Introduction

Tumor progression is a complex and dynamic process emerging from the interplay between malignant cells, immune surveillance, and the tumor microenvironment. The immune system plays a pivotal role in preventing and controlling tumor growth through continuous recognition and elimination of transformed cells—a process known as tumor immunosurveillance1,2,3. Through cytotoxic mechanisms, immune effector cells can eradicate nascent neoplasms before they become clinically detectable. However, many tumors evolve strategies to evade or suppress immune responses via mechanisms such as immune exhaustion, regulatory suppression, or metabolic reprogramming within the tumor microenvironment4,5,6. Consequently, therapeutic outcomes vary widely among patients depending on tumor type, mutational burden, microenvironmental context, and immune competence7,8,9. These observations underscore the need for quantitative and personalized approaches to enhance immune-mediated tumor control while minimizing toxicity and treatment resistance.

Hyperthermia therapy (HT), the controlled elevation of tissue temperature to 39-45 °C, has emerged as a potent adjuvant capable of reshaping tumor-immune interactions and enhancing therapeutic efficacy. Unlike ablative heating, mild HT (\(39-42~^{\circ }\)C) does not induce direct cytotoxicity but profoundly modulates immune function. Experimental and clinical evidence shows that it improves tumor perfusion and oxygenation, enhances the expression of heat shock proteins (HSP70, HSP90), promotes antigen presentation and dendritic cell maturation, facilitates lymphocyte trafficking, and triggers the release of immunogenic and danger-associated signals that amplify antitumor immunity6,10,11,12,13,14,15,16,17,18,19,20,21. These effects position HT as a powerful modulator of the tumor-immune ecosystem with strong potential for synergy with immunotherapy6,19,20,22,23. Nevertheless, the underlying biophysical mechanisms and temperature thresholds that govern immune activation remain only partially understood, motivating the use of quantitative modeling to elucidate how HT alters tumor-immune dynamics and to identify optimal regimes for immune-mediated tumor control.

In this context, mathematical modeling provides a rigorous framework for integrating and analyzing the coupled dynamics of tumor growth, immune response, and therapeutic interventions. By formalizing biological processes through systems of nonlinear differential equations or agent-based simulations, such models enable in silico exploration of treatment strategies, outcome prediction, and optimization of therapeutic protocols24,25,26. Among the seminal works in cancer immunology is the model proposed by Kuznetsov et al.26,27, inspired by Lotka–Volterra predator-prey dynamics, which captures the bidirectional interactions between tumor cells (prey) and cytotoxic lymphocytes (predators). Despite its simplicity—two state variables and few parameters—it reproduces key biological processes such as tumor proliferation, immune-mediated killing, and immune inactivation, and exhibits rich nonlinear behavior including multistability, oscillations, and bifurcations that correspond to clinically relevant outcomes such as tumor elimination, dormancy, or escape26,28. Subsequent studies have built upon this framework through parameter fitting, bifurcation analysis, or inclusion of additional biological mechanisms such as immune exhaustion, tumor heterogeneity, and treatment timing28,29. For example, Bekker et al. applied the Kuznetsov model to interpret tumor-immune responses under low-dose radiotherapy, whereas Kuznetsov et al. introduced a biophysical extension incorporating immune cell activation kinetics. More recent modeling efforts by Byrne et al. 30,31 developed multiphase continuum models that integrate macrophage polarization and T-cell exhaustion into tumor growth dynamics, showing that these regulatory mechanisms can induce bifurcations separating tumor progression from immune control. Building on this line of research, a few mathematical models have been proposed to investigate the effects of hyperthermia on tumor progression and immune regulation. Ibrahim et al. developed a comprehensive control-theoretic framework coupling heat diffusion with tumor-immune-vascular interactions32. Scheidegger et al. introduced an artificial immune-tumor ecosystem model incorporating adaptive immune learning and combined radio-hyperthermia effects 33. Lin et al. proposed a phenomenological model in which heat-induced stress promotes heat shock protein (HSP) expression, indirectly enhancing immune activation through signals generated by thermally damaged cells34. A detailed comparison of these models and ours is provided in Appendix A. Despite these advances, a unified framework linking temperature-dependent immune modulation to tumor-immune dynamics under mild (non-cytotoxic) HT remains lacking, underscoring the need for models that can guide future therapeutic optimization.

The present study develops a generalized dynamical model to investigate how local temperature elevation modulates tumor-immune population dynamics. It focuses on the regime of mild hyperthermia, where temperature acts primarily as an immunomodulatory stimulus rather than a cytotoxic stressor, aiming for a simplified yet clinically predictive modeling approach. Extending the classical framework of Kuznetsov et al.26,27, we incorporate temperature-dependent modulation of key biological processes, specifically the reduction of tumor proliferation and the enhancement of immune effector recruitment and cytotoxic activity14,21,35,36,37. This formulation enables the quantitative assessment of how mild HT can alter the nonlinear structure of tumor-immune interactions. Through detailed stability and bifurcation analyses, we demonstrate that thermal modulation can reshape the separatrix dividing tumor control from tumor persistence, thereby inducing qualitative transitions in system dynamics. These results provide mechanistic insight into the immunological benefits of HT and establish a theoretical foundation for optimizing temperature-based adjuvant strategies that enhance immune-mediated tumor eradication.

