Nanobot Algorithms for Treatment of Diffuse Cancer

We assume $c$ distinct cancer sites numbered $j=0,\mathrm{\dots },c-1$, with site $j$ at location $y_j\in {ℝ}^2$. We assume that there are endogenous chemical gradients of a specific attractive signal chemical – hereafter referred to as chemical M – centered at each cancer site. We assume (for simplicity) that the concentration of M at all locations is fixed over time. The strength of an individual cancer site’s surrounding chemical M signal, which is represented by $P_{M_j}$ for each site $j$, respectively, is directly proportional to the amount of treatment that the cancer site requires. Regarding the total concentration of M at a certain location, we assume that different sites’ M-gradients simply interact additively. The concentration of M at position $x\in {ℝ}^2$ is given by

where $m\in {ℝ}^{+}$ and $P_{M_j}\in {ℝ}_{\ge 0}$ are parameters.

We now formalize the notion of different cancer sites requiring different amounts of treatment. We fix a (minimum) ratio $r_{K,M}\in {ℝ}^{+}$ of K to M that represents successfully treating all of the cancerous matter at site $j$: where $K_j^{(t)}$ is the number of K-payloads dropped at cancer site $j$ up to time $t$, the condition $K_j^{(t)}/P_{M_j}\ge r_{K,M}$ corresponds to the event in which cancer site $j$ has been completely treated by time $t$.

The other specific signal chemicals we consider are both artificial chemicals that can be carried and electively dropped as payloads by agents: firstly, an attractive signal chemical we call chemical A; and secondly, a repellent signal chemical we call chemical R. Unlike for chemical M, A and R are not persistent; they dynamically dissipate and diffuse through the colloidal environment over time. We model the concentration of A and R across the space over time via instantaneous point-source diffusion.

Beyond K, agents can also carry additional payloads of A and R which they can choose to drop and release. An individual agent can only drop its payload(s) at one cancer site in total. After an agent has completed dropping all of the payloads it elected to drop, it then for all practical purposes effectively ceases to exist in the environment. Let $A_j^{(t)}$ be the multiset containing the timesteps for which all of the unique chemical A payloads dropped at cancer site $j$ before time $t$ were indeed dropped, where $A_j^{(0)}$ is the empty set, and the maximum value over all of the elements in $A_j^{(t)}$ is less than $t$. $R_j^{(t)}$ is analogously defined.

Imagine one agent releases its chemical A payload of size $P_A\in {ℝ}_{\ge 0}$ at cancer site $j$ at time $t^{\ast }$. The payload will immediately begin diffusing, with the concentration of this individual payload, at location $x$ at time $t$, being a function of the distance from the cancer site ${\Vert x-y_j\Vert }_2$ and the time since its release $(t-t^{\ast })$: $\frac{P_A}{4\pi D_A(t-t^{\ast })}\text{exp}\left(-\frac{{({\Vert x-y_j\Vert }_2)}^2}{4D_A(t-t^{\ast })}\right)$ [18], where $D_A\in {ℝ}^{+}$ is the diffusion coefficient for A. We simplify to assume that the diffusion of each agent’s payload is independent, i.e., additive. Thus, the concentration of A at time $t$ at position $x\in {ℝ}^2$ is given by

where $t_{j,i}^{\ast }$ is the time at which the $i$’th A-payload to be dropped at cancer site $j$ was indeed dropped. The concentration of R at time $t$ at position $x\in {ℝ}^2$, ${\gamma }_R^{(t)}(x)$, is defined analogously to ${\gamma }_A^{(t)}(x)$ with the appropriate $P_R$, $D_R$, $R_j^{(t)}$ instead of their A versions.

We now describe the update step for the locomotion of an individual agent, i.e., the movement model. This model is most closely inspired by the actual nanoparticles in [13]. Agents take into account the local makeup of signal chemicals M, A, and R, following (either attractively or repellently, depending) the gradient(s) by biasing their orientation accordingly. All steps, taken in the direction of the current orientation, have a displacement distance of $\alpha \in {ℝ}^{+}$.

