A Mathematical Model for Tumor–Immune Dynamics in Multiple Myeloma

Abstract

We propose a mathematical model that describes the dynamics of multiple myeloma and three distinct populations of the innate and adaptive immune system: cytotoxic T cells, natural killer cells, and regulatory T cells. The model includes significant biologically- and therapeutically-relevant pathways for inhibitory and stimulatory interactions between these populations. Due to the model complexity, we propose a reduced version that captures the principal biological aspects for advanced disease, while still including potential targets for therapeutic interventions. Analysis of the reduced two-dimensional model revealed details about long-term model behavior. In particular, theoretical results describing equilibria and their associated stability are described in detail. Consistent with the theoretical analysis, numerical results reveal parameter regions for which bistability exits. The two stable states in these cases may correspond to long-term disease control or a higher level of disease burden. This initial analysis of the dynamical system provides a foundation for later work, which will consider combination therapies, their expected outcomes, and optimization of regimens.

Hearn Jay Cho and Helen Moore contributed equally to the building of the model and the conceptual framework.

Appendix

In this section, we include formal calculations that were omitted in the main text.

1.1 Existence and Uniqueness of Solutions

Here, we discuss an existence and uniqueness result for the full model given by Eqs. (2) through (5), and the reduced model given by system (6).

Proposition 4

Given initial conditions $(M^0,T_C^0,N^0,T_R^0)$ in the closed first quadrant

$$\displaystyle \begin{aligned} Q=\{(M,T_C,N,T_R): M \geq 0, T_C \geq 0, N \geq 0, T_R \geq 0\}, \end{aligned}$$

the differential equations defined by Eqs.(2) through (5) have a unique solution which exists for all times t ≥ 0 and lies in Q.

Proof

The right-hand side of this differential equation is continuously differentiable in a neighborhood of Q and thus for initial conditions in Q locally there exists a unique solution. As T _C ≡ 0 and T _R ≡ 0 are equilibrium solutions and as $\left .\frac {dM}{dt}\right |{ }_{M=0}$ and $\left .\frac {dN}{dt}\right |{ }_{N=0}$ have source terms and thus are positive, it follows that Q is positively invariant, i.e., solutions that start in Q will remain in Q throughout their interval of existence. Finally, since all the Michaelis–Menten type expressions of the form $\frac {ax}{b+x}$ with x ∈{M, T _C, N, T _R} are bounded, it follows that there exist constants C ₁, C ₂, C ₃, and C ₄ such that

$$\displaystyle \begin{aligned} \left| \frac{dM}{dt} \right| \leq s_M + C_1 M, \qquad \left| \frac{dT_C}{dt} \right| \leq C_3 T_C, \end{aligned}$$

and

$$\displaystyle \begin{aligned} \left| \frac{dN}{dt} \right| \leq s_N + C_2 N, \qquad \left| \frac{dT_R}{dt} \right| \leq C_4 T_R, \end{aligned}$$

hold. Hence each of these quantities grow at most exponentially and thus solutions exist for all times. $\square$

Analogous to the full model, we have the following existence and uniqueness result for the reduced system (6):

Proposition 5

Given initial conditions $(M^0,T_C^0)$ in the closed first quadrant Q = {(M, T _C) : M ≥ 0, T _C ≥ 0}, the differential equations (6) have a unique solution which exists for all times t ≥ 0 and lies in Q.

1.2 Scaled Model

To reduce the number of unknown parameters, we perform the following scaling: M ^∗ = M∕K _M, N ^∗ = N∕K _N, $T_C^*= T_C/K_C$, and $T_R^* = T_R/K_R$.

$$\displaystyle \begin{aligned} \frac{dM^*}{dt} &= \frac{s_M}{K_M} + r_M(1-M^*)M^* - \delta_M\left[1+ \left(\frac{a_{NM} N^*}{b_{NM}/K_N + N^*} + \frac{a_{CM}T_C^*}{b_{CM}/K_C + T_C^*} \right.\right.\\ &\quad \left.\left.+ a_{CNM}\frac{N^*}{b_{NM}/K_N + N^*} \cdot \frac{T_C^*}{b_{CM}/K_C + T_C^*}\right)\right.\\ &\quad \times\left. \left(1-\frac{a_{MM} M^*}{b_{MM}/K_M +M^*}-\frac{a_{RM} T_R^*}{b_{RM}/K_R + T_R^*}\right) \right]M^* \\ \frac{d T_C^*}{dt} &= r_C(1-T_C^*)\left(1-\frac{a_{MC} M^*}{b_{MC}/K_M+M^*}+\frac{a_{NC} N^*}{b_{NC}/K_N + N^*}\right)T_C^* - \delta_C T_C^* \\ \frac{dN^*}{dt} &= \frac{s_N}{K_N} + r_N(1-N^*)\left(1+\frac{a_{CN} T_C^*}{b_{CN}/K_C+T_C^*}\right)N^* - \delta_N N^* \\ \frac{dT_R^*}{dt} &= r_{R}(1-T_R^*)\left(1+\frac{a_{MR} M^*}{b_{MR}/K_M+M^*}\right)T_R^*-\delta_R T_R^* \end{aligned} $$