Materials and methods

Lotka–Volterra model and stability analysis

The Kuznetsov model is a system of nonlinear ordinary differential equations developed to describe the dynamic interaction between immune effector cells and tumor cells as follows27:

$$\begin{aligned} \frac{dx}{d\tau }&= \sigma + \frac{\rho xy}{\eta + y} - \mu xy - \delta x, \nonumber \\ \frac{dy}{d\tau }&= \alpha y(1 - \beta y) - xy. \end{aligned}$$
(1)

In this framework, the variable \(x(\tau )=E/E_0\) represents the population of cytotoxic immune effector cells, while \(y(\tau )=T/T_0\) denotes the tumor cell population, both normalized to their initial value, and evolving over nondimensionalized time \(\tau\) (see reference27 for more details). The model incorporates key biological processes: immune cell recruitment at a constant rate \(\sigma\), tumor-stimulated activation of immune cells with saturating kinetics governed by parameters \(\rho\) and \(\eta\), and immune-mediated cytotoxicity modulated by \(\mu\). Immune cells decay at rate \(\delta\). The tumor growth is modeled by a logistic term with proliferation rate \(\alpha\) and carrying capacity parameter \(\beta\), counterbalanced by immune surveillance via a bilinear interaction term xy. The resulting coupled equations capture essential features of tumor-immune dynamics, including possible tumor elimination, immune escape, and stable coexistence, depending on the parameter regime26,27.

The parameter values used in this model correspond to those originally estimated by Kuznetsov and collaborators27 through multiparametric fitting to experimental data describing the progression of B-lymphoma BCL1 in the spleen of chimeric mice38. Specifically, the parameter values are: \(\sigma = 0.1181, \rho = 1.131, \eta = 20.19, \mu = 0.00311, \delta = 0.3743, \alpha = 1.636, \gamma =1\), and \(\beta = 2.0 \times 10^{-3}\). The use of these parameter values enables the model to capture key dynamical features of tumor-immune interactions observed in vivo.

To determine the fixed points of the model, which describes the interaction between tumor cells y and immune effector cells x, one must analyze the steady-state conditions of system described by Eq. (1). Those fixed points correspond to the values \((x^*, y^*)\) for which both time derivatives vanish. There are two common approaches to determine these equilibria. The first consists in expressing the immune cell population \(x\) as a function of the tumor cell population y by solving the equation for \(\frac{dy}{d\tau } = 0\), which yields:

$$\begin{aligned} x = \alpha (1 - \beta y). \end{aligned}$$
(2)

Substituting this expression into the equation for \(\frac{dx}{d\tau }\) results in a single equation involving only y. This leads to a cubic expression

$$\begin{aligned} A y^3 + B y^2 + C y + D = 0, \end{aligned}$$
(3)

whose real, non-negative solutions correspond to the tumor cell populations at equilibrium, and The corresponding effector cell values can then be obtained from Eq. (2).

The coefficients of the third degree polynomial are given by

$$\begin{aligned} A&= \mu \beta ,\end{aligned}$$
(4)
$$\begin{aligned} B&= -[\mu +\beta (\rho -\delta -\mu g)], \end{aligned}$$
(5)
$$\begin{aligned} C&= \rho -\delta -\mu g+\delta \beta g+\frac{\sigma \gamma }{\alpha }, \end{aligned}$$
(6)
$$\begin{aligned} D&= g\left( \frac{\sigma \gamma }{\alpha }-\delta \right) . \end{aligned}$$
(7)

These resulting cubic equation encodes the nonlinear interactions between tumor and immune cells and determines at most three equilibrium points (plus the trivial solution \((y=0,x=\sigma /\delta )\)), whose numbers and nature (real and biologically feasible) depend on the model parameters.

An alternative approach is based on finding the nullclines of the system, that is, the curves along which each of the derivatives vanishes. Solving \(\frac{dx}{d\tau } = 0\) and \(\frac{dy}{d\tau } = 0\) independently yields two expressions for y(x):

$$\begin{aligned} y_1(x)&= \frac{1}{\beta }(1 - \frac{x}{\alpha }), \nonumber \\ y_2(x)&= \frac{(\sigma +x(\rho -\delta -\mu g)) \pm \sqrt{(\sigma +x(\rho -\delta -\mu g))^2+4\mu x(\delta x-\sigma )}}{2\mu x}. \end{aligned}$$
(8)

Alternatively, x can be formulated in terms of y as

$$\begin{aligned} x_1(y)&= \frac{\alpha }{\gamma }(1 - \beta y), \nonumber \\ x_2(y)&= \frac{\sigma }{\mu y+\delta -\frac{\rho y}{g+y}}. \end{aligned}$$
(9)

The fixed points correspond to the intersections of these two curves in the positive quadrant. These intersection points can be identified numerically or graphically and provide the biologically feasible steady states of the system. This analysis facilitates both analytical and numerical exploration of the system’s equilibrium behavior, which is essential to understanding possible scenarios of tumor control, persistence, or immune escape.

To analyze the stability of the fixed points in the Kuznetsov model, one first computes the Jacobian matrix of the system evaluated at each equilibrium point. The Jacobian matrix J(x, y) is derived from the partial derivatives of the right-hand sides of the system’s differential equations with respect to the state variables x and y. At a fixed point \((x^*, y^*)\), the Jacobian takes the form:

$$\begin{aligned} J(x^*, y^*) = \begin{bmatrix} -\delta + \dfrac{\rho y^*}{\eta + y^*} - \dfrac{\rho \eta x^* y^*}{(\eta + y^*)^2} - \mu y^* & \dfrac{\rho x^* \eta }{(\eta + y^*)^2} - \mu x^* \\ -y^* & \alpha (1 - 2 \beta y^*) - x^*. \end{bmatrix} \end{aligned}$$
(10)

