The Optimal Balance between Oncolytic Viruses and Natural Killer Cells: A Mathematical Approach
<p>The relative population dynamics of cancer cells, infected cancer cells, virus and NK cells with respect to the relative time with two different initial conditions. (<b>a</b>) <span class="html-italic">x</span>(0) = 0.5, <span class="html-italic">y</span>(0) = 0, <span class="html-italic">v</span>(0) =0.3, <span class="html-italic">z</span>(0) = 0.1 and (<b>b</b>) <span class="html-italic">x</span>(0) = 0.9, <span class="html-italic">y</span>(0) = 0, <span class="html-italic">v</span>(0) = 0.7, <span class="html-italic">z</span>(0) = 0.2. We used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <mo> </mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.036</mn> </mrow> </semantics></math>.</p> "> Figure 2
<p>The relative population dynamics of cancer cells, infected cancer cells, virus and NK cells with respect to the relative time with two different initial conditions. (<b>a</b>) <span class="html-italic">x</span>(0) = 0.5, <span class="html-italic">y</span>(0) = 0, <span class="html-italic">v</span>(0) = 0.3, <span class="html-italic">z</span>(0) = 0.1 and (<b>b</b>) <span class="html-italic">x</span>(0) = 0.9, <span class="html-italic">y</span>(0) = 0, <span class="html-italic">v</span>(0) = 0.7, <span class="html-italic">z</span>(0) = 0.2. The population <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>x</mi> <mfenced> <mi>t</mi> </mfenced> <mo>,</mo> <mo> </mo> <mi>y</mi> <mfenced> <mi>t</mi> </mfenced> <mo>,</mo> <mo> </mo> <mi>v</mi> <mfenced> <mi>t</mi> </mfenced> <mo>,</mo> <mo> </mo> <mi>z</mi> <mfenced> <mi>t</mi> </mfenced> </mrow> </mfenced> </mrow> </semantics></math> converges to the set (0.5, 0.09, 1.8, 0). We used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <mo> </mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.036</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>The relative population dynamics of cancer cells, infected cancer cells, virus and NK cells with respect to the relative time with two different initial conditions. (<b>a</b>) <span class="html-italic">x</span>(0) = 0.5, <span class="html-italic">y</span>(0) = 0, <span class="html-italic">v</span>(0) = 0.3, <span class="html-italic">z</span>(0) = 0.1 and (<b>b</b>) <span class="html-italic">x</span>(0) = 0.9, <span class="html-italic">y</span>(0) = 0, <span class="html-italic">v</span>(0) = 0.7, <span class="html-italic">z</span>(0) = 0.2. The population <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>x</mi> <mfenced> <mi>t</mi> </mfenced> <mo>,</mo> <mo> </mo> <mi>y</mi> <mfenced> <mi>t</mi> </mfenced> <mo>,</mo> <mo> </mo> <mi>v</mi> <mfenced> <mi>t</mi> </mfenced> <mo>,</mo> <mo> </mo> <mi>z</mi> <mfenced> <mi>t</mi> </mfenced> </mrow> </mfenced> </mrow> </semantics></math> converges to the set (0.5516, 0.0514, 1.0078, 0.1685). We used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <mo> </mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.036</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>The equilibrium populations of the relative NK cells (<b>a</b>) and the relative cancer cells (<b>b</b>) over <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> ranged from 0 to 2 with the bursting size of the virus <span class="html-italic">b</span> = 0, 5, 10 and 15. The NK cells are stimulated or activated within different ordered parameter sets (<math display="inline"><semantics> <mi>b</mi> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math>). The equilibrium population of NK cells increases gradually at <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.4, 0.58 and 0.8 when <math display="inline"><semantics> <mi>b</mi> </semantics></math> = 5, 10 and 15, respectively. The result illustrates that there are necessary conditions in choosing parameters where NK cells activate and affect the efficacy of oncolytic virotherapy: Once the NK cells are activated, the cancer cell population increases, which leads to the reduction in efficacy of OV. We used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.36</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>The change of dynamic structure of equilibrium populations at the critical value, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <msubsup> <mi>r</mi> <mn>2</mn> <mo>∗</mo> </msubsup> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. The equilibrium populations of cancer cells (blue), infected cancer cells (red), virus (green) and NK cells (magenta) are changed at <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <msubsup> <mi>r</mi> <mn>2</mn> <mo>∗</mo> </msubsup> </mrow> </semantics></math>. The equilibrium populations of both cancer cells and NK cells gradually increase, but the other populations decay for <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>></mo> <msubsup> <mi>r</mi> <mn>2</mn> <mo>∗</mo> </msubsup> </mrow> </semantics></math>, in other words, increasing <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> reduces the infected cancer cell and virus population, but it increases the cancer cell and NK cell population. We used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.36</mn> </mrow> </semantics></math>.</p> "> Figure 6
<p>As <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> increases, the equilibrium cancer cell increases and shows oscillation at a higher value of bursting rate. The vertical arrow shows the change of the equilibrium cancer cell population and the horizontal arrow shows the change of the virus bursting rate to show oscillatory pattern in the equilibrium cancer cell population from <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>. The equilibrium cancer cell and NK cell over the bursting rate of virus <span class="html-italic">b</span> with different values of <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> are shown in (<b>a</b>,<b>b</b>), respectively. We measured the relative equilibrium population of cancer cells with various values of <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0, 0.5, 1.0 and 1.5 when <span class="html-italic">b</span> varies from 0 to 30. Here, we used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.036</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>Two-dimensional (2D) bifurcation diagram of the equilibrium cancer cell population with respect to the bifurcation parameter <span class="html-italic">b</span> with or without the existence of NK cells. The bifurcation values in <span class="html-italic">b</span> can be created or destroyed depending on the activation of NK cells. We simulated and calculated the equilibrium cancer cell population over b, which ranged from 0 to 30, with two scenarios: (<b>a</b>) no activation of NK cells when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and (<b>b</b>) activation of NK cells when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. Here, we used the parameters <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <mi>a</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.36</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.48</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>v</mi> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.036</mn> </mrow> </semantics></math>.</p> "> Figure 8
<p>Stability region of equilibrium points with respect to two parameters <span class="html-italic">(b</span> and <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math>). <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> is asymptotically stable in the dark blue region, <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> is asymptotically stable in the light blue region, <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics></math> is asymptotically stable in the green region, three populations (<span class="html-italic">x</span>, <span class="html-italic">y</span>, <span class="html-italic">v</span>) oscillate over time in the orange region, and all populations oscillate over time in the yellow region. The relative cancer cell population shows minimum population at the border between the light blue and orange-colored regions. The result illustrates a whole picture for the number of bifurcation points when either <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </semantics></math> or <span class="html-italic">b</span> varies.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Model
2.2. Equilibrium Points
- (1)
- If , then from the second equation in Equation (6), , which implies since . It leads to and z . Therefore, we have an equilibrium point .
