and, for simplicity, take it to be 1/2, so that

$$\begin{aligned} K_X=X^0. \end{aligned}$$

1.1.2 Estimates for $C^0,\ C_p^0,\ O^0$ and $O_p^0$

Osteoclasts are giant multinucleated cells with diameter in the range of 10–300 $\mu$m (Gardner 2007; Tiedemann et al. 2017); we take their average diameter to be 200 $\mu$m, which is about 10 times the diameter of human macrophages ($\sim$21 $\mu$m). Osteoblasts are morphologically diverse cylindrical cells, with a diameter of 20–50 $\mu$m (Qiu et al. 2019); we take their average diameter to be 40 $\mu$m, which is twice the diameter of human macrophages. By correlating the mass with volume, we estimate that the mass of one osteoclast and of one osteoblast are, respectively, $10^2=100$ and $2^2=4$ times that of a macrophage, so that, with a mass of macrophage estimated at $5\times 10^{-10}$ g,

$$\begin{aligned} m_{\text {OC}}=&\ 100\times 5\times 10^{-10}=5\times 10^{-8}\ \text {g},\\ m_{\text {OB}}=&\ 4\times 5\times 10^{-10}=2\times 10^{-9}\ \text {g}. \end{aligned}$$

In mice experiments, Volk et al. (2014) found that the number of OCs in the bone is in the range 5–15 OCs per 1 mm bone surface. We take 10 OCs per 1 mm of bone surface, or equivalently 1,000 OCs per 1 cm of bone surface. Assuming that the bone matrix is a thick-less box, we get

$$\begin{aligned}C^0=K_C=1000^2\times m_{OC}=5\times 10^{-2}\ \text {g/cm}^3.\end{aligned}$$

A “suitable” OB/OC ratio for the study of static human co-culture system was found, in Jolly et al. (2018), to be 2:1, or OB $=$ 2OC. Since $O^0=K_O=m_{OB}O=m_{OB}\times 2C=(m_{OB}/m_{OC})\times 2C^0$, we get

$$\begin{aligned} O^0=K_O=2\times 1000^2\times m_{OB}=4\times 10^{-3}\ \text {g/cm}^3. \end{aligned}$$

Based on ex vivo experiment in bone resorption, it was found that the ratio in cell numbers, OC$_p$/OC $=$ 3 (Zhang et al. 2008) (Fig. 1); assuming that the mass of one OC$_p$ is $5\times 10^{-8}$ g, we get, in terms of densities, that $C_p/C=3\times (1/25)=0.12$, hence

$$\begin{aligned}C_p^0=6\times 10^{-3}\ \text {g/cm}^3.\end{aligned}$$

Based on in vivo experiments in bone formation, it was found that, in terms of cells numbers, OB$_p$/OB=2.5 (Zouani et al. 2013) (Fig. 4A) and that, in terms of volume, $v_{\text {OB}}=26v_{\text {OB}_p}$ ( (Zouani et al. 2013) Abstract). Hence, in terms of densities, $O_p/O=2.5\times (1/26)=0.096$, so that

$$\begin{aligned} O_p^0=3.84\times 10^{-4}\ \text {g/cm}^3. \end{aligned}$$

1.1.3 Estimate for $T_\beta ^0$

In Scheiner et al. (2013), the concentration of active TGF-$\beta$ was measured as $5.63\times 10^{-4}$ pM, where 1 pM$=35.5\times 10^{-12}$ g/L; hence,

$$\begin{aligned} T_\beta ^0=K_{T_\beta }=2\times 10^{-17}\ \text {g/cm}^3. \end{aligned}$$

1.1.4 Estimates for $\mu _C,\ \mu _O,\ \mu _{C_p}$ and $\mu _{O_p}$

The life span of OC is approximately 2 weeks, and the life span of active OB is 3 months (Manolagas 2000). We take $t_{1/2}^C=7$ days and $t_{1/2}^O=45$ days. Hence,

$$\begin{aligned} \mu _C=&\ \dfrac{\ln 2}{\text {7 d}}=0.1\ \text {d}^{-1},\\ \mu _O=&\ \dfrac{\ln 2}{\text {45 d}}=1.54\times 10^{-2}\ \text {d}^{-1}. \end{aligned}$$

