Identifying viable radiation dose ranges to balance competing objectives of tumor response and off-target toxicity

Identifying viable radiation dose ranges to balance competing objectives of tumor response and off-target toxicity | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Identifying viable radiation dose ranges to balance competing objectives of tumor response and off-target toxicity Daniel Glazar , Ryan Werthmann , Aileen Chen , View ORCID Profile Renee Brady-Nicholls doi: https://doi.org/10.1101/2025.10.13.682060 Daniel Glazar 1 Department of Integrated Mathematical Oncology, Moffitt Cancer Center & Research Institute , Tampa, FL Find this author on Google Scholar Find this author on PubMed Search for this author on this site Ryan Werthmann 2 High School Internship Program in Integrated Mathematical Oncology, Moffitt Cancer Center & Research Institute , Tampa, FL Find this author on Google Scholar Find this author on PubMed Search for this author on this site Aileen Chen 3 Department of Thoracic Radiation Oncology, Division of Radiation Oncology, MD Anderson Cancer Center , Houston, TX Find this author on Google Scholar Find this author on PubMed Search for this author on this site Renee Brady-Nicholls 1 Department of Integrated Mathematical Oncology, Moffitt Cancer Center & Research Institute , Tampa, FL Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Renee Brady-Nicholls For correspondence: Renee.Brady{at}moffitt.org Abstract Full Text Info/History Metrics Preview PDF Abstract Ionizing radiation is an effective localized therapeutic strategy that is used to treat approximately 50% of all cancer patients. However, off-target effects in the radiation field vicinity may induce toxicities that exacerbate patient symptoms, leading to diminished quality of life. In patients receiving radiotherapy (RT)—the standard of care for head and neck cancer (HNC)—a major toxicity of particular concern in feeding tube dependence. This is often due to the proximity of the tumor to vital organs such as the esophagus and larynx. Treatment in these areas can cause adverse events, such as difficulty swallowing and dry mouth, increasing the need for a feeding tube. Such symptoms can be measured using patient-reported outcomes (PROs), which provide a measure of a patient’s health, symptoms, and overall wellbeing, as perceived and reported by the patients themselves. To capture both on-target tumor response and off-target toxicities of RT in HNC, we developed a mathematical model to inform acceptable radiation doses that maximize tumor response while minimizing off-target toxicities. The classical linear-quadratic dose-response model was employed to describe tumor response to RT with exponential growth. To model off-target toxicities, we introduce a novel concept of radiation exposure analogous to drug exposure. We then employed an inhomogeneous continuous-time Markov chain model with radiation exposure as a time-varying covariate to describe time-to-feeding tube dependence. We then define two thresholds to be selected by the clinician in consultation with the patient to derive minimum efficacious dose (MED) and maximum tolerable dose (MTD). We observe that RT is either viable (MED≤MTD) or unviable (MED>MTD). To enhance the model’s applicability, we simulated administration of radiosensitizing agents and provision of symptom management therapy by altering model parameters. Overall, this model offers adaptable, data-informed treatment decisions by integrating both tumor control and quality of life considerations into a singular model. 1 Introduction Ionizing radiation is an effective localized therapeutic strategy in killing tumors, which is used to treat approximately 50% of all cancer patients. 1 Radiotherapy (RT) works by delivering high-energy beams concentrated on the tumor in a small, localized region in the body. Radiation causes DNA damage to tumor cells either directly or indirectly via the creation of reactive oxygen species (ROS), resulting in tumor cell death. However, off-target effects are common in the localized radiation field, causing normal healthy tissue damage and exacerbating patient symptoms, leading to diminished quality of life. In the case of head-and-neck cancer (HNC), for which RT serves as standard of care, tumor proximity to vital organs such as the esophagus and larynx may increase the risk of adverse events, such as difficulty swallowing, necessitating interventions such as insertion of a feeding tube. 2 Tumor response is measured using tumor growth inhibition (TGI) metrics, such as change in tumor size, time-to-nadir, and tumor growth rate. 3 , 4 However, to make clinical decisions, tumor response for HNC and other solid tumors is evaluated using discrete categories using the Response Evaluation Criteria In Solid Tumors (RECIST). 