Bilateral Bayesian Network for Lymphatic Progression

• Bilateral Bayesian Network is a probabilistic graphical model that represents lymphatic tumor spread on both sides of the neck with latent and observed states.
• It integrates anatomical drainage patterns and Bayesian learning to compute patient-specific risks of occult nodal metastases.
• The framework supports tailored clinical target volume decisions by using risk-threshold rules to balance irradiation coverage and residual disease risk.

Searching arXiv for papers on bilateral Bayesian networks, especially in head-and-neck lymphatic progression and related Bayesian-network usages. A bilateral Bayesian Network (BN) is a probabilistic graphical model that explicitly represents lymphatic tumour spread on both sides of the neck, from a central oropharyngeal primary, and uses this representation to compute patient-specific risks of occult nodal metastases and guide elective clinical target volume definition. In the formulation proposed for head and neck radiotherapy, the network distinguishes latent nodal involvement from observed imaging findings, models ipsilateral and contralateral lymph node levels I, II, III, and IV in parallel, and supports risk-threshold-based construction of individualized CTV-E definitions with the potential to reduce irradiated volumes relative to standard clinical protocols while maintaining a low estimated probability of undetected nodal involvement in excluded levels (Moos et al., 26 Sep 2025).

1. Core definition and state representation

The network is bilateral because it includes parallel sets of nodes for the ipsilateral and contralateral sides of the neck, with sides $s \in \{i,c\}$ and lymph node levels $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$. For each side $s$ and level $\ell$, the model contains two random variables. The latent state is $x_\ell^s \in \{0,1\}$, where $x_\ell^s=1$ denotes metastatic involvement and $x_\ell^s=0$ denotes no involvement. The observed imaging state is $z_\ell^s \in \{0,1\}$, where $z_\ell^s=1$ denotes radiologically positive and $z_\ell^s=0$ denotes radiologically negative. A primary tumour variable $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$0 indicates absence or presence of a primary tumour at the site of interest.

This separation between latent and observed states permits explicit representation of occult metastases, defined by $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$1 and $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$2. The model therefore distinguishes three clinically relevant nodal states: healthy, with $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$3; macroscopic or detected disease, with $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$4 and $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$5; and microscopic or occult disease, with $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$6 and $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$7. For compact notation, the microscopic involvement pattern on side $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$8 is written as $$\ell \in \mathcal{L}=\{\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}\}$$9, and the imaging pattern as $s$0.

Although the cohort is stratified by T-stage into early-stage and advanced-stage tumours, the BN itself uses $s$1 as the primary tumour source node and learns separate parameter sets of base and transition probabilities for early-stage and advanced-stage tumours. This gives the model a stage-dependent but structurally fixed representation of bilateral lymphatic progression (Moos et al., 26 Sep 2025).

2. Directed acyclic graph and probabilistic factorization

The BN is a directed acyclic graph designed to reflect anatomical lymphatic drainage. The tumour node connects to lymph node levels on both sides through “base” spread arcs, $s$2 and $s$3. On each side, the model encodes stepwise progression through $s$4. Each latent nodal variable then connects to an observation node through $s$5.

A recurrent source of confusion is the meaning of “bilateral” in this model. The BN does not contain direct arcs between ipsilateral and contralateral lymph node level nodes; there is no edge such as $s$6. Bilateral coupling appears only through the shared tumour source and through joint posterior inference. The model is therefore bilateral in representation and inference, but not through explicit cross-neck transmission arcs.

The prior over microscopic states on both sides is

$s$7

Conditional on $s$8, the ipsilateral and contralateral sides factorize. With the observation model, the full joint distribution becomes

$s$9

In generic BN notation, this is an instance of

$\ell$0

This factorization encodes two essential modeling commitments. First, lymphatic progression is represented as a combination of direct spread from the primary and stepwise drainage along anatomical chains. Second, observed radiological findings are treated as consequences of latent microscopic or macroscopic states rather than as direct surrogates for them (Moos et al., 26 Sep 2025).

3. Parameterization, likelihood, and Bayesian learning

The BN is parameterized by a set of base and transition probabilities,

$\ell$1

The base probability $\ell$2 is the probability of direct spread from the tumour $\ell$3 to lymph node level $\ell$4, while $\ell$5 is the transition probability of stepwise spread from parent level $\ell$6 to child level $\ell$7, given parent involvement. The model samples a 14-dimensional parameter space consisting of 4 ipsilateral base probabilities, 4 contralateral base probabilities, 3 ipsilateral transition probabilities, and 3 contralateral transition probabilities. Separate parameter sets are learned for early-stage and advanced-stage tumours.