Monocyte progenitors give rise to osteoclasts (Lösslein et al. 2021) and the main OC$_p$ cells are the classical monocytes (Kylmäoja et al. 2018). The half-life of classical monocytes is 20–22 h (van Furth and Cohn 1968; Patel et al. 2017); we take $t_{1/2}^{C_p}=0.88$ days (21 h). Hence,

$$\begin{aligned} \mu _{C_p}=\dfrac{\ln 2}{0.88\ \text {d}}=0.79\ \text {d}^{-1}. \end{aligned}$$

An intermediate stage in bone formation by OB cell differentiation, after mesenchymal cells condensing, is the formation of cartilage with chondrocytes cells (Aghajanian and Mohan 2018); we assume that the half-life of OB$_p$ cells is the same as that of chondrocytes. The life span of chondrocytes ranges between 23 and 37 days (Wati et al. 2019); we take $t_{1/2}^{O_p}=15$ days. Hence,

$$\begin{aligned} \mu _{O_p}=\dfrac{\ln 2}{15\ \text {d}}=4.6\times 10^{-2}\ \text {d}^{-1}. \end{aligned}$$

1.1.5 Estimate for $\mu _G$

The half-life of circulating OPG ranges between 10 and 20 minutes (Tomoyasu et al. 1998); we take $t_{1/2}^G=15$ minutes (0.01 days). Hence,

$$\begin{aligned} \mu _G=\dfrac{\ln 2}{0.01\ \text {d}}=69.31\ \text {d}^{-1}. \end{aligned}$$

1.1.6 Estimates for $M_s^0$ and $M_p^0$

The concentration of MSCs in bone is 0.001% (Brozovich et al. 2021), and the density of bone marrow is 1.05 g/cm$^3$ (Rantalainen et al. 2013). Hence,

$$\begin{aligned}M_s^0=0.001\%\times 1.05=1.05\times 10^{-5}\ \text {g/cm}^3.\end{aligned}$$

MDSC (myeloid-derived suppressor cells) suppress PBMC (peripheral blood mononuclear cells) that protect from harmful invaders (e.g., lymphocytes, macrophages, dendritic cells). Experiments in Perico et al. (2022) show that as the ratio MDSC/MSC decreases, suppression of PBMC decreases and it becomes less than 20% when MDSC/MSC $=0.5$. Assuming that this situation arises when MDSC form 20% of myeloid precursor cells, we get

$$\begin{aligned}\dfrac{M_p^0}{M_s^0}=5\times 0.5=2.5, \end{aligned}$$

so that

$$\begin{aligned} M_p^0=2.625\times 10^{-5}\ \text {g/cm}^3. \end{aligned}$$

1.1.7 Equation. (2.1): Estimates for $\mu _{M_s},\ M_{s_0},\ \lambda _{M_sT_\beta },\ \lambda _{O_pO}$ and $\lambda _{M_sO_p}$

The half-life of MSCs is approximately 12 hours (0.5 days) (Eggenhofer et al. 2012; Hoogduijn and Lombardo 2019). Hence,

$$\begin{aligned} \mu _{M_s}=\dfrac{\ln 2}{0.5\ \text {d}}=1.4\ \text {d}^{-1}. \end{aligned}$$

We assume that $M_{s0}$ is determined by the steady-state equation (with no TGF-$\beta$ and no differentiation to OB$_p$), that is,

$$\begin{aligned} M_{s0}-\mu _{M_s}M_s^0=0. \end{aligned}$$

Hence,

$$\begin{aligned} M_{s0}=1.47\times 10^{-5}\ \text {g/cm}^3 \text {d}^{-1}. \end{aligned}$$