5 For example, for target lesions, partial response is defined as ≥30% reduction in sum of longest diameters (SLD) relative to baseline, whereas progressive disease is defined as ≥20% increase in SLD relative to nadir with ≥5mm absolute increase in SLD. Treatment-related adverse events are measured by clinician reporting using the Common Terminology Criteria for Adverse Events (CTCAE) developed by the US National Cancer Institute (NCI). However, these reports lack reliability 6 and underestimate patient symptoms. 7 - 9 Therefore, the US NCI has further developed a patient-reported outcomes (PRO) version of the CTCAE (PRO-CTCAE) to more directly measure patient symptoms. 10 , 11 Patients are prompted to complete a questionnaire to report symptom presence, frequency, severity, and/or interference with daily life on a Likert scale over a 7-day recall period. Other PRO instruments exist that measure general (e.g., FACT-G, 12 EORTC-QLQ-C30 13 ), as well as disease-specific (e.g., FACT-HN 14 , EORTC-QLQ-HN43 15 ) and treatment-specific (e.g., FACT-HN-RAD 16 ) symptoms. There has been a rich history for more than 30 years developing mathematical models to describe tumor response dynamics to RT from ordinal and partial differential equations to stochastic processes, agent-based models, and machine learning. 17 - 19 These mathematical models have been developed to explore a variety of biological mechanisms, such as damage and repair processes, 20 oxygen depletion and reoxygenation, 21 and cancer stem cell radioresistance. 22 Perhaps the most popular model has been the classical linear-quadratic (LQ) model, 23 , 24 which we will employ in this study. A big clinical question of interest remains how best to balance the trade-offs between competing objectives of tumor response and managing toxicity. Therefore, there is a need for integrative mathematical models that both quantify and balance these such competing objectives. Indeed, mathematical models form the backbone for the development of tumor control probability (TCP) 28 , 29 and normal tissue complication probability (NTCP) 30 , 31 models. However, current NTCP models do not fully leverage information that can be gathered from PROs. As a relatively new field of study, considerably less attention has been paid to the development and application of mathematical models to describe PRO dynamics. Nevertheless, a number of important developments have been made in correlating PRO dynamics to tumor response dynamics and clinical outcomes, 25 linking PRO dynamics with latent symptom constructs using dynamic factor models and Ornstein-Uhlenbeck processes, 26 and tumor response dynamics using inhomogeneous continuous-time Markov chain (ICTMC) models. 27 In this study, we will explore the trade-offs between tumor response and toxicity in HNC patients undergoing RT using in silico simulations. We model tumor response using the classical LQ model. For toxicity, we introduce a novel concept of radiation exposure borrowing from the field of pharmacokinetics. We then employ an ICTMC model with radiation exposure as a time-varying covariate. We then define target tumor response and toxicity outcomes and define minimum effective dose (MED) and maximum tolerable dose (MTD), respectively, and explore the viability of RT and effect of administration of a radiosensitizing agent and provision of symptom management therapy. 2 Materials and methods 2.1 Modeling tumor response to radiotherapy We modeled the tumor response to RT using the classical linear-quadratic (LQ) dose-response model 23 with exponential growth: where B ( t ) denotes the normalized tumor burden at time t [day], such that B 0 = B (0) = 1 ; k denote the tumor growth rate [day -1 ] ; α,β denote radiosensitivity parameters [Gy 1 , Gy -2 ] d ( t ) denotes bolus radiation dose [Gy] administered at time t and B − ( t ) B + ( t ) denote tumor burden immediately before and after administration of radiation dose at time t , respectively. The tumor survival fraction upon administration of radiation dose at time t is given by . Solving equations 1 and 2 explicitly then gives us: where B 0 = B (0) = 1 as above and and are the total cumulative and squared physical dose, respectively, at time t , with N t being the total number of RT doses up to time t . Finally, assuming constant dose per fraction d , we can derive explicit expressions for the summations: The standard of care for RT in HNC is about five to eight weeks, given five days/week. Here, we simulate RT at a constant dose d over 6 weeks for 30 fractions ( Figure 1A ). We take tumor burden at the end of treatment (day 42) B ( t = 42 ; d ) as our endpoint for tumor response ( Figure 1B ). We then define minimum efficacious dose (MED) to be the minimum radiation dose needed to achieve a certain tumor burden at this endpoint relative to baseline, which can be computed simply as the inverse of B ( t = 42 ; d ) ( Figure 1C ). Download figure Open in new tab Figure 1 Modeling tumor response to radiotherapy. A) Tumor response dynamics as a function of RT dose (color). Untreated tumor growth dynamics are shown in dashed curve. The dotted vertical line denotes end of therapy (42 days) at which tumor response is evaluated as an endpoint. B) Tumor response at end of RT (42 days) as a function of RT dose (Gy). The horizontal dotted line denotes negligible tumor response relative to tumor burden at treatment initiation ( B (42) = B (0)). C) Minimum effective dose (MED, Gy) as a function of the target tumor response endpoint. 2.2 Modeling radiation-induced toxicity 2.2.1 Radiation exposure Although RT uses high-energy rays to affect an acute tumor response, off-target biological effects accumulate over the course of treatment. Therefore, in order to model the effects of RT on PROs, we introduce a novel concept of radiation exposure analogous to drug exposure. Radiation exposure R ( t ) [Gy] is modeled using a one-compartment pharmacokinetic (PK) model with bolus dose d ( t ) [Gy] and a constant first-order elimination rate, λ [day] -1 ( Figure 2A ): Download figure Open in new tab Figure 2 Modeling radiation-induced toxicity. A) Radiation exposure dynamics as a function of RT dose (color). No treatment (R(t)=0) is shown as a dashed line. B) Markov chain model schematic. C) Probability of feeding tube-dependence dynamics as a function of RT dose (color). The dotted vertical line denotes end of therapy (42 days) at which toxicity is evaluated as an endpoint. D) Feeding tube-free survival as a function of RT dose (color). The dotted vertical line denotes end of therapy (42 days) at which toxicity is evaluated as an endpoint. E) Toxicity (feeding tube-free survival at end of RT (42 days)) as a function of RT dose (Gy). F) Maximum tolerable dose (MTD, Gy) as a function of the target toxicity endpoint (feeding tube-free survival at end of RT (42 days)). Solving equations 5 and 6 explicitly then gives us: for RT physical dose d i administration i at time t i with N t being the total number of RT doses up to time t . Finally, assuming constant dose per fraction d , we can simplify the above equation: 2.2.2 Markov chain model We model PRO dynamics by employing an n-state inhomogeneous continuous-time Markov chain (ICTMC) model ( Figure 2B ). Here, we take n = 2 with state 0 denoting low symptom (e.g., low difficulty swallowing) or symptom absence (e.g., no feeding tube) and state 1 denoting high symptom (e.g., high difficulty swallowing) or symptom presence (e.g., feeding tube dependence). The transition rate matrix Q ( t ) is modeled with radiation exposure R ( t ) as a time-varying covariate: where Q ij ( t ) denotes the transition rate [day -1 ] from state i to state j at time t, q ij denotes the baseline transition rate from state i to state j in the absence of RT ( R ( t ) =0), and γ ij denotes the strength of effect of radiation exposure [Gy -1 ] on the transition rate from state i to state j . The transition probability matrix P ( t ) is then defined as the solution to the following Kolmogorov forward equation: where P ij ( t ) denotes the probability of transitioning from state i to state j from time 0 to time t . Finally, the transient state probabilities p ( t ) ( Figure 2C ) are then defined as: where p i ( t ) denotes the probability of being in state i at time t , and p 0 = [ p 0,0 p 0,1 ] denotes the initial state probabilities at time 0. Here, we assume the absence of symptom at treatment initiation, so that p 0,0 = 1 p 0,1 = 0. As such, we define the transition from state 0 to state 1 as a clinical event and take the probability of event at the end of RT (day 42) as our toxicity endpoint. Taken together, Equations 8 – 11 form a dose-exposure-symptom response model for radiation-induced toxicity. 2.2.3 Cox survival model Note that this endpoint is independent of the transition from state 1 to state 0. As such, we can simplify the above Markov chain model by considering the Cox survival model defined by the hazard rate: Event-free survival S ( t ) ( Figure 2D ) is then defined as: Here, we solve the integral numerically by computing left Riemann sums with Δ t = ¼ day. As in the case of defining MED above, we define maximum tolerable dose (MTD) to be the maximum dose needed to achieve a certain probability of event at the end of therapy (day 42 Figure 2E ), which can be computed simply as the inverse of S ( t ; D ) for a constant dose D ( Figure 2F ). Table 1 shows all model parameters. Tumor growth rate k = 0.05 was chosen assuming a 14-day doubling time, which is within the range typically observed for HNC. 32 Radiosensitivity parameters α = 0.05 β = 0.005 were chosen from literature. 