Dropping asterisks and introducing the notation $\bar {s}_M = s_M/K_M$, $\bar {b}_{NM} = b_{NM}/K_N$, $\bar {b}_{CM} = b_{CM}/K_C$, etc., we obtain

$$\displaystyle \begin{aligned} \frac{dM}{dt} &= \bar{s}_M + r_M(1-M)M - \delta_M\left[1+ \left(\frac{a_{NM} N}{\bar{b}_{NM} + N} + \frac{a_{CM}T_C}{\bar{b}_{CM} + T_C} \right.\right.\\ &\quad \left.\left.+ a_{CNM}\frac{N}{\bar{b}_{NM} + N} \cdot \frac{T_C}{\bar{b}_{CM} + T_C}\right)\left(1-\frac{a_{MM} M}{\bar{b}_{MM} +M}-\frac{a_{RM} T_R}{\bar{b}_{RM} + T_R}\right) \right]M \\ \frac{d T_C}{dt} &= r_C(1-T_C)\left(1-\frac{a_{MC} M}{\bar{b}_{MC}+M}+\frac{a_{NC} N}{\bar{b}_{NC} + N}\right)T_C - \delta_C T_C \\ \frac{dN}{dt} &= \bar{s}_N + r_N(1-N)\left(1+\frac{a_{CN} T_C}{\bar{b}_{CN}+T_C}\right)N - \delta_N N \\ \frac{dT_R}{dt} &= r_R(1-T_R)\left(1+\frac{a_{MR} M}{\bar{b}_{MR}+M}\right)T_R-\delta_R T_R \end{aligned} $$

Based on this scaling, we restrict the values of all scaled threshold parameters $\bar {b}_{pq}$ (p, q = M, N, C, or R), to be between 0 and 2 (i.e., much smaller or twice the population carrying capacity).

1.3 Stability of Trivial Equilibria for M < M _c

Recall that P(M) = (b _MM + M)f ₁(M, 0) and thus we have that

$$\displaystyle \begin{aligned} P'(M^*) = f_1(M^*,0) + (b_{MM}+M^*) \frac{\partial f_1}{\partial M}(M^*,0) = (b_{MM}+M^*) \frac{\partial f_1}{\partial M}(M^*,0). \end{aligned}$$

Hence the second eigenvalue has the same sign as P′(M ^∗).

If there is only one positive real root, then this equilibrium point is locally asymptotically stable. Ignoring the positive factor $\frac {r_M}{K_M}$, if the complex roots are α ± iβ, then we have that

$$\displaystyle \begin{aligned} P(M) = - (M - M^*) \left[ (M-\alpha)^2 + \beta^2 \right], \end{aligned}$$

and thus

$$\displaystyle \begin{aligned} P'(M^*) = - \left[ (M^*-\alpha)^2 + \beta^2 \right] < 0 . \end{aligned}$$

If there are three distinct real roots $0 < M_1^* < M_2^* < M_3^*$,

$$\displaystyle \begin{aligned} P(M) = - (M-M_1^*) (M-M_2^*) (M-M_3^*), \end{aligned}$$

then the low and high equilibrium points are locally asymptotically stable and the intermediate one is unstable. This simply follows from

$$\displaystyle \begin{aligned} \begin{array}{rcl} P'(M_1^*) &\displaystyle = &\displaystyle - (M_1^* - M_2^*) (M_1^* - M_3^*) < 0, \\ P'(M_2^*) &\displaystyle = &\displaystyle - (M_2^* - M_1^*) (M_2^* - M_3^*) > 0, \\ P'(M_3^*) &\displaystyle = &\displaystyle - (M_3^* - M_1^*) (M_3^* - M_2^*) < 0. \end{array} \end{aligned} $$

In particular, the trivial equilibrium closest to the critical value M _c from below is always locally asymptotically stable except when there exists a double root M ^∗. Then the corresponding eigenvalue is 0 and this equilibrium point is a saddle node while the other equilibrium point is locally asymptotically stable.