Consider some agent $i$ at time $t$. Its position is given by $x_i^{(t)}\in {ℝ}^2$. We first define ${\gamma }_{\text{TOT}}^{(t)}\left(x_i^{(t)}\right)≔{\gamma }_M\left(x_i^{(t)}\right)+{\gamma }_A^{(t)}\left(x_i^{(t)}\right)-{\gamma }_R^{(t)}\left(x_i^{(t)}\right)\in {ℝ}_{\ge 0}$. If ${\gamma }_{\text{TOT}}^{(t)}\left(x_i^{(t)}\right)$ is identically zero, let $\beta \sim U(-\pi ,\pi )$, and without loss of generality let $\mu =(1,0)$. Otherwise, let

which we then force to be within $[-\pi ,\pi ]$ by updating $\beta =[(\beta +\pi )\text{mod}(2\pi )]-\pi$. We define agent $i$’s orientation vector $\theta \in {ℝ}^2$ such that $\beta$ is the (scalar) angle formed between $\mu$ and $\theta$. Note that larger $b$ values yield movement that is more biased toward following the given chemical signal(s) (i.e., farther from standard Brownian motion), or greater orientation-bias; we thus refer to $b$ as the “orientation-bias parameter” hereafter. Agent $i$’s position is then updated as follows, taking a step of length $\alpha$ in the direction of its current orientation vector:

To summarize, if the attractive chemical signal (M plus A) dominates the repellent signal (R) in steepness, then the agent is biased to move roughly towards the nearest local maximum of the attractive chemical gradient (e.g., the nearest cancer site). If instead the repellent signal dominates, then the agent is biased to move roughly away from the nearest local maximum of the repellent gradient, which is, again, often the nearest cancer site.

We assume a bounded space $[0,{\varphi }_{\text{max}}]\times [0,{\varphi }_{\text{max}}]\subseteq {ℝ}^2$. At each timestep, each agent follows the above update step repeatedly until its new location falls within the given boundary. Despite space being bounded, note that we model the diffusion of chemicals in an unbounded space. We also define a maximum runtime cutoff $T^{\ast }$ (in the number of discrete timesteps). The reality of nanobots moving within the human body forces a finite timeline: after some amount of time, the nanobots will naturally dissolve or disintegrate, becoming useless for our purposes of delivering drug treatment. This is often referred to as the clearance time and is what the value $T^{\ast }$ represents. To be concrete, the values of all parameters that involve distance or time are in SI units unless otherwise specified. To be consistent with [13], we can think of one timestep in our model to be equivalent to one real-time second.

We define two performance metrics, a success metric that measures the effectiveness of the treatment and a treatment time metric that measures efficiency. Our first metric $S$, measuring what proportion of the total cancer has been killed by the clearance time $T^{\ast }$ (i.e., how successful or effective the treatment is), is given by

Note that after the first $\lceil P_{M_j}\cdot r_{K,M}\rceil$ K-payloads are dropped by agents at a given cancer site $j$, all additional K-payloads dropped there are useless; once a site’s demand is satisfied, further treatment is unnecessary. Hereafter, “success” refers to this metric $S$.

Our second performance metric $T_{\text{fin}}$ measures how long it takes for the treatment to stabilize and (mostly) finish (i.e., how efficient the treatment is, ignoring its success). Let $S^{\prime }(t)≔\frac{S(t+\delta )-S(t)}{\delta }$, where $\delta \in {\mathbb{Z}}^{+}$ is a parameter. We then define $T_{\text{fin}}≔\min \left\{t|S^{\prime }(t)\le D\right\}$, where $D\in {ℝ}_{\ge 0}$ is a parameter with some small value. That is, $T_{\text{fin}}$ is the first point in time for which the average rate of change in $S$ is less than or equal to $D$. When the success metric flattens out and stops increasing any further, we can assume that all of the agents are either already planted at their final site, or stuck, unable to find a site in reasonable time; it is at this point in time in which we deem the treatment to have (mostly) finished. Consider an experiment with $x\in {\mathbb{Z}}^{+}$ repeated trials. We define $S_{\text{avg}}(t)$ to be the average value of $S(t)$ over all $x$ trials. For small $n$, $S(t)$ can fail to be particularly smooth, causing the notion of its rate of change to become misleading. As a result, we instead use the smoother $S_{\text{avg}}$ when calculating $T_{\text{fin}}$. Analogously to $S^{\prime }$, let $S_{\text{avg}}^{\prime }(t)≔\frac{S_{\text{avg}}(t+\delta )-S_{\text{avg}}(t)}{\delta }$. We then redefine

For this work, we fix $\delta =30000$ and $D=3\cdot {10}^{(-7)}$. Note that $T_{\text{fin}}$ is not guaranteed to exist/be defined, since our simulation runs are finite and some algorithms such as RW have slow progressing treatment. Hereafter, “treatment time” refers to this quantity $T_{\text{fin}}$.