The stability of each fixed point is determined by analyzing the eigenvalues of the Jacobian matrix. If both eigenvalues have negative real parts, the fixed point is locally asymptotically stable; if at least one eigenvalue has a positive real part, the fixed point is unstable. In the case of complex conjugate eigenvalues with non-zero imaginary parts, their real part determines whether the fixed point behaves as a stable or unstable spiral. This linear stability analysis provides insight into the local dynamics near each equilibrium configuration of the tumor-immune system. A particularly important scenario occurs when a fixed point is a saddle point, which is characterized by one positive and one negative real eigenvalue. In this case, the linearized dynamics near the equilibrium are dominated by a one-dimensional unstable manifold and a one-dimensional stable manifold. The stable manifold acts as a separatrix in the phase space, delineating the boundary between trajectories that converge to different long-term outcomes—for example, tumor clearance versus uncontrolled tumor growth39. The precise geometry of this separatrix plays a crucial role in understanding the bistability and therapeutic thresholds of the tumor-immune system dynamics.

Phase portraits

Phase diagrams are constructed by numerically integrating the system of differential equations for a wide range of initial conditions \((x_0, y_0)\), where x and y denote the populations of immune effector cells and tumor cells, respectively. Each trajectory in the phase plane represents the temporal evolution of the system starting from a specific initial state. By plotting multiple such trajectories, one can visualize how the system behaves under different initial immune-tumor configurations (blue points in Fig. 1 connected to the trajectories represented as green continuous lines). The direction of motion along each trajectory is determined by the vector field defined by the differential equations. Therefore, the system’s evolution is highly sensitive to the choice of parameters, and initial conditions, resulting in markedly different dynamical outcomes. Figure 1 illustrates how small variations in the tumor proliferation rate \(\alpha\) and the immune recruitment rate \(\rho\) reshape the qualitative dynamics of the system. When \(\alpha\) increases slightly (from 0.9816 to 1.014), a saddle-node bifurcation occurs: two additional fixed points emerge, corresponding to the separatrix (SP) and the tumor escape (TE) state, which coexist with the tumor control (TC) attractor. In contrast, when \(\rho\) decreases (from 1.1310 to 0.6789), no new fixed points appear; instead, the existing ones are rearranged, shifting the separatrix toward lower tumor cell numbers and thereby reducing the basin of attraction of the TC state. Critical points, such as stable nodes, spirals, or saddle points, appear as convergence or divergence centers in the diagram. Of particular interest is the presence of separatrices, which are special trajectories that emerge from saddle points (denoted with black triangles in Fig.1a,b) and partition the phase space into distinct dynamical regimes39,40. These separatrices identify the boundary between initial conditions leading to tumor eradication and those resulting in tumor persistence or escape, thereby providing insight into the thresholds that govern the outcome of tumor-immune interactions. Fixed points, stability analysis, and phase portraits were numerically calculated using self-made Python programs.

Fig. 1
figure 1

Phase portraits showing the effect of small parameter variations on tumor-immune dynamics. (a,b) Increasing \(\alpha\) from 0.9816 to 1.0143 induces a saddle-node bifurcation, leading to the emergence of the separatrix (SP) and tumor escape (TE) states. (c,d) Decreasing \(\rho\) from 1.1310 to 0.6789 rearranges the existing fixed points (TC, SP, TE), shifting the separatrix and reducing the basin of attraction of the tumor control (TC) state. When one parameter changes, all other parameters remain: \(\alpha =1.636\), \(\rho =1.131\), \(\sigma = 0.1181, \eta = 20.19, \mu = 0.00311, \delta = 0.3743, \gamma =1\), and \(\beta = 2.0 \times 10^{-3}\).

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Separatrix between control and tumor escape

To identify the separatrix that divides the phase space into regions of tumor control and escape (when both exist), it is necessary to find the existence of a saddle point in the phase portrait. This point is identified as one of the intersection points between the straight line \(x_1(y)\) and the nonlinear curve \(x_2(y)\), corresponding to the steady-state conditions of the tumor-immune dynamics. Among the real-valued intersections, the saddle point is characterized by its dynamical instability: it is stable along one direction in phase space and unstable along the other.

Saddle-node bifurcation

The saddle node bifurcation occurs when \(x_1(y)\) becomes tangent to \(x_2(y)\), indicating the critical parameter values at which the saddle and one of the stable equilibria merge and annihilate. In our case, this tangency marks the point at which TC and TE emerge. The tangency point between the line \(x_1(y)\) and the curve \(x_2(y)\) is found by first equating the two functions (Eq. 9) and solving for the parameter \(\rho\) as a function of y and \(\alpha\), yielding \(\rho =\rho ^*(y,\alpha )\). Next, the tangency condition is enforced by differentiating both functions and requiring their derivatives to be equal. This allows us to solve for \(\alpha\) as a function of y, denoted \(\alpha =\alpha ^*(y)\). Substituting this expression back into the previous result provides \(\rho =\rho ^*(y,\alpha ^*(y))\), fully determining the values of \(\alpha\) and \(\rho\) that ensure tangency at the given point y.