33 Finally, radiation exposure and toxicity model parameters were chosen by experimentation and visual inspection. View this table: View inline View popup Download powerpoint Table 1 Model parameters. 3 Results 3.1 ldentifying viable dose ranges There is an observable tradeoff between tumor response and off-target toxicity with respect to RT dose ( Figures 1B , 2E ). For example, a very low dose may result in negligible toxicity, but at the cost of marginal tumor response. The dose-response profile thus defines the minimum effective dose (MED Figure 1C blue surface in Figure 3A ). Alternatively, a very high dose may result in complete tumor response, but at the cost of high toxicity. The dose-toxicity profile thus defines the maximum tolerable dose (MTD Figure 2F red surface in Figure 3A ). Download figure Open in new tab Figure 3 Assessing viability of RT. A) Minimum effective dose (MED) and maximum tolerable dose (MTD) as functions of target tumor response and toxicity endpoints. The volume in the region MTD≥MED (red surface above blue surface) denotes the viable RT doses for meeting both target endpoints. B) Projection of panel A onto the plane of target tumor response versus toxicity endpoints. The black curve denotes the separatrix such that only a single RT dose is viable to meeting both target endpoints. Below (grey, MTDMED) this curve are the regions of target endpoints where RT is unviable and viable, respectively. C) Administering a radiosensitizer boosts the tumor response and shifts the separatrix (panel B) downwards, increasing the region of RT viability. D) Providing symptom management therapy alleviates radiation-induced toxicity and shifts the separatrix (panel B) to the left, increasing the region of RT viability. By combining these two relationships, we can define the range of viable RT doses for pre-specified tumor response and toxicity endpoints ( Figure 3A ). In this case, specifying a high (low) tumor response endpoint denotes optimistic (pessimistic) expectation/hope, whereas high (low) toxicity endpoint denotes risk prone (risk averse) attitudes towards treatment. For a pre-specified tumor response and toxicity endpoints, viable RT doses are defined as the region between MED≤MTD. If MED<<MTD, then there are many viable RT dose, in which case RT is robust. If MED=MTD, then there is precisely one viable RT dose, in which case RT may be risky to the patient as meeting the bare minimum endpoints for tumor response and toxicity. Finally, if MED>MTD, then RT is inviable, since there is no single RT dose that will satisfy both tumor response and toxicity endpoints. The intersection between dose-response and dose-toxicity surfaces (i.e., MED=MTD) can be projected onto the tumor response-toxicity plane, which defines a separatrix between viable and non-viable RT ( Figure 3B ). The area below the curve is the region of nonviable RT demonstrating the trade-off between tumor response and toxicity. The area above and including the curve is the region of viability. Points along the separatrix indicate a single RT dose to meet the bare minimum endpoints for tumor response and toxicity. As one moves further into the region of viability and away from the separatrix, the range of viable radiation doses increases at the cost of worse tumor response and toxicity endpoints. 3.2 Radiosensitizers and symptom management Clinicians may potentially administer a radiosensitizer to induce an improved tumor response to radiation. Here, we model administration of a radiosensitizer by increasing the radiosensitivity parameter α from 0.05 to 0.1, which increases the α/β ratio from 10 to 20. This has the effect of increasing tumor response and shifting the separatrix down, increasing the viable RT region ( Figure 3C ). Alternatively, clinicians can potentially provide symptom management therapy to decrease risk of feeding tube-dependence. Here, we model symptom management therapy by decreasing the baseline hazard rate q 01 from 0.002 to 0.001. This has the effect of decreasing the risk of feeding tube-dependence at end of therapy and shifting the separatrix to the left, increasing the viable RT region ( Figure 3D ). 