Each agent has a payload of chemical K only. There is no chemical A or R anywhere – i.e., ${\gamma }_A^{(t)}$ and ${\gamma }_A^{(t)}$ are both zero, or equivalently, $A_j^{(t)}$ and $R_j^{(t)}$ are the empty sets. Though there still exists the endogenous, time-constant chemical M gradients. When an agents reaches a cancer site, it always, and immediately, releases its K-payload to deliver its treatment.

Each agent has a payload of chemical K and a payload of chemical A. When an agent reaches a cancer site, it always, and immediately, releases both K- and A-payloads to deliver its marginal treatment and amplify the current/existing chemical signal, respectively. That is, there are chemical M gradients centered at each cancer site, with this attractive signal being amplified and boosted over time whenever an agent plants at the respective site and releases chemical A. There is no chemical R anywhere – i.e., ${\gamma }_R^{(t)}$ is zero and $R_j^{(t)}$ is the empty set.

Each agent has a payload of chemical K, a payload of chemical A, and a payload of chemical R. When an agent reaches some cancer site $j$ at time $t^{\ast }$, it always, and immediately, releases its K-payload to deliver its marginal treatment. Then, if

the agent also releases its A-payload, but does not release its R-payload. Otherwise, if ${\gamma }_A^{(t^{\ast })}(y_j)/P_{M_j}\ge r_{A,M}$, the agent instead releases its R-payload, but does not release its A-payload. That is, when the chemical A signal at a cancer site is too strong, agents will release chemical R (instead of A) at that site in an effort to encourage agents to explore and administer treatment elsewhere. $r_{A,M}$ is a parameter of the model specific to this algorithm.

Intuitively, or rather, ideally, we want to have agents find the cancer sites efficiently by leveraging chemical A just as in Algorithm KMA, but once a given cancer site is fully treated, we want to immediately stop having agents come there. More reasonably within the constraints of our model, once a given cancer site is fully treated, we want chemical R to be released there by any further agents that happen to still reach that site in order to start gradually encouraging agents to move away from the already treated site. However, this requires agents being able to sense the current amount of treatment already administered which we assume to be impossible. As such, we must use chemical A as a proxy to estimate the current amount of treatment (K) already administered since K and A payloads are often dropped in conjunction with each other.

We define $r_{A,M}≔k\cdot r_{K,M}\cdot P_A$, where $k\in {ℝ}^{+}$. Recall that while (8) still holds true (it is true initially, trivially), agents will always drop both chemicals K and A when they reach a cancer site, so $A_j^{(t)}=K_j^{(t)}$ holds in this case. Up to significant error – due to diffusion, different A-payloads being dropped at different points in time, and different sites’ payloads interacting additively – ${\gamma }_A^{(t)}(y_j)$ is roughly proportional to $A_j^{(t)}\cdot P_A=K_j^{(t)}\cdot P_A$ (with this final equality holding while (8) still holds true). In order for ${\gamma }_A^{(t)}(y_j)/P_{M_j}\ge r_{A,M}$ to be somewhat estimating/coinciding with the event $K_j^{(t)}/P_{M_j}\ge r_{K,M}$, then $r_{A,M}$ should be proportional to $r_{K,M}\cdot P_A$; hence the above formulation. In [11], we vary the multiplicative factor $k$, i.e., we vary $r_{A,M}$, to both tune our estimation as well as vary how early agents shift from dropping A-payloads to dropping R-payloads.

Lastly, as a baseline of comparison, we also consider the simple random walk, RW, where agents are completely blind to any chemical signals/gradients including chemical M. Recalling the movement model as defined in Section 2.3, RW is characterized by letting $\beta \sim 𝒰(-\pi ,\pi )$ and w.l.o.g. letting $\mu =(1,0)$. All agents have a payload of chemical K which they immediately release upon reaching a cancer site.