Bifurcation analysis

The bifurcation analysis is performed in order to understand how the qualitative behavior of the system changes as the parameters in the vary. As more parameters are involded in a dynamical system the more it will be subject to changes in its behavior. As we are interested in how the parameters \(\alpha\) and \(\rho\) affect the behavior of the system. We begin by ruling out limit cycles. That is, isolated closed trajectories. The idea of ruling out these sort of trajectories, is that we know that once a bifurcation point is achieved, the qualitative behavior of the system will remain after passing this bifurcation value (i.e. if a an equibrium point is created, it will remain and it will not be destroyed or if it is destroyed it will not be created). In order to rule out closed orbits, we follow the same argument as that of Kutznetsov at all in27. In order to apply this criterion. Set \(g(x,y)=x^{-a}y^{-b}\); this function is continuous in the simply connected domain determined by \(x>0\), \(y>0\). Furthermore, by computing \(\nabla \cdot (g\left\langle dx/d\tau , dy/d\tau \right\rangle )\) we get:

$$\begin{aligned} \nabla \cdot (g\left\langle dx/d\tau , dy/d\tau \right\rangle )&=\partial _{x}\left( g(x,y)dx/d\tau \right) + \partial _{y}\left( g(x,y)dy/d\tau \right) \end{aligned}$$
(11)
$$\begin{aligned}&=x^{-1-a}y^{-b}\left( -a\sigma +\frac{(1-a)\rho xy}{\eta + y}-\mu (1-a)xy-(1-a)\delta x \right) \end{aligned}$$
(12)
$$\begin{aligned}&+ x^{-1-a}y^{-b}\left( \alpha (1-b)x-\alpha \beta (2-b)yx-(1-b)x^{2}\right) . \end{aligned}$$
(13)

Since we must guarantee \(\nabla \cdot (g\left\langle dx/d\tau , dy/d\tau \right\rangle )\) has one sign, we choose \(a=1, b=1\). Therefore,

$$\begin{aligned} \nabla \cdot (g\left\langle dx/d\tau , dy/d\tau \right\rangle ) = -x^{-2}y^{-1}(\sigma + \alpha \beta yx)<0. \end{aligned}$$
(14)

Thus, by Dulac’s criterion; it follows that the system does not have closed trajectories. For a friendly version of Dulac’s criterion, see Chapter 7 in39. To get a better understanding of the type of bifurcation we have with respect to the parameters \(\rho\) and \(\alpha\). We must consider how the behavior of the system changes as we change both of these paremeters. In order to get such behavior, we must consider how the \(2-dimensional\) space determined by \(\rho\) and \(\alpha\), is partioned in regions where different numbers of steady states are present. In Sections 3.1 and 3.1, we further explain how Figs. 4 and 5c illustrate the mechanism and implications of this type of bifurcation.

Results and discussion

Sensitivity analysis

For each parameter set, the fixed points of the system are determined either by computing the real roots of the cubic equation (Eq. 3) or by identifying the intersections between the functions \(x_1(y)\) and \(x_2(y)\) (Eq. 9). These fixed points are then classified according to their stability—stable, unstable, or saddle. Stable fixed points associated with a low tumor cell count are referred to as “tumor control” (TC), while those with a high tumor cell count are referred to as “tumor escape” (TE). Next, the system of ODEs (Eq. 1) is numerically integrated using various initial conditions \(y_0 = y(t=0)\) and \(x_0 = x(t=0)\), represented by green and red points in Fig. 2. Initial conditions that evolve toward tumor control are marked in green, while those leading to tumor escape are marked in red. Notably, small changes in the parameters displace the limit between the two regions, and may lead to unexpected scenarios where initial points near to the TC, end up in tumor escape as shown in panel (b), when \(\rho\) is reduced to 60% or 70% of its original value. This scenario is further illustrated in Appendix B, where representative time evolution curves demonstrate how certain initial conditions—despite being close to the tumor control state—can still lead to tumor escape when immune recruitment is insufficient, preventing effector cell recovery and allowing the tumor to rebound.

Finally, we define the Tumor Control Likelihood Index (TCLI) as the ratio of green to red points, which serves as a quantitative indicator of the likelihood that a patient will achieve tumor control, assuming their disease progression state is unknown at the start of treatment. As shown in Fig. 2, modifying only the value of the parameter \(\alpha\) alters the phase diagram such that, as the tumor growth rate increases, the region of initial conditions leading to tumor escape expands and encroaches upon the region associated with tumor control. This results in a more unfavorable scenario for the patient, as expected for a faster-growing tumor.

Fig. 2
figure 2

Variation in tumor control (TC) versus tumor escape (TE) regions with respect to \(\alpha\) and \(\rho\). The ratio between initial conditions that evolve towards the TC state (green dots) and TE (red dots) is affected by changes in tumor progression rate \(\alpha\) (panel a) and immune recruitment rate \(\rho\) (panel b). In panels (a,b), only one parameter was varied at a time—\(\alpha\) in (a) and \(\rho\) in (b)—while all others were kept constant.

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Recognizing that HT may affect multiple parameters in the Kuznetsov model (as summarized in Table 1), we conducted a sensitivity analysis to determine which parameters most strongly influence the likelihood of immune-mediated tumor control. To this end, we generated phase diagrams across a range of parameter values and computed the Tumor Control Likelihood Index (TCLI) as a quantitative success metric. This index reflects the patient’s capacity to achieve tumor control via immune response. Based on this measure, we systematically identified the parameters that exert the greatest influence on therapeutic outcomes. The result of this analysis (presented in Fig. 3) shows that the dynamic system is more sensitive to changes in the parameters \(\rho\) and \(\alpha\), such that the TCLI increases more abruptly when the value of alpha (tumor growth rate) is reduced or when the value of rho (immune cell recruitment rate) is increased.

Table 1 Impact of hyperthermia and improved oxygenation on model parameters.
Full size table
Fig. 3
figure 3

Sensitivity analysis of model parameters using the tumor control likelihood index (TCLI). Each curve represents the TCLI value when varying a specific model parameter while keeping others fixed at baseline values. The TCLI quantifies the proportion of initial conditions that evolve toward TC rather than TE. The analysis reveals that the model is most sensitive to changes in tumor proliferation rate (\(\alpha\)) and immune recruitment rate (\(\rho\)).

