Conditional Success of Adaptive Therapy: The Role of Treatment-Holiday Thresholds and Non-Existence of Optimal Strategies Revealed by Mathematical Modelling and Optimal Control

Conditional Success of Adaptive Therapy: The Role of Treatment-Holiday Thresholds and Non-Existence of Optimal Strategies Revealed by Mathematical Modelling and Optimal Control | 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 Conditional Success of Adaptive Therapy: The Role of Treatment-Holiday Thresholds and Non-Existence of Optimal Strategies Revealed by Mathematical Modelling and Optimal Control Lanfei Sun , View ORCID Profile Haifeng Zhang , Kai Kang , Xiaoxin Wang , Leyi Zhang , Yanan Cai , View ORCID Profile Changjing Zhuge , View ORCID Profile Lei Zhang doi: https://doi.org/10.1101/2025.02.13.638061 Lanfei Sun 1 School of Mathematics Statistics and Mechanics, Beijing University of Technology , Beijing 100124, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site Haifeng Zhang 2 School of Mathematical Sciences, Jiangsu University , Zhenjiang, 212013, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Haifeng Zhang Kai Kang 1 School of Mathematics Statistics and Mechanics, Beijing University of Technology , Beijing 100124, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site Xiaoxin Wang 1 School of Mathematics Statistics and Mechanics, Beijing University of Technology , Beijing 100124, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site Leyi Zhang 1 School of Mathematics Statistics and Mechanics, Beijing University of Technology , Beijing 100124, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site Yanan Cai 3 College of Science and Engineering, James Cook University , Queensland, 4814, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site Changjing Zhuge 1 School of Mathematics Statistics and Mechanics, Beijing University of Technology , Beijing 100124, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Changjing Zhuge For correspondence: zhuge{at}bjut.edu.cn zhanglei44{at}pumch.cn Lei Zhang 5 Department of Liver Surgery, Peking Union Medical College Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College , Beijing 100730, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Lei Zhang For correspondence: zhuge{at}bjut.edu.cn zhanglei44{at}pumch.cn Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract Adaptive therapy improves cancer treatment by controlling the competition between sensitive and resistant cells through treatment holidays. This study highlights the critical role of treatmentholiday thresholds in adaptive therapy for tumors composed of drug-sensitive and resistant cells. Using a Lotka-Volterra model, adaptive therapy outcomes are compared with maximum tolerated dose therapy and intermittent therapy outcomes, showing that adaptive therapy success depends critically on the threshold for pausing and resuming treatment and on competitive interactions between cell populations. Three comparison scenarios between adaptive therapy and other therapies emerge: uniform-decline where adaptive therapy underperforms regardless of threshold, conditional-improve where efficacy requires threshold optimization, and uniform-improve where adaptive therapy consistently outperforms alternatives. Tumor composition including initial burden and resistant cell proportion influences outcomes. Threshold adjustments enable adaptive therapy to suppress resistant subclones while preserving sensitive cells, extending progression-free survival. Crucially, this work establishes an optimal control problem for time-to-progression and mathematically proves that under biological constraints like neutral competition or low initial burden, the theoretically optimal strategy is unrealizable as it requires infinitely many treatment holidays, rendering it clinically impractical. These findings emphasize personalized treatment strategies for enhancing long-term therapeutic outcomes. Mathematics Subject Classification 92C50,92C42. 1. I ntroduction Cancer remains one of the leading causes of death worldwide [ 9 ], with its biological complexity and heterogeneity posing significant challenges to conventional therapies such as chemotherapy, radiotherapy, and even immunotherapy [ 34 , 39 , 56 ]. Therapeutic resistance, a central obstacle in cancer treatment, arises from the therapeutic selection pressure of pre-existing or induced resistant tumor cells [ 29 , 47 , 48 , 58 , 66 ]. This process drives the evolution of tumor tissue, ultimately leading to the dominance of primary or induced refractory subpopulations, and subsequent relapse and treatment failure [ 25 , 38 , 60 ]. The inherent heterogeneity and plasticity of tumor cells often render complete eradication exceedingly difficult, highlighting the necessity of strategies that address the entire tumor population and its evolutionary dynamics, rather than targeting specific subpopulations [ 26 , 35 , 39 ]. Recently, ecological theory has provided an innovative framework for understanding tumor progression, conceptualizing tumors as ecosystems where drug-sensitive and drug-resistant cells compete for resources [ 1 , 6 , 8 , 10 , 11 , 21 , 22 , 28 , 71 ]. Within this ecological framework, cancer cells under drug treatment activate regenerative or protective mechanisms to survive near-lethal stress, enabling them to enter a stem cell-like state and transmit resistance to their progeny [ 4 , 62 ]. This adaptation imposes costs, such as reduced proliferation rates, decreased environmental carrying capacity, and intensified interclonal competition [ 44 , 72 ]. The conventional maximum tolerated dose (MTD) therapy aggressively eliminates sensitive cell populations. However, this strategy inadvertently disrupts competitive equilibria, allowing resistant subclones to proliferate uncontrollably [ 3 , 15 , 22 , 33 , 67 ]. This unintended consequence of MTD therapy suggests the limitations of conventional eradication-focused paradigms and highlights the need for therapeutic strategies that preserve sensitive cells to maintain ecological suppression of resistant populations [ 19 , 21 , 22 , 69 ]. Adaptive therapy (AT) is a dynamic treatment strategy that employs a feedback-controlled approach, adjusting therapeutic intensity and scheduling in response to real-time tumor burden [ 15 , 16 , 22 ]: treatment is paused when tumor burden declines to a certain lower threshold and resumed upon regrowth to an upper threshold. Adaptive therapy can be classified into dose-skipping (AT-S) and dose-modulating (AT-M) regimens, depending on whether more than two dose rates are used during treatment phases. By retaining a reservoir of sensitive cells, this intermittent dosing modality suppresses resistant subclone expansion and prolongs progression-free survival, as evidenced by clinical successes in prostate cancer and other malignancies. Despite significant advancements [ 37 , 51 , 67 , 72 , 73 ], further research is needed to refine the optimization of treatment-holiday thresholds, a core parameter governing AT efficacy, as most existing studies focus on the efficiency of adaptive therapy under predefined thresholds. Several studies have explored the effects of varying thresholds on AT outcomes across different scenarios [ 30 , 37 , 51 , 53 , 57 , 64 ], providing valuable insights that underscore the importance of threshold adjustments and establishing a solid foundation for further investigation. However, a deeper understanding of the dynamic mechanisms driving context-dependent outcomes under