4 Discussion Here, we developed a mathematical model exploring the trade-off between tumor response and toxicity of RT in patients with HNC. For tumor response to RT, we employed a classical LQ dose-response model 23 with exponential growth. To account for both acute and chronic RT toxicities, we introduced a novel concept of radiation exposure borrowing from the field of pharmacokinetics. We then described the dose-exposure-symptom response relationship using a Markov chain model with radiation exposure as a time-varying covariate. Using our model, we identified two dose thresholds, MED and MTD, that can be used together to determine the viability of RT in maximizing tumor response while minimizing toxicity. A big question remains as to the biological interpretation of such a concept. Delivery of localized ionizing radiation beams cascades into DNA damage (e.g., double strand breaks) either directly or indirectly via the creation of reactive oxygen species (ROS), leading to cell death. 34 However, such biological effects and molecules are usually confined to the local site of treatment, in contrast with pharmaceutical drugs, which circulate in a central blood compartment and extravasate into peripheral tissue. An alternative approach may be to consider a dose-normal tissue response-symptom response relationship by employing an LQ for normal healthy tissue. In this case, if we assume negligible impact of tumor on the surrounding normal tissue homeostasis, then we could set the initial condition at or near steady state. Following normal tissue complication by RT, we could model tissue repair mechanisms and return to homeostatic conditions. Finally, we could employ a Markov chain model with normal tissue deviation from homeostasis, rather than radiation exposure, as a time-varying covariate. In this study, we considered only a single dichotomous symptom, which can be interpreted according to context variously as symptom presence vs. absence, low vs. high symptom, feeding tube independence vs. dependence, etc. We could easily extend our model to consider multiple polytomous PROs. Of course, this will come at the cost of computational complexity and potential model non-identifiability. We can then simplify this model by then taking the upper and lower diagonal to be the same rates, respectively, simplifying to an M/M/1/K queue model for K = n states with time-varying arrival and departure rates for each PRO item. 35 In this study, we approximated the transient state distribution of the inhomogeneous Markov chain model using a stepwise homogeneous Markov chain model for which there is a known analytic solution. However, further model extensions may render this approximation computationally infeasible, especially when estimating model parameters from patient data. Alternative computational approaches exist, such as simulation, Runge-Kutta methods, 36 Inoue’s method, 37 and equation learning (e.g., SINDy-PI 38 ). Since our toxicity endpoint was risk of feeding tube-dependence at end of treatment, we simplified our Markov chain model into a Cox survival model. However, we could consider other toxicity endpoints, such as duration of feeding tube-dependence, in which case we could leverage the full Markov chain model to explore those endpoints. Additionally, because of the generalizability of the model, any time-to-event endpoint could be considered, such emergency department (ED) visits, hospitalization, progression-free survival, and overall survival. The toxicity model also only considered treatment-related symptoms. We could further extend the model to consider cancer-related symptoms by introducing the concept of tumor pressure, whereby tumor burden serves as a time-varying covariate for the transition rates. In turn, we could employ a tumor growth inhibition (TGI) model to describe these tumor burden dynamics. 39 , 40 Additionally, we could further consider comorbidities by including other relevant covariates (e.g., diabetes, depression, weight, blood pressure, immune function, blood cell counts) In the case of many PRO items, another potential avenue for further model development is dimensionality reduction via a dynamic factor model, whereby latent constructs (e.g., cancer-related, treatment-related, and comorbidity-related symptoms) are inferred to explain time-to-event outcomes. 