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It is worth noting that the effects summarized in Table 1 also encompass the role of perfusion and oxygenation as modulators of tumor–immune dynamics under mild HT. Hypoxia is a hallmark of most solid tumors and significantly influences tumor biology. It leads to metabolic reprogramming, where cells favor glycolysis over oxidative phosphorylation, even in the presence of oxygen, to meet energy demands. This metabolic shift supports rapid cell proliferation and survival under low-oxygen conditions. Additionally, hypoxia activates hypoxia-inducible factors (HIFs), which regulate the expression of genes involved in angiogenesis, metastasis, and immune evasion. These adaptations contribute to tumor progression and resistance to therapies. The hypoxic microenvironment also affects the extracellular matrix and immune cell infiltration, further complicating treatment strategies. Understanding these mechanisms is crucial for developing therapies that target the hypoxic tumor microenvironment to improve patient outcomes 42. Improved vascular function and oxygen supply can relieve hypoxia and transiently enhance tumor cell proliferation by restoring oxidative metabolism, while simultaneously facilitating immune cell trafficking and recruitment through upregulation of adhesion molecules and chemokines 14,15,21,37. Although these processes may influence multiple biological parameters in the Kuznetsov model, namely \(\alpha\), \(\rho\), and \(\sigma\), the sensitivity analysis shown in Fig. 3 revealed that the system dynamics are primarily governed by variations in the tumor proliferation rate (\(\alpha\)) and the immune recruitment rate (\(\rho\)). As depicted in Fig. 3, changes in the immune cell influx rate (\(\sigma\)) exerted a comparatively minor effect on tumor control. Therefore, our subsequent analysis focuses on the temperature-dependent modulation of \(\alpha\) and \(\rho\), which represent the dominant pathways through which hyperthermia alters tumor-immune population dynamics.

Tumor evolution under changes of \(\alpha\) and \(\rho\)

Among the various effects of heat on both tumor cells and the immune system, two parameters have been identified as having the most significant influence on the probability of tumor control. As demonstrated in the previous section, tumor proliferation rate (\(\alpha\)) and tumor-induced immune cell recruitment rate (\(\rho\)) critically determine patient outcomes. The modulatory effects of HT on these parameters are well supported by consistent evidence from in vitro and in vivo studies. Accordingly, the present study focuses on examining how variations in (\(\alpha\)) and (\(\rho\)) affect tumor-immune dynamics, as described in the following sections.

***Reduction in the proliferation rate tumor cells \(\alpha\). Moderate HT (\(39-42^\circ\)C) has been shown to inhibit tumor cell proliferation through multiple mechanisms. Elevated temperatures can induce cell cycle arrest at specific checkpoints, particularly in the G1 and G2/M phases, thereby delaying or halting cell division37,50,51. Additionally, HT disrupts cellular homeostasis by increasing oxidative stress and promoting the denaturation or aggregation of proteins essential for DNA replication and cell cycle progression. This stress may activate cellular repair mechanisms or trigger apoptotic pathways in proliferating cells. Moreover, the thermal environment sensitizes tumor cells to concurrent treatments such as radiotherapy and chemotherapy, enhancing overall cytotoxicity52). These antiproliferative effects, while not directly cytotoxic at moderate temperatures, contribute to slowing tumor growth and creating a therapeutic window for immunomodulation or combination therapy. We analyzed several data sets from the literature that reported growth curves for cell populations treated with HT at different temperatures36,37. Cancer cell proliferation rates showed a linear dependence on temperature, which is why we explored values of \(\alpha\) proportional to the original value of the parameter.Further details are provided in Appendix C, where tumor growth curves from three different studies at varying temperatures are fitted to a logistic growth model.

Figure 4a shows the steady-state solutions of the Kuznetsov dynamical system (i.e., the fixed points), obtained as the intersections of equations 8. By varying the parameter \(\alpha\) from 60 to 120% of its baseline value, the displacement of the three fixed points—tumor escape (TE), saddle point (SP), and tumor control (TC)—can be clearly observed. As the tumor proliferation rate \(\alpha\) increases, the TE point shifts toward larger tumor burdens and a slightly less effective immune response. Correspondingly, the saddle point (SP), which defines the position of the separatrix, moves to the right, enlarging the basin of attraction associated with tumor escape. Likewise, reaching the TC state requires a higher initial number of effector immune cells. These results suggest that even modest increases in tumor proliferation can markedly reduce the likelihood of immune-mediated tumor control, underscoring the potential clinical relevance of interventions—such as mild HT—that transiently suppress tumor growth and thereby shift the balance toward favorable immune dominance.