Illustrative values reported for early-stage tumours include ipsilateral base probabilities $\ell$8, $\ell$9, $x_\ell^s \in \{0,1\}$0, and $x_\ell^s \in \{0,1\}$1. For advanced-stage tumours, the contralateral base probability for level II is reported as $x_\ell^s \in \{0,1\}$2 with bounds $x_\ell^s \in \{0,1\}$3, indicating high contralateral II risk for advanced tumours.

The learning data consist of observed bilateral nodal patterns for $x_\ell^s \in \{0,1\}$4 patients, written as $x_\ell^s \in \{0,1\}$5. During parameter learning, the imaging model is simplified by assuming perfect imaging, with sensitivity $x_\ell^s \in \{0,1\}$6 and specificity $x_\ell^s \in \{0,1\}$7, so that $x_\ell^s \in \{0,1\}$8 deterministically. Under that assumption, the per-patient likelihood is

$x_\ell^s \in \{0,1\}$9

and the total log-likelihood is

$x_\ell^s=1$0

Bayesian parameter estimation uses a uniform prior over all parameters, $x_\ell^s=1$1 for $x_\ell^s=1$2, and Markov Chain Monte Carlo with the affine-invariant ensemble sampler “emcee,” 500 walkers, Differential Evolution / Snooker proposals, 6000 burn-in steps, and 3000 retained steps per walker, yielding 1.5 million posterior samples. These samples are subsequently used to propagate parameter uncertainty into risk estimates. The principal simplifying assumptions are that ipsilateral and contralateral sides are conditionally independent given $x_\ell^s=1$3, and that imaging findings at different lymph node levels are independent given their true state. These assumptions reduce model complexity and mitigate sparse data issues, but they also limit direct encoding of cross-neck dependencies (Moos et al., 26 Sep 2025).

4. Posterior risk estimation and CTV-E decision rules

For risk assessment rather than parameter learning, the observation model uses realistic imaging performance. The reported values are sensitivity $x_\ell^s=1$4 and specificity $x_\ell^s=1$5 for FDG-PET/CT, from Guedj et al. This replaces the deterministic training assumption with a probabilistic misclassification model, allowing the BN to estimate occult involvement probabilities in each lymph node level given observed nodal findings.

The joint posterior over bilateral latent states, conditioned on imaging findings and tumour stage, is

$x_\ell^s=1$6

From this posterior, the marginal risk for a specific lymph node level is obtained by summing over all bilateral latent configurations in which that level is involved. These quantities are the patient-specific risk estimates for each lymph node level, ipsilateral and contralateral.

Elective coverage can be defined by a per-level threshold rule:

$x_\ell^s=1$7

The framework, however, goes beyond independent per-level decisions by defining a joint missed-disease risk over all excluded levels, $x_\ell^s=1$8, equal to the posterior probability that at least one spared lymph node level harbors microscopic disease. Parameter uncertainty is incorporated by evaluating $x_\ell^s=1$9 across posterior samples and comparing the upper credible bound with the risk threshold:

$x_\ell^s=0$0

CTV-E is then constructed by iteratively including lymph node levels in descending marginal risk order until this criterion is satisfied.

The clinical significance of this decision rule is the explicit trade-off it creates. Low thresholds such as $x_\ell^s=0$1 lead to broader elective coverage and lower residual risk. Higher thresholds such as $x_\ell^s=0$2 or $x_\ell^s=0$3 lead to smaller irradiated volume but higher residual risk of occult disease. The bilateral BN turns this trade-off into a quantitatively specified decision problem rather than a fixed population rule (Moos et al., 26 Sep 2025).

5. Representative bilateral scenarios

The framework was evaluated in four representative scenarios. In the first scenario, N0 early-stage disease with no macroscopic nodal involvement on either side, ipsilateral risk remained non-trivial and contralateral risk was lower but non-zero, especially in level II. At $x_\ell^s=0$4, the CTV-E included ipsilateral I-IV and contralateral II, with residual joint missed-disease risk $x_\ell^s=0$5. At $x_\ell^s=0$6, the CTV-E was ipsilateral II-III plus contralateral II, described as very close to standard guidelines, with residual risk $x_\ell^s=0$7. At $x_\ell^s=0$8, only ipsilateral II-III were included and all contralateral levels were omitted, with residual risk in omitted levels $x_\ell^s=0$9.

In the second scenario, early-stage disease with radiologically positive ipsilateral level II and all other levels radiologically negative, ipsilateral III and IV risks increased because of stepwise progression from II, while contralateral II risk rose but remained lower than ipsilateral risk. At $z_\ell^s \in \{0,1\}$0, the recommendation was ipsilateral I-IV and contralateral II, with residual risk $z_\ell^s \in \{0,1\}$1. At $z_\ell^s \in \{0,1\}$2, the recommendation was ipsilateral II-III plus contralateral II, with residual risk $z_\ell^s \in \{0,1\}$3. At $z_\ell^s \in \{0,1\}$4, the recommendation was ipsilateral II-III only, with residual risk $z_\ell^s \in \{0,1\}$5.