The different site and demand arrangements we consider in our experiments are the following: (a) Two well spaced sites with equal demand: positions $(0.0015,0.0025),(0.0035,0.0025)$, demands both $25$; (b) Five well spaced sites with equal demand: positions $(0.0025,0.0025),(0.001,0.004),(0.001,0.001),(0.004,0.004),(0.004,0.001)$, demands all $10$; (c) One major site with nearly all of the total demand and two outliers, all sites well spaced: positions $(0.0025,0.0025),(0.001,0.002),(0.004,0.003)$, demands $46,2,2$; (d) Two well spaced sites with unequal demand: positions $(0.0015,0.0025),(0.0035,0.0025)$, demands $15,35$; (e) Four sites clustered together and one other site faraway, all with equal demand: positions $(0.0008,0.0027),(0.0012,0.0023),(0.0008,0.0023),(0.0012,0.0027),(0.004,0.0025)$, demands all $10$. While this selection of specific arrangements is clearly inexhaustive with regards to all possibilities, they are intended to be representative of larger classes of common cancer patterns which are all mutually distinct. Arrangements (a) and (b) are representative of typical diffuse cancer, where (a) and (b) are the sparse and dense versions of this, respectively. On the other hand, (c) is representative of typical concentrated cancer. Arrangement (d) is somewhere in between these diffuse and concentrated categories, with the discrepancy in demand between the sites in (b) being less extreme than for (c). Similarly, arrangement (e) is a much less extreme version of concentrated cancer with the area of concentration being larger for (e) than for (c), spread out over a cluster of multiple sites. Within all Figures 1 through 4, the site and demand arrangements (a)–(e) above correspond to subfigures (a)–(e), respectively.

For simulation results, we begin with Algorithm KM, the basic case of only having the time-constant, endogenous chemical M signals for agents to follow. For the different cancer arrangements listed in Section 4.1 and for different orientation-bias parameter $b$ values, we plot the success metric $S$ over time ($t$/$T^{\ast }$). See Figure 1, in which cancer site arrangements are depicted in the small scatterplots with sites being points labeled with their demand.

For reasonable orientation-bias and sufficiently short clearance time, KM outperforms the simple random walk RW in both success and treatment time for all cancer arrangements other than (b). The strength of a cancer site’s M-signal is directly proportional to its demand. As such, the observed high success of Algorithm KM is reasonable and intuitive: having agents (which start at uniformly random initial locations) follow the chemical M gradients yields reasonably good allocation of treatment in expectation, because more agents tend towards those sites that require more agents as those sites’ M-signals are stronger.

For arrangement (a), in which there are two cancer sites with equal demand, we observe that greater orientation-bias corresponds to significantly faster progressing treatment. Under greater orientation-bias, agents follow the M-signal(s) more closely and thus reach a final site faster. The (average) maximum achieved success rate of treatment is approximately equivalent and preserved across all levels of orientation-bias for arrangement (a). With that said, the speed at which the treatment progresses is also relevant, since a clearance time of $T^{\ast }=50000$ (approximately $14$ hours), for example, yields $95$ percent success for the $b=6\cdot {10}^{(-11)}$ setting, compared to only $45$ percent success for the lowest $b$ settings.

For arrangement (d), in which the cancer sites now have uneven demand, the effect of the orientation-bias parameter on performance is different. While greater orientation-bias still yields faster treatment time by the same reasoning that agents will reach some site faster in expectation, here, when the orientation-bias is too strong, the maximum success of treatment achieved decreases. With an uneven demand arrangement, as the orientation-bias increases, the M-signal belonging to the site with greater demand will begin to dominate the weaker M-signal; consequently, agents will converge towards the site with greater demand, leaving the other site untreated. For this uneven demand arrangement across sites, sending all of the agents to the site that requires more treatment (which is what often happens under the greatest orientation-bias setting) still ends up yielding a reasonably high success metric value. But this treatment scenario completely ignores the cancer site with lower demand, which could arguably be unacceptable in application. There is a point, though, before which increasing $b$ does not reduce the maximum success achieved; as such, the greatest $b$ value before success decreases is the optimal parameter setting, as higher $b$ settings speed up treatment time. Here, that is $b\approx 4\cdot {10}^{(-11)}$, though this optimal $b$ value is likely to be dependent on the specific site and demand arrangement, among other things. Further experiments are needed to investigate this, testing algorithm performance for the same site locations as arrangement (d) but with a different demand arrangement.