***Increased rate of immune cell recruitment \(\rho\). Mild HT exerts a range of immunomodulatory effects that can significantly enhance antitumor immunity. One of the primary mechanisms involves the upregulation of heat shock proteins (HSPs), which facilitate tumor antigen presentation and promote the activation of antigen-presenting cells (APCs), leading to enhanced priming of cytotoxic T lymphocytes10,11,12,18,46. HT also increases the expression of adhesion molecules and chemokines, thereby improving the infiltration and migration of immune effector cells, such as \(\text {CD8}^+\)T cells and natural killer (NK) cells, into the tumor microenvironment15,16. In addition, it modulates the tumor stroma by reducing the presence and suppressive activity of regulatory T cells (Tregs), myeloid-derived suppressor cells (MDSCs), and M2-type tumor-associated macrophages (TAMs), while simultaneously promoting a proinflammatory immune milieu10. Hyperthermia-induced vascular remodeling can also improve tumor perfusion, facilitating immune cell access and enhancing drug delivery47. Furthermore, heat stress can induce immunogenic cell death (ICD), releasing damage-associated molecular patterns (DAMPs) that stimulate robust immune responses17. Collectively, these effects synergize with immunotherapies by reversing immunosuppression and potentiating the functional activity of antitumor immune cells13. The relationship between recruitment rate and HT can be approximated by the leukocyte migration rates at different temperatures, which are reported in the work of Ahl et al.35 for neutrophils and CX3CR1 positive cells. In this work, a linear relationship is observed between the migration rate of immune cells and temperature during heating periods, which leads us to explore \(\rho\) values which are proportional to its original value at room temperature.

Figure 4b illustrates the steady-state behavior of the system under variation of the parameter \(\rho\), which represents the rate at which tumor cells recruit effector immune cells. When \(\rho\) is increased from 50 to 150% of its baseline value, a clear shift in the fixed point structure is observed. The TC point becomes reachable from a wider set of initial conditions, while the TE point recedes, reflecting improved immune surveillance. Notably, the saddle point that determines the position of the separatrix moves to the left, reducing the basin of attraction associated with tumor escape. These results indicate that enhanced immune recruitment, as might occur under immunostimulatory HT, significantly improves the likelihood of tumor control. As \(\rho\) increases, both the TC and TE fixed points shift slightly, but the overall change is modest as shown in Fig. 4b. What changes substantially is the size of the attraction basins—i.e., the region of initial conditions leading to each outcome—driven by the displacement of the saddle point, which alters the position of the separatrix. These findings suggest that therapies capable of enhancing immune cell recruitment—such as localized HT—may broaden the range of initial clinical conditions under which tumor control can be achieved, leading to a more favorable and robust prognosis for the patient.

It is important to note that for certain values of \(\rho\), not all steady-state solutions are present. When \(\rho\) decreases or \(\alpha\) increases, the clinically relevant transition is the emergence of the TE point. Prior to this bifurcation, only the trivial and TC solutions exist; beyond certain critical values of \(\rho\) and \(\alpha\), the TE point appears, marking the onset of tumor escape. For instance, when \(\rho\) is increased to 130% or 150% of its baseline value, only the TC fixed point remains, reflecting robust immune recruitment that prevents tumor escape. In contrast, when \(\rho\) is reduced to 50% of its original value, the system no longer permits tumor control, and all trajectories lead to TE. This phenomenon, in which a pair of fixed points (typically one stable and one saddle) merge and annihilate as a parameter varies, is known as a saddle-node bifurcation. These bifurcation-driven transitions in system dynamics will be analyzed in detail in the following section.

From a clinical standpoint, the existence of such bifurcations suggests the presence of threshold-like behaviors in tumor-immune interactions, where small variations in immune efficacy or tumor proliferation can trigger a sudden loss of tumor control. Identifying and maintaining patient-specific parameters above these critical thresholds—through immunostimulatory HT—could therefore be essential to prevent irreversible tumor escape.

Fig. 4
figure 4

Steady-state solutions of the model. Fixed points as intersections of \(y_1(x)\) and \(y_2(x)\) for increasing values of tumor proliferation rate \(\alpha\) in (a), and immune recruitment rate \(\rho\) in (b). Panels on the right illustrate the displacement of the fixed points: tumor escape (TE), saddle point (SP), and tumor control (TC). In panels (a,b), only one parameter was varied at a time—\(\alpha\) in (a) and \(\rho\) in (b)—while all others were kept constant. \(\alpha =100\%=1.636\) and \(\rho =100\%=1.131\).

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Bifurcation analysis under HT

A transcritical bifurcation is a local bifurcation in which the stability of two fixed points is exchanged as a parameter crosses a critical value, typically involving the trivial and a non-trivial equilibrium intersecting and exchanging stability. Transcritical bifurcations are not the focus of our analysis, as they describe transitions from the trivial steady state (\(y=0\) and \(x=\sigma /\delta\)), to a regime in which the immune system controls a small malignancy (tumor control). In our parameter regime, the trivial point remains fixed and always exists; however, it acts as a saddle that organizes the system’s dynamics toward the TC state once it becomes available, depending primarily on the values of \(\alpha\) and \(\rho\).

Rather than transcritical bifurcations, our focus is on the saddle-node bifurcation, where both the saddle point and the tumor escape equilibrium emerge. This bifurcation plays a crucial role in shaping the patient’s outcome by defining the boundary between immune control and immune evasion. Identifying the ranges of \(\alpha\) and \(\rho\) values at which the TE state appears or disappears is therefore essential for the design of cancer therapies aimed at limiting tumor growth and enhancing the patient’s chances of recovery.

Figure 5 displays the critical saddle-node bifurcation surface obtained from the tangency condition between the functions \(y_1(x)\) and \(y_2(x)\), which defines the emergence of the tumor escape state, as described in the methods section. The combined information provided in panels (a) to (c) characterizes the separatrix geometry in parameter space, delineating the boundary between effective immune control and uncontrolled tumor progression.

