# Game Theory of Tumor–Stroma Interactions in Multiple Myeloma: Effect of Nonlinear Benefits

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## Abstract

**:**

## 1. Introduction

#### 1.1. From Intra-Tumor Cooperation to Tumor–Stroma Interactions

#### 1.2. From Two-Player Games to Collective Interactions with Nonlinear Effects

#### 1.3. Multiple Myeloma as a Modelling Case Study

## 2. Results

#### 2.1. Stability (Bistability) Depends on the Shape of the Benefit Functions

_{i,j}→0 (linear benefits) in our model, and is consistent with previous models of linear benefits [23]. Observe, however (Figure 2), that if benefits are nonlinear this is not necessarily the case: the OB–OC equilibrium can persist irrespective of the number of MM cells introduced in the population or disappear entirely, depending on the shape of the benefit functions (in Figure 2 the position of the inflection point). While a model with linear benefits predicts a bistable system, nonlinear benefits can have both or only one of the two stable equilibria observed in linear models. Further examples are shown in Figure 3 and Figure 4, where additional mixed equilibria, not observed with linear benefits, arise with nonlinear benefits; in other cases, instead, the dynamics are similar (Figure 5). Unfortunately, given the huge number of combinations of parameters in the nonlinear model, any useful generalization is impossible. To understand and predict the dynamics of the system we need to know the shape of the benefit functions.

#### 2.2. Nonlinear Benefits Can Lead to the Coexistence of Three Types and Cyclical Dynamics

#### 2.3. Therapies That Target Growth Factors May Be More Effective Than Chemotherapy

## 3. Discussion

## 4. Materials and Methods

_{OC}, x

_{OB}, and x

_{MM}the frequencies of three types of cells, respectively OC (osteoclasts), OB (osteoblasts), and MM (malignant plasma cells), in the population (x

_{OC}+ x

_{OB}+ x

_{MM}= 1). N is the number of cells within the diffusion range of the growth factors (hence, equal to group size), which we assume is the same for all types and all growth factors. Fitness is calculated by considering the payoffs obtained in the randomly formed groups weighted by the probability that such groups occur. In a well-mixed population, the probability that a group contains n

_{OC}, n

_{OB}, and N − n

_{OC}− n

_{OB}individuals (excluding the focal cell itself) of type OC, OB, and MM, respectively, is given by

_{OC}= b

_{OC,OC}(n

_{OC}+ 1) + b

_{OB,OC}(n

_{OB}) + b

_{MM,OC}(N − 1 − n

_{OC}− n

_{OB}) − c

_{OC}

_{OB}= b

_{OC,OB}(n

_{OC}) + b

_{OB,OB}(n

_{OB}+ 1) + b

_{MM,OB}(N − 1 − n

_{OC}− n

_{OB}) − c

_{OB}

_{MM}= b

_{OC,MM}(n

_{OC}) + b

_{OB,MM}(n

_{OB}) + b

_{MM,MM}(N − n

_{OC}− n

_{OB}) − c

_{MM}

_{i}is the cost of producing growth factors by type i and b

_{i,j}is a function that indicates the effect on type j of the growth factors produced by type i. Note that each player’s strategy has also an effect on itself. The fitness of type i is given by

_{OC}′ = x

_{OC}(W

_{OC}− W *)

_{OB}′ = x

_{OB}(W

_{OB}− W *)

_{MM}′ = x

_{MM}(W

_{MM}− W *)

_{OC}W

_{OC}+ x

_{OB}W

_{OB}+ x

_{MM}W

_{MM}. To introduce nonlinearities, we assume a benefit function defined by

_{i}is the number of cells of type i; the parameters define the effect of the growth factors produced by type i on type j: B

_{i,j}defines the maximum benefit; h

_{i,j}defines the position of the inflection point: h

_{i,j}→1 gives increasing returns and h

_{i,j}→0 diminishing returns; s

_{i,j}defines the steepness of the function at the inflection point (s

_{i,j}→∞ models a threshold public goods game; s

_{i,j}→0 models a linear benefit (Figure 10); and the normalization in (9) in prevents the logistic function (10) from becoming constant for s

_{i,j}→0).