26 By introducing latent constructs, we could directly model their dynamics using an Ornstein-Uhlenbeck process, 26 rather than indirectly from the Markov chain model for observed item responses via the dynamic factor model. In our study, radiation exposure and toxicity model parameters were chosen by experimentation and visual inspection. Future implementation of this model will require calibration to patient data (e.g., PROs, tumor burden, time-to-event). Frequentist (e.g., maximum likelihood estimation (MLE)) or Bayesian (e.g., sampling by Hamiltonian Monte Carlo (HMC) simulations 41 ) approaches could be used to estimate these parameters. In this study, we assumed that RT plans contain just one degree of freedom (i.e., physical dose) with constant radiation field and profile, just scaled by dose. However, we recognize that RT plans are more complex than just dose. Different RT plans may be drawn by the radiation oncologist according to the geometry of the tumor and its surrounding tissues to explore the trade-off between various toxicities. For example, a tumor in the larynx will force the radiation oncologist to consider off-target effects in the esophagus versus the trachea, giving two different toxicity profiles. We suggest that these toxicity profiles may be differentially tolerated by the patient. A potential future model development may include introducing radiation exposure compartments for each anatomical structure of interest, which will in turn produce differential effects on various symptoms, recapitulating toxicity-toxicity trade-offs. We envision that future developments of this model will aid the clinician in selecting a RT plan that will optimize the trade-offs between target tumor response and toxicity endpoints selected by the patient in consultation with their healthcare providers. The model may determine which RT plans are viable or unviable. In the case of unviable RT, the patient may be viable for either alternative therapies (e.g., chemo-therapy, immunotherapy) or (neo-)adjuvant therapies to prime the tumor for RT by increasing the tumor’s radiosensitivity by administering a radiosensitizing agent or to manage treatment-related toxicity via target-specific symptom management therapies. Furthermore, one could consider “adaptive therapy” if one symptom is predicted to become too intolerable at a certain point in time to spare normal tissue while trying to maintain as much tumor response as possible. Alternating different RT plans to alternate different off-target healthy tissues to spare as much normal tissue cumulatively as possible while trying to maintain as much tumor response as possible. 5 References 1. ↵ Baskar R , Lee KA , Yeo R , Yeoh KW . Cancer and radiation therapy: Current advances and future directions . lnt J Med Sci . 2012 9 ( 3 ). doi: 10.7150/ijms.3635 OpenUrl CrossRef PubMed 2. ↵ Van den Bosch L , van der Laan HP , van der Schaaf A , et al. Patient-Reported Toxicity and Quality-of-Life Profiles in Patients With Head and Neck Cancer Treated With Definitive Radiation Therapy or Chemoradiation . lnt J Radiat Oncol Biol Phys . 2021 111 ( 2 ). doi: 10.1016/j.ijrobp.2021.05.114 OpenUrl CrossRef PubMed 3. ↵ Bruno R , Mercier F , Claret L. Evaluation of tumor size response metrics to predict survival in oncology clinical trials . Clin Pharmacol Ther . 2014 95 ( 4 ). doi: 10.1038/clpt.2014.4 OpenUrl CrossRef PubMed 4. ↵ Bruno R , Bottino D , De Alwis DP , et al. Progress and opportunities to advance clinical cancer therapeutics using tumor dynamic models . Clinical Cancer Research . 2020 26 ( 8 ). doi: 10.1158/1078-0432.CCR-19-0287 OpenUrl Abstract / FREE Full Text 5. ↵ Eisenhauer EA , Therasse P , Bogaerts J , et al. New response evaluation criteria in solid tumours: Revised RECIST guideline (version 1.1) . Eur J Cancer . 2009 45 ( 2 ). doi: 10.1016/j.ejca.2008.10.026 OpenUrl CrossRef PubMed Web of Science 6. ↵ Atkinson TM , Li Y , Coffey CW , et al. Reliability of adverse symptom event reporting by clinicians . Quality of Life Research . 2012 21 ( 7 ). doi: 10.1007/s11136-011-0031-4 OpenUrl CrossRef PubMed Web of Science 7. ↵ Basch E. The Missing Voice of Patients in Drug-Safety Reporting . New England Journal of Medicine . 2010 362 ( 10 ). doi: 10.1056/nejmp0911494 OpenUrl CrossRef 8. Fromme EK , Eilers KM , Mori M , Hsieh YC , Beer TM . How accurate is clinician reporting of chemotherapy adverse effects? A comparison with patient-reported symptoms from the Quality-of-Life Questionnaire C30 . Journal of Clinical Oncology . 2004 22 ( 17 ). doi: 10.1200/JCO.2004.03.025 OpenUrl Abstract / FREE Full Text 9. ↵ Pakhomov S V. , Jacobsen SJ , Chute CG , Roger VL . Agreement between patient-reported symptoms and their documentation in the medical record . American Journal of Managed Care . 2008 14 ( 8 ). 10. ↵ Basch E , Reeve BB , Mitchell SA , et al. Development of the national cancer institute’s patient-reported outcomes version of the common terminology criteria for adverse events (PRO-CTCAE) . J Natl Cancer lnst . 