***Multiplicity of bifurcation conditions. A noteworthy feature common to all panels is the presence of parameter regions where multiple bifurcation solutions exist. Specifically, for small values of \(\rho\) (approximately \(\rho < 0.71\)), two distinct values of \(\alpha\) yield tangency between the straight line \(y_1(x)\) and the nonlinear curve \(y_2(x)\) due to the change in concavity of \(y_2(x)\). These two values of \(\alpha\) produce distinct pairs (x, y) at the bifurcation, as observed in Fig. 5c. Similarly, for relatively small values of \(\alpha \lesssim 1.8\) and some isolated higher values—corresponding to steep slopes of \(y_1(x)\)—there exist two values of \(\rho\) that lead to tangency with different \(y_2(x)\) curves. Consequently, certain low values of \(\alpha\) correspond to two separate saddle-node bifurcations at different (x, y) coordinates. These multiplicities follow the trends described next for the individual panels and reflect the complex geometry of the bifurcation surface in parameter space.

Panel (a): Tumor burden y at the saddle-node bifurcation as a function of \(\alpha\) and \(\rho\). The plots in panel (a) show how the tumor cell population y behaves at the point of saddle-node bifurcation, as a function of tumor proliferation rate \(\alpha\) and immune recruitment rate \(\rho\). In both cases, the value of y at bifurcation increases with the parameter, although with different curvature. As \(\alpha\) increases, the critical tumor burden at which immune escape becomes dynamically possible also rises. This suggests that while faster-growing tumors reach this threshold more quickly in time, the system requires a larger tumor mass to trigger the bifurcation and lose immune control. Similarly, as \(\rho\) increases, the bifurcation occurs at higher tumor burdens, indicating that with stronger immune recruitment, the tumor must grow larger to overcome immune control. As a result, the saddle-node bifurcation—marking the loss of immune dominance—occurs at higher values of y. Importantly, these bifurcations occur within a relatively narrow range of tumor sizes (\(y \approx 260\)–460), underscoring the system’s sensitivity and the importance of precise parameter control.

Panel (b): Effector cell population x at the saddle-node bifurcation as a function of \(\alpha\) and \(\rho\). The plots in panel (b) show how the number of effector immune cells x behaves at the saddle-node bifurcation, as a function of tumor proliferation rate \(\alpha\) and immune recruitment rate \(\rho\). As \(\alpha\) increases, x rises approximately linearly, indicating that more rapidly growing tumors require a proportionally greater immune response to delay or prevent escape. In contrast, the dependence of x on \(\rho\) is nonlinear and increasing for both branches of the bifurcation surface: as immune recruitment improves, the tumor is able to grow larger before triggering the bifurcation, and this shift is accompanied by a higher required number of effector cells. This result reflects the dynamic coupling between tumor-induced recruitment and effector expansion—suggesting that while enhanced immune responsiveness can delay immune escape, it also raises the threshold of immune effort needed to preserve tumor control.

Panel (c): Bifurcation surface in the \((\alpha ,\rho )\) parameter space. This panel shows the values of the immune recruitment rate \(\rho\) at which a saddle-node bifurcation occurs, plotted against the tumor proliferation rate \(\alpha\). The curve consists of two branches, each corresponding to distinct tangency conditions between the functions \(y_1(x)\) and \(y_2(x)\) in regions where multiple bifurcations are possible. The upper branch defines the boundary at which the system transitions from having a single steady-state solution (tumor control, TC) to acquiring two additional fixed points: a saddle point (SP) and a tumor escape state (TE). In contrast, the lower branch—visible only for small values of \(\rho\) (approximately \(\rho < 0.71\))—marks the reverse transition: the loss of the tumor control (TC) and saddle states, leaving only the tumor escape solution. These two branches thus enclose a region in \((\alpha ,\rho )\) space where three steady states coexist, corresponding to a bistable regime. Beyond this region, the system reverts to monostability. The existence of this bistable zone underscores the nonlinear sensitivity of the tumor-immune dynamics and highlights the importance of precise modulation of both tumor growth and immune recruitment parameters in therapy design. Below the lower branch, the system exhibits only a single equilibrium point corresponding to tumor escape. In this regime, the TC and SP solutions no longer exist, indicating that the immune system cannot maintain control regardless of the immune recruitment rate \(\rho\).

The complementary view of panels (a), (b) and (c) highlights how the number of effector cells required for bifurcation changes with \(\alpha\) and \(\rho\), and helps identify clinically relevant parameter regimes where immune escape can be prevented or reversed by therapeutic modulation. Together, these trends suggest that heat-induced modulation of \(\alpha\) and \(\rho\) can shift the critical conditions for immune failure, potentially enabling favorable therapeutic reprogramming of the system’s state.

Fig. 5
figure 5

Critical saddle-node bifurcation surfaces for tumor and effector cell populations. Panel (a) shows the normalized tumor cell population y at which the saddle-node bifurcation occurs, as a function of the tumor proliferation rate \(\alpha\) and the immune recruitment rate \(\rho\). The curve defines the critical tumor burden at which a new pair of fixed points—TC and TE—emerge. The three views provide complementary perspectives of the bifurcation surface geometry in \((\alpha , \rho , T)\) space. Panel (b) displays the corresponding normalized effector cell population x at bifurcation, delineating the minimum immune response required to reach this critical transition under varying tumor and immune parameters. Panel (c) shows curves representing codimension-one bifurcations in the two-dimensional parameter space \((\alpha ,\rho )\), where changes in either parameter lead to qualitatively distinct dynamic regimes. Between the two branches, three equilibria coexist: two stable (TC and TE) separated by one unstable saddle, indicating a bistable region where the outcome depends on initial conditions. To the right of the second (lower) branch, only the tumor escape equilibrium remains stable. This structure illustrates how small thermal-induced variations in \(\alpha\) and \(\rho\) can shift the system between tumor control, bistability, and escape regimes.