heterogeneous physiological conditions remains essential. It requires a systematic analysis of the underlying dynamical mechanisms by which different thresholds give rise to distinct effects, enhancing conventional empirical threshold selection to achieve an optimal balance between tumor suppression and resistance mitigation. Moreover, it can further facilitate the development of personalized strategies tailored to different conditions. Consequently, quantitative modelling of threshold-driven tumor evolutionary dynamics is essential for advancing precision in adaptive therapy design. Therefore, this study leverages a mathematical model of tumor evolutionary dynamics to analyze and explain the effects of the thresholds on the outcomes of AT under various conditions of competitive interactions between sensitive and resistant subpopulations, elucidating the biophysical principles underlying threshold-mediated resistance control. The findings are expected to provide theoretical validation for personalized adaptive therapy protocols, facilitating a paradigm shift toward dynamically optimized cancer management. In summary, this study, as a state-of-the-art work, systematically investigates the impact of treatmentholiday thresholds on adaptive therapy outcomes using a Lotka–Volterra model that simulates the competitive dynamics between drug-sensitive and drug-resistant tumor cell populations. By quantifying the relationship between treatment-holiday threshold and employing time-to-progression (TTP) as an indicator of progression-free survival, the analysis demonstrates that AT efficacy critically depends on the competition between two subpopulations of cancer cells. Strong competition can allow for the indefinite delay of disease progression, while weak competition requires the benefits of AT. Furthermore, this work also reveals that AT is not uniformly superior to MTD, as its efficiency depends on tumor composition (initial conditions) and patients’ personal conditions (parameters). By comparing the TTP between AT and MTD, three distinct scenarios are identified:1) uniform-decline , where AT consistently underperforms MTD regardless of the threshold; 2) conditionalimprove , where the effectiveness of AT depends on the specific threshold selected; and 3) uniform-improve , where AT consistently outperforms MTD. These findings highlight the necessity of precise threshold tuning to optimize treatment outcomes. Collectively, these findings provide a theoretical foundation for refining empirical threshold selection and advancing personalized adaptive therapy protocols, thereby paving the way for dynamically optimized cancer management. 2. MODEL DESCRIPTION To investigate the effects of different treatment strategies, the tumor dynamics under therapeutic conditions are modelled by the widely used Lotka-Volterra (LV)-type equations [ 1 , 5 , 18 , 23 , 36 , 37 , 54 ] because the cancer cells behave more similarly to unicellular organisms than to normal human cells [ 10 , 13 , 17 , 43 ]. The cancer cells are further assumed to be classified into two types: drug-sensitive cells ( S ), which can be killed by drugs, and drug-resistant cells ( R ), which cannot be killed by drugs. These two cell types compete within the cancer microenvironment ( Figure 1A ), and therefore the LV model is described as the following differential equations (1) – (3) . where D ( t ) denotes the dose curve of therapeutic agents and S and R represent the burdens of the two types of cells respectively. Equation (1) describes the growth and death of sensitive cells, which means that in the absence of therapy, their proliferation is regulated by intrinsic rate r S , logistic competition and the death rate is d S . The drug-dependent suppression of sensitive cells is represented by the factor where γ is the killing strength of therapeutic agents on sensitive cells. Although this factor is multiplied with the proliferation terms, it is the integrated effective term taking both suppression of proliferation and increasin g death rate for simplicity. Equation (2) governs resistant cells, which obey logistical growth at rate r R and die at constant rate d R , unaffected by the drug. K is the carrying capacity of the cancer microenvironment, indicating the maximum number of cells that can be supported. Download figure Open in new tab FIGURE 1. Schematic diagram of cancer dynamics models and illustrations of different treatment strategies and their outcomes. (A) A schematic representation of the Lotka-Volterra competition between sensitive cancer cells ( S ) and resistant cancer cells ( R ). The interactions between these two cell types are influenced by the carrying capacity K and the competition coefficients α and β . Additionally, in the presence of therapeutic agents, sensitive cells are killed by the drugs. (B) Schematic illustration of four typical therapeutic strategies: the maximal tolerance dosage (MTD), intermittent therapy (IT), dose-skipping adaptive therapy (AT-S), and dose-modulating adaptive therapy (AT-M). (C) Schematic illustration of tumor burden and compositional evolution under distinct therapeutic strategies. Moreover, the model equations (1) – (2) incorporate asymmetric competition between drug-sensitive and drugresistant cells, characterized by the parameters α and β . Here, α quantifies the inhibitory effect of drug-resistant cells on the growth of sensitive cells, while β represents the impact of drug-sensitive cells on the growth of resistant cells. Specifically, a higher value of α implies that resistant cells more strongly reduce the growth potential of sensitive cells, possibly due to resource competition or indirect suppression via secreted factors (e.g., growth factors, cytokines). Similarly, a higher value of β indicates that sensitive cells exert a stronger influence on the growth of resistant cells, potentially through competition for space, nutrients, or other resources within the tumor microenvironment. These competitive interactions determine the dominance relationships between the cell populations. For example, in the absence of therapy, coexistence of the two populations is possible under weak competition (i.e., α < 1 and β 1 and β < 1), the sensitive cell population will eventually become extinct; conversely, if α 1, the resistant cell population will be eliminated [ 37 ]. In addition, the competition coefficients also significantly influence treatment outcomes, with a competitive advantage for sensitive cells being crucial for eradicating resistance [ 14 , 32 , 32 , 45 ]. Thus, it is necessary to take these competitive dynamics into account in order to design better strategies. Different therapeutic strategies are characterized by distinct profiles of the dosing function D ( t ). Two widely used therapeutic approaches and two regimens of adaptive therapy are considered in this work, including maximum tolerated dose, intermittent therapy (IT), dose-skipping adaptive therapy (AT-S) and dose-modulating adaptive therapy (AT-M) ( Figure 1B ). The conventional maximum tolerated dose approach seeks to maximize cancer cell eradication by administering the highest dose that a patient can tolerate based on acceptable toxicity levels, which can be defined by continuous administration as (4). As continuous high-intensity dosage leads to the development of therapeutic resistance through Darwinian selection pressures [ 15 ], as well as that high dosage of therapeutic agents causes severe toxicity, the intermittent therapy regimens were