Immunological experiments during the last two decades have answered many important questions related to the causal relationship between chronic inflammation and carcinogenesis. The presence of inflammatory cells in the cancer milieu raises the question of the tumor progression despite a likely immune system reaction to tumor antigens. This aspect is particularly important since untreated tumors grow according to non-linear, macroscopic, laws as the Gompertz law (GL) Gompertz (1); Norton (2); Vaghi et al. (3) or the logistic one (LL) Verhulst (4); Vaghi et al. (3). Therefore those growth patterns emerge, at a larger level of magnification, from many microscopic biological factors, which turn out to be summarized by simple mathematical descriptions.

Since prolonged inflammation is a hallmark of cancer Hiam-Galvez et al. (5), initiating tumor genesis or supporting tumor growth, and the global immune response is significantly altered during tumor progression, immunotherapy is becoming a valid option in cancer treatment. However, the immune response can be detrimental rather than helpful [see, for example, Lin et al. (6)]: individual auto-antibodies play an antagonist role in cancer, but the agonist auto-antibody in some cancer patients turned out to be deleterious and harmful.

Some preclinical and clinical evidence confirm the synergistic action of radiotherapy (RT) and immunotherapy against the tumor cells Zhao and Shao (7). Although The intrinsic sensitivity to radiation is patient-specific Puglisi et al. (8); Puglisi et al. (9) and may depend on different factors, RT is able to ablate cancer cells not only by directly induced necrosis or apoptosis but also by triggering an immune response that actively recruits immune cells within the tumor microenvironment. For example, RT promotes the release of tumor-associated antigens, which, once processed by antigen-presenting cells (APCs), prime CD8⁺ and CD4⁺ T cells in the draining lymph nodes. These lymphocytes attack both primary tumor and metastatic sites, posing the biological basis of the in situ vaccination driving the so-called abscopal effect Ngwa et al. (10); Mole (11); Demaria et al. (12): RT induces a systemic behavior that can activate the immune response against metastasis, i.e., in locations that are far from the RT-treated primary tumor.

The involvement of the immune system has been demonstrated in different experimental models such as melanoma Twyman-Saint Victor et al. (13), colorectal Dovedi et al. (14) and breast cancers Demaria et al. (15), but the clinical presentation where considered anecdotal or at least rare.

Its rarity in clinical practice is likely due to the simultaneous engagement of immune escape mechanisms, such as the recruitment of regulatory CD4⁺ (Treg) and myeloid-derived suppressor cells counterbalancing the anti-tumor CD8⁺ T cell-mediated effects, and the tumor release of hypoxia-inducible factors with pro-survival activity Ji et al. (16).

More generally, susceptibility to the abscopal effect has been associated with several biological factors, such as tumor size or oxygen levels in tumor tissues. Indeed, the presence of hypoxic regions results in both increased resistance to the lethal effect of radiation mediated by reactive oxygen species (ROS) production and an immune suppressive tumor landscape ruled by Treg-recruiting chemokines and impaired APC function McNamee et al. (17); Castorina et al. (18)

Mathematical modeling approaches have become increasingly abundant in describing immunotherapy and its synergy with RT Dewan et al. (19) Agur and Vuk-PavlovićCheck that all equations and special characters are displayed correctly. (20) Walker and Enderling (21); Ng and Dai (22); Serre et al. (23) Gong et al. (24) Marconi et al. (25); Vanpouille-Box et al. (26); Chakwizira et al. (27); Kosinsky et al. (28); Liu et al. (29); Valentinuzzi et al. (30); Friedrich et al. (31); Malinzi et al. (32); Bekker et al. (33). Indeed, the complexity of cancer provides challenges and opportunities for new developments, and mathematical formulations contribute by helping to elucidate mechanisms and by quantitative predictions that can be validated experimentally Agur and Vuk-Pavlović (20); Altrock et al. (34); Brady and Enderling (35) .

A large part of mathematical models on tumor growth and therapies are based on sets of coupled differential equations. The number of equations increases together with the details of the biological description and this implies a large number of parameters and initial conditions to be specified. The detailed analyses are often so complex to require surrogate models for a reliable determination of the parameters Browning and Simpson (36).

On the other hand, more economical models introduce a macroscopic evolution of tumor growth and therapy with fewer parameters and a coarse-grain dynamical evolution Norton (2); Wheldon (37); Vaghi et al. (3); Guiot et al. (38); Castorina et al. (39); Castorina et al. (40).

