KMAR: Distributed Nanobot Chemotaxis

• KMAR is a distributed chemotaxis-based algorithm that coordinates nanobots for diffuse cancer treatment by integrating cancer-killing drugs with natural tumor markers, attractive, and repellent signals.
• The algorithm uses local chemical sensing and a dynamic switch between attractive and repellent payloads to balance treatment across multiple cancer sites based on site-specific demand.
• Simulation studies demonstrate that KMAR achieves faster treatment times and high success rates across diverse spatial patterns, highlighting its robustness and adaptive load-balancing.

Searching arXiv for the cited papers and closely related usage of the acronym. KMAR most commonly denotes the third nanobot control algorithm introduced for treatment of diffuse cancer in "Nanobot Algorithms for Treatment of Diffuse Cancer" (Harasha et al., 8 Sep 2025). In that setting, KMAR is a fully distributed chemotaxis-based coordination rule for a swarm of motile nanosized particles that must locate multiple distinct cancer sites, deliver a cancer-killing payload, and allocate that payload approximately in proportion to site-specific demand before agent clearance time $T^*$. The acronym expands to K for the cancer-killing drug, M for the natural tumor marker, A for an attractive amplifying signal, and R for a repellent signal. The same acronym also appears in environment-aware wireless communications, where it abbreviates knowledge-map-assisted radio (Li et al., 2021); however, in the present sense KMAR refers to the nanobot algorithmic framework of chemical amplification plus chemically induced site avoidance (Harasha et al., 8 Sep 2025).

1. Definition, objective, and algorithmic position

KMAR is designed for diffuse cancer, meaning cases with multiple spatially distinct tumor sites, each requiring a different amount of treatment. The operational objective has three coupled parts: agents must find all sites using only local chemical sensing, allocate their limited drug payloads proportionally to each site’s demand, and do so rapidly enough that treatment completes before finite clearance time $T^*$ (Harasha et al., 8 Sep 2025). Each site $j$ has a demand parameter encoded by the strength of its natural marker, and successful treatment requires enough delivered $K$-payloads to satisfy

$\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$

where $K_j^{(t)}$ is the cumulative number of $K$-payloads dropped at site $j$ by time $t$.

Within the paper’s three-algorithm progression, KMAR is the most elaborate control law. KM uses only the natural marker $M$: agents follow naturally existing chemical signals and drop $T^*$0 upon arrival at a site. KMA augments KM by having agents additionally drop an attractive signal $T^*$1, which amplifies whichever site was found and accelerates treatment, but can create overconcentration on a single site. KMAR retains the speed benefit of amplification while introducing a repellent signal $T^*$2: once a site appears sufficiently treated, subsequent agents reaching it drop $T^*$3 instead of $T^*$4, so that the site begins to repel later arrivals and the swarm redistributes toward under-treated sites (Harasha et al., 8 Sep 2025).

A common misconception is to interpret KMAR as a centralized allocation scheme. It is not. The agents do not communicate directly and have no explicit global state; coordination is intended to emerge purely from local responses to the superposed chemical field. This suggests that KMAR is best understood as a field-mediated load-balancing mechanism rather than as a deliberative routing or assignment algorithm.

2. Chemical fields, demand encoding, and motion law

The natural marker $T^*$5 is modeled as a static Gaussian field generated by all cancer sites: $T^*$6 where $T^*$7 is the location of site $T^*$8, $T^*$9 is the marker strength, and $j$0 is a spread parameter. The paper states that $j$1 is proportional to treatment demand, so the same quantity simultaneously defines therapeutic need and chemoattractive strength (Harasha et al., 8 Sep 2025). This dual role is central: higher-demand sites are intended to pull more agents in expectation even before any amplification occurs.

The attractive field $j$2 and repellent field $j$3 are both modeled as sums of diffusion kernels created by instantaneous point-source payload releases. If an $j$4-payload of size $j$5 is dropped at site $j$6 at time $j$7, then for $j$8 its contribution is

$j$9

and summing all past such drops yields $K$0. The repellent field $K$1 is defined analogously with payload size $K$2 and diffusion coefficient $K$3 (Harasha et al., 8 Sep 2025).

Agent motion is governed by the combined effective field

$K$4

At time $K$5, agent $K$6 at position $K$7 moves a fixed distance $K$8 in either Explore mode or Follow mode. In Follow mode, the agent computes the gradient

$K$9

samples angular noise

$\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$0

with truncation to $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$1, rotates $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$2 by $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$3, and steps in the resulting direction. The parameter $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$4 controls orientation bias: larger $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$5 reduces angular variance and yields stronger alignment to the gradient; smaller $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$6 makes motion closer to Brownian motion. If $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$7, the agent instead performs a random-walk step (Harasha et al., 8 Sep 2025).