For the dense diffuse arrangement (b), treatment is fast and successful across all amounts of orientation-bias. However, RW also already performs well here because agents, with initial locations distributed uniformly at random, are already close to some cancer site from the start in expectation. For the most concentrated arrangement (c), greater orientation-bias corresponds with faster as well as slightly more successful treatment, though these differences in achieved success are minimal. Arrangement (c) is the arrangement that produces the best performance, generally, across all orientation-bias settings, as the major site has a strong M-signal which agents ascend efficiently and reliably. For arrangement (e) where there is a cluster of several cancer sites close together in addition to one outlier site far away from the cluster, we see a similar trend relative to orientation-bias as for arrangement (d) in which greater orientation-bias can sometimes yield suboptimal success. Compared to Figure 1(d) though, the drop in success under greater orientation-bias is less extreme here. Greater orientation-bias still again yields faster progressing treatment for arrangement (e).

Importantly, for weak orientation-bias, our results show that KM has generally slow treatment time. As weak orientation-bias is essentially equivalent to weak chemical M signals, a characteristic of the natural environment that is out of our control in implementation, it is worth considering further strategies for improving treatment time; hence Algorithm KMA.

In Figure 2, for KMA, we plot the success metric over time for the same set of cancer arrangements and for different orientation-bias parameter $b$ values. Our results show that KMA has fast treatment time with A-payloads continually amplifying the existing attractive chemical signals, but, for many site and demand arrangements, suffers from poor success as all agents tend to converge on a singular site. Similar to KM, greater orientation-bias yields faster treatment time but also possibly suboptimal allocation of treatment, and thus lower total success. The optimal $b$ value for overall performance varies across cancer patterns.

When the first agent finds a cancer site, it drops its chemical A payload, amplifying the given site’s attractive M-signal. The other agents will then be more likely, in expectation, to find this site next with its amplified signal. These agents will drop their A-payloads as well, amplifying the attractive signal even more. This process continues, only speeding up and intensifying over time. Essentially, KMA can induce a sort of cascade effect in which a couple of agents finding a particular site sparks a process where many agents also head towards that same site. Such a cascade effect is extremely advantageous for fast treatment time, but, when there are many separate cancer sites all needing treatment, sending a large majority of the total nanobot swarm towards a single site is not ideal for success.

Sending almost all of the agents to one of two total cancer sites is better for the success metric than sending almost all of the agents to one of five sites, which is why success is greater for arrangements (a) and (d) compared to (b) and (e). In the case of arrangement (c) in which one site has almost all of the total demand, sending all of the agents to that demanding site will still yield close to perfect success. Our results show that KMA indeed achieves very high success of treatment for this arrangement, across all orientation-bias settings. In fact, compared to the results of Algorithm KM for this same arrangement (c), both algorithms achieve similar levels of success but KMA is much faster, demonstrating a clear advantage in KMA over KM for arrangement (c) (as opposed to the tradeoff present for the other arrangements between the efficiency of KMA versus the effectiveness of KM).

Ultimately, we show that for very concentrated arrangements like (c), greater orientation-bias ($b=6\cdot {10}^{(-11)}$) yields both higher success and faster treatment time. For the other arrangements, there is a trade-off between success and treatment time when choosing $b$; moderate orientation-bias, $b\approx 4\cdot {10}^{(-11)}$, yields intermediate performance in both metrics.

In Figure 3, for KMAR, we plot the success metric over time for the same set of cancer arrangements and for different orientation-bias parameter $b$ values. Ultimately, KMAR is shown to have great performance – in both success and treatment time – across all cancer arrangements, assuming well-chosen parameter values. We show that, generally, greater orientation-bias $b$ yields better performance for KMAR in both metrics. Under greater orientation-bias, agents in KMAR follow A-signals more closely to reach sites faster as well as follow R-signals more closely to find untreated sites faster, appropriately when tasked to do each. Recall from the movement model defined in Section 2.3 that the orientation-bias parameter $b$ determines how closely (in expectation) agents follow all of the various chemical gradients, both attractive and repellent. However, the results in Figure 3 show that the net effect of greater orientation-bias is in fact faster treatment times.