This is the key point to obtain an effective quantitative control of the tumor progression. Indeed, microscopic models, which have the advantage of a deep understanding of the biological dynamics, of the physiologically-based pharmacokinetic [see for example Maaß et al. (41)] and of the possible translation to different populations/diseases, require many parameters and, although some of them can be determined by previous analyses, the parametric error propagation will produce a large band of fluctuation in the prediction of the quantitative evolution of the disease. Therefore we prefer to apply macroscopic growth laws of the sigmoid family with two parameters. Our choice of the GL is due to the result that untreated tumor growth has been better described by it (see Vaghi et al. (3) for a recent study). Moreover, in a transplantable rat tumor, it was shown that control and regrowth curves after radiotherapy could be fitted by the same Gompertzian law, provided adjustments for the initial lag and the estimated number of clonogens immediately after irradiation were performed [Jung et al. (42)]. Gompertzian growth has been assumed to describe human tumor repopulation during fractional radiotherapy also in Hansen et al. (43) and by O’Donoghue (44).

The main motivation of the proposed approach is, in our opinion, its complementary role in the clinical evaluation of disease progression, often based on macroscopic variables, and the better parameter identification, thus increasing model verifiability Braakman et al. (45).

Finally, this method offers a clear description of the complex interplay between radiotherapy, immunotherapy, and tumor progression, providing insights for advancing cancer treatment strategies that harness the abscopal effect.

The paper is structured as follows: the mathematical model, based on the GL and on the definition of the effective GL parameters, is recalled in the next section (Appendices A, B and C contain the corresponding calculations). Different macroscopic growth laws (as the LL) can be applied, without changing the underlying method. The emerging phenomenological approach, based on the suitable redefinition of the two GL parameters to describe the data, is reported in Section 3. The final sections are devoted to discussion and to the possible clinical use of the phenomenological model.

The proposed method is based on the mathematical model reported in detail in Appendices A, B and C in the Supplementary Materials. In what follows, the assumptions and some exact results are reported. Then, the phenomenological model is discussed.

According to the mathematical approach in the previous section a phenomenological, simplified, method of analysis of the experimental data emerges. Indeed, a large part of the therapy and immune response effects can be assimilated to a redefinition of the GL parameters, with time and treatment dependence, which turns out to be detailed enough to compare with data. This phenomenological, simpler, procedure facilitates the validation of the model [Braakman et al. (45)] with respect to more complex analytical approaches, as discussed later.

Moreover, the function I(t),W(t),Y (t) in the previous differential equations are largely unknown, therefore the data fits of the effective GL parameters (see the general formulas in Mathematical Formalism) give model-independent information about the role of IT and RT.

The phenomenological approach, based on the previous mathematical formulation, consists in fitting the specific rate data by GL with effective parameters. Let us recall that the specific rate is much more reliable than the volume tumor variation, in determining its progression and the phase of growth or decrease.

We are aware that the coarse-grain proposed approach misses the detailed dynamics and can be considered an oversimplified description since the underlying pathways, Ng and Dai (22); Serre et al. (23); Marconi et al. (25); Liu et al. (29); Valentinuzzi et al. (30); Friedrich et al. (31); Bekker et al. (33) are summarized by the macroscopic Equation 8. However, one has to recall that, independently of the microscopic conditions, a large part of untreated tumors follow the GL (see Vaghi et al. (3) for a recent review) and that with a small number of parameters, one gets quantitative clinical indications for personalized treatments. If, for example, a patient gets a small specific rate by RT on the primary tumor, then a small contribution of IT might be able to result in a complete recovery.

The abscopal effect has been described by a macroscopic coupling between RT and immune system response, where the initial progression of the metastasis follows the GL with untreated primary tumor parameters. This is a reasonable assumption although the in-situ conditions can produce different results.

