Therapeutic Resistance as a Manifestation of Therapeutic Trajectory Evolution: A New Research Paradigm

1. Introduction

Therapeutic resistance remains a major unresolved problem in cancer therapy and a major cause of treatment failure and disease progression. Resistance can be present from the beginning of treatment or emerge after an initial response, and it is associated with a broad and continually expanding range of genetic, epigenetic, signaling, metabolic, cellular, immune, microenvironmental, and pharmacological mechanisms [1,2,3,4,5,6].

Previous work formulated cancer therapy as a continuous process in which a therapeutic input $\mu$ acts within an initial tumor-host state-space organization ${\theta }_0$, leading to therapeutic trajectory evolution $T$, attractor formation $A$, and an observable therapeutic outcome $y$ [7]. State-space modulation was subsequently introduced as the intentional modification of state-space organization to redirect therapeutic evolution [8]. A probabilistic extension represented possible trajectories, attractors, and outcomes under a given initial state-space organization and therapeutic input by a joint conditional probability distribution [9], and a hierarchical formulation further made the hierarchical conditional organization of this continuous therapeutic process [10].

The present work brings these formulations together and applies them to therapeutic resistance. It develops an analytical definition of resistance as both a manifestation of therapeutic trajectory evolution and a conditional-probability outcome, places therapeutic sensitivity within the same probabilistic organization of therapeutic evolution, and derives therapeutic implications through state-space modulation.

2. Resistance as a Manifestation of Therapeutic Trajectory Evolution and a Conditional-Probability Outcome

Within the state-space and trajectory-evolution formulation, therapeutic outcome does not arise directly from therapeutic input. The therapeutic input acts within a given initial state-space organization, and the treated system evolves through an intermediate therapeutic process [7]:

$$({\theta }_0,\mu )\to T\to A\to y$$

Here, ${\theta }_0$ denotes the initial state-space organization of the tumor-host system, $\mu$ the therapeutic input, $T$ therapeutic trajectory evolution, $A$ attractor formation, and $y$ the observable therapeutic outcome.

Under a given initial state-space organization and therapeutic input, multiple therapeutic realizations are dynamically accessible. Let $T$, $A$, and $y$ be random variables describing therapeutic trajectory, attractor, and observable outcome. Their possible realizations are represented by the joint conditional probability distribution (9,10)

$$P(T_i,A_j,y_k\mid {\theta }_0,\mu )$$

The chain rule decomposes this joint distribution without requiring additional conditional-independence assumptions:

$$\begin{gathered} P(T_i,A_j,y_k\mid {\theta }_0,\mu )=P(T_i\mid {\theta }_0,\mu ) \\ ·P(A_j\mid T_i,{\theta }_0,\mu ) \\ ·P(y_k\mid A_j,T_i,{\theta }_0,\mu ) \end{gathered}$$

Let ${\mathcal{Y}}_R$ denote the set of observable outcomes classified as resistant under the therapeutic objective and evaluation criteria, and let $R$ denote the event that the realized outcome belongs to this set:

$$R=\{y\in {\mathcal{Y}}_R\}$$

The probability of resistance under a specified initial state-space organization and therapeutic input is then

$$P(R\mid {\theta }_0,\mu )=P(y\in {\mathcal{Y}}_R\mid {\theta }_0,\mu )$$

and, for discrete realizations, can be written as the probability mass assigned to all trajectory-attractor-outcome combinations terminating in the resistant outcome region:

$$\begin{gathered} P\left(R\right|{\theta }_0,\mu )={\sum }_i{\sum }_j \\ {\sum }_{\mathrm{k}:y_k\in {\mathcal{Y}}_R}P(T_i,A_j,y_k\mid {\theta }_0,\mu ) \end{gathered}$$

Resistance therefore has two connected analytical meanings. At the process level, it is a manifestation of therapeutic trajectory evolution. At the probabilistic level, its occurrence is represented by a conditional probability within the probabilistic organization of therapeutic evolution under ${\theta }_0$ and $\mu$. The joint conditional probability distribution describes the possible therapeutic realizations and their relative probabilities; clinically observed resistance is one realized outcome from this distribution.