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Bistability with linear benefits. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 10; c

_{OC}= 0.1; c

_{OB}= 0.2; c

_{MM}= 0.3; B

_{OC,OC}= 0; B

_{OC,OB}= 1; B

_{OC,MM}= 1.1; B

_{OB,OC}= 1; B

_{OB,OB}= 0; B

_{OB,MM}= 0; B

_{MM,OC}= 1.1; B

_{MM,OB}= −0.3; B

_{MM,MM}= 0; s

_{i,j}→0.

**Figure 2.**Bistability with nonlinear benefits. The top panels show the benefit function for different values of h

_{i,j}. In the bottom panels, the distance from each vertex is inversely proportional to the frequency of each type; arrows show the direction of the dynamics; colors show the average fitness of the population. N = 10; c

_{OC}= 0.1; c

_{OB}= 0.2; c

_{MM}= 0.3; B

_{OC,OC}= 0; B

_{OC,OB}= 1; B

_{OC,MM}= 1.1; B

_{OB,OC}= 1; B

_{OB,OB}= 0; B

_{OB,MM}= 0; B

_{MM,OC}= 1.1; B

_{MM,OB}= −0.3; B

_{MM,MM}= 0; s

_{i,j}= 20.

**Figure 3.**Nonlinear benefits can lead to an additional mixed equilibrium on the OB–OC edge. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 25; c

_{i}= 0.1; B

_{OC,OC}= 0.6; B

_{OC,OB}= 1.1; B

_{OC,MM}= 2; B

_{OB,OC}= 1; B

_{OB,OB}= 0.6; B

_{OB,MM}= 0; B

_{MM,OC}= 3; B

_{MM,OB}= −0.5; B

_{MM,MM}= 1; h

_{OC,OC}= 0; h

_{OC,OB}= 0.01; h

_{OC,MM}= 0.2; h

_{OB,OC}= 0.05; h

_{OB,OB}= 0.05; h

_{OC,MM}= 0.2; h

_{MM,OC}= 0.5, h

_{MM,OB}= 0.5; h

_{MM,MM}= 0.5. Nonlinear: s

_{OC,OC}= 50; s

_{OC,OB}= 30; s

_{OC,MM}= 50; s

_{OB,OC}= 30; s

_{OB,OB}= 30; s

_{OB,MM}= 30; s

_{MM,OC}= 5; s

_{MM,OB}= 20; s

_{MM,MM}= 50; Linear: s

_{i,j}→0.

**Figure 4.**Nonlinear benefits can lead to an additional mixed equilibrium on the MM–OC edge. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 20; c

_{OC}= 1.5; c

_{OB}= 0.5; c

_{MM}= 2; B

_{OC,OC}= 0.6; B

_{OC,OB}= 1.1; B

_{OC,MM}= 2.3; B

_{OB,OC}= 1; B

_{OB,OB}= 0.5; B

_{OB,MM}= 0; B

_{MM,OC}= 3; B

_{MM,OB}= −0.5; B

_{MM,MM}= 1.5; h

_{OC,OC}= 0; h

_{OC,OB}= 0; h

_{OC,MM}= 0; h

_{OB,OC}= 0; h

_{OB,OB}= 0; h

_{OB,MM}= 0; h

_{MM,OC}= 0.3, h

_{MM,OB}= 0.5; h

_{MM,MM}= 0.1. Nonlinear: s

_{OC,OC}= 10; s

_{OC,OB}= 10; s

_{OC,MM}= 100; s

_{OB,OC}= 10; s

_{OB,OB}= 10; s

_{OB,MM}= 20; s

_{MM,OC}= 10; s

_{MM,OB}= 10; s

_{MM,MM}= 100; Linear: s

_{i,j}→0.