According to the specific conditions recalled in the introduction, choosing the best dose fractionation and timing with respect to immunotherapy is difficult. Notoriously, the use of protracted RT schedules (standard fractionation or slight hypofractionation) is discouraged since radiosensitive lymphocytes are cleared out from tumor tissues at each fraction delivery, thus preventing their anti-tumor function, Filatenkov et al. (61). Conversely, large doses per fraction are effectively immunogenic, Muraro et al. (62). In particular, doses below 12 Gy are the most suitable for enhancing the anti-tumor immune response as over such a threshold there is the degradation of immunogenic cytosolic DNA by an exonuclease, Trex, whose expression, as evaluated in preclinical experiments, is cell line-dependent and increases with increasing radiation dose Dewan et al. (19); Vanpouille-Box et al. (26). On the other side, even very low doses per fraction (< 1 Gy) seem to activate macrophages against cancer cells and stimulate T-cell immunity Klug et al. (63). Doses over 12 Gy are involved in the damage of tumor vasculature by activation of acid sphingomyelinase and production of ceramides, which culminate in vessel obliteration with subsequent tumor regression for insufficient nutrient and oxygen supply Song et al. (64). Therefore, the entire dose range used in clinical practice may be useful to control tumors and all RT fractionations combined with immunotherapy deserve clinical investigations.

Recently, old RT techniques simultaneously combining very different doses within the tumor, namely spatially fractionated radiation therapy (SFRT), are gaining new interest because of the assumption that tumor tissues spanning a wide dose range may benefit from multiple immune activation mechanisms, which eventually could be further boosted by ICI administration Ferini et al. (65); Tubin et al. (66). Given the ability of new instrumental exams to “map” tumor areas with different metabolisms, there is the possibility of modulating the dose distribution according to the oxygenation patterns inside the tumor to maximize both direct and indirect (immune-mediated) lethal effects of radiation Ferini et al. (67); Ferini et al. (68). With the latter approach, complete responses have been documented earlier than with classic homogeneously-delivered stereotactic RT doses and before ICI administration, likely implying rapid immune intervention enhanced and maintained by the addition of IT Ferini et al. (69).

All of the above considerations require the modeling of tumor response to RT and IT to help predict the best combination strategy, also given the inadequacy of current radiobiological mathematical models to comprehensively explain the results deriving from this association Ferini et al. (69).

In the proposed mathematical and computational approach, the fractionization effects are taken into account by the functions D n f and W ¯ t n f , t 0 and a spatially non-homogeneous behavior can be easily implemented. However, a complete discussion requires a forthcoming devoted analysis.

A comprehensive scope of the combined impact of radiotherapy and immunotherapy is vital for clinical decision-making. Yet, numerous mathematical models in existing literature prove overly complex, characterized by an abundance of differential equations, parameters, and initial conditions, rendering their practical implementation quite challenging.

In this study, we use a macroscopic mathematical approach that does not rely on the underlying microscopic dynamics. Instead, we propose a simplified model of the tumor progression using a Gompertz law, which involves just two parameters. Moreover, utilizing numerical solvers of the equations provided in the appendices is a straightforward process.

Examining how control influences system dynamics under radiation and/or drug therapy sheds light on the disease’s temporal progression. This approach holds promise in assisting clinicians to make informed decisions by providing a clearer understanding of treatment outcomes, namely, assessing whether the therapy administered results in full recovery.

Besides its clinical relevance, our approach shows potential for additional experimental validation of the synergistic impact of the abscopal phenomenon on treatment outcomes.

We recognize the importance of conducting rigorous experimental investigations to solidify the theoretical basis of our approach. By performing targeted studies and gathering more empirical data, we aim to validate the approach in different conditions and the significance of the abscopal effect in influencing therapy outcomes. This experimental validation will provide a deeper understanding of the interplay between the administered treatment, tumor response, and the abscopal effect.

Moreover, our future research endeavors will focus on elucidating the mechanisms underlying the abscopal effect and quantifying its impact on treatment efficacy. By combining computational modeling with comprehensive experimental studies, we strive to enhance our understanding of this phenomenon and optimize therapeutic strategies accordingly. Ultimately, we aim to improve patient-oriented outcomes by harnessing the full potential of the abscopal effect in cancer therapy.

Publicly available datasets were analyzed in this study. This data can be found here: https://figshare.com/articles/dataset/autoantibody_discovery/10302227; data on AB93 and AB641, platform figshare.

PC: Conceptualization, Formal analysis, Methodology, Writing – original draft. FC: Conceptualization, Data curation, Investigation, Validation, Writing – review & editing. GF: Methodology, Validation, Writing – original draft. SF: Funding acquisition, Methodology, Resources, Validation, Writing – review & editing. EM: Data curation, Investigation, Software, Validation, Writing – review & editing. DG: Funding acquisition, Resources, Supervision, Writing – review & editing.