3. Sensitivity and Resistance Within the Probabilistic Organization of Therapeutic Evolution

Resistance occupies a defined region of the therapeutic outcome space. Let ${\mathcal{Y}}_S$ denote a set of outcomes defined as favorable relative to the therapeutic objective, and let $S$ denote the event that the realized outcome lies within this region. The corresponding conditional probability is

$$P(S\mid {\theta }_0,\mu )=P(y\in {\mathcal{Y}}_S\mid {\theta }_0,\mu )$$

Both $P(S\mid {\theta }_0,\mu )$ and $P(R\mid {\theta }_0,\mu )$ arise from the same joint conditional probability distribution of trajectories, attractors, and outcomes. Sensitivity corresponds to probability assigned to favorable therapeutic realizations and, at the observed level, to a favorable realized outcome; resistance corresponds to probability assigned to resistant therapeutic realizations and, at the observed level, to a resistant realized outcome. The joint distribution, the probability assigned to a defined outcome region, and the outcome that is realized are distinct analytical objects.

This formulation provides a direct connection to clinical response categories. Complete response (CR), partial response (PR), stable disease (SD), and progressive disease (PD) are observable categories within the therapeutic outcome space. The joint conditional probability distribution describes the probabilities of the therapeutic realizations from which these observed categories emerge.

4. Sensitivity and Resistance Are Not Intrinsically Deterministic

The conditional-probability formulation leads directly to a central analytical implication: sensitivity and resistance are not intrinsically deterministic properties of a tumor-host system. For a given initial state-space organization ${\theta }_0$ and therapeutic input $\mu$, multiple therapeutic trajectories are dynamically accessible, leading through different attractors to different possible outcomes. These possibilities are represented by the joint conditional probability distribution $P(T_i,A_j,y_k\mid {\theta }_0,\mu )$.

Sensitivity and resistance correspond to different regions of this distribution. If $S$ and $R$ denote favorable and resistant outcome events, respectively, their probabilities are $P(S\mid {\theta }_0,\mu )$ and $P(R\mid {\theta }_0,\mu )$. The clinically observed response represents one realization from this set of possible therapeutic evolutions.

A patient who ultimately exhibits sensitivity also carries resistance-associated factors. These factors are present within the state-space organization, yet the therapeutic evolution that is realized proceeds through a favorable trajectory and attractor. Conversely, a patient who ultimately exhibits resistance is not deterministically resistant; the resistant outcome is the realization of a resistant therapeutic trajectory from the set of trajectories accessible under the given initial state-space organization and therapeutic input. Observed sensitivity and resistance therefore reflect different realized therapeutic evolutions.

Accordingly, the presence of resistance-associated factors does not by itself determine the realized therapeutic outcome. Each factor is embedded within the broader state-space organization and contributes, together with other variables and their relationships, interactions, feedbacks, dependencies, compensations, and constraints, to the probabilistic organization of therapeutic evolution. Its therapeutic effect is expressed through the probabilities of possible therapeutic realizations rather than through a fixed factor-to-resistance correspondence.

possible therapeutic realizations

$$P(T_i,A_j,y_k\mid {\theta }_0,\mu )$$

↓

one realized sensitive or resistant outcome

Thus, $P(S\mid {\theta }_0,\mu )$ and $P(R\mid {\theta }_0,\mu )$ describe the probabilities of sensitive and resistant therapeutic realizations, whereas the clinically observed sensitivity or resistance is one realized outcome from the joint conditional probability distribution.

5. Therapeutic Implications of the Trajectory-Probability Formulation

Treating ${\theta }_0$ as an integrated therapeutic variable makes state-space organization an operational object of intervention. To modulate this organization, state-space modulation operates through state-space modulating operators, which act through system-level reorganization of the conditions under which therapeutic trajectory evolution occurs. Such reorganization can be achieved through modulation of fundamental organizing variables that serve as modulating interfaces for global state-space modification [8]. The therapeutic consequence of an intervention is therefore defined by the resulting reorganization of the state space and its effect on subsequent trajectory evolution, rather than by the elimination of individual resistance-associated factors.