proposed, which strategically suspend drug administration during predefined recovery periods and emerged as a clinically utilized strategy for balancing therapeutic efficacy with toxicity management [ 36 ]. The IT strategies can be characterized as deterministic treatment cycles as (5). where T is the fixed period of one cycle of treatment, T D is the duration of drug administration and n = 1, 2, … represents the number of cycles. As described before, AT dynamically adjusts cancer treatment based on tumor response to prolong efficacy and manage resistance. Two primary approaches within adaptive therapy are dose-skipping and dose-modulating regimens. In dose-skipping adaptive therapy, treatment is administered at the maximum tolerated dose until the tumor burden shrinks to a predetermined threshold ( C TH0 ), for example, 50% of the baseline burden; therapy is then paused, allowing the tumor to regrow to a specific threshold ( C TH1 ), usually, 100% of the baseline, before resuming treatment. In dose-modulating adaptive therapy, treatment doses are adjusted at regular intervals based on tumor response: increasing the dose if the tumor grows and decreasing it if the tumor shrinks. This strategy seeks to tailor therapy dynamically to tumor behavior, potentially reducing toxicity and managing resistance more effectively. Both approaches exploit the competitive dynamics between treatment-sensitive and resistant cancer cells to enhance long-term treatment outcomes and have already succeeded in trials or experiments [ 15 , 46 , 50 , 67 , 73 ]. Based on the above description, the two AT regimens can be formulated as following equations (6) and (7) respectively. where N 0 = S 0 + R 0 is the initial tumor burden and N ′ ( t ) is the derivative of N ( t ) representing whether the tumor burden is growing or shrinking. The dose-modulating thresholds satisfy C TH0 < C TH1 < C TH2 and C TH0 < 1 < C TH2 . Notably, the dosing strategies (6) and (7) represent state-dependent switching mechanisms rather than closed-form mathematical definitions. This implementation induces history-dependent therapeutic decision and consequently, the governing equations (1) – (3) constitute delay differential equations with historydependent terms D ( t ). As a state-of-the-art investigation, this study provides comprehensive parameter analysis rather than casespecific parameter estimation. So, in this work, the parameter values are taken in ranges according to established experimental and clinical studies [ 24 , 37 , 55 , 70 , 73 ]. The proliferation rates of sensitive and resistant cells, i.e. r S and r R , are derived from in vitro experimental data [ 73 ], while the drug-induced killing strength γ is sourced from the estimation based on several experiments of cell lines [ 70 ]. Given the critical clinical risk associated with large tumor burdens, the initial tumor burden in this study is set as 75% of the environmental carrying capacity [ 55 ]. The initial proportion of resistant cells is taken from existing dynamic models [ 24 ]. All the baseline parameter values are summarized in Table 1 . View this table: View inline View popup Download powerpoint TABLE 1. Model parameters and their ranges. 3. RESULTS 3.1 The treatment-holiday threshold is crucial for the efficiency of adaptive therapy To investigate the effect of treatment-holiday thresholds on the outcomes of adaptive therapy, a simplified assumption of neutral competition ( α = β = 1) is considered. The outcomes are primarily characterized by the time to progression (TTP), which is typically defined as the time at which the tumor burden grows to a certain percentage greater than its initial burden [ 24 , 37 , 55 , 70 , 73 ]. For simplicity, this percentage is set to 120% in this study. As illustrated in the example of tumor evolution under adaptive therapy ( Figure 2A ), during the initial phase of adaptive therapy, drugs are administered to suppress tumor growth primarily through the depletion of sensitive cells. During this stage, the proportion of sensitive cells is high, which aligns with real-world cases. Following the principles of adaptive therapy, treatment is paused when the tumor burden reaches a predefined treatment-holiday threshold C TH0 N 0 , conventionally set at 50% of the initial tumor burden, i.e., C TH0 = 0.5. The treatment holidays allow sensitive cells to repopulate, leading to tumor regrowth. When the tumor burden returns to the initial level N 0 , treatment resumes. The treatment-holiday threshold C TH0 thus defines a cycle, from the start of treatment to its suspension. These cycles continue until resistant cells dominate the tumor burden, ultimately leading to disease progression. Download figure Open in new tab FIGURE 2. Threshold-mediated therapeutic outcomes under neutral competition ( α = β = 1).A representative example of tumor evolution under adaptive therapy (AT), where the initial proportion of resistant cells f R = 0.01 and the initial tumor burden as a fraction of carrying capacity is n 0 = 0.75. The purple dashed line represents the time to progression (TTP) under the AT strategy with the threshold C TH0 = 0.5; the red, green, and black lines represent resistant cells, sensitive cells, and total tumor burden, respectively. Shaded areas indicate periods of active treatment. (B1-B2) The phase diagram and the vector field of tumor evolution with and without treatment. Light blue trajectories illustrate the trajectories with different tumor initial states. The upper right blank region represents situations beyond environmental carrying capacity ( K ). Panel (B1) illustrates the dynamics without treatment, while panel (B2) shows the cases with maximum tolerated dose strategy. The dark blue trajectory in (B1) shows the dynamics of the total tumor burden without treatment, while the red curve illustrates dynamics under AT as shown in panel (A). (C1)-(C3) The impact of changing the treatmentholiday threshold on TTPs under different initial tumor compositions. The numbers on the plots indicate the total number of treatment cycles experienced before disease progression, with 0 cycles of AT corresponding to the MTD strategy. Panel (C1) represents a uniform-decline scenario of AT, where the TTPs with AT-S are consistently worse than those with MTD; panel (C2) shows a conditional-improve scenario, where AT-S outperforms MTD only for certain threshold values; and panel (C3) demonstrates a uniform-improve scenario, where all AT outperforms MTD for all threshold values. (D) The dependence of the three scenarios on the initial tumor burden and composition. (E) Probability of the three outcome scenarios versus the proliferative fitness ratio r S / r R with randomly initial tumor states, demonstrating that a sensitive cell growth advantage ( r S / r R > 1) enhances the AT success probability. To further elucidate this phenomenon, phase-plane analysis reveals distinct dynamical regimes with and without treatment ( Figure 2B1-B2 ). In untreated systems, tumor populations evolve toward coexistence equilibria, as indicated by the nullcline, which mathematically corresponds to the diagonal in Figure 2B1 . In the presence of treatment ( Figure 2B2 ), the long-term steady state shifts to the upper-left corner, where resistant cells dominate the tumor, and sensitive cells eventually disappear. From a dynamical perspective, treatment exerts selective pressure, pushing the tumor toward a resistant cell-dominant state. Therefore, the significance of treatment holidays lies in allowing