From the steady-state equations:

$$\begin{aligned} M_{s0}(1+\frac{\lambda _{M_sT_\beta }}{2})-\lambda _{M_sO_p}M_s-\mu _{M_s}M_s&=0\\ \lambda _{M_sO_p}M_s-\lambda _{O_pO}O_p-\mu _{O_p}O_p&=0\\ \lambda _{O_pO}O_p-\mu _OO&=0. \end{aligned}$$

We get,

$$\begin{aligned} \lambda _{M_sT_\beta }=10.78,\quad \lambda _{O_pO}=0.16\ \text {d}^{-1}\quad \text {and}\quad \lambda _{M_sO_p}=7.55\ \text {d}^{-1}. \end{aligned}$$

1.1.8 Equations. (2.4) and (2.5): Estimates for $\lambda _{C_pC}$ and $\lambda _{M_pC_p}$

We solve simultaneously the following steady-state equations:

$$\begin{aligned} \lambda _{M_pC_p}M_p-\lambda _{C_pC}C_p(1/2)-\mu _{C_p}C_p=0\quad \text {and}\quad \lambda _{C_pC}C_p(1/2)-\mu _CC=0. \end{aligned}$$

Hence,

$$\begin{aligned} \lambda _{M_pC_p}=371.05\ \text {d}^{-1}\quad \text {and}\quad \lambda _{C_pC}=1.67\ \text {d}^{-1}. \end{aligned}$$

1.1.9 Equation. (2.6): Estimates for $\lambda _{OB_f}$ and $\lambda _{CB_r}$

The bone formation rate, in normal control case, was found, in Heaney and Whedon (1958), to be 9.53 mg/Kg calcium per day. Hence,

$$\begin{aligned} \lambda _{OB_f}=9.53\times 10^{-6}\ \text {d}^{-1}. \end{aligned}$$

From the steady-state equation

$$\begin{aligned} \lambda _{OB_f}O=\lambda _{CB_r}C \end{aligned}$$

we get,

$$\begin{aligned} \lambda _{CB_r}=7.624\times 10^{-7}\ \text {d}^{-1}. \end{aligned}$$

1.1.10 Equation. (2.7): Estimate for $\lambda _{CT_\beta }$

From the steady-state equation

$$\begin{aligned} \lambda _{CT_\beta }C-\mu _{T_\beta }T_\beta /2=0, \end{aligned}$$

we find that

$$\begin{aligned} \lambda _{CT_\beta }=8\times 10^{-14}\ \text {d}^{-1}. \end{aligned}$$

1.1.11 Equations. (2.8) and (2.9): Estimates for $\alpha _R,\ \alpha _G,\ \mu _{Q_R}$ and $\mu _{Q_G}$

The half-life of $Q_R=$ RANK/RANKL is approximately 3.3 seconds (Warren et al. 2015) (Table 1), or equivalently $t_{1/2}^{Q_R}=3.85\times 10^{-5}$ days. Hence,

$$\begin{aligned} \mu _{Q_R}=\dfrac{\ln 2}{t_{1/2}^{Q_R}}=1.8\times 10^4\ \text {d}^{-1}. \end{aligned}$$

In Warren et al. (2015) (Table 1), the dissociation constant of $Q_R$ is 45 times that of $Q_G=$ OPG/RANKL, which implies that $t_{1/2}^{Q_G}=t_{1/2}^{Q_R}/45=8.56\times 10^{-7}$ days. It follows that

$$\begin{aligned} \mu _{Q_G}=8.1\times 10^5\ \text {d}^{-1}. \end{aligned}$$

Table 1 in Scheiner et al. (2013) gives the numerical values of $\alpha _R=K_{a,\text {RANKL-RANK}}=3.4\times 10^{-2}$ pM$^{-1}$d$^{-1}$, where 1 pM $=3.55\times 10^{-14}$ g/cm$^3$. Hence,

$$\begin{aligned} \alpha _R=9.6\times 10^{11}\ \text {cm}^3\text {/g d}^{-1}. \end{aligned}$$