**Figure 5.**Nonlinear benefits and linear benefits can lead to similar dynamics. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 20; c

_{OC}= 0.1; c

_{OB}= 0.2; c

_{MM}= 0.3; B

_{OC,OC}= 0.55; B

_{OC,OB}= 1.1; B

_{OC,MM}= 0.6; B

_{OB,OC}= 1; B

_{OB,OB}= 0.5; B

_{OB,MM}= 0; B

_{MM,OC}= 0.8; B

_{MM,OB}= −0.5; B

_{MM,MM}= 1.5; h

_{OC,OC}= 0; h

_{OC,OB}= 0; h

_{OC,MM}= 0; h

_{OB,OC}= 0; h

_{OB,OB}= 0; h

_{OB,MM}= 0; h

_{MM,OC}= 0.3, h

_{MM,OB}= 0.02; h

_{MM,MM}= 0.1. Nonlinear: s

_{OC,OC}= 10; s

_{OC,OB}= 10; s

_{OC,MM}= 100; s

_{OB,OC}= 10; s

_{OB,OB}= 10; s

_{OB,MM}= 20; s

_{MM,OC}= 10; s

_{MM,OB}= 10; s

_{MM,MM}= 100; Linear: s

_{i,j}→0.

**Figure 6.**Nonlinear benefits can lead to an interior mixed equilibrium. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 20; c

_{i}= 0.1; B

_{OC,OC}= 0.45; B

_{OC,OB}= 0.9; B

_{OC,MM}= 2; B

_{OB,OC}= 0.9; B

_{OB,OB}= 0.3; B

_{OB,MM}= 0; B

_{MM,OC}= 2; B

_{MM,OB}= −0.3; B

_{MM,MM}= 0.9; h

_{OC,OC}= 0.05; h

_{OC,OB}= 0.05; h

_{OC,MM}= 0.5; h

_{OB,OC}= 0.05; h

_{OB,OB}= 0.05; h

_{OB,MM}= 0.5; h

_{MM,OC}= 0.5, h

_{MM,OB}= 0.5; h

_{MM,MM}= 0.5. Nonlinear: s

_{OC,OC}= 10; s

_{OC,OB}= 10; s

_{OC,MM}= 50; s

_{OB,OC}= 10; s

_{OB,OB}= 10; s

_{OB,MM}= 10; s

_{MM,OC}= 50; s

_{MM,OB}= 50; s

_{MM,MM}= 50; Linear: s

_{i,j}→0.

**Figure 7.**Nonlinear benefits can produce oscillations leading to an interior mixed equilibrium. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 10; c

_{OC}= 0.1; c

_{OB}= 0.12; c

_{MM}= 0.14; B

_{OC,OC}= 1; B

_{OC,OB}= 1; B

_{OC,MM}= 1; B

_{OB,OC}= 0.7; B

_{OB,OB}= 0.7; B

_{OB,MM}= 0.7; B

_{MM,OC}= 0.9; B

_{MM,OB}= 0.9; B

_{MM,MM}= 0.9; h

_{OC,OC}= 0.4; h

_{OC,OB}= 0.7; h

_{OC,MM}= 0.1; h

_{OB,OC}= 0.7; h

_{OB,OB}= 0.4; h

_{OB,MM}= 0.2; h

_{MM,OC}= 0.4, h

_{MM,OB}= 0.3; h

_{MM,MM}= 0.7. Nonlinear: s

_{OC,OC}= 20; s

_{OC,OB}= 20; s

_{OC,MM}= 5; s

_{OB,OC}= 10; s

_{OB,OB}= 10; s

_{OB,MM}= 50; s

_{MM,OC}= 10; s

_{MM,OB}= 5; s

_{MM,MM}= 5; Linear: s

_{i,j}→0.