the tumor to remain longer in a state dominated by sensitive cells, thereby delaying the transition to resistance. In this regard, increasing the treatment-holiday threshold can slow the transition of tumor composition to a resistant state, which implies it is possible to improve adaptive therapy outcomes by modulating the threshold and preventing continuous treatment from driving the tumor toward resistance via Darwinian selection [ 22 ]. Therefore, the effects of varying the treatment-holiday threshold are examined under different initial tumor states, characterized by the tumor burden ( n 0 ) and the fraction of resistant cells ( f R ) (ref. Table 1 ). As shown in Figures 2C1-C3 , altering the treatment-holiday threshold significantly impacted TTP. Notably, when the threshold is too low, meaning the tumor never reached the treatment-holiday threshold even during the initial cycle, adaptive therapy lacks treatment holidays and effectively becomes equivalent to MTD therapy. Consequently, as C TH0 is around 0, TTP of AT is de facto that of MTD therapy, as shown in the leftmost segments of Figures 2C1-C3 . Importantly, for different initial tumor states, three distinct scenarios emerged when comparing adaptive therapy with MTD. The first scenario, uniform-decline ( Figure 2C1 ), occurs when adaptive therapy consistently performs worse than MTD, regardless of the threshold. This scenario typically arises in tumors with a low initial tumor burden. The second scenario, conditional-improve ( Figure 2C2 ), occurs when adaptive therapy outperforms MTD only at specific threshold values, suggesting that careful threshold selection is required. The third scenario, uniform-improve ( Figure 2C3 ), occurs when adaptive therapy is always superior to MTD. The findings highlight the complex relationship between treatment-holiday thresholds and TTP. Moreover,as shown in Figures 2C2-C3 , the effect of the threshold is neither monotonic nor continuous, exhibiting discontinuities as the number of treatment cycles changes discretely (Figure S1). Specifically, in tumors with a low initial burden ( Figures 2C1 and C2 ), increasing the number of treatment holidays reduces drug exposure, which in turn weakens tumor suppression because under these conditions, the initial tumor burden is low, and therefore, environmental resources remain abundant which makes the changes in the burden of sensitive cells have little impact on the growth rate of resistant cells. In contrast, for tumors with a higher initial burden ( Figure 2C3 ), where environmental resources are constrained, fluctuations in sensitive cell numbers significantly affect the growth of resistant cells. In such cases, increasing the number of treatment holidays enables sensitive cells to grow and exert stronger suppression on resistant cells, leading to a marked increase in TTP. This aligns with clinical observations in prostate cancer, where intermittent dosing preserved therapeutic sensitivity [ 73 ]. To further investigate the impact of initial tumor states on these three scenarios, we examined how different initial tumor burdens and initial resistant cell fractions influence adaptive therapy outcomes ( Figure 2D ). When the initial tumor burden exceeds 0.6 K , adaptive therapy uniformly improves TTP, regardless of the initial proportion of resistant cells. However, when the initial tumor burden is below 0.3 K , adaptive therapy can only conditionally improve TTP, implying that careful selection of the treatment-holiday threshold is crucial. In the worst scenario, when the initial tumor burden is very low and the proportion of resistant cells is relatively high ( Figure 2D , top-left), adaptive therapy fails to improve TTP at any threshold, indicating primary resistance. These findings extend prior theoretical work [ 54 ] by quantifying phase-transition-like phenomena in thresholdmediated outcomes. Furthermore, clinical implementation faces inherent uncertainty in tumor state assessment and threshold selection, implying that both the treatment-holiday threshold and the initial tumor burden and resistance fraction exhibit random variability. To address this, Monte Carlo simulations were performed to evaluate the probabilities of the three outcome scenarios under parameter variability ( Figure 2E ). Inspired by the NortonSimon Hypothesis [ 49 , 59 ], which posits that sensitive cells typically have a greater fitness advantage than resistant cells (i.e., sensitive cells have a higher intrinsic proliferation rate than resistant cells), we further explored the effect of the relative growth rate of sensitive cells ( r S / r R ) on the probabilities of the three scenarios. By randomly varying the treatment threshold and, while keeping r S / r R fixed, randomly altering r S and r R , we found that the probability of the uniform-improve scenario increases as the relative fitness of sensitive cells increases. When r S / r R is small, adaptive therapy is less likely to improve TTP relative to MTD. Conversely,when r S / r R is high, indicating that sensitive cells have a significant growth advantage over resistant cells, adaptive therapy is more likely to enhance TTP. These results confirm that the treatment-holiday threshold has a profound impact on adaptive therapy outcomes and patient-specific tumor characteristics, including initial tumor burden and the relative fitness of sensitive and resistant cells, significantly influence treatment outcomes. These findings align with existing research [ 72 ] and demonstrate that r S / r R dependence provides quantitative criteria for patient stratification, providing insights for addressing a key challenge in empirical threshold selection [ 67 ]. 3.2 Context-dependent superiority of adaptive therapy across treatment strategies This subsection further compares the efficacy of adaptive therapy and intermittent therapy under the uniformimprove scenario, extending the previous comparison with maximum tolerated dose. As indicated in prior results, the effectiveness of different treatment strategies is influenced by treatment parameters. Therefore, to ensure a fair comparison, treatment parameters must be appropriately aligned. A reasonable approach is to compare strategies based on their best achievable outcomes ( Figure 3 ). Specifically, for intermittent therapy, the optimal treatment cycle and duration should be selected, while for adaptive therapy, the optimal threshold values should be determined. Download figure Open in new tab FIGURE 3. Comparison of optimal outcomes under different therapeutic strategies and success landscapes of adaptive therapy. (A1) - (A4) Tumor evolution dynamics under four therapeutic strategies with optimal treatment-holiday thresholds, given n 0 = 0.75 and f R = 0.01. The purple dashed line indicates the TTP with optimal treatment parameters, while the orange solid line in (A2) represents the TTP with the conventional 50% threshold ( C TH0 = 0.5). Red, green, and black curves represent resistant cells, sensitive cells, and total tumor burden, respectively. Shaded regions denote periods of active treatment. For IT, the treatment interval ( T ) is 15, and the treatment duration ( T D ) is 6. For AT-S, the optimal threshold is close to 1, so here C TH0 is set to 0.98. Similarly, for AT-M, the middle threshold is C TH1 = 1, and the upper threshold is C TH2 = 1.05. (B1) Cumulative drug toxicity and the corresponding TTP for the four therapeutic strategies. The cumulative toxicity is quantified by the total drug dosage