Within the present conditional-probability formulation, the purpose of state-space modulation can be specified directly. A state-space modulating operator $\upsilon$ transforms the initial state-space organization ${\theta }_0$ into a modulated state-space organization ${\theta }_{\upsilon }$:

$${\theta }_0\stackrel{\upsilon }{\to }{\theta }_{\upsilon }$$

Because state-space organization conditions the accessibility and probability of therapeutic trajectories, this transformation redistributes the joint conditional probability distribution of subsequent therapeutic realizations:

$$\begin{gathered} p_{ijk}^0=P(T_i,A_j,y_k\mid {\theta }_0,\mu ) \\ \downarrow \\ {p}_{ijk}^{\upsilon }=P(T_i,A_j,y_k\mid {\theta }_{\upsilon },\mu ) \end{gathered}$$

The therapeutic purpose of state-space modulation is therefore to redistribute probability mass toward favorable therapeutic realizations and away from resistant realizations.

Bicarbonate-mediated pH modulation provides an empirical example of state-space modulation applied to therapeutic resistance. Bicarbonate has previously been formulated as a state-space modulating operator acting through pH, a fundamental organizing variable [8]. In hepatocellular carcinoma, bicarbonate-integrated TACE markedly increased therapeutic response in both controlled and real-world studies [11,12], and intratumoral bicarbonate combined with PD-1 blockade achieved an objective response rate of 93.3%, including 53.3% complete responses, in a prospective study of 30 patients [13]. Within the present formulation, these therapeutic shifts are represented as a redistribution of conditional probabilities toward sensitive realizations and away from resistant realizations, corresponding to an increase in $P(S\mid {\theta }_{\upsilon },\mu )$ and a decrease in $P(R\mid {\theta }_{\upsilon },\mu )$.

6. A Different Research Paradigm for Therapeutic Resistance

Traditional resistance research is primarily organized around identifying resistance-associated factors and mechanisms, determining their associations with or contributions to resistant phenotypes or outcomes, and developing interventions, biomarkers, or predictive strategies on that basis [1,2,3,4,5,6]. This analytical structure represents a specific form of the more general micro-to-macro mapping problem described previously [10]. Resistance-associated factors and mechanisms constitute microscopic information, whereas therapeutic resistance is identified as an observable macroscopic outcome. The intrinsic bottleneck arises because the microscopic variable space of the tumor-host system is open-ended and effectively unbounded, and its variables do not operate independently. Their composition, relationships, interactions, feedbacks, dependencies, and constraints cannot be exhaustively enumerated, and no finite microscopic description can completely represent their collective organization. Current resistance research therefore necessarily selects or derives a restricted subset of resistance-associated information and relates it to resistant outcomes. Such factor-to-outcome mapping can be analytically tractable and clinically informative, but it remains a partial representation: most microscopic variables are excluded, and the organization among variables is not fully preserved. The limitation therefore lies not in an insufficient number of identified resistance factors or available data, but in the analytical structure of direct mapping from selected microscopic information to a macroscopic resistant outcome [10].

Addressing this bottleneck requires a different research paradigm. The trajectory-probability formulation provides such a paradigm by following a different analytical logic. It places state-space organization, therapeutic trajectory evolution, attractor formation, and conditional outcome probability within the same therapeutic process, and asks how resistance emerges through this process and what determines the conditional probability of a resistant outcome.

7. Conclusions

Within the trajectory-probability formulation, therapeutic resistance is defined analytically as a resistant outcome that emerges as a manifestation of therapeutic trajectory evolution and whose occurrence is represented by $P(R\mid {\theta }_0,\mu )$. Sensitivity and resistance belong to the same probabilistic organization of therapeutic evolution. Sensitivity and resistance are therefore not intrinsically deterministic: $P(S\mid {\theta }_0,\mu )$ and $P(R\mid {\theta }_0,\mu )$ describe the probabilities of possible therapeutic realizations, whereas the clinically observed sensitive or resistant outcome is one realized outcome from the corresponding conditional probability distribution. On this basis, state-space modulation through a state-space modulating operator $\upsilon$ can, in theory, redistribute the joint conditional probability distribution of therapeutic realizations, with the therapeutic objective of increasing $P(S\mid {\theta }_{\upsilon },\mu )$ and decreasing $P(R\mid {\theta }_{\upsilon },\mu )$.