- (2)
- If and from the fourth equation in Equation (6), we obtain from the second equation and from the third equation in Equation (6), which leads to . If , then we obtain and from the first equation in Equation (6). Thus, we have an equilibrium point .
- (3)
- If and , then, from the second and third equation in Equation (6), . Since , we have . From the first and second equation in Equation (6), . From the second and third equations in Equation (6), , which is . Thus, we have an equilibrium point , where ,and .
- (4)
- If and , from the fourth equation in Equation (6), . From the second and third equations in Equation (6), we obtain . It leads to . From the first equation in Equation (6), we obtain . Let us define
2.3. Stability of Equilibrium Points
2.4. Numerical Simulation
3. Results
3.1. Existence and Stability of the Equilibrium Points
3.2. The Effect of NK Cell Activation on Population Dynamics
3.3. Activation of NK Cells Reduces the Efficacy of Oncolytic Virotherapy, Requiring a Higher Bursting Rate of Virus to Generate Oscillations in the Cancer Cell Population
3.4. The Stability of Equilibrium Points Depends on the Existence of NK Cells
3.5. Two Parameters (b and r2) Bifurcation Diagram
4. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | Description | Value | Units | References |
---|---|---|---|---|
Cancer growth rate | [27] | |||
Infection rate of the virus | virus | [27] | ||
Killing rate of cancer cells by NK cells | NK cell | [20] | ||
Killing rate of infected cancer cells by NK cells | NK cell | [20] | ||
Death rate of infected cancer cells | [27] | |||
b | Burst rate of the virus | 50 | Viruses/cell | [27] |
Clearance rate of the virus | 0.0118 | [20] | ||
Stimulation (or activation) rate of the NK cells by infected cancer cells | infected cancer cell | [20] | ||
Clearance rate of NK cells | [20] |
Interval of b | Stability of | Stability of | Stability of | |
---|---|---|---|---|
0.43 | Stable | Unstable | Unstable | |
Unstable | Stable | Unstable | ||
Unstable | Unstable | Stable | ||
Unstable | Stable | Unstable | ||
Unstable | Limit cycle | Unstable | ||
0.5 | Stable | Unstable | Unstable | |
Unstable | Stable | Unstable | ||
Unstable | Unstable | Stable | ||
Unstable | Stable | Unstable | ||
Unstable | Limit cycle | Unstable | ||
0.6 | Stable | Unstable | Unstable | |
Unstable | Stable | Unstable | ||
Unstable | Unstable | Stable | ||
Unstable | Stable | Unstable | ||
Unstable | Limit cycle | Unstable | ||
0.7 | Stable | Unstable | Unstable | |
Unstable | Stable | Unstable | ||
Unstable | Unstable | Stable | ||
Unstable | Stable | Unstable | ||
Unstable | Limit cycle | Unstable | ||
0.8 | Stable | Unstable | Unstable | |
Unstable | Stable | Unstable | ||
Unstable | Unstable | Stable | ||
Unstable | Stable | Unstable | ||
Unstable | Limit cycle | Unstable |
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Kim, D.; Shin, D.-H.; Sung, C.K. The Optimal Balance between Oncolytic Viruses and Natural Killer Cells: A Mathematical Approach. Mathematics 2022, 10, 3370. https://doi.org/10.3390/math10183370
Kim D, Shin D-H, Sung CK. The Optimal Balance between Oncolytic Viruses and Natural Killer Cells: A Mathematical Approach. Mathematics. 2022; 10(18):3370. https://doi.org/10.3390/math10183370
Chicago/Turabian StyleKim, Dongwook, Dong-Hoon Shin, and Chang K. Sung. 2022. "The Optimal Balance between Oncolytic Viruses and Natural Killer Cells: A Mathematical Approach" Mathematics 10, no. 18: 3370. https://doi.org/10.3390/math10183370