administered before disease progression, given by equation (8) . (B2) Therapeutic efficiency index, defined as the TTP-to-toxicity ratio as equation (8) . Here the efficiency indices are displayed as normalized to that of MTD. (C1) - (C2) Effects of varying the initial tumor burden on TTP and the efficiency of the four treatment strategies. (C3) - (C4) Effects of varying the initial proportion of resistant cells on TTP and the efficiency of the four treatment strategies. (D1) - (D2) Success probability landscapes of AT-S as a function of the initial tumor state and treatment threshold. The color at each point represents the probability of AT-S improving treatment outcomes compared to MTD. (D1) illustrates the dependence of success probability on the initial tumor burden and treatment-holiday threshold with randomly varying f R , r S , and r R , which simulates the uncertainty in tumor proliferation and treatment decisions; while (D2) shows its dependence on the initial fraction of resistant cells and initial tumor burden with randomly varying C TH0 , r S , and r R . The optimal parameter combination is determined based on different initial tumor states. Since this study does not focus on identifying these optimal parameters of intermittent therapy, computational details are omitted. However, as previous results suggest that under adaptive therapy in the conditional-improve and uniformimprove scenarios, a threshold closer to 1 generally leads to a longer TTP ( Figure 3 ). In practical applications, a threshold near 1 would necessitate frequent initiation and cessation of treatment, requiring highly precise monitoring of tumor status, which is unrealistic. Similarly, for intermittent therapy, frequent treatment pauses and resumptions are impractical. Given these practical constraints, achieving the true optimal outcome is infeasible. Consequently, only suboptimal treatment outcomes that closely approximate the optimal results are considered as benchmarks for comparison. For intermittent therapy, the treatment cycle period ( T ) is examined to range from 7 to 63 time units, with drug administration lasting from 7 to 21 time units ( T D ) per cycle. Since this study does not focus on identifying these optimal parameters of IT, computational details are omitted. The result shows that the suboptimal period is T = 15 with T D = 9, simulating a regimen in which the patient undergoes both treatment and rest periods within a feasible schedule ( Table 1 and equations (5) ). For dose-skipping adaptive therapy, the treatment-holiday threshold is set to C TH0 = 0.98, while for dose-modulating adaptive therapy, C TH1 = 1 and C TH2 = 1.05 are selected as the suboptimal treatment parameters ( Figure 3A1-A4 ). In the uniform-improve scenario for adaptive therapy, a comparison of the suboptimal outcomes across the four treatment strategies ( Figure 3A1-A4 ) reveals that both adaptive therapy strategies result in a longer TTP than intermittent therapy, while intermittent therapy still outperforms MTD. Furthermore, when comparing the two adaptive therapy strategies, AT-M slightly outperforms AT-S. However, considering the operational complexity of AT-M, AT-S is primarily considered in subsequent discussions to balance efficacy with practical feasibility. From an efficiency perspective, due to factors such as drug side effects, a higher total drug dosage is not necessarily preferable. Therefore, based on the principle of maximizing TTP with the minimum amount of drug, the contribution of unit dosage to TTP is considered a measure of treatment strategy efficiency ( equation (8) , Figure 3B1-B2 ): It is observed that MTD requires the least total drug dosage ( Figure 3B1 ), but this is primarily due to its short TTP. Consequently, MTD’s efficiency is the lowest among the four treatment strategies ( Figure 3B2 ). Notably, when efficiency is used as the evaluation criterion, intermittent therapy performs the best ( Figure 3B2 ). This is because intermittent therapy, compared to adaptive therapy, has longer treatment holidays with a relatively smaller decrease in TTP, resulting in a higher contribution of unit drug dosage to TTP ( Figure 3B2 ). This suggests that intermittent therapy is a reasonable option, particularly for patients with lower drug tolerance, which is consistent with clinical practice. Furthermore, to compare the robustness of the four treatments, the stability of TTP and treatment efficiency under uncertainties in patient physiological conditions and tumor status monitoring is examined. The impact of different initial tumor states on TTP and efficiency under the suboptimal parameters for the four strategies is analyzed ( Figure 3C1-C4 ). For different initial tumor burdens, the trends in TTP for the four treatment strategies under the suboptimal parameters are similar; TTP increases with the tumor burden, which is related to the criteria for determining tumor progression. Since progression is determined based on the initial tumor burden, a larger initial tumor burden necessitates a higher burden for progression to occur, leading to this seemingly counterintuitive result ( Figure 3C1 ). From this perspective, some researchers have proposed a “modifying adaptive therapy” strategy, which delays treatment until the tumor reaches a certain size [ 27 ]. Moreover, intermittent therapy shows significant variability with different initial tumor burdens ( Figure 3C2 ). When the tumor burden is small, intermittent therapy is less efficient than adaptive therapy. As the tumor burden increases, this relationship reverses. Therefore, the robustness of intermittent therapy’s efficiency is inferior to that of adaptive therapy. This suggests that when adopting intermittent therapy, a more precise assessment of the patient’s specific condition is necessary. On the other hand, both adaptive therapy strategies and intermittent therapy show similar trends in TTP and efficiency with varying initial proportions of resistant cells ( Figure 3C3-C4 ), indicating that the optimal TTP is more influenced by cell composition when the proportion of resistant cells is very low. Notably, when the initial proportion of resistant cells is high, the TTP and efficiency of different treatment strategies are close to those of MTD. This is because, when the initial proportion of resistant cells is high, the determining factor for TTP is no longer the competition between sensitive and resistant cells but rather the proliferation of resistant cells themselves, which reduces the differences between treatment strategies. Furthermore, similar to the previous results, considering the uncertainty in real-world scenarios, the probability of adaptive therapy being the best strategy is explored under conditions of random tumor growth rates and status monitoring. The probability that adaptive therapy results in a longer TTP than both MTD and intermittent therapy is shown in Figure 3D1-D2 . In real scenarios, since the tumor growth rate is unknown and tumor composition is difficult to obtain, the dependence of TTP on treatment-holiday thresholds, initial tumor burden, and initial resistant cell proportion is explored under random variations of other factors. Figure 3D1 shows the dependence of the probability of adaptive therapy outperforming other treatment strategies on different thresholds ( C CH0 and initial tumor burden ( n 0 ), with the other parameters, including the initial resistant cell proportion ( f R ) and the intrinsic growth rates of tumor cells ( r S and r R ) varying randomly. Figure 3D2 shows the dependence of the probability on the initial resistant fraction ( f R ) and the initial tumor burden ( n 0 ), with other parameters, C TH0 , r S and r R , varying randomly. It can be observed that the probability of selecting adaptive therapy increases significantly when the initial tumor burden is larger. However, when the initial tumor burden is smaller, adaptive therapy does not consistently improve TTP, which aligns with previous findings. Notably, the selection of the treatment-holiday threshold and tumor composition has a weaker impact on the selection of adaptive therapy than the initial tumor burden, indicating that when detailed information about tumor components is unavailable, parameters such as the treatment threshold have less influence, with the initial tumor burden playing a key role. On the other hand, TTP is highly sensitive to the selection of the threshold, among other factors ( Figures 2 and S3-S7). In other words, the selection of the optimal threshold is also dependent on various tumor states. Therefore, when other parameters are randomly varied, the impact of the threshold on TTP becomes overshadowed by fluctuations in these parameters. These findings are consistent with clinical observations, emphasizing the importance of a precise understanding of tumor dynamics, particularly intrinsic growth rates. If such tumor information is available, treatment strategy selection can be significantly improved, highlighting the critical importance of accurate tumor characterization, especially regarding tumor dynamics. 3.3 Conditional superiority landscapes of adaptive therapy compared to other strategies under non-neutral competition The competitive interaction between sensitive ( S ) and resistant ( R ) cells, governed by coefficients α and β , fundamentally determines tumor evolution dynamics under therapeutic stress [ 20 , 41 , 42 , 65 ]. Four distinct regimes emerge based on relative competition strengths: 1) weak competition ( α < 1, β 1, β > 1), 3) asymmetric competition with α < 1 < β, or 4) β < 1 < α . Detailed analyses of α < 1 < β and β < 1 1, β > 1), untreated tumors evolve toward exclusive dominance of either S or R , depending on initial composition ( Figure 4A ), mathematically equivalent to equations (1) – (3) possessing two stable steady states. Continuous MTD administration drives the tumor toward a stable steady state where sensitive and resistant cells maintain a fixed ratio with unchanged tumor burden ( Figure 4B ). In the absence of treatment, the tumor’s final state depends on its initial condition, mathematically determined by the basin of attraction in which the system starts. Thus, sensitive cell dominance at initiation leads to resistant cell elimination without treatment, requiring only therapeutic containment to prevent long-term progression. Conversely, initial resistant cell dominance under MTD results in a stable tumor burden with constant cellular proportions. The initial tumor burden determines whether disease progression occurs. Even when exceeding progression thresholds, tumor burden stabilizes below carrying capacity. This property renders the “uniformlyimprove” scenario inapplicable for strong competition systems, as therapeutic outcomes primarily depend on initial tumor state rather than threshold calibration ( Figure 4C ). That is, as long as the long-term steady state under treatment remains acceptable ( Figure 4B ), treatment strategy selection becomes less critical for strong competition cases. Download figure Open in new tab FIGURE 4. Tumor evolution characteristics under strong competition in the presence of therapy. Phase diagram and vector fields of tumor evolution without treatment. The blue lines indicate that the long-term tumor state depends on the initial conditions. The blank region indicates that the effective tumor burden exceeds the carrying capacity ( S + αR > K or βS + R > K ). (B) Phase diagram and vector fields of tumor evolution with treatment. The long-term tumor dynamics converge to a state characterized by a heterogeneous composition of sensitive and resistant cells. The other features are similar to those in panel (A). (C) The dependence of the three scenarios on the competition coefficients α and β , where, in fact, only two scenarios occur: uniform-decline (red) and conditional-improve (green), with the uniformimprove scenario absent. Here the initial tumor burden is taken as n 0 = 0.75 and the fraction of resistant cells is taken as f R = 0.01 with the other parameters as shown in Table 1 . Unlike neutral or strong competition, weak competition ( α < 1, β < 1) permits coexistence equilibria regardless of initial states ( Figure 5A ), while continuous treatment induces resistant dominance through Darwinian selection ( Figure 5B ). Similar to neutral competition, weak competition exhibits three distinct therapeutic response scenarios ( Figure 5C ), necessitating strategic treatment selection. Download figure Open in new tab FIGURE 5. Effectiveness of adaptive therapy under conditions of non-neutral competition. (A) Phase diagram and vector field of the dynamics of sensitive and resistant cells in the absence of treatment under weak competition. The blank region represents scenarios where the effective tumor burden exceeds the carrying capacity ( S + αR > K or βS + R > K ). The blue lines represent tumor evolution trajectories with different initial tumor states. (B) Phase diagram and vector field of the dynamics with treatment. The blue lines show that tumor evolution trajectories all move toward a resistant-dominant state. The other elements in the figure are similar to those in panel (A). (C) Dependence of the three scenarios on the competition coefficients, with the initial tumor burden n 0 = 0.75, the fraction of resistant cells f R = 0.01, and other parameters set as in Table 1 . (D1-D8) Dependence of the probability of selecting adaptive therapy on competition coefficients, under different initial tumor burdens. Here, the selection of adaptive therapy refers to the condition where the TTP of adaptive therapy is better than that of other strategies. The fraction of resistant cells f R , cell division rates r S and r R , and treatment-holiday thresholds C TH0 are randomly sampled to represent various individual conditions. The sample size is 1200 for each pair of competition coefficient values α and β . (D0) Overall dependence of the probability of selecting adaptive therapy on the initial tumor burden. Boxplots show the aggregated probabilities calculated from panels (D1)-(D8). Note that panels (D0)-(D8) include both weak and strong competition conditions. Given clinical uncertainties in weak competition systems, comprehensive parameter-space exploration proves unnecessary and inefficient. The analysis therefore focuses on measurable initial tumor burden ( n 0 ) to evaluate adaptive therapy superiority probability under clinical uncertainty, following previous subsections ( Figure 3D1-D2 ). Less accessible parameters including resistant cell fraction ( f R ), thresholds ( C TH0 ), and growth rates ( r S , r R ) are randomized ( Figure 5D0-D8 ). Analysis reveals enhanced adaptive therapy superiority with decreasing α and increasing β ( Figure 5D1-D8 ), consistent with its premise of leveraging sensitive cells to suppress resistant populations. This relationship weakens when resistant cells evade suppression. Notably, large initial tumor burdens in strong competition systems reduce adaptive therapy selection probability ( Figure 5 upper-right quadrant), primarily due to: 1) bistability-mediated tumor burden homeostasis across treatments negating adaptive advantages, and 2) extended progression timelines for large burdens due to progression criteria, diminishing therapeutic differentiation. Surgical intervention may become preferable when tumor burdens approach carrying capacity. Crucially, adaptive therapy selection probability increases with initial tumor burden except at extreme values ( Figure 5D0 ), paralleling neutral competition observations ( Figures 2E and 3D1-D2 ). These findings confirm tumor burden’s primacy as the clinically actionable determinant while suggesting benefits from precise competition-strength measurements. 