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If the thermal effects of HT—namely, the transient reduction of \(\alpha\) and increase of \(\rho\)—last only a short time compared to the system’s characteristic evolution timescale (approximately 200 days), these changes may act as perturbations to the effective initial conditions. In such scenarios, immune stimulation through enhanced effector cell recruitment could shift the patient’s state rightward in the phase diagram, increasing the likelihood of reaching the TC basin of attraction. A single or repeated hyperthermia treatment may thus allow the patient to cross the separatrix into a favorable outcome. Alternatively, if heat also kills tumor cells or slows their proliferation during the early stages of progression, the effective tumor cell count at treatment onset decreases, shifting the initial condition downward in the phase plane. In either case, the combined transient effects of reducing \(\alpha\) and increasing \(\rho\) may enable therapeutic transitions across the separatrix. Therefore, if patient-specific parameter values are known and initial tumor and effector cell counts can be assessed before and after HT, it becomes feasible to design effective personalized combined therapies involving immunotherapy and hyperthermia.

It is important to note that all effects predicted by our bifurcation and phase space analyses are limited to the specific values of the model parameters that were held fixed throughout the study. These parameters were originally estimated in the seminal work by Kuznetsov et al.27, based on the experimental data reported by Siu et al., which involved H-2 chimeric mice bearing BCL1 murine lymphoma38. Despite this limitation, the same methodological framework can be readily applied to other datasets—whether animal or clinical derived—once such data become available. Currently, there is a critical lack of systematic clinical datasets reporting immune cell counts, tumor burden, and signaling molecules as functions of both time and hyperthermia treatment conditions. The acquisition and integration of such data would enable the application of this modeling approach to the rational design of hyperthermia-based immunomodulatory cancer therapies. In fact, overcoming this data gap represents the main outlook and challenge posed by the present study. In the meantime, and in the absence of suitable experimental datasets (in vitro, in vivo, or clinical), we are developing agent-based simulations that incorporate biologically realistic variables and parameters. These synthetic data platforms will allow us in the near future to extend, explore, and validate the bifurcation-based analysis under a broader spectrum of plausible tumor-immune scenarios.

The novelty of this work lies in providing a quantitative and interpretative framework to explore how mild, non-cytotoxic hyperthermia influences tumor-immune population dynamics. In contrast to previous formulations32,33,34, which describe thermal effects through more complex, indirect, or predominantly cytotoxic mechanisms, our model directly links temperature to tumor proliferation and immune recruitment—based on experimental evidence showing approximately linear temperature dependence of these parameters within the mild hyperthermia range—within a streamlined population-level structure. By performing a bifurcation analysis on these two temperature-sensitive parameters, the model reveals how thermal modulation can alter stability regimes and favor immune-mediated tumor control. Its simplicity, inherited from the classical Kuznetsov model, facilitates accessibility and potential translational relevance while retaining key nonlinear tumor-immune interactions. This makes it a useful reference for interpreting experimental findings, benchmarking future modeling efforts, and guiding the design of combination strategies that integrate hyperthermia with immunotherapy.

In the proposed model, clinical personalization can be achieved through two main axes of parameterization. The first involves immune-related biomarkers such as \(\text {CD8}^+\) T cell counts, cytokine profiles, or tumor immune infiltration indices, which can be used to calibrate the immune recruitment rate (\(\rho\)) and represent patient-specific immune competence. Hyperthermia has been shown to modulate the tumor microenvironment by increasing IL-6 levels, heat shock proteins (HSPs), and lymphocyte infiltration, thereby enhancing immune activity23,53. The second axis focuses on tumor-specific parameters such as the intrinsic proliferation rate (\(\alpha\)), which can be estimated from proliferation biomarkers like the Ki-67 index54 or from imaging-based tumor growth rate measurements that have demonstrated prognostic value in predicting therapy response and survival55. These biomarkers and indices, together with each patient’s disease progression conditions, are key determinants of malignancy evolution. Therefore, personalized treatment design and outcome prediction require dynamic models in which these individualized data serve as quantitative inputs for parameter calibration.

Conclusions

This work presents a mathematical analysis of how moderate hyperthermia modulates tumor-immune dynamics by affecting two key biological processes: tumor proliferation and immune cell recruitment. Extending the Kuznetsov model, we performed a detailed bifurcation analysis and phase space characterization, identifying how variations in the parameters \(\alpha\) (tumor growth rate) and \(\rho\) (immune recruitment rate) shape the qualitative behavior of the system. In particular, we show that saddle-node bifurcations delineate the transition between tumor control and immune escape, with the position of the separatrix highly sensitive to thermal modulation of these parameters.

Our results demonstrate that moderate changes in \(\alpha\) or \(\rho\) can restructure the basins of attraction in phase space, enlarging the domain associated with tumor control. This highlights how external perturbations, such as HT, can serve as dynamic modulators capable of altering the stability structure of the system. These findings provide a mechanistic understanding of how thermal interventions influence nonlinear tumor-immune dynamics, offering a mathematical framework for studying treatment-induced transitions in cancer-immune ecosystems. Given the current lack of comprehensive experimental datasets reporting tumor and immune cell dynamics under HT, it becomes essential to validate and refine these predictions through agent-based models that incorporate spatial heterogeneity, stochastic effects, and mechanistic cell-cell interactions. These insights support the design of personalized combination therapies, where HT is used to shift the patient’s effective state toward immune-mediated tumor elimination. The present study is formulated generically and can be applied to alternative parameter sets calibrated to other experimental systems. Future work should extend this approach to include transient thermal effects, stochasticity, spatial dynamics, and data-driven parameter identification from preclinical or clinical studies.