**Figure 8.**Nonlinear benefits can lead to a limit cycle. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. N = 20; c

_{OC}= 1.2; c

_{OB}= 1.0; c

_{MM}= 1.8; B

_{OC,OC}= 1.1; B

_{OC,OB}= 1.1; B

_{OC,MM}= 1.1; B

_{OB,OC}= 0.95; B

_{OB,OB}= 1.1; B

_{OB,MM}= 1.5; B

_{MM,OC}= 1.8; B

_{MM,OB}= −0.35; B

_{MM,MM}= 0.35; h

_{OC,OC}= 0; h

_{OC,OB}= 0; h

_{OC,MM}= 0; h

_{OB,OC}= 0; h

_{OB,OB}= 0; h

_{OB,MM}= 0; h

_{MM,OC}= 0.2, h

_{MM,OB}= 0.2; h

_{MM,MM}= 0.2. Nonlinear: s

_{OC,OC}= 4; s

_{OC,OB}= 4; s

_{OC,MM}= 40; s

_{OB,OC}= 4; s

_{OB,OB}= 4; s

_{OB,MM}= 4; s

_{MM,OC}= 6; s

_{MM,OB}= 6; s

_{MM,MM}= 1000; Linear: s

_{i,j}→0.

**Figure 9.**Therapies that target growth factors may be more effective than chemotherapy targeting MM cells. The distance from each vertex is inversely proportional to the frequency of each type. Arrows show the direction of the dynamics; colors show the average fitness of the population. Changes in frequency due to therapies that reduce the amount of MM cells (left) with constant h

_{i,j}= 0.3; or therapies that increase the value of h

_{i,j}from 0.3 to 0.7 (right). Circles show the original population before MM mutants arise (white), unstable states (gray), and stable states (black); colored curves show frequency changes driven by clonal selection; dotted black lines show frequency changes driven by therapy. At the OB–OC equilibrium (0), a mutant MM clone invades the population (1) and leads to an MM–OC equilibrium (2); if h

_{i,j}does not change, a therapy that reduces the fraction of MM cells (3, left) suddenly reduces the average fitness of the population, but the population immediately bounces back to the previous equilibrium (4, left). A therapy that increases h

_{i,j}to h*

_{i,j}(3, right) instead leads to a spontaneous extinction of the MM types and a reduction in fitness (4, right). Other parameters are: N = 10; c

_{OC}= 0.1; c

_{OB}= 0.2; c

_{MM}= 0.3; B

_{OC,OC}= 0; B

_{OC,OB}= 1; B

_{OC,MM}= 1.1; B

_{OB,OC}= 1; B

_{OB,OB}= 0; B

_{OB,MM}= 0; B

_{MM,OC}= 1.1; B

_{MM,OB}= −0.3; B

_{MM,MM}= 0; s

_{i,j}= 20.

**Figure 10.**Sigmoid benefits. The benefit of the growth factors produced by type i on type j is described as a normalized logistic function (of the number n

_{i}of cells of type i), with inflection point h

_{i,j}; here h

_{i,j}= 0.4; B

_{i,j}= 1. Multiple curves are shown, with increasing steepness s

_{i,j}(increasing opacity) from s

_{i,j}→0 (linear benefits) to s

_{i,j}→∞ (a step function).

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**MDPI and ACS Style**

Sartakhti, J.S.; Manshaei, M.H.; Archetti, M.
Game Theory of Tumor–Stroma Interactions in Multiple Myeloma: Effect of Nonlinear Benefits. *Games* **2018**, *9*, 32.
https://doi.org/10.3390/g9020032

**AMA Style**

Sartakhti JS, Manshaei MH, Archetti M.
Game Theory of Tumor–Stroma Interactions in Multiple Myeloma: Effect of Nonlinear Benefits. *Games*. 2018; 9(2):32.
https://doi.org/10.3390/g9020032

**Chicago/Turabian Style**

Sartakhti, Javad Salimi, Mohammad Hossein Manshaei, and Marco Archetti.
2018. "Game Theory of Tumor–Stroma Interactions in Multiple Myeloma: Effect of Nonlinear Benefits" *Games* 9, no. 2: 32.
https://doi.org/10.3390/g9020032