In the third scenario, early-stage disease with radiologically positive ipsilateral II and III, ipsilateral IV risk was elevated because of the III$z_\ell^s \in \{0,1\}$6IV transition, while contralateral II risk increased but contralateral III and IV remained relatively lower. At $z_\ell^s \in \{0,1\}$7, the CTV-E included ipsilateral I-IV plus contralateral II, with residual risk $z_\ell^s \in \{0,1\}$8; this was broader ipsilaterally than guidelines, while contralateral coverage remained restricted. At $z_\ell^s \in \{0,1\}$9, the CTV-E was ipsilateral II-IV plus contralateral II, with residual risk $z_\ell^s=1$0. At $z_\ell^s=1$1, the CTV-E was ipsilateral II-IV only, with residual risk $z_\ell^s=1$2.

In the fourth scenario, advanced-stage disease with radiologically positive ipsilateral II and contralateral II, bilateral involvement in II propagated risk bilaterally to more distal levels. Ipsilateral III and IV risk was high, ipsilateral I risk increased because of elevated base probability in advanced stage, and contralateral III and IV risk was also substantial. At $z_\ell^s=1$3, the CTV-E included ipsilateral I-IV and contralateral II-IV, with residual risk $z_\ell^s=1$4. At $z_\ell^s=1$5, the CTV-E included ipsilateral I-IV and contralateral II-III, with residual risk $z_\ell^s=1$6. At $z_\ell^s=1$7, the CTV-E included ipsilateral I-III and contralateral II-III, with residual risk $z_\ell^s=1$8.

These scenarios show the asymmetry and stage dependence of bilateral risk estimation. They also clarify an important modeling point: because the BN does not explicitly model $z_\ell^s=1$9 arcs, contralateral risk is not driven by ipsilateral findings per se, but by stage-dependent contralateral base and transition probabilities together with observed contralateral involvement. Bilateral interaction therefore arises at the level of joint posterior inference and joint risk of missed disease (Moos et al., 26 Sep 2025).

6. Methodological significance, limitations, and terminological scope

Relative to standard Danish guidelines (DAHANCA), which use relatively coarse, pattern-based recommendations and often electively irradiate bilateral II-III or II-IV for many oropharyngeal presentations, the bilateral BN implements a data-driven workflow. It computes patient-specific marginal risks for each lymph node level, constructs CTV-E by including levels in descending risk order while controlling joint residual risk, and compares the resulting coverage and residual risk with guideline-based volumes. Across representative scenarios, at moderate thresholds of at least $z_\ell^s=0$0, BN-derived CTV-E often matches or slightly de-escalates guideline volumes. At higher thresholds of $z_\ell^s=0$1 to $z_\ell^s=0$2, the approach allows further ipsilateral or contralateral omission while keeping residual risk in the $z_\ell^s=0$3 to $z_\ell^s=0$4 range. This supports discussion of more patient-specific elective nodal target volumes and potential de-escalation of irradiated elective volumes.

The methodological advantages identified for this framework are an anatomically grounded probabilistic structure, patient-specific risk estimates, joint bilateral risk accounting, and transparent threshold-based trade-offs with uncertainty quantification via credible intervals. The principal limitations are equally explicit. The BN does not contain direct ipsilateral-contralateral dependency arcs; parameter learning assumes perfect imaging; later risk assessment uses fixed sensitivity and specificity estimates that are not level-specific; the model is limited to lymph node levels I-IV and previously untreated necks; and it does not include imaging-based covariates or geometry. Proposed extensions include adding more lymph node levels, introducing ipsilateral-contralateral dependency arcs, incorporating additional clinical or imaging variables such as HPV status and volumetric tumour burden, using time-dependent or dynamic BN/HMM hybrids similar to the trinary HMM of Pérez Haas et al. 2025, and adapting the framework to other cancer sites. The model is being evaluated in a prospective trial, DeEscO, NCT06563362 (Moos et al., 26 Sep 2025).

The phrase “bilateral BN” also has a broader, field-dependent use. In multi-label learning, a BN has been described as a bilateral dependency model when each directed edge $z_\ell^s=0$5 encodes a pairwise label dependency, quantified by $z_\ell^s=0$6, and topological sorting of the resulting DAG yields a label order for classifier chains. In that setting, “bilateral” refers to directed pairwise dependency relations rather than to an anatomical left-right representation. This suggests that the term is not intrinsically tied to a single graph topology; in head and neck radiotherapy, however, its most specific meaning is the explicit bilateral modeling of ipsilateral and contralateral lymphatic spread (Wang et al., 2019).