The negative chemotaxis in KMAR is not separately parameterized. It arises because $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$8 enters $\frac{K_j^{(t)}}{P_{M_j}} \ge r_{K,M},$9 with a minus sign. When the repellent contribution dominates locally, gradient ascent on $K_j^{(t)}$0 sends the agent away from the treated site. This is a formal consequence of the effective-field definition, not an independent motion rule.

3. Payload-release policy and the approximation of completion

All algorithms in the paper use a common event structure. Agents are initialized uniformly at random in the bounded square $K_j^{(t)}$1. At each time step all agents move according to the Explore/Follow rule. If an agent enters within distance $K_j^{(t)}$2 of a cancer site $K_j^{(t)}$3, that agent is deemed to have reached the site (Harasha et al., 8 Sep 2025).

Upon arrival, KMAR always increments treatment by dropping $K_j^{(t)}$4: $K_j^{(t)}$5 The algorithm then decides whether to reinforce the site’s attractiveness or to convert it into a source of repulsion. This decision is made by comparing the local $K_j^{(t)}$6-field at the site against the demand parameter $K_j^{(t)}$7. If

$K_j^{(t)}$8

the site is treated as still under-advertised, and the agent drops $K_j^{(t)}$9. If

$K$0

the site is treated as sufficiently advertised, and the agent drops $K$1 instead. The agent is then terminated and does not move or release any further payloads (Harasha et al., 8 Sep 2025).

The paper is explicit that agents cannot measure $K$2 or treatment completion directly. KMAR therefore uses the local $K$3-field as a proxy. Because most arrivals at a site initially drop $K$4 whenever they drop $K$5, the authors argue that $K$6, and consequently $K$7 roughly scales with the number of arrivals. To align the $K$8-based switch with the actual treatment threshold, they choose

$K$9

for a tunable factor $j$0. In the paper’s interpretation, $j$1 begins repulsion before full treatment, $j$2 begins repulsion around true completion, and larger $j$3 delays repulsion so that behavior approaches KMA (Harasha et al., 8 Sep 2025).

This switch rule clarifies an important point: “sufficiently treated” in KMAR is operational rather than directly observed. The trigger is not exact therapeutic completion but the event that normalized local $j$4-concentration exceeds the threshold $j$5. A plausible implication is that KMAR trades exact treatment-state estimation for a chemically implementable surrogate that can still induce useful redistribution.

4. Distributed coordination and relation to KM and KMA

The allocation problem in the paper is to send approximately $j$6 agents to each site $j$7, up to a small global slack due to excess agents. KMAR attempts to realize that objective through three coupled mechanisms: demand-proportional natural markers, amplification, and dynamic repulsion (Harasha et al., 8 Sep 2025). The natural marker $j$8 is stronger at higher-demand sites, so the swarm already has a bias toward demand-proportional allocation. Amplification through $j$9 accelerates convergence after discovery by creating a positive-feedback cascade: early arrivals make a site more attractive, drawing in further arrivals. Repulsion through $t$0 adds negative feedback: once the normalized $t$1-level is high enough, further arrivals make the site less attractive, eventually pushing subsequent agents elsewhere.

The distinction among the three algorithms can be summarized as follows:

Algorithm	Chemicals used	Effective field
KM	$t$2	$t$3
KMA	$t$4	$t$5
KMAR	$t$6	$t$7

KM exhibits only positive chemotaxis toward the static natural marker. KMA preserves the same motion law but adds a dynamic attractive field, which substantially improves speed but can cause a cascade lock-in effect in which the swarm over-treats whichever site first accumulates $t$8. KMAR preserves KMA’s acceleration mechanism while introducing negative feedback through $t$9, so that a site can transition from being an attractor to being a repellor once its normalized $M$0-level crosses the threshold (Harasha et al., 8 Sep 2025).

The global success metric is defined by

$M$1

This metric caps each site’s contribution at its demand, so over-treatment at one site cannot compensate for under-treatment at another. That choice is significant because it makes balanced allocation, rather than raw total delivery, the relevant performance criterion. By design, KMAR aims to keep $M$2 close to $M$3, reduce over-treatment through repulsion, and reduce under-treatment through the combination of amplification and redistribution (Harasha et al., 8 Sep 2025).

5. Simulation framework and empirical behavior

The simulations are conducted in a 2D square domain $M$4 with $M$5 m, a one-second time step, agent step size $M$6 m, $M$7, $M$8 agents, total demand $M$9, $T^*$00, and lifetime $T^*$01 s, or approximately $T^*$02 hours (Harasha et al., 8 Sep 2025). Five hand-crafted spatial demand patterns are used: two-site equal-demand, two-site unequal-demand, five-site dense diffuse, four-site cluster plus outlier, and one major site plus two outliers. Each configuration is run for 20 trials.