Table 1 in Scheiner et al. (2013) gives the numerical value of $\alpha _G=K_{a,\text {RANK-OPG}}=10^{-3}$ pM$^{-1}$. Hence,

$$\begin{aligned} \alpha _G=2.82\times 10^{10}\ \text {cm}^3\text {/g d}^{-1}. \end{aligned}$$

1.1.12 Equations. (2.13) and (2.14): Estimates for $\mu _A,\ \mu _D,\ \mu _{CA}$ and $\mu _{R_LD}$

Fosamax (alendronate) is administered as tablet of 70 mg weekly and has an elimination half-life of approximately 10 years [5]. Hence,

$$\begin{aligned} \mu _A=\dfrac{\ln 2}{10\times 365\ \text {d}}=1.9\times 10^{-4}\ \text {d}^{-1}. \end{aligned}$$

The half-life of Xgeva is 32 days (Narayanan 2013). Hence,

$$\begin{aligned} \mu _D=\dfrac{\ln 2}{32\ \text {d}}=2.17\times 10^{-2}\ \text {d}^{-1}. \end{aligned}$$

We assume that the depletion of A through blocking resorption (at $C=C^0$) is more than twice the degradation of A, taking

$$\begin{aligned} \mu _{CA}=8.4\times 10^{-3}\ \text {cm}^3\text {/g d}^{-1}; \end{aligned}$$

indeed, since $C^0=5\times 10^{-2}$ g/cm$^3$,

$$\begin{aligned} \mu _{CA}C^0=4.2\times 10^{-4}>1.9\times 10^{-4}=\mu _A. \end{aligned}$$

Simulations of the model in health (Fig. 2) yield an estimate of $R_L\sim 9\times 10^{-11}$ g/cm$^3$. Assuming, as above, that $\mu _{R_LD}R_L$ is more than twice the degradation of D, we take

$$\begin{aligned} \mu _{R_LD}=3.38\times 10^8\ \text {cm}^3\text {/g d}^{-1}. \end{aligned}$$

1.2 Parameter Sensitivity Analysis

We performed sensitivity analysis with respect to the bone density B without the effect of miRNAs and with no drugs, for the production/activation parameters $\lambda _{M_ST_\beta }$, $\lambda _{M_sO_p}$, $\lambda _{O_pO}$, $\lambda _{M_pC_p}$, $\lambda _{C_pC}$, $\lambda _{OB_f}$, $\lambda _{CB_r}$ and $\lambda _{CT_\beta }$, and the binding rates $\alpha _R$ and $\alpha _G$.

Fig. 8

Parameter sensitivity analysis for the bone density after 2 years. The p values are less than 0.01

The computations were done using Latin hypercube sampling/partial rank correlation coefficient (LHS/PRCC) with a MATLAB package by Kirschner (2007), Marino et al. (2008) (Fig. 8). The range for the parameters in the sensitivity analysis was between $\pm 50\%$ of their baseline values in Table 2.

Expectedly, increasing any one of the osteoblastic-formation-prone parameters $\lambda _{M_ST_\beta }$, $\lambda _{M_sO_p}$, $\lambda _{O_pO}$, $\lambda _{OB_f}$ and $\lambda _{CT_\beta }$ results in increased bone density; and increasing any one of the osteoclastic-resorption-prone parameters $\lambda _{M_pC_p},\ \lambda _{C_pC}$ and $\lambda _{CB_r}$ results in decreased bone density.

An increase in $\alpha _R$ results in increased concentration of the complex $Q_R=R$-$R_L$, and thus, in increased density C (Eq. (2.5)); and by increased resorption (Eq. (2.6)); as we see, $\alpha _R$ is negatively correlated.

On the other hand, $\alpha _G$ is positively correlated. Indeed, if $\alpha _G$ increases then $Q_G$ will increase, which means that more $R_L$ will be captured by decoy receptor G. Hence, the number of $R_L$ available to attach to R will decrease, which means that $Q_R$ will decrease and bone density will increase.