2014 106 ( 9 ). doi: 10.1093/jnci/dju244 OpenUrl CrossRef PubMed 11. ↵ Dueck AC , Mendoza TR , Mitchell SA , et al. Validity and reliability of the us national cancer institute’s patient-reported outcomes version of the common terminology criteria for adverse events (PRO-CTCAE) . JAMA Oncol . 2015 1 ( 8 ). doi: 10.1001/jamaoncol.2015.2639 OpenUrl CrossRef PubMed 12. ↵ Cella DF , Tulsky DS , Gray G , et al. The Functional Assessment of Cancer Therapy scale: development and validation of the general measure . Journal of Clinical Oncology . 1993 11 ( 3 ): 570 – 579 . doi: 10.1200/JCO.1993.11.3.570 OpenUrl Abstract / FREE Full Text 13. ↵ Bjordal K , De Graeff A , Fayers PM , et al. A 12 country field study of the EORTC QLQ-C30 (version 3.0) and the head and neck cancer specific module (EORTC QLQ-H and N35) in head and neck patients . Eur J Cancer . 2000 36 ( 14 ). doi: 10.1016/S0959-8049(00)00186-6 OpenUrl CrossRef PubMed Web of Science 14. ↵ D’Antonio LL , Zimmerman GJ , Cella DF , Long SA . Quality of life and functional status measures in patients with head and neck cancer . Archives of Otolaryngology - Head and Neck Surgery . 1996 122 ( 5 ). doi: 10.1001/archotol.1996.01890170018005 OpenUrl CrossRef PubMed Web of Science 15. ↵ Singer S , Amdal CD , Hammerlid E , et al. International validation of the revised European Organisation for Research and Treatment of Cancer Head and Neck Cancer Module, the EORTC QLQ-HN43: Phase IV . Head Neck . 2019 41 ( 6 ): 1725 – 1737 . doi: 10.1002/hed.25609 OpenUrl CrossRef PubMed 16. ↵ Gharzai LA , Mierzwa ML , Peipert JD , et al. Monitoring Adverse Effects of Radiation Therapy in Patients with Head and Neck Cancer: The FACT-HN-RAD Patient-Reported Outcome Measure . JAMA Otolaryngol Head Neck Surg . 2023 149 ( 10 ). doi: 10.1001/jamaoto.2023.2177 OpenUrl CrossRef 17. ↵ Zheng D , Preuss K , Milano MT , et al. Mathematical modeling in radiotherapy for cancer: a comprehensive narrative review . Radiation Oncology . 2025 20 ( 1 ): 49 . doi: 10.1186/s13014-025-02626-7 OpenUrl CrossRef PubMed 18. Bekker RA , Kim S , Pilon-Thomas S , Enderling H. Mathematical modeling of radiotherapy and its impact on tumor interactions with the immune system . Neoplasia . 2022 28 : 100796 . doi: 10.1016/j.neo.2022.100796 OpenUrl CrossRef PubMed 19. ↵ Enderling H , Alfonso JCL , Moros E , Caudell JJ , Harrison LB . Integrating Mathematical Modeling into the Roadmap for Personalized Adaptive Radiation Therapy . Trends Cancer . 2019 5 ( 8 ). doi: 10.1016/j.trecan.2019.06.006 OpenUrl CrossRef PubMed 20. ↵ Zhao J , Wei X , Tian JP . Modeling of tumor radiotherapy with damage and repair processes . Eur Phys J Plus . 2022 137 ( 5 ). doi: 10.1140/epjp/s13360-022-02568-z OpenUrl CrossRef 21. ↵ Zhou S , Zheng D , Fan Q , et al. Minimum dose rate estimation for pulsed FLASH radiotherapy: A dimensional analysis . Med Phys . 2020 47 ( 7 ): 3243 – 3249 . doi: 10.1002/mp.14181 OpenUrl CrossRef PubMed 22. ↵ Lopez Alfonso JC , Jagiella N , Nunez L , Herrero MA , Drasdo D. Estimating Dose Painting Effects in Radiotherapy: A Mathematical Model . PLoS One . 2014 9 ( 2 ): e89380 . doi: 10.1371/journal.pone.0089380 OpenUrl CrossRef PubMed 23. ↵ Fowler JF . The linear-quadratic formula and progress in fractionated radiotherapy . British Journal of Radiology . 1989 62 ( 740 ). doi: 10.1259/0007-1285-62-740-679 OpenUrl Abstract / FREE Full Text 24. ↵ McMahon SJ . The linear quadratic model: Usage, interpretation and challenges . Phys Med Biol . 2019 64 ( 1 ). doi: 10.1088/1361-6560/aaf26a OpenUrl CrossRef PubMed 25. ↵ Bhatt AS , Schabath MB , Hoogland AI , Jim HSL , Brady-Nicholls R. Patient-Reported Outcomes as Interradiographic Predictors of Response in Non-Small Cell Lung Cancer . Clinical Cancer Research . 2023 29 ( 16 ): 3142 – 3150 . doi: 10.1158/1078-0432.CCR-23-0396 OpenUrl CrossRef PubMed 26. ↵ Tran TD , Lesaffre E , Verbeke G , Duyck J. Latent Ornstein-Uhlenbeck models for Bayesian analysis of multivariate longitudinal categorical responses . Biometrics . 2021 77 ( 2 ). doi: 10.1111/biom.13292 OpenUrl CrossRef 27. ↵ Glazar D , Jim H , Schabath M , Hoogland A , Brady-Nicholls R. Patient-reported outcomes to inform patient-specific tumor growth inhibition parameters . In: Population Approach Group Europe 33 . 2025 :Abstr 11446. Accessed October 11, 2025 . www.page-meeting.org/?abstract=11446 28. ↵ Deasy JO . Comments on the use of the Lyman-Kutcher-Burman model to describe tissue response to nonuniform irradiation (multiple letters) . lnt J Radiat Oncol Biol Phys . 2000 47 ( 5 ). doi: 10.1016/S0360-3016(00)00500-9 OpenUrl CrossRef PubMed 29. ↵ Niemierko A , Goitein M. Modeling of normal tissue response to radiation: The critical volume model . lnt J Radiat Oncol Biol Phys . 