1.4 Algebraic Simplification for Positive Equilibria

We verify Eq. (14):

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{K_C \left( 1 - \frac{1}{\frac{r_C}{\delta_C} \left(\eta + \frac{a_{MC}M}{b_{MC} + M} \right) } \right)} {b_{CM}+ K_C \left( 1 - \frac{1}{\frac{r_C}{\delta_C} \left(\eta + \frac{a_{MC}M}{b_{MC} + M} \right) } \right) } = \frac{K_C \left( \frac{r_C}{\delta_C} \left(\eta + \frac{a_{MC}M}{b_{MC} + M} \right) -1 \right)} {(b_{CM}+K_C) \frac{r_C}{\delta_C} \left(\eta + \frac{a_{MC}M}{b_{MC} + M} \right) - K_C} \\ &\displaystyle &\displaystyle \qquad = \frac{K_C \left( \frac{r_C}{\delta_C} \left(\eta (b_{MC}+M) + a_{MC}M \right) - (b_{MC}+M) \right)} {(b_{CM}+K_C) \frac{r_C}{\delta_C} \left(\eta (b_{MC}+M) + a_{MC}M \right) - K_C (b_{MC}+M)} \\ &\displaystyle &\displaystyle \qquad = \frac{K_C \left\{ \left[ b_{MC} \left( \frac{r_C}{\delta_C} \eta -1 \right) \right]+ \left[ \frac{r_C}{\delta_C}(\eta +a_{MC}) -1 \right]M \right\}} {b_{MC} \left[ (b_{CM} + K_C) \frac{r_C}{\delta_C} \eta - K_C \right] + \left[ (b_{CM}+K_C) \frac{r_C}{\delta_C} (\eta + a_{MC}) - K_C \right] M} \\ &\displaystyle &\displaystyle \qquad = \frac{\zeta_0 + \zeta_1M}{\alpha_0 + \alpha_1 M} \end{array} \end{aligned} $$

with α ₀, α ₁, ζ ₀, and ζ ₁ defined by this relation. It follows that

$$\displaystyle \begin{aligned} a_{NM}\rho + (a_{CM}+a_{CNM}\rho) \frac{K_C \left( 1 - \frac{1}{\frac{r_C}{\delta_C} \left(\eta + \frac{a_{MC}M}{b_{MC} + M} \right) } \right)} {b_{CM}+ K_C \left( 1 - \frac{1}{\frac{r_C}{\delta_C} \left(\eta + \frac{a_{MC}M}{b_{MC} + M} \right) } \right) } = \frac{\beta_0+\beta_1M}{\alpha_0 + \alpha_1 M} \end{aligned} $$

(21)

with

$$\displaystyle \begin{aligned} \begin{array}{rcl} \alpha_0 &\displaystyle = &\displaystyle b_{MC} \left[ (b_{CM} + K_C) \frac{r_C}{\delta_C} \eta - K_C \right], \\ \alpha_1 &\displaystyle = &\displaystyle (b_{CM}+K_C) \frac{r_C}{\delta_C} (\eta + a_{MC}) - K_C , \\ \beta_0 &\displaystyle = &\displaystyle a_{NM} \rho \alpha_0 + (a_{CM}+a_{CNM}\rho) \zeta_0 \\ &\displaystyle = &\displaystyle b_{MC} \left\{ a_{NM} \rho \left[ (b_{CM} + K_C) \frac{r_C}{\delta_C} \eta - K_C \right] + (a_{CM}+a_{CNM}\rho) K_C \left( \frac{r_C}{\delta_C} \eta {-}1 \right) \right\} \\ &\displaystyle = &\displaystyle b_{MC} \left\{ a_{NM} \rho b_{CM} \frac{r_C}{\delta_C} \eta + (a_{NM}\rho + a_{CM}+a_{CNM} \rho) K_C \left( \frac{r_C}{\delta_C} \eta -1 \right) \right\} , \\ \beta_1 &\displaystyle = &\displaystyle a_{NM} \rho \alpha_1 + (a_{CM}+a_{CNM}\rho) \zeta_1 \\ &\displaystyle = &\displaystyle a_{NM} \rho \left( (b_{CM}+K_C) \frac{r_C}{\delta_C} (\eta + a_{MC}) - K_C \right)\\ &\displaystyle &\displaystyle + (a_{CM}+a_{CNM}\rho) K_C \left[ \frac{r_C}{\delta_C}(\eta +a_{MC}) -1 \right] \\ &\displaystyle = &\displaystyle a_{NM} b_{CM} \rho \frac{r_C}{\delta_C} (\eta + a_{MC})\\ &\displaystyle &\displaystyle + ( a_{NM}\rho + a_{CM}+a_{CNM}\rho) K_C \left[ \frac{r_C}{\delta_C}(\eta +a_{MC}) -1 \right] . \end{array} \end{aligned} $$