3.4 The nonexistence of practical optimal therapeutic strategies As shown in Figure 2C 3, TTP is positively correlated with the number of treatment holidays. Motivated by this observation, this subsection presents theoretical analysis of the optimal control problem that maximizes TTP through optimization of the dose curve D ( t ), as formalized in equations (9) – (15) : subject to with parameters satisfying the following constraints (16)-(19). This optimal control problem aims to maximize Time-to-Progression (TTP) through an optimized dosing schedule, with tumor growth governed by Lotka-Volterra dynamics. For this formulation, Theorem 1 establishes the nonexistence of practically implementable optimal controls under the specified conditions. Theorem 1. Consider the optimal control problem defined by (9) - (19), where D * ( t ) denotes an optimal control maximizing J [ D (·)]. Assume that S ′(0) > 0 and R ′(0) > 0 when D ( t ) ≡ D min = 0 and D * ( t ) is an optimal control. Then any optimal control D * ( t ) maximizing J [ D (·)] necessarily has infinitely many points of discontinuity if either of the following conditions is satisfied. α = β = 1, or 1 < β 0 and R ′(0) > 0 under D ( t ) ≡ D min = 0 are biologically reasonable, as they indicate that the tumor burden remains below the microenvironment’s carrying capacity, enabling tumor growth in the absence of therapy. Furthermore, Theorem 1 combined with Pontryagin’s Maximum Principle implies that any optimal control (if it exists) must undergo infinitely many switches between maximum and minimum dosing levels, rendering it clinically impractical. Before proving the theorem, some typical calculations for the optimal control problem and lemmas are needed. Firstly, the necessary condition for the optimal control problem should be calculated using Pontryagin’s Maximum Principle. The Hamiltonian ℋ is defined as equation (20) . Thus, the equations for the costates are given as following eqautions (21) and (22) . with the transversality conditions as boundary conditions ( equations (23) – (25) ). where ν is the Lagrange multiplier associated with the terminal constraint N ( T ) = (1 + ε) n 0 K , obtained from the terminal condition of the Hamiltonian as given by equation (25) . So the ν should satisfies the equation According to Pontryagin’s Maximum Principle, since the Hamiltonian ℋ is linear in D ( t ) ( Equation (20) and D * ( t ) subjects to (18), hence, D * ( t ) is determined by the sign of the switching function Φ( t ) given by the equation (26) . where In summary, the equations for the optimal solution D * ( t ) of the optimal control problem (if it exists) are summarized in the following Lemma 2. Lemma 2. If the optimal control problem (9) - (19) has a solution with optimal control D * ( t ), then D * ( t ) satisfies the following equations: The Lemma 2 also provides a numerical method based on a forward-backward sweep scheme if the optimal control exists. For simplicity and considering biological realism, hereafter in this section, we make the following assumptions (28)-(29). and Biologically, these assumptions indicate that the tumor burden is within the carrying capacity of the microenvironment, enabling tumor growth. In add ( ition, to sim plify notation, the following symbols are introduced throughout this section. 1) let denote the factor in (10). 2) let S * ( t ), R * ( t ), N * ( t ) = S * ( t ) + R * ( t ) be the state solution for the optimal control problem (9)-(19). And 3) let T * be the optimal terminal time (objective value), i.e., the time point such that N * ( T * ) = (1 + ε) n 0 K if the optimal control exists. Based on the necessary condition, more properties of the optimal control D * ( t ) can be obtained as Lemma 3-5. Lemma 3. Under the condition of Theorem 1, if there exists an optimal control for the optimal control problem (10) - (19), the costate corresponding to the S variable must be negative at the terminal point T * , λ S ( T * ) must be negative and the derivative must be positive. Proof. Under the condition of D ( t ) ≡ D min = 0, it is straightforward to verify that S ′ ( t ) > 0 and R ′ ( t ) > 0 for all t ∈ [0, T ] except possibly at isolated points if S ′ (0) > 0 and R ′ (0) > 0 using a standard argument for ordinary differential equations by contradiction as follows. First, for the case that S ′ ( t ) = R ′ ( t ) = 0 it will reach a contradiction by the uniqueness theorem of ordinary differential equations, as the constant solution is also a solution. For the case that S ′( t ) = 0 while R ′( t ) > 0 (without loss of generality). Assume that t ′ = inf{ t > 0| S ′ ( t ) ≤ 0} and t″ = inf{ t> 0| R ′( t ) ≤ 0}, then because S ( t ) and R ( t ) is continuously differentiable and S ′ (0) > 0, R ′ (0) > 0, we have t ′ > 0, t ″ > 0. So without loss of generality, we can assume that t ′ 0, R ′ ( t ) > 0 for t 0 in some semi-neighbourhood of t′, t ∈ ( t′, t ′ + δ′). So, it is impossible that S ′ ( t ) = 0 on any neighbourhood of t ′ because the right-hand side of (10) cannot keep zero when R ′ ( t ) > 0 under the assumption that S ( t ) ≠ 0. According to (26), as S ′( t ) > 0 without drugs ( D ( t ) ≡ 0), it follows that So it is impossible that the case Φ( t ) ≡ 0 on some interval happens. Thus, the optimal control (if it exists) can be written as (30) as follows. Therefore, by (23)-(25), the boundary value of the costate λ S ( t ) must satisfy And this completes the proof. □ The Lemma 3 is consistent with the fact that at the time of progression, the total tumor burden is growing and progression always occurs during drug administration. Otherwise, intuitively, drug administration can always slow down the growth of tumor within a small time interval as sensitive cells are being eliminated. Lemma 4. Assume the optimal control problem has a solution that is piecewise constant with finitely many discontinuities. If α = β = 1, there exists T 0 < T * such that holds for every t ∈ [ T 0 , T * ]. Proof. For simplicity, the asterisk in the superscript are omitted in this proof. According to Lemma 3, in some neighbourhood of T * , the following holds. so and therefore Lemma 5. Assume that 1 ≤ β ≤ α and . If the optimal control problems (9) - (19) has a piecewise constant optimal control according to PMP, there exists such that for every Proof. Similar to the proof of Lemma 4, the asterisks are also omitted in this proof. And, in a neighbourhood of T * , the following holds. and thus, Moreover, since , the threshold of progression is thus . So the final state of R ( t ) should satisfies . Therefore, Also because that 1 < β < α , so we have that which, according to the simplification of the notations, implies that And the result of the lemma is followed by a common argument of calculus. Now the lemmas