1993 25 ( 1 ). doi: 10.1016/0360-3016(93)90156-P OpenUrl CrossRef PubMed Web of Science 30. ↵ Eisbruch A , Ten Haken RK , Kim HM , Marsh LH , Ship JA . Dose, volume, and function relationships in parotid salivary glands following conformal and intensity-modulated irradiation of head and neck cancer . lnt J Radiat Oncol Biol Phys . 1999 45 ( 3 ). doi: 10.1016/S0360-3016(99)00247-3 OpenUrl CrossRef PubMed Web of Science 31. ↵ Marks LB , Yorke ED , Jackson A , et al. Use of Normal Tissue Complication Probability Models in the Clinic . lnt J Radiat Oncol Biol Phys . 2010 76 ( 3 SUPPL .). doi: 10.1016/j.ijrobp.2009.07.1754 OpenUrl CrossRef PubMed 32. ↵ Zumer B , Pohar Perme M , Jereb S , Strojan P. Impact of delays in radiotherapy of head and neck cancer on outcome . Radiation Oncology . 2020 15 ( 1 ): 202 . doi: 10.1186/s13014-020-01645-w OpenUrl CrossRef PubMed 33. ↵ van Leeuwen CM , Oei AL , Crezee J , et al. The alfa and beta of tumours: a review of parameters of the linear-quadratic model, derived from clinical radiotherapy studies . Radiation Oncology . 2018 13 ( 1 ): 96 . doi: 10.1186/s13014-018-1040-z OpenUrl CrossRef PubMed 34. ↵ Srinivas US , Tan BWQ , Vellayappan BA , Jeyasekharan AD . ROS and the DNA damage response in cancer . Redox Biol . 2019 25. doi: 10.1016/j.redox.2018.101084 OpenUrl CrossRef PubMed 35. ↵ Shortle JF , Thompson JM , Gross D , Harris CM . Fundamentals of Queueing Theory . Wiley 2018 . doi: 10.1002/9781119453765 OpenUrl CrossRef 36. ↵ Butcher JC . Numerical Methods for Ordinary Differential Equations . 2016 . doi: 10.1002/9781119121534 OpenUrl CrossRef 37. ↵ Inoue Y. A NEW APPROACH TO COMPUTING THE TRANSIENT-STATE PROBABILITIES IN TIME-INHOMOGENEOUS MARKOV CHAINS . Journal of the Operations Research Society of Japan . 2022 65 ( 1 ): 48 – 66 . doi: 10.15807/jorsj.65.48 OpenUrl CrossRef 38. ↵ Kaheman K , Kutz JN , Brunton SL . SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics . Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences . 2020 476 ( 2242 ). doi: 10.1098/rspa.2020.0279 OpenUrl CrossRef 39. ↵ Bernard A , Kimko H , Mital D , Poggesi I. Mathematical modeling of tumor growth and tumor growth inhibition in oncology drug development . Expert Opin Drug Metab Toxicol . 2012 8 ( 9 ). doi: 10.1517/17425255.2012.693480 OpenUrl CrossRef PubMed 40. ↵ Ribba B , Holford NH , Magni P , et al. A review of mixed-effects models of tumor growth and effects of anticancer drug treatment used in population analysis . CPT Pharmacometrics Syst Pharmacol . 2014 3 ( 5 ). doi: 10.1038/psp.2014.12 OpenUrl CrossRef PubMed 41. ↵ Betancourt MJ , Byrne S , Livingstone S , Girolami M. The Geometric Foundations of Hamiltonian Monte Carlo . Published online October 19, 2014. View the discussion thread. Back to top Previous Next Posted October 14, 2025. Download PDF Email Thank you for your interest in spreading the word about bioRxiv. NOTE: Your email address is requested solely to identify you as the sender of this article. Your Email * Your Name * Send To * Enter multiple addresses on separate lines or separate them with commas. You are going to email the following Identifying viable radiation dose ranges to balance competing objectives of tumor response and off-target toxicity Message Subject (Your Name) has forwarded a page to you from bioRxiv Message Body (Your Name) thought you would like to see this page from the bioRxiv website. Your Personal Message CAPTCHA This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. Share Identifying viable radiation dose ranges to balance competing objectives of tumor response and off-target toxicity Daniel Glazar , Ryan Werthmann , Aileen Chen , Renee Brady-Nicholls bioRxiv 2025.10.13.682060; doi: https://doi.org/10.1101/2025.10.13.682060 Share This Article: Copy Citation Tools Identifying viable radiation dose ranges to balance competing objectives of tumor response and off-target toxicity Daniel Glazar , Ryan Werthmann , Aileen Chen , Renee Brady-Nicholls bioRxiv 2025.10.13.682060; doi: https://doi.org/10.1101/2025.10.13.682060 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Cancer Biology Subject Areas All Articles Animal Behavior and Cognition (7643) Biochemistry (17717) Bioengineering (13910) Bioinformatics (42017) Biophysics (21480) Cancer Biology (18628) Cell Biology (25537) Clinical Trials (138) Developmental Biology (13392) Ecology (19935) Epidemiology (2067) Evolutionary Biology (24356) Genetics (15617) Genomics (22530) Immunology (17755) Microbiology (40438) Molecular Biology (17200) Neuroscience (88705) Paleontology (667) Pathology (2840) Pharmacology and Toxicology (4832) Physiology (7657) Plant Biology (15171) Scientific Communication and Education (2046) Synthetic Biology (4304) Systems Biology (9828) Zoology (2272)