The parameters $r_{A,M}={10}^7$ and $(P_R/P_A)=5$ are fixed here; further simulated experiments are needed in order to test whether the negative correlation between $b$ and $T_{\text{fin}}$ is a general property of KMAR, or rather dependent on these specific parameter values. Regarding the positive correlation between orientation-bias and success, this could be viewed as evidence for our chosen fixed values for $r_{A,M}$ and $(P_R/P_A)$ being fairly optimal. Assuming favorable settings of $r_{A,M}$ and $(P_R/P_A)$, increasing the amount of orientation-bias results in – to put it simply – agents doing what we want them to do more often, or more deterministically, thus yielding greater success (as is indeed observed in the results). The degree of correspondence between orientation-bias and success varies slightly among the different cancer site and demand arrangements. For example, for arrangements (c) and (e), the two largest $b$ settings yield approximately the same success rate, while for the other arrangements, there is a nonegligible difference in success between these two $b$ values. There are also certain anomalies such as for arrangement (c) in which the $b=2\cdot {10}^{(-11)}$ and $b=3\cdot {10}^{(-11)}$ settings have similar treatment times (within $10000$ seconds, or three hours) but very different success rates (nearly $35$ percent difference). Generally, for the arrangements with fewer total cancer sites, (a), (c), and (d), the effect of $b$ on success $S$ is more dramatic; the difference in success across $b$ settings is the least dramatic for arrangement (b). While our results demonstrate that increasing the amount of orientation-bias improves overall KMAR performance, the choice to vary $b$ may be limited in implementation by the constraints of nanotechnologies.

We fix a specific parameter setting for each algorithm, motivated by simulation results in Sections 4.2–4.4 and in [11], and proceed to compare their performance. For Algorithms KM and KMAR, we choose the parameter setting which yields the highest success. For KMA, there is a tradeoff when varying parameter values in which faster treatment times correspond with lower success rates and vice versa; as such, we choose a parameter setting for KMA here which is both moderately fast as well as moderately successful. For each distinct algorithm, including the simple random walk RW, we plot the success metric $S$ over time ($t/T^{\ast }$), again for the same set of cancer arrangements. See Figure 4. Overall, across all site and demand arrangements, our results show that Algorithms KM and KMAR both generally achieve the highest success, though KMAR has faster treatment time. KMA has fast treatment time, but outside of very concentrated cancer arrangements, it has lower success compared to all other algorithms, at least for sufficiently long clearance times. For dense site arrangements, RW performs just as well as Algorithms KM and KMAR.

For arrangements (a) and (d), the trend in relative performance between algorithms is similar. KMAR and KM both achieve the same, highest eventual success rate, though KMAR is faster. KMA has a lower final success rate than KMAR and KM, but, because of its fast progressing treatment (still not faster than KMAR), it outperforms RW and KM below certain thresholds of clearance times. For arrangement (d), the more concentrated cancer arrangement between (a) and (d), KMA’s relative performance is better than it is for (a). The improvement in treatment time of KMAR over KM is also significantly more dramatic for the more unbalanced (demand) arrangement (d). For an arrangement like (d) in which there is a site with dominant demand, KMAR’s and KMA’s use of amplifying A-payloads is shown to be particularly beneficial. For the dense diffuse arrangement (b), KMA achieves lower eventual success than the other algorithms, but has greater success than RW and KM for short clearance times because of its fast treatment speed. While RW, KM, and KMAR all achieve around the same final success rate here, KMAR outperforms RW and KM for many clearance times due to its fast treatment time. For the most concentrated cancer arrangement (c), all of the algorithms perform similarly very well in both success and treatment time. Here, our three algorithms demonstrate the largest improvement in performance over the simple random walk RW. Concentrated cancer arrangements yield the strongest chemical signals in magnitude (surrounding the major site); our algorithms that follow these chemical signals are thus able to significantly outperform RW, which is blind to any chemical signals. Lastly, for arrangement (e) in which there is a small cluster of sites plus one faraway outlier site, KM actually outperforms KMAR in achieved success, though KM’s treatment time is slower. KMA has the lowest eventual success, but has higher success than RW for sufficiently short clearance times.

Although KM achieves high success across all arrangements, it is arguably less reasonable to assume to have choice over the strength of M-signals compared to having choice over the strength of A- and R-signals, as the former is a feature of the natural environment while the latter is under the control of nanobot engineers. Assuming extremely weak M-signals, KM would likely have extremely slow progressing treatment, to the point at which KMAR and even KMA would significantly outperform KM in achieved success for finite clearance times. This point helps to illustrate the benefit of KMA’s fast treatment time. Despite the improvements of KMA over KM in treatment time, and the improvements in KMAR over KM and KMA in both success and treatment time, we acknowledge that each more involved algorithm is more speculative than the previous regarding individual nanobot capabilities.