are sufficient to prove the main Theorem 1. Proof of Theorem 1. By contradiction, assume the optimal control D * ( t ) has only finitely many discontinuities. Then, by Pontryagin’s Maximum Principle (PMP), D * ( t ) is a piecewise constant function whose range has only two values. Let T 1 denote the last discontinuity point of the optimal control. So we have that D * ( t ) = D max for t ∈ [ T 1 , T * ] and moreover, from Lemma 4 and Lemma 5, we can choose T 2 ∈ [ T 1 , T * ) such that N ′( t ) > 0 and hold for all t ∈ [ T 2 , T * ]. Now we are going to prove that there exists Δ t > 0 such that the state solution has a better termination time than T * with D Δ t ( t ) defined as the following (36). Let S Δ t ( t ), R Δ t ( t ), N Δ t ( t ) = S Δ t ( t ) + R Δ t ( t ) denote the solution with this D Δ t ( t ). Then N * ( t ) and N Δ t ( t )obey the same differential equations for all t except t ∈ [ T * ™ 2Δ t, T * ™ Δ t ]. Moreover, by denoting t i = T * ™ (2 ™ i )Δ t, i = 0, 1, 2 and , , then we have equations (37) – (44) . And therefore, we have that Hence, for the case 1, α = β = 1, it has already been obtained that and according to Lemma 4. And since the o (Δ t ) only depends on the properties of the (assumed) optimal solution N , * ( t ), there exists a such that for Δ t ∈ (0, δ ′ ), and therefore N * ( T * ) ™ N Δ t ( T * ) > 0, implying that N ( t ) reaches (1 + ε) n 0 K at some t > T * , contradicting the assumption that T * is the optimal objective value. For the case 2, 1 ≤ β ≤ α and , it follows that N * ( T * ) ™ N Δ t ( T * ) > 0 by Lemma 5, which also contradicts with the assumption that N * ( t ) is the optimal state solution. So the proof has been completed. □ 4. DISCUSSION Adaptive therapy crucially leverages competitive dynamics to maintain therapeutic-sensitive cell reservoirs that suppress resistant clones, strategically delaying drug resistance evolution while minimizing tumor progression [ 22 ]. By preserving sensitive cells through threshold-modulated treatment holidays, adaptive therapy sustains ecological suppression of resistant populations—an advantage over traditional maximum tolerated dose regimens, which inadvertently accelerate resistance [ 15 ]. The clinical success of adaptive therapy hinges on its paradigm-shifting principle: harnessing intra-tumor competition to transform therapy-sensitive cells into permanent controllers of resistant cell expansion [ 69 , 74 ]. However, the rule-of-thumb thresholds used in clinical practice may require further research and optimization as they might not fully account for the complex, dynamic interactions between sensitive and resistant cell populations, and may vary across different tumor types and patient profiles [ 30 , 37 , 51 , 52 , 57 , 64 ]. This study employs a Lotka-Volterra model to investigate how treatment-holiday thresholds influence adaptive therapy outcomes in tumors composed of drug-sensitive and resistant cells. By simulating tumor dynamics under varying thresholds, initial tumor burdens, and competition coefficients, the analysis identifies three distinct therapeutic scenarios: uniform-improve (where AT consistently outperforms MTD regardless of threshold selection), conditional-improve (AT efficacy depends on threshold calibration), and uniform-decline (MTD superiority). When considering clinical uncertainties in measurement data, generally, the greater the initial tumor burden, the more likely adaptive therapy will outperform other treatment strategies. However, in cases where the tumor burden is excessively high, the advantages of adaptive therapy become irrelevant, and alternative approaches, beyond pharmacological treatment, may be required. The competitive dynamics between sensitive and resistant cells is also a key factor in tumor evolution under therapeutic stress. In strong competition environments, tumors tend to evolve toward a state dominated by either sensitive or resistant cells, depending on the initial composition. Continuous MTD therapy drives the tumor toward a stable state where the ratio of sensitive to resistant cells remains constant. Thus, the critical factor in treatment outcomes is not the treatment-holiday threshold but rather the impact of the initial tumor state on the final result. In weak competition environments, sensitive and resistant cells can coexist, and adaptive therapy may lead to resistant cell dominance through Darwinian selection. Treatment effectiveness is influenced by the competition coefficients, α and β , with lower α and higher β favoring adaptive therapy. Adaptive therapy prolongs TTP most effectively when sensitive cells maintain a competitive advantage and when initial tumor burden is high, enabling ecological suppression of resistant cells. This study advances our understanding of adaptive therapy by systematically quantifying the role of treatmentholiday thresholds in modulating tumor evolution under treatment. By integrating ecological competition principles into therapeutic design, the current work provides a mechanistic basis for optimizing intermittent dosing strategies. The findings highlight that adaptive therapy is not universally superior to MTD; its success depends critically on tumor initial state and the treatment-holiday threshold selection. These insights bridge theoretical ecology and clinical oncology, emphasizing the need for dynamic, patient-specific treatment protocols rather than static dosing regimens. This work establishes a quantitative relationship between treatment-holiday thresholds and progressionfree survival. By categorizing outcomes into three distinct scenarios, the model offers clinicians a potential framework to stratify patients based on measurable parameters such as initial tumor burden and resistant cell fraction. Furthermore, the demonstration that strong competition can stabilize tumor burden indefinitely under specific thresholds has profound implications for managing advanced cancers. These results extend prior work [ 22 , 37 , 54 , 64 ] by incorporating threshold-driven mechanism in adaptive therapy. Despite these advances, the study has several limitations. First, the binary classification of tumor cells overlooks the phenotypic plasticity and continuous heterogeneity that are critical drivers of therapeutic resistance [ 7 , 26 , 35 ]. Second, model parameters such as the competition coefficients ( α, β ) are based on population-level assumptions rather than molecular data, necessitating further investigation into the microscopic foundations of these assumptions [ 61 , 63 ]. Third, the Lotka-Volterra model does not account for the spatial structure of tumors or the microenvironmental constraints that alter tumor dynamics [ 31 , 68 ]. Future studies should integrate single-cell and spatial omics data to mechanistically parameterize competition coefficients and explore hybrid models that combine continuum and spatial interactions. Additionally, clinical validation through adaptive therapy trials, incorporating real-time biomarker monitoring, will be crucial for translating these insights into practice. DATA AND CODE AVAILABILITY This study did not use any real-world data; all data were generated through computational simulations. All original code has been deposited on GitHub ( https://github.com/zhuge-c/AT-0 ). Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request. ACKNOWLEDGMENTS This work was funded by The National Natural Science Foundation of China (NSFC) via grant 11801020. 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