Besides success $T^*$03, the paper defines a treatment-time summary $T^*$04 through a smoothed average success curve $T^*$05, a derivative window $T^*$06 s,

$T^*$07

and threshold $T^*$08, with

$T^*$09

This identifies the first time at which average success is essentially flat (Harasha et al., 8 Sep 2025).

The KMAR parameter sweeps examine orientation bias $T^*$10, switch threshold $T^*$11, and the relative repulsion strength $T^*$12. The reported behavior is pattern dependent but systematic. For diffuse patterns, moderate $T^*$13 values, around $T^*$14, form a sweet spot: treatment time remains reasonable and final success remains high, whereas very low thresholds induce repulsion too early and very high thresholds make the algorithm behave too much like KMA. For the strongly concentrated pattern, larger $T^*$15 improves success because the major site benefits from prolonged amplification before repulsion begins (Harasha et al., 8 Sep 2025). The authors therefore fix $T^*$16 as a good general choice for later KMAR experiments.

Increasing $T^*$17 generally reduces treatment time and increases success in KMAR. The paper’s interpretation is that strong gradient following helps in both phases: it accelerates convergence to newly advertised sites during the $T^*$18-dominated phase and also accelerates escape from completed sites once $T^*$19 becomes significant. This contrasts with KM and KMA, where increasing $T^*$20 can worsen allocation because too many agents become trapped in a single basin of attraction (Harasha et al., 8 Sep 2025). Varying $T^*$21 shows that stronger repulsion often improves final success by pushing agents away from completed sites, though it can increase treatment time and, in extreme cases, can lead to boundary effects or oscillatory behavior.

Across the paper’s direct comparisons, the qualitative summary is consistent. KMAR and KM typically achieve similarly high final success on sparse diffuse patterns, but KMAR does so much faster. KMA is often faster than KM but less successful because of over-focusing. In dense diffuse settings, KMAR is the fastest among the high-success methods. In the cluster-plus-outlier pattern, KM attains the highest success but is about twice as slow as KMAR, whereas KMA is faster but less successful. In the strongly concentrated pattern, KM, KMA, and KMAR all perform very well, with KM having the highest success, KMA the fastest time, and KMAR intermediate yet still strong (Harasha et al., 8 Sep 2025). The paper’s general conclusion is that KMAR shows great performance across all types of cancer patterns, demonstrating robustness and adaptability.

6. Theory status, assumptions, limitations, and context

The paper does not provide formal convergence theorems or closed-form performance guarantees for KMAR. There are no results of the form “all sites are visited with probability 1” and no analytical bounds on success or treatment time. The analysis is explicitly simulation-based, supplemented by heuristic reasoning about the proportionality between $T^*$22 and $T^*$23, the threshold choice $T^*$24, and the interplay between positive-feedback cascades and negative feedback from repulsion (Harasha et al., 8 Sep 2025). For a technically minded reader, this places KMAR in the category of constructive algorithmic proposals with empirical validation rather than in that of analytically characterized stochastic processes.

Several modeling assumptions delimit the scope of the results. Diffusion is idealized as instantaneous point-source diffusion in an effectively unbounded medium, even though agent motion occurs in a bounded square. Agents are assumed to sense local concentrations and gradients perfectly, with no sensing noise. There is no direct communication among bots. The environment is two-dimensional and omits blood flow, obstacles, vasculature, and organ boundaries. Tumor-marker fields are static, even though real markers could decay or evolve during treatment. Most importantly, KMAR’s switching rule uses local $T^*$25-concentration as a proxy for treatment progress, not a direct measure of delivered therapy (Harasha et al., 8 Sep 2025).

The paper also characterizes KMAR as more speculative than KM and KMA with respect to implementation. Carrying multiple payload types, deciding precisely when to drop $T^*$26 versus $T^*$27 from the sensed ratio $T^*$28, and engineering sufficiently strong repellent signals in a biological setting are all identified as practical challenges. Negative chemotaxis is described as inspired by experimental observations in other nanoparticle systems rather than experimentally validated for the exact mechanism assumed here (Harasha et al., 8 Sep 2025).

Within the broader research landscape, KMAR sits at the intersection of nanomedicine, swarm robotics, and chemotaxis-based navigation. Its conceptual contribution is to show how purely chemical, field-based interactions can implement nontrivial task allocation across multiple targets without centralized control or heavy onboard computation. The paper suggests several natural extensions: alternative repelling mechanisms, more complex agent behaviors, responses to external flows, multi-pass treatment strategies, multi-species swarms, and formal analysis of how $T^*$29, $T^*$30, and $T^*$31 shape allocation and convergence (Harasha et al., 8 Sep 2025). A plausible implication is that KMAR functions less as a finalized biomedical protocol than as a blueprint for self-organizing, chemically coordinated nanobot treatment in which